Music and Numbers

Through The

Looking Glass

Lui Di Martino

Volume One

COPYRIGHT – This work is copyright 2009 by Lui Di Martino. Permission must be

sought from the author before any of the contents of this work are used elsewhere.

Dedicated to my soul mate and research queen Lynda Brasier

Table of Contents

Introduction – Unbroken thread

1 – The birth of Triangles 1

2 – Tonal Fountain 23

3 – The Vedic Square 27

4 – The Lambdoma 41

5 – Fibonacci Mode Boxes 49

6 – The invisible aspect of the Triangle of Keys 57

7 – Where else? 61

8 – Fibonacci numbers 68

9 – More number sequence partners found within divisions 73

10 – Both sides number flows – building a mirror universe 79

11 – Opposing forces 83

12 – Cycling a chord 88

13 – Swings four ways 93

14 – Dorian symmetry – explanation of the Keys 96

15 – Shades of Dark and Light 106

16 – In and out of the mirror 117

17 – Chords through the triangles 136

18 – Triangles in and out 142

19 – A link between triangle of frequencies and body's response 146

20 – Working conclusions 158

Introduction - The Unbroken Thread

The purpose of this book is to convey the results obtained from symmetrical reflection of various

formulas, both music and number based. Once a music scale has been cycled, and the mirror

cycle has also been exposed, there is a focus on the symmetrical relationships between the

various positions. The law of position and the clockwise/anti-clockwise flows go on to show that

there is a mirror side that is not an isolated system, but one that integrates its own information with

the information this side of the mirror. Both sides of these mirrored formulas will be seen to need

each other in order to function as a whole unit.

The tri-tone relationships within any music scale will also be focused on. It will be seen that these

tri-tone positions are catalyst for swapping information over to the other side of the mirror. The

interplay that exists between two tri-tones within the major scale system will be exposed. A mirror

structure is built on the logic of the swapping-over process at the tri-tones. This same mirror

structure is then shown to occur in many ways using music theory and also natural number cycles.

The mirror structure itself performs the role of uniting both sides of the mirror into one whole unit of

information.

In effect, the results shown in the proceeding chapters have yet to be pondered in the way shown,

and the obvious question is can we exploit this information in various ways? Therefore at the end

of the book there is speculation on how this mirror information exposed can be utilized. Does it add

another angle to the idea of a holographic universe, for example? Is it another useful insight into

the nature of consciousness itself? Can it help thinkers concoct various experiments in order to

travel in and out of the mirror universe? Teleporting? String theory?

Information contained within various grids composed of symmetrically reflected music and number

formulas will be seen to travel in and out of the mirror sides, and one asks if setting up or

mimicking the results of these dual grids of frequency information can lead to a real penetration

into the mirror side, in a way that will not be based on having to create masses of energy in order

to travel vast distances into space. In other words do universes exist in mirror pairs that share

information, and can we “ride” on that wave as it enters the mirror side at one specific axis point?

There is much speculation about creating warp speeds in order to travel through space. But is

there a shorter way, as shown within the results of dual formula grids, and how the information

interacts either side of the mirror?

If these sorts of statements seem a little far fetched, please consider deferring any final judgement

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until the results shown in this book have been allowed to digest. If string theory has reality as its

basis, then the information in this book will support the idea of mirror strings, based on overtone

relationships. From there, there is a real basis for building a mirror version of this universe. High

powered laser set ups can run experiments to test if it is possible to create the right frequency

based environment where objects can be placed within it and the information contained in these

objects can be teleported into another environment, at mirror interval frequency to the original.

The results in this book can be used as a basis for experiments in Consciousness related fields of

study. Again, it is a question of devising appropriate experiments that will test the data. A few ideas

are presented in the last chapter.

The reason such questions are asked in the first place is because of the nature of the dual cycles

of information being unearthed. With a little pondering, what at first seems extremely simple music

and number cycles, one can perhaps see that the relationships unearthed can become applicable

to natural process that are going on, both inside us and outside.

Of course in science there is a vast body of knowledge, built up and including many wonderful

formulas. So why should simple information based on cycles have anything worthwhile to offer the

science world? This is perhaps a question answerable by any scientist that takes the time to focus

on the results in this book, and perhaps on pondering the results will see a relationship there with

the many more apt scientific formulas that exists, or perhaps inspire a new science formula that

describes reality. If there is found a cross-over from music theory/number cycles to science, then

the book will have been a worthwhile venture.

Are you for real?

There is a "hidden" process, that we seldom take into consideration. That process is the

symmetrical reflection of certain formulas. It highlights a mirror side. It begs the question as to

which side is real, given the results exposed.

What results do show, when the mirror side of a formula is plotted, is that a whole unit of

information is present. The examples may be relatively simple to date, but they nevertheless do

cover such things as Fibonacci numbers, the Phi ratio, music scales, overtones, and various other

grids.

It is too uncanny that, firstly, a whole unit of information should be exposed by symmetrical

reflection of formula (as opposed to just a jumble of unrelated material on the mirror side). For it to

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then produce the exact same structure in various types of grids, that bear no relation to each other,

is truly uncanny, and is a basis for further experimenting.

All musicians are well aware of musical scales. Almost everyone is aware of the sound of a musical

scale, especially when one hums the doh-reh- meh-fah-sol-lah-teh-doh notes. A large majority of

the music we listen to in our western society is based on the major (and minor) scales. This type of

scale is also a cycle, in that one can repeat it over and over, into lower and lower sounds or higher

and higher sounds. This is possible mainly because of the nature of certain intervals. The most

prolific interval in a musical cycle is the Octave.

The Major scale (the doh reh meh fah sol lah teh doh sound) is a cycle that has additional inner

cycles, which musicians call Modes. The Major scale cycle itself is a Mode. Beginning the scale

from Reh, instead of Doh , will begin a new cycle that ends on the Reh one octave higher, or lower.

All these seven possible individual cycles have a mirror partner. It's uncertain when this was

discovered, but I finally gained the confirmation that music scales could be mirrored from reading a

book by Vincent Persichetti, called “Twentieth century Harmony”. As is often the case in music, I'd

discovered the mirroring of these cycles for myself some four years previously. I have found that

there is a difference between my own angle on these cycles, and the traditional musical use of

them. There is no need to view the examples in this book as a musical exercise. As mentioned we

will be pondering on the law of Position, and focusing on any inner structure evident between the

two sides of the mirror. As far as the music scale examples go, there is a list of animations

available for the non-musician, which I hope will clarify the whole process involved.

In the western music system there are twelve major scales, all evolved from what is known as the

circle of 5ths. These individual major scales are based on a system of sharps and flats that

distinguish them one from the other. The major scale can be written as a series of seven letters, to

replace the doh reh meh symbols. The very start of the circle of 5ths, and the twelve major scales

is on the letter C. This creates the C major scale, which , as yet, contains no sharps (#) or flats (b)

in its make-up.

C D E F G A B C

It would be beneficial for any non-musician to search various sources for information on the circle

of 5ths, and other basic music theory.

These seven letters (pitches) can also be roots of their own seven note scales. Here another scale

is produced from the above information, merely by starting at the note D:

D E F G A B C D

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This is the second cycle, or second Mode, known as the Dorian Mode.

E F G A B C D E

This is the third Mode, known as the Phrygian Mode, and so on.

Each of the cycles are built using strict ratios. The symmetrical reflection of these mathematical

ratios is what opens up a mirror side. And this mirror side is not a disjointed affair, but actually the

missing half of the complete unit of information.

One will find that the results apparent within the musical experiments, are also the very same

results that emerge within other types of cyclic phenomena, for example, the Fibonacci numbers.

Therefore, one is led to believe that there is an intrinsic structure that binds the two sides of the

mirror together, which is expressed through different approaches and experiments. It is this mirror

structure that becomes the main focus within the book.

One argument that criticizes the idea of mirroring formulas is that it the results are merely a

reflection, and not exposing a real life situation or solution. This however, is not the truth of the

matter. What is really brought into such a question is the assumption that only one side of the

mirror is valid. Yet the formulas in question will show their dual nature, that is, they unite to produce

one overall formula based on the two sides of the mirror. Which side can rightfully say "you are just

a reflection"? It isn't that simple. The claim is that these dual formulas that are being exposed are

one formula in its overall expression, the WHOLE UNIT.

Therefore, if one side of the mirror is considered an illusion, then so is the other side. Both sides

are inextricably linked together, and are seen to share information, which transfers to the other side

at specific axis points.

The rule of symmetry used within this book is that of equally mirrored moves around an axis point.

If a musical note moves forward/upward by one tone to another note, then that move is simply

mirrored by a reverse move of one tone from the same note.

mirror point

Bb – C - D

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The same note here is C, because it is the axis point. C to D is an ascending movement of one

tone. C to Bb is a reverse descending movement of one tone. And now the note D has a mirror

partner, Bb, in that they exist at a symmetrical point either side of the Root C mirror point. This

relationship will carry far more meaning than at first may seem the case here. Regardless of the

ratio one employs for finding this D note, that ratio is simply mirrored and one will find the

appropriately tuned Bb, according to which tuning system is used. The C to D move may have

been accomplished using a 9:8 ratio, or 200 cents (used in Equal temperament). The mirror move

will simply be an 8:9 ratio or 200 cents, from C down to Bb. Any further moves are always mirrored

in the same manner, by equal movement relative to an axis/mirror point.

Obviously, the most apparent thing here is that no musical note is ever alone. Well, how about the

C note, that is the same both sides of the mirror? The axis point is rather different, and it does have

a partner, an axis-point partner. This other axis point is not a visible aspect of the two scales being

mirrored, but it still reflects the very same pairs that are reflected around the C note. It was this fact

that became the driving force to experimenting with the mirror side of formulas much more,

because they led to the uncanny result of only one unifying structure, even though totally unrelated

systems were being used. It will be seen that what is visible information will swap over to the

“invisible” mirror side, and back again continually.

As this book is predominantly about symmetrically reflecting (mirroring) scale formulas and number

cycles it would be wise perhaps to look at one vital difference between various concepts of what a

mirror image actually is.

It is said that one would put a scale to the mirror:

C D E F G A B C

And in the mirror it would become:

C B A G F E D C

And joining the two together:

Mirror point

C D E F G A B C/C B A G F E D C

Here it is on the music stave:

v

This music stave will be known as a ‘treble clef’, which ascends and descends in movements of

the seven letters used in music, be it a line to a space to a line etc, or vice versa. Composers who

are looking for something different have mirrored music this way often. The art of placing music

manuscript paper in front of a mirror and then playing the tune “backwards” was pretty rampant in

Brahms’ day. It is certainly a very valid form of mirroring when it comes to manuscript paper.

Holding a real set of frequencies based on a given formula to the mirror, however, is quite different.

Mirroring the actual frequencies that each note possesses means that one applies equal

proportions on either side of an axis point (the mirror point). This would mean that any ratio must

be reciprocal. 1:2 must become 2:1 in the mirror. It is the ratios, or scale formulas that are

mirrored. In music there are scales, musical ladders of pitches that move in a sequential order.

These scales are put together by the use of musical Formulas. On the whole the formulas can be

defined by a number of symbols. Whatever the symbols used it is always the overall formula that is

symmetrically reflected.

vi

The scale reflecting the C major scale in the mirror above is a little different to the one shown

previously. At first one is entitled to think that surely this is not a true mirror image. And that is one

of the points, in some ways, that is worth pondering on. This other form of mirroring, the formulas

themselves, has opened up questions that cannot be ignored. This is especially true once an

investigator realizes that there is actual structure of kind within the mirror environment.

The mirror point has the note C twice. Musically what this means is that the mirror note of the note

C remains as C. From this axis point at the note C there is a leap upwards in pitch to the note D.

The letter T over this note is part of the formula, or instruction as to how far to move upwards in

pitch. The T simply means TONE. This has come to mean an upward movement by the ratio of 9:8,

or in equal temperament it is a move of 200 cents (an octave covering 1200 cents).

T T Bb 9:8 C 9:8 D

The mirror note is that of Bb. It is not the same as the note B. The note B is a white note on the

piano, whilst the note Bb is the next black note downward in pitch (travelling to the left of the piano)

The C is in the center acting as an axis. To the right is one tone's movement to the note D, and

reciprocally the mirror movement is one tone to the left, equalling the new mirror note Bb.

Here is the full scale of C Major again

C D E F G A B C

We have already stuck the C in the center and moved to the D, and we already know that the Bb is

not in the above scale. So what is being exposed? Let's continue with the next note of the C major

scale:

C D E

From D to E is yet another move of one tone. Therefore the mirror move must be one tone to the

left of the Bb.

T T T T Ab Bb C/C D E

Are symmetry groups related to the symmetric reflection of a musical scale's formula? Symmetry

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groups deal mainly with geometric objects, such as the platonic solids. Comparing this to a

symmetrically reflected musical scale formula is not the same thing.

There are various animations provided with this book. They are labelled according to the

corresponding chapter within the book. It is best not to watch all of them first, but rather read a

chapter and then watch the relevant animation.

There are also a few additional animations that cover material not in the book. Again these are

labelled accordingly.

viii

Chapter One

The birth of Triangles

The following diagrams will show that symmetrically reflecting music scale formulas does not

produce haphazard results, but actually leads to only one overall 'mirror' structure that will

consistently appear throughout all the examples in this book. And it will also be shown that any

information in these diagrams is seen to swap from one side of the mirror to the other, at one

particular axis. What this means is that any information that exists this side of the mirror, reaches

an axis point and is then transferred to the other side of the mirror, reaches the same type of axis

again, and again transfers to this side of the mirror.

The information in question will be based on such things as frequencies, which invariably emerge

in sets of three (I call them triangles of frequencies), number relationships that are cyclic in nature

(including Fibonacci number cycles), and other phenomena related to things like the natural

overtones. Obviously a key question is whether this information swapping between mirror sides

can be exploited in any way.

There are a number of applications that this information can be seen to be useful in. These will all

be discussed at the end of the book, when the results of the mirroring of various cyclic sequences,

both numerical and music scale based, have been shown.

In this chapter we will follow one thread. The formula of the Major scale will be reversed, that is,

symmetrically reflected (mirrored). By applying this mirroring technique to the major scale, and

then also to the modal system contained therein, it will be shown that a hidden structure is at play,

which can be seen to merge both sides of the duality/mirror together and represent it as a whole

unit. The resulting whole unit of information is the main focal point.

The catalyst responsible for this structure, that will merge both sides of the mirror into one whole,

will be shown to be at a position that is dead center of the Major scale, an invisible/in-between axis

point. That distance, from the note C, for example, is known as a Tri-tone, because it is three

whole tones away. This is the center point of the scale.

1

b5

C d e f F#/Gb g a b C

Other names for this interval are flat 5th (b5) or diminished 5th, or augmented 4th (#4). It’s all the

same thing when dealing with equal temperament. The note that occupies this special axis like

point in the center of the C Major scale is F#, also known as Gb.

It should also be noted right from the beginning that this mirror structure will remain the result of

symmetrically reflecting scale formula regardless of the tuning used.

The notes F#/Gb do not belong to the C major scale. It originally occurred to me to call the center

of the major scale an invisible-like axis point. It was later that I realized that information, in terms of

frequency relationships, was being shared between the visible axis and the invisible axis. What

this eventually came to mean was that information was appearing in and out of the mirror. Or one

may call the interplay between visible and invisible axis points “involved” and “uninvolved”. Much

more will be said on this after the first experiment of mirroring the major scale formula.

As you can see this b5 center axis occurs in between the major scale formula. It occurs between

the 4th and 5th note of the scale.

If you would like to see an actual animation of how to mirror a musical scale formula, you will find a

series of animations in the folder that came with this pdf book. The animations cover the

construction of some of the following grids as well. This may help to visualize the process.

The first structure that will emerge in one unbroken 'mirror' thread can be referred to as a Triangle of Keys. These triangles can be seen as a hidden element within any major or minor scale. The

formula that builds the scale of C major will be used to highlight how it is part of a “mirror seed”,

where one can mirror a formula three times in all, before returning to the beginning again. In other

words:

C D E F G A B C

The above scale does not convey the whole picture. What we see here is a fragment of a greater

whole, of which this part is only this side of the mirror's visible aspect. Observe in this next

2

diagram how the same looking formula flows in both directions and creates two different scales.

The T signifies two steps on a piano, and the S signifies a 1 step movement.

C Major having its formula reversed (Mirrored)

mirror point

S T T T S T T T T S T T T S

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

(C Phrygian) C Db Eb F G Ab Bb C D E F G A B C (C Ionian)

If you have viewed the first few animations that come with this book, you will have seen how this

mirrored dual scale has been achieved.

Start at the center C, and follow the logic of the tones (T) and semitones (S) as they are mirrored

around that C axis note. For the Tone, imagine the 9:8 ratio, and for the Semitone imagine the

16:15 ratio. And in terms of Equal Temperament then the Tone is 200 cents, and the semi-tone is

100 cents.

One can see the red formula is a mirror of the black formula. A few minutes on an instrument like a

piano and this should pose no problem in understanding. One can mentally imagine the center C,

and simple utter that D is in symmetrical reflection to Bb, E is in symmetrical reflection to Ab, and

so on. The scale to the right of the central C would be ascending on a piano, note for note (it is in

fact all the white notes of the piano), whilst the scale to the left of the central C would be

descending on the piano keys (these are a mixture of black and white notes).

These are two cycles that coexist on one formula. By applying the formula leftwards as well as

rightwards from a center axis, one is cutting through into a mirror world tonally speaking, and it will

be seen that there is structure within this other world of information, which has an umbilical link

with this side, and often swaps its information to this visible side of the mirror. Even though, at first,

this may seem like a purely musical exercise, it should be born in mind that sound is a natural

occurrence, and that harmonics play a big part in that. Should the journey into a mirror world prove

to be structural, it may open up further avenues for exploration.

Here is this scale mirrored, shown on the music stave. C is the mirror point in the center:

3

C Phrygian C Major (Ionian)

As you can see C Ionian has mirrored to C Phrygian. C is the central note (in red). The first

partnership is then 1 IONIAN/3 PHRYGIAN. The number 1 refers to the fact that the Ionian mode

is the first mode of any major scale. This is the normal doh-reh-meh-fah-sol-lah-teh-doh sound that

most people are familiar with. The number 3 refers to the fact that the Phrygian mode is the 3rd

cycle within any major scale. The Phrygian is a Minor sounding scale, often associated with

Spanish Flamenco or dark metal music. These cycles within a major scale are referred to as

Modes.

This mirroring of the Major scale (Ionian Mode) equates to a major sounding scale mirroring to a

minor sounding scale (Phrygian). Eventually it will be seen that a major scale, chord, or even

single note always mirrors to a minor scale or chord or single note, or vice versa, according to the

law of Position.

The numbers above the scales represent the note position from the Root of the scale (the root

being the 1 of the major scale) as well as the Modal positions that each note occupies. The major

scale possesses seven modes, and here are their names and positions within the main parent

major scale:

1 – Ionian 2 – Dorian 3 – Phrygian 4 – Lydian 5 – Mixolydian 6 – Aeolian 7 - Locrian

The notes and modal positions are symmetrically reflecting through the mirror like this:

1/3 2\2 3/1 4/7 5/6 6/5 7/4 C/C D/Bb E/Ab F/G G/F A/Eb B/Db

Notice how the 1 2 3 4 5 6 7 sequence has mirrored to an anti-clockwise sequence, starting at

number 3 – 3 2 1 7 6 5 4. This in itself already shows that the two sides of the mirror are inter-

related, and have structure of some kind, with contrary flows being evident. Generally speaking,

musicians are taught the clockwise sequence of modes, but rarely are told that these formulas are

really dual formulas. It would have been far easier for nature to create gibberish when sets of

4

formulas are simply symmetrically reflected. When the whole Mode Box is shown, in the next

section, it should become quite clear that both sides of the mirror are really one structure. The

search here is to find what it is that unites both sides of this duality into one whole.

Step one shows that the C Ionian Mode (the Major Scale) becomes the C Phrygian Mode (a Minor

scale) when the flow of Tones and Semi-tones are reversed. Yet there is a seed/process occurring

here that will allow one to continue with this mirroring. The seed we are looking for is the Major

scale that this mirror Phrygian Mode belongs to (nowadays all the Greek modes belong to a parent

Major/Ionian scale). The modes of any major scale always flow in the same order, regardless of

the major scale in question. The 3rd mode is always Phrygian, and it is always part of a parent

major scale that carries the number 1, which is always known as an Ionian mode. This will become

quite clear shortly.

Because we know that the Phrygian mode is a member of a parent major scale we are in effect

searching for the root position major scale that houses it. It is seen that C Phrygian resides in the

scale of Ab Major. It is the 3rd mode of that scale as you can see above (the Major scale carries

the number 1).

Formula - T T S T T T S

Position - 1 2 3 4 5 6 7 1

Mode - Ion Dor Phr Lyd Mix Aeo Loc

Parent scale - Ab Bb C Db Eb F G Ab = Ab Major

5

This next chart is for reference, and it will show all twelve major scales being mirrored. One may

use this chart to check which major scale mirrors to which Phrygian mode. Do not spend too much

time here, as you may want to check back once there is a grasp of the results shown further on:

The twelve Major Keys mirrored

Mirror Point

Phrygian Modes Major scales (Ionian Modes)

S T T T S T T T T S T T T S

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1C Db Eb F G Ab Bb C D E F G A B C

C# D E F# G# A B C# D# E# F# G# A# B# C#

D Eb F G A Bb C D E F# G A B C# D

Eb Fb Gb Ab Bb Cb Db Eb F G Ab Bb C D E

E F G A B C D E F# G# A B C# D# E

F Gb Ab Bb C Db Eb F G A Bb C D E F

F# G A B C# D E F# G# A# B C# D# E# F#

G Ab Bb C D Eb F G A B C D E F# G

Ab Bbb Cb Db Eb Fb Gb Ab Bb C Db Eb F G Ab

A Bb C D E F G A B C# D E F# G# A

Bb Cb Db Eb F Gb Ab Bb C D Eb F G A Bb

B C D E F# G A B C# D# E F# G# A# B

All the Phrygian modes on the left hand side of the above twelve major scales mirrored are really

just an arrangement of the same twelve major scales themselves, offset and commencing from the

3rd mode. To find which major scale each Phrygian mode belongs to it is just a question of looking

for the number 1 on the left hand side. The C Phrygian, for example, is from the Ab major parent

scale, the D Phrygian is from the Bb major scale, and so on.

6

From mirroring C major we have ventured into the scale of Ab Major. Therefore the Ab is now the

new axis point, and the major scale will be mirrored once more from that new axis.

S T T T S T T T T S T T T S 3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1 Phr Ab Bbb Cb Db Eb Fb Gb Ab Bb C Db Eb F G Ab Ion (from Fb Major). Replace these mirror notes on the left hand side with their enharmonic equivalent:

(G#) (A ) (B) (C#) (D#) (E) (F#) (G#) = G# Phrygian, from E Major

The above would look extremely complicated for a non-musician. As with the scale of C major, all

that is occurring here is that the Ab major scale is being reflected across the mirror point. You

should already have noticed this result within the twelve major scales mirrored diagram. There is

no need to really play these scales, as we are dealing with position of notes according to cyclic

events and how they hide at a deeper level one consistent structure throughout, which will become

evident as we go along.

There are two ways to represent the above mirror scale. It is still a Phrygian Mode, because that is

always the result one obtains when mirroring an Ionian Mode. Strictly speaking, the major scale in

question (that houses the particular Phrygian Mode) is known as Fb major. Ab Phrygian, one of its

modes, is more commonly known as G# Phrygian because hardly anyone thinks in the key of Fb

Major. It is easier to think of Fb as the note E, its enharmonic equivalent (the Fb connection will

also be analyzed in volume two). As you can see G# Phrygian is the third mode of the E Major

scale (E- F- G#, again a Phrygian/Ionian partnership).

Numerically this will look similar to the first example of C major and its mirror scale. That is the

beauty of a single formula (and the equal temperament system). You may not have seen the

double flat sign before (bb), but it is simply an instruction to lower the note in question by two semi-

tones (each “b” representing one semi-tone). In the above example, Bbb equates to the pitch for

the note A (lower the B note by two semitones). This way of writing scales is to keep things tidy, in

order to not use the same letter name twice.

The note partners around the Ab/G# axis (in red) are obtained in a similar fashion to those found

around the C axis previously. We will use the G# as the axis in order to highlight the note partners:

7

1/3 2/2 3/1 4/7 5/6 6/5 7/4

G#/G# Bb/F# C/E Db/D# Eb/C# F/B G/A (Ab/Ab) (Bb/Gb) etc

If you study the scales above, you should have no trouble seeing that these note pairs are at

symmetrical distances from the root axis. Again notice there is a contrary flow between either side

of the mirror. The whole process has jumped up a minor 6th (or down a Major 3rd) interval. C

Ionian/Phrygian and Ab/G# Ionian/Phrygian share what may be called a `mirror link'. Symmetrical

association links them. This link also agrees with the Lambdoma chart, as devised by Pythagoras

and we will compare the results of this emerging mirror structure with the Lambdoma in the

chapter four.

Again within this G# (or Ab) Phrygian mode there is the seed for continued mirroring. The new

major scale that has emerged is that carrying the number 1 on the mirror side, so step three is to

mirror this new major key, which is that of E Major. If we do not continue and complete this thread

the full potential of the cycle has not been exposed. Because a major scale has emerged behind

the mirror, through one of its modes, the implication is that some inbuilt hidden mirror cycle is

occurring. This inbuilt cycle is further confirmed in the section on Mode Boxes, a little further on. In

fact, practically every experiment in this book leads to the triangles of keys, which is about to be

exposed. The formula and numbers for mirroring E major will read the same as the previous two.

S T T T S T T T T S T T T S

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

Phr. E F G A B C D E F# G# A B C# D# E Ion (from C major)

As you can see the mode of E Phrygian is the result. This is the third mode of the C Major scale,

and we are back to the beginning. We have established a thread where three major keys relate to

each other through the mirror via the Phrygian modes up a minor 6th or down a Major 3rd. As the

movement is anti-clockwise we can assume that it is the latter movement that is occurring.

8

Three Mirrors

C

Maj 3rd Maj 3rd

Ab/G# E Maj 3rd

Here’s another way of looking at it:

Three Mirrors in One

C major mirrors to Ab major. Ab Major mirrors to E major, and E major mirrors back to C major.

The link is through the third mode of each major scale, the Phyrygian. Contrary cyclic sequences

have been seen to emerge, rather than no system at all.

9

This is only one of quite a few ways of achieving this triangular effect that is evident when musical

cycles are mirrored. This is still not the whole structure however, as will be shown in the next

section on Mode Boxes.

Every Mode belonging to the major scale will now be mirrored. Each mode is built according to a

formula comprising a different flow of Tones and Semi-tones. Here is a list of the traditional seven

modes, as they are constructed from the parent major scale

C D E F G A B C = Ionian Mode (parent major scale)

D E F G A B C D = Dorian Mode

E F G A B C D E = Phrygian Mode

F G A B C D E F = Lydian Mode

G A B C D E F G = Mixolydian Mode (Dominant scale)

A B C D E F G A = Aeolian Mode (Relative Minor AKA Natural Minor)

B C D E F G A B = Locrian Mode

All twelve major keys have the same flow of modal relationships. Here they are in D major.

D E F# G A B C# D = Ionian Mode

E F# G A B C# D E = Dorian Mode

F# G A B C# D E F# = Phrygian Mode

G A B C# D E F# G = Lydian Mode

A B C# D E F# G A = Mixolydian Mode

B C# D E F# G A B = Aeolian Mode

C# D E F# G A B C# = Locrian Mode

The same flow of modes is due to the same major scale formula being applied at the start. The

only difference is that the formula begins on a different note to C.

The next diagram is a representation of all these seven modes, together with their mirror partners.

One will find the mirror triangles even more prevalent when the results are analyzed.

10

MODE BOXES

Each Mode of the Major Scale is based on its own unique formula and it is the formula which is

reversed as if put through a mirror. One simple major scale will produce all the information below,

which I call a Mode Box. Use this mode box as a reference. You will need to check back to it often

so it would be best to draw out your own. It is amazing how it will start to make sense if you do this

mirroring for yourself! Instead of using the T and the S, I have opted to use numbers throughout

this box. Mainly because it will be easier for the eye to relate to the flow of numbers rather than a

series of T and S letters. The 2 refers to the Tone, and the 1 refers to the semitone. This can be

equated to the movement of one or two steps on a piano, or one or two frets on a guitar, for

example.

11

The right hand side of the above box consists of the seven Modes of the Major scale. Most

musicians who have studied a little music theory will recognize these seven different scales, all

generated from the parent major scale. For non-musicians, it should be relatively clear that the

right hand side consists of seven different orders of the notes C D E F G A B C. The seven cycles

actually sound very different to each other, even though the same notes are used. Yet we are not

so much concerned with how these scales sound, as this is not strictly a book about music. The

important thing is to understand that these seven cycles are being individually mirrored, using the

axis positions at the center.

The left hand side, being the mirror side, instantly looks different, made up of different notes to

their counterparts on the other side of the mirror.

To confirm the results within the above Mode Box, it is best to take each formula in turn and mirror

it. Using a piano to work out how the mirroring is done, will of course allow the reader to fully

understand how the mirror scales are emerging.

Take each individual line separately and firstly notice how the same formula of tones(2) and

semitones(1) is flowing both left and right from the center of the grid. Then take note of the modal

names and how they pair up across the mirror, the flow of numbers vertically and the individual

note partners from each line that partner each other.

One always begins in the center of this mode box. From this viewpoint it should be easy to spot

that each formula is reflected from here as if a mirror were put to it. Yet the mirror is not reflecting

the pitches themselves, because these change in the mirror, as you can see. The note D, which is

on the C Ionian line, becomes Bb by an equal movement of a tone away from the red C/C axis

point. Therefore it is the proportions or ratios that are being mirrored. The top line would look like

this if only the ratios according to the Lambdoma were used:

1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:1 9:1 5:1 2:3 3:2 5:3 15:1 1:1

The Lambdoma, which is a chart of representing Overtones (also known as harmonics) would fully

comply with this set of ratios.

12

If one study the numbers within the mode box occurring vertically it will be seen that the two sides

of the mirror move contrary to each other, as shown by the arrows either side. There is a clockwise

flow on the right, being the modes 1 through to 7, and an anti-clockwise flow of Modes on the left.

The anti-clockwise flow begins at the 3rd Mode, the Phrygian, and moves the opposite way to the

right hand side’s flow. After 3 will come the 4, which is actually at the bottom left of the box, and

then the 5, 6, 7, 1, 2 and finally the 3.

Modal partnerships emerge as ‘dual’ in nature. The Ionian mode mirrors to the Phrygian mode and

vice versa, for example. This is known as Double Reflection. The Dorian Mode does not behave

this way as it seems to contain the duality within itself and is perfectly symmetrical. This carries

many implications and will be looked at much more in depth as we explore musical and numerical

symmetry.

Let’s concentrate only on the left hand side. These seven mirror modes do not belong to only one

major scale as the seven modes on the right hand side do. There are in fact six major scales

within this section of the Mode Box (if we come to count Gb and F# Major as one). We have seen

the first major scale was that of Ab Major, home of C Phrygian above, on the first line. We also see

that D Dorian, the next mirror scale down, is from the same C major scale opposite.

Moving down the list to E we see that the mirror scale is that of E Ionian, which is the actual E

Major scale itself. Here are all the mirror major scales that have emerged:

Ab C E Gb Bb D F#

These major scales are expressing a particular mode/cycle from within their own seven cycles.

When one digests the many relationships living within this Mode box they will see how the mirror

not only converts the C Major scale and its modes into other tonalities and keys but it is also busy

establishing its own inbuilt structure.

This is how the Modes are paired when their formulas are mirrored, as shown in the mode box

above. This table below holds true for all twelve Major keys, and within those keys one may begin

their mirroring from any Mode. The names of the modes will be abbreviated, as is general

procedure:

13

1 3 2 2 3 1 4 7 5 6 6 5 7 4

Ion/Phr Dor/Dor Phr/Ion Lyd/Loc Mix/Aeo Aeo/Mix Loc/Lyd

It's the same dual flow as seen when only C major was mirrored. One is dealing with the major

scale both sides of the mirror, with one vital difference; the non-mirror side is a major scale with its

seven modes, and the mirror side is seven modes belonging to seven different major scales. This

means that definite structure has appeared. We are looking at a similar Major scale type structure

on the left but all its logic is opposite and it commences from the 3rd position. All the mirror modes

are similar types of modes as on the right hand side. If you observe the two flows of numbers

above, the red on the right and the black on the left, you will notice one is clockwise and one is

anti-clockwise. They are like two wheels in motion:

Flow of Mode box

1 3

7 2 2 4

6 3 1 5

5 4 7 6

The left hand circle starts at the number 3, signifying the Phrygian Mode, and it flows anti-

clockwise. The right hand circle starts at the number 1, signifying the Ionian mode, and it flows

clockwise. Also of note is the fact that the Ionian is a major scale and the Phrygian is a Minor

scale. All the relationships share this major/minor duality together. The 4/7, for example, signifies a

Lydian Mode (major sounding) and a Locrian Mode (minor sounding). Again there is the 5/6, which

signifies a Mixolydian Mode (major) and an Aeolian Mode (minor) pair.

One important aspect to understanding the inbuilt mirror structure lies at the 45-degree angle on

the left hand side of the mode box. In the following representation of the C Major Mode Box,

observe the notes - C E G# Bb D F# A# - along the 45-degree angle on the mirror side.

14

Left hand side of Mode Box All different Major keys Seven Modes of C Major

Ab Major

C Major

E Major

Gb Major

Bb Major

D Major

F# Major

The first thing to notice here is the numerical relationships between modes from either side of the

mirror. A 3/1 is consistent throughout the mode box, for example, as indeed the other number

pairs are. Each number signifies the position the note occupies within the parent major scale. One

can also see how all the numbers on either side of the mode box are moving in contrary flow, and

all is very systematic.

We have seen that by simply mirroring and re-mirroring the Major scale, three times in all, we find

one triangle. The keys that emerged were C E Ab major, which make create cyclic triangular flow.

What is interesting about the arrangement at the 45-degree angle of the left hand side of the mode

box is that two of these triangles have appeared. The second augmented triangle is Bb D F#: The

note A# is the same as the Bb so one can even consider that the two triangles are hovering

around another axis. The sharp (#) replacing the flat (b) is actually highly significant to the way

musical contraction and expansion interact in a mirroring environment, and will be focused on fully

toward the end of the book.

15

3

C 4

Db

5

Eb

6

F

7

G

1

Ab

2

Bb

3/1

C2

D

3

E

4

F

5

G

6

A

7

B

1

C2

D

3

E4

F

5

G

6

A

7

B

1

C

2/2

D3

E

4

F

5

G

6

A

7

B

1

C

2

D1

E

2

F#

3

G#4

A

5

B

6

C#

7

D#

1/3

E4

F

5

G

6

A

7

B

1

C

2

D

3

E7

F

1

Gb

2

Ab

3

Bb4

Cb

5

Db

6

Eb

7/4

F5

G

6

A

7

B

1

C

2

D

3

E

4

F6

G

7

A

1

Bb

2

C

3

D4

Eb

5

F

6/5

G6

A

7

B

1

C

2

D

3

E

4

F

5

G5

A

6

B

7

C#

1

D

2

E

3

F#4

G

5/6

A7

B

1

C

2

D

3

E

4

F

5

G

6

A4

B

5

C#

6

D#

7

E#

1

F#

2

G#

3

A#4/7

B1

C

2

D

3

E

4

F

5

G

6

A

7

B

Be aware of all these things as we move deeper through the mirror. We have already found the

notes C E G# (Ab) in the previous experiment so this 45-degree angle must be of significance. If

you have been in accord with things so far then this angle will be focused on further as more

evidence of its peculiarities are exposed. Also, remember that these triangles repeat linearly this

way, with an axis in between them. The dual note/major key, Gb and F# for example, are the most

important aspect of all, and will be seen as the catalysts that swap information from one side of the

mirror to the other.

However, these seven mirror modes belong to other major scales. These other major scales

create exactly the same triangular structures as at the 45-degree angles. Vertically the major

scales are Ab C E – Gb – Bb D F#. The Major scale always carries the number 1 above it, so

together with the twelve major scale diagram, the non-musician should be able to see the logic of

the Mode box.

These triangles can also be written as an ascending scale – C D E F# Ab Bb.

There are no semitone movements within this scale, each movement being that of one tone. There

is a scale within music called the whole tone scale, but to distinguish the fact that these notes are

representative of major scales themselves, I have called this series of notes the Circle of Tones Structure.

The triangles along the 45-degree angle have rearranged themselves compared to those along the

vertical line – Ab C E – Gb – Bb D F#. These different groupings are really a dance between the

visible and the invisible nature of axis points. By the end of this book I am hoping to show that

these triangular frequencies travel in and out of the mirror, meaning they exist this side one

instance and the ‘other’ side the next. To make this possible another axis point is employed. That

axis point happens to exist at the 45-degree angle, which is why the same triangles/circle of tones

exists there. Also, the F#/Gb position is associated with the Poles within music, so this is another

indication that the poles are swapping at this in-between point. A later chapter will deal with this “in

and out” of the mirror phenomena.

It is because the mirror cycle is started off-base commencing from the third position that the

modes are sent on some kind of a tonality journey through a series of major scales. The ensuing

loop that is created sees the emergence of this triangular-based structure. Every major scale

comes with this structure.

16

This next example is the mode box with the note names removed, showing only the numerical

positions. Notice how 1/3 and 3/1 are always symmetrically linked at every point of the mode box.

This is true for the other pairs 4/7 and 7/4, 5/6 and 6/5 , and 2/2.

The number positions of the Dual Modal partnerships

The next example of the Mode Box of C Major contains numbers and colours. The numbers signify

the calculated frequency of each note (cycles per second). There are of course differences of

opinion when attaching colour to frequency. However, the colours shown overleaf are the most

common ones that are associated with these musical notes.

17

Another view of Mode Box

One can see here how the frequencies fit in within a major scale formula, and also the formulas for

each of its modes. Ratios are used in order to ascertain each step of the scale. As there are many

web sites that convey this information on how to build scales and the frequencies involved, the

reader is urged to further explore this for themselves. One could view the very center of the major

scale as 374 (the mean between the F and the G) on one side, and 187 on the mirror side.

To get the mirror frequencies here it is a question of Reciprocity of movement, that is, any ratio is

simply divided instead of multiplied. For example, the 9/8 ratio used to find the note D, to the right

of the note C is made into an 8/9 instead. The multiplied ratio leads to the note D, and the 8/9

leads to the note Bb.

18

Triangles tree of note pairs

C

B Db

Bb D

A Eb

Ab E G F# F

The mirror note partners are opposite to each other here. In the key of C we find, for example, that

the note B mirrors to Db, as indeed it does in this diagram. D mirrors to Bb and A to Eb, E to Ab, G

to F. Only C and F# are axis points and therefore have no mirror partner. That is because they are

both axis point partners. The C is the visible axis, and the F# is the invisible axis at the 4.5 position

of the C major scale. This twin axis partnership between the root and the tri-tone position will be

discussed fully further on.

There are only four possible triangles within the twelve Key Major and Minor scale system. Each

Major scale can mirror 3 times in an unbroken link and produces a triangle relationship (like C Ab

E). As well as this, the seven modes of each individual major scale can be mirrored and there are

found two triangles within the result. In the case of C major the two triangles are – C Ab E and D

Bb F#. Each note of the triangle is separated by the Minor 6th interval, or its inversion, the major

3rd interval.

C min 6th Ab min 6th E min 6th C

D min 6th Bb min 6th F# min 6th D

C maj 3rd Ab maj 3rd E maj 3rd C D maj 3rd Bb maj 3rd F# maj 3rd D

19

These two triangles will all be related to C Major, even though there are six other major scales

involved. One will find that the scale of D Major will also have these particular two triangles on the

mirror side of its formulas, when a mode box for D major is produced. In fact it is easy to know

which major scales house the same two triangles. The triangles themselves are telling us which

major scales they belong to, as everything is symmetrical. Therefore E, Ab, Bb will also contain

these very same two triangles when a mode box is drawn for those particular key.

Every note is separated by a tone. The circle of tones are also the six major scales where the

triangles of C Ab E and D Bb F# belong. Therefore if two triangles of keys belong to six major

scales, then four triangles of keys belong to twelve major scales.

C Db D Eb

Ab E A F Bb F# B G

It is these same four triangles that we will consistently find by using various musical and number

approaches. These triangles of frequency can also be plotted on a circle, as they are cyclic (one

can repeat the major scale over and over and these triangles will appear vertically and at the 45-

degree angle on the left hand side of the mode box)

20

Notice that all the notes are a Tone apart form each other. C to D is one tone, or 9:8 ratio. D to E is

also one tone, all the way round the circle. These two symbols can be called two Circles or Cycles

of Tones.

A Pythagorean tuning for the major scale would look like this:

9/8 9/8 256/243 9/8 9/8 9/8 256/243

To this tuning would be attributed a series of pitches, shown as alphabetical symbols, for example:

C D E F G A B C

256/243 9/8 9/8 9/8 256/243 9/8 9/8......9/8 9/8 256/243 9/8 9/8 9/8 256/243

C Db Eb F G Ab Bb C D E F G A B C

Here we see the logic that, regardless of the tuning used, a mode box will look rather similar.

However, it would also be a mistake to think any tuning delivers the same kind of result on one's

mind. Obviously, when experimenting on the properties of the mode box, different tunings will

produce different perceptual results.

When seen as a mode box made up of equal temperament tuning, the notes will be tuned in cents,

with each semitone being 100 cents. Different temperaments will give slightly different intervals,

between each note. The perfect 5th, for example is an interval of 702 cents according to Just

Intonation (Pythagorean tuning). whereas in Equal Temperament it is 700 cents.

One rather psychedelic looking example of the mirroring of the major scale formula is this “leaf

effect” caused when all seven cycles are continually mirrored. The color-coding should help one

trace these modal formulas. The Dorian, for example begins on the central yellow, with the

Phrygian commencing on the red etc.

21

In this section, hopefully this mirror structure has stuck in the mind, and also that you agree the

results of mirroring scales do produce these triangular key relationships, being the groups of the

separate major scales within a whole mode box, and the mirroring of the major scale three times in

all to produce one triangle of keys. The real key to understanding how the information is part of a

whole unit is to focus on the tri-tone area and then to bring in the Dorian mode connection, which

is like an axis position as well (it carries a 2/2 relationship in the mode box, and it will be seen to

be related to the F# tri-tone position in the mode box, at the 4.5). Before that, these triangle

relationships will be shown to emerge within other examples, that are not always music theory

based.

22

Chapter two

A Tonal Fountain

Upon mirroring the major scale formula, it is noticed that the two mirror sides are

disjointed. At first glance there is no more than asymmetry. To find a point that unites

both sides of the mirror is no easy task, but one that a non-musician can undergo,

due to the fact we are talking of positions rather than the art of actually making music

from all this. The animations should help. Even this is really only the beginning,

because, further contemplation may lead to the conclusion that this easy looking info

is implying something rather fundamental in terms of unification of things like duality

itself, the positive/negative or expansion/contraction principles evident within nature..

Do not all structures possess a frequency, a number relationship, both inward

relating towards its own components, and outwards towards other objects?

The mirroring of a major scale begins with a 3/1 relationship. The first mode of the

major scale mirrors to another major scale that begins from that mirror major scale's

3rd mode. No access point here. Brick wall. One could say that this asymmetrical

relationship stops the manifestation sliding back into invisible oblivion. Next, one

would think of plotting down all seven modes to see where there may be an access

point to the other side of the mirror. However, the access point is actually evident

within the first 3/1 relationship itself:

3 - C Db Eb F G Ab Bb C/C D E F G A B C - 1

C is the visible axis. It pairs up D/Bb, E/Ab etc, either side of the mirror. The only note

missing from the twelve chromatics here is the F#, between the 4th and 5th position

of both scales. It already qualifies as another type of axis point, 4.5/4.5. This means

that at the 4.5 either side of the mirror there is the same note. F#/F#. Because this is

a similar type of relationship to the C/C (the original mirror axis point), it qualifies as

another axis point.

At that in-between axis point the dual mirror partners are in perfect symmetry. Here is

how the two F# notes either side of the mirror perform their symmetrical association

with the same mirror note pairs that are around the C axis:

23

C Db Eb F (f#) G Ab Bb C/C D E F (f#) G A B C

The F, descending from the f# on the left, mirrors to the G ascending from the f# on

the right. This G/F mirror pair also happens around the C axis (F/G are in bold). The

F/G both represent a semitone move away from the f#.

After the F/G, the next note on the left from f# is the Eb, which mirrors to the A on the

right. Again A/Eb are mirror pairs around the C axis. There is a visible/invisible axis

partnership occurring here, and it is the tri-tone that creates it.

Db will mirror to B, around the two f# positions, and one returns to C/C again. The

rest of the symmetrical relationships ascend from the f# on the left, and descend

from the f# on the right. Here is the sub-section in order to show this:

(f#) G Ab Bb C/C D E F (f#)

One can begin with the F to the left of the f# on the right hand side. This

symmetrically pairs up with the G, to the right of the f# on the right hand side. Then

rest of the dual mirror relationships are E/Ab, D/Bb and C/C again. Again, these are

exactly the same pairs as around the C/C axis.

The disjointedness at the start actually shows that, amongst its positions is an axis

that creates symmetry at a 4.5 position either side of the mirror..

In these examples, the F# point can be seen to be the axis for all the modal pairs, as

seen in the Mode Box:

1)

F Aeolian G mixolydian

C Db Eb F (f#) G Ab Bb C/C D E F (f#) G A B C

24

2)

Eb Mixolydian A Aeolian

C Db Eb F (f#) G Ab Bb C/C D E F (f#) G A B C

As we can also see both Mix/Aeo and Aeo/Mix appear in symmetry around the 4.5

axis. All the other pairs will emerge in likewise fashion.

One can see, for example, the Ion/Phr pair (red arrows). The Ionian will be on the left

hand side of the mirror, at Ab, and the Phrygian will be on the right hand side, at E.

Both will be in symmetry around the F# axis at the 4.5 positions. Then when the

modal pair swaps over to a Phr/Ion there will be the C Phrygian one side of the mirror

mirroring to the C Ionian on the other side. This accounts for the Ab/E and C/C

positions.

The Lyd/Loc pair is another example. This will equate to the notes and modal

positions of B/Db (Lyd/Loc) and G/F (Loc/Lyd)

Another way to see this symmetry born around the F# is to view one of the

animations that accompanies this book. The animation will be called “F# Tonal

fountain”. It will be far easier to see visually.

25

The DOR above the two F# positions is short for DORIAN. It will be argued shortly

that these F# positions are actually manifestations of another Dorian mode, which

carries the 2/2 relationship. It should be evident that if a relationship is equal on both

sides, it is also an axis point relating to something somewhere. Where exactly that is

will be seen to be in relation to one of the triangles of frequencies.

26

Chapter three

The Vedic Square

The Vedic Square is a mathematical tool popular in countries like India, and is becoming

increasingly popular in the west, mainly because of the speed one can learn to mentally attain the

results to fairly complicated multiplications, divisions, subtraction, addition, and even algebraic

equations. In effect , a Vedic square is no more than a 9 by 9 multiplication table, with the totals

reduced to single digit, as will be seen.

What will be shown in this chapter is a series of clockwise and anti-clockwise sequences of

numbers. They work for the general number line, and obviously other bases will throw up other

sequences. The reason for focusing on the nine number sequences that emerge with this

particular base system is that it throws up the same triangles of frequencies as in the music theory

examples. highlights the 4.5 axis of perfect symmetry away from musical examples, confirms the

Dorian mode connection using number as a basis, and also shows just how one little seed (the

nine sequences) can create an infinity of relationships. Without the reason for investigating this

inbuilt triangle of frequencies structure further, then the following exercise would justifiably be seen

as numerology. Sometimes it can be simplicity itself that goes on to lay a solid foundation. Bear

with this approach, and it will be shown that the Vedic square confirms the results in the Mode box.

One should not underestimate the power of breaking down totals to single digit. It has long aided

mathematicians in the Far East, who can realize complicated maths results in very fast time

without the need to write anything down. For the purposes this book, exposing the nine individual

single digit number sequences is in order to highlight a consistently emerging mirror structure

based on triangles of frequencies. Once shown that the Vedic Square produces these, there will

be other examples , such as the Fibonacci numbers, which also expose the same triangles of

frequencies.

All numbers can be broken down to a number between one and nine. Here is an example:

37 = 3 + 7 = 10 = 1 + 0 = 1

27

Therefore the number 37 can also be broken down to and represent the number 1. The number 1

will be seen as the beginning of a cycle of numbers based on 37. What we do now is add 37 to 37

and break it down again to a number 1 to 9. This is like multiplication tables.

37 + 37 = 74 = 7 + 4 = 11= 1 + 1 = 2

If we keep adding on the original number to the total we will uncover a "number sequence" which

will be related to the number 37 and its natural cycle, that is, doubled, tripled, quadrupled etc. The

sequence will cycle over and over like this:

37 = 3 + 7 = 10 = 1 + 0 = 1 74 = 7 + 4 = 11 = 1 + 1 = 2

111 = 1 + 1 + 1 = 3 148 = 1 + 4 + 8 = 13 = 1 + 3 = 4 185 = 1 + 8 + 5 = 14 = 1 + 4 = 5 222 = 2 + 2 + 2 = 6 259 = 2 + 5 + 9 = 16 = 1 + 6 = 7 296 = 2 + 9 + 6 = 17 = 1 + 7 = 8 333 = 3 + 3 + 3 = 9 370 = 3 + 7 = 10 = 1 +0 = 1 etc.

As you can see the number sequence that has emerged here is - 1 2 3 4 5 6 7 8 9, which is

obtained by cycling the number 37. Other numbers that share this same ‘cyclic’ sequence are 1,

10, 19, 28,46,55 etc. Try this yourself. Take, say, the number 19 and cycle it in the above manner.

You will see the same sequence emerge. Not all sequences are as straightforward as this and you

will see that some will be mirrors of others.

In fact there are only nine possible number sequences and these sequences represent every

number into infinity. It is the first nine numbers that house all the possible sequences (the zero is

teamed with the number 9 as will become apparent). The number 10 would obviously represent

the number sequence 1; the number 11 would represent the sequence 2 and so on. Here is the

number two cycled.

28

2 = 2 4 = 4 6 = 6 8 = 8 10 = 1 + 0 = 1 sequence = 2 4 6 8 1 3 5 7 9

12 = 1 + 2 = 3 14 = 1 + 4 = 5 16 = 1 + 6 = 7 18 = 1 + 8 = 9 20 = 2 + 0 = 2 etc.

Other numbers that share this same sequence will be – 11 20 29 38 etc. If we follow this

procedure with the number three we uncover another sequence:

3 = 3 6 = 6 9 = 9 sequence = 3 6 9

12 = 1 + 2 = 3 15 = 1 + 5 = 6

18 = 1 + 8 = 9 etc.

When this is done with the first nine numbers we will uncover all the sequences. Instead of listing

each number in the above manner I will only give the sequence that the first nine numbers

uncover, the first nine numbers being all that is needed:

1 2 3 4 5 6 7 8 9 2 4 6 8 1 3 5 7 9 3 6 9 3 6 9 3 6 9 4 8 3 7 2 6 1 5 9 5 1 6 2 7 3 8 4 9 6 3 9 6 3 9 6 3 9 7 5 3 1 8 6 4 2 9 8 7 6 5 4 3 2 1 9 9 9 9 9 9 9 9 9 9

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One can spend half an hour or so satisfying themselves that any number cycled in a similar way

will only produce these nine number sequences.

Producing a 9 by 9 multiplication table would yield the same grid. For example, 1*1 = 1, 1*2 = 2,

1*3 = 3 etc, forming the first number sequence of the Vedic Square. 1*10 = 10, but would be

broken down to the number 1 when cross adding the digits, and so the sequence would repeat all

over again. Then 2*1= 2, 2*2 = 4, 2*3 = 6 etc, forming the second number sequence of the Vedic

square, and so on.

The number sequences emerge horizontally as well as vertically. Four of the number sequences

mirror the flow of another four; the first flows contrary to the eighth, the second flows contrary to

the seventh, the third flows contrary to the sixth, and the fourth flows contrary to the fifth - 1/8 2/7

3/6 4/5 . At this point there is a swap over at the 4.5, as the pairs work their way back to the

beginning, 5/4, 6/3, 7/2, 8/1.

1/8

2/7

3/6

4/5

-----------4.5 axis

5/4

6/3

7/2

8/1

9/0

One flow is constant and you can see that it is that of the 9. The zero needs to accompany the 9,

and is its own constant, yet cannot stand on its own after its initial beginning. After the 9 will come

the 10, which is the return of the 1, and therefore the zero has occurred “within” the 9, which too

carries the duality within itself. Buckminster-Fuller showed how the 9 is also a zero by using indig

numbers, which will be shown shortly.

Observe how the first number sequence mirrors the eighth sequence (flow in opposite directions

to each other).

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1 2 3 4 5 6 7 8 9

8 7 6 5 4 3 2 1 9

One sequence flows left to right (clockwise) whilst the other sequence is in contrary flow. As you

can see the numbers are mirrors of each other both vertically and horizontally. The numbers 1 and

8, for example, are mirrors of each other up/down and left/right. Vertically the result is always a 9

when the two numbers are added together. This is true when all number sequence partners are

displayed as above.

2 4 6 8 1 3 5 7 9

7 5 3 1 8 6 4 2 9

Using such a simple set of number sequences allows access to seemingly unrelated numbers due

to their clockwise/anti-clockwise relationship. Numbers like 776567 can be related to a number like

47 just because of their hidden number sequence cycles. In this case they both belong to the 2 4

6 8 1 3 5 7 9 sequence. Therefore both these numbers have a clockwise spin inherent within

their doubling and tripling etc. An anti-clockwise partner to these numbers would need to break

down to a 7, and be part of the 7 5 3 1 8 6 4 2 9 sequence. Something like a 99999799999

perhaps?! Cycling this number will produce the 7th number sequence, so there is a relationship

there.

What struck me first is that when you add up a number from both sides of the mirror the total is

always a 9. 1/8, for example, or 2/7, 3/6, they all add up to 9 together. This is precisely the same

numerical relationship that is found within musical inversions. Here is an example of how to invert

an interval in music:

C to G is a 5th interval

G to C is a 4th interval

C to D = 2nd

D to C = 7th

These too are adding up to 9.

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These number sequences of the Vedic Square also correspond to the indig numbers +/-1 +/-2 +/-3

+/-4 ++/--4, as devised by Buckminster-Fuller. It is reckoned that every number can be made up of

these indig numbers. It will be seen that between numbers 4 and 5, at the 4.5 position, there is a

swapping over of partnerships, or clockwise and anti-clockwise flows. This is what Buckminster-

Fuller has to say about these four possible plus/minus integers:

‘one produces a plus oneness

two produces a plus twoness

three produces a plus threeness

Four produces a plus fourness

Where after we reverse:

Five produces a minus fourness

Six produces a minus threeness

Seven produces a minus twoness

Eight produces a minus oneness

1 2 3 4 5 6 7 8

+1 +2 +3 +4 -4 -3 -2 -1

We can see that a 2, for example, produces a plus twoness when the number is cycled – 2 4 6 8 1

3 5 7 9. One can see straight away that the cross over point was at the 4.5, between the fourth

and fifth number. Yet it was from a clockwise positive state to an anti-clockwise negative state, so

to speak. This 4.5 is a 9 halved after all. Also the plus/minus signs refer to the dual flows,

clockwise and anti-clockwise, as seen in the number sequences.

Every number can then be associated with one of these two flows when in direct proportion to

another thing. For example, the oneness is plus at the number 1 and minus at the number 8. The

1 and 8 are number sequence partners and flow in an opposite direction to each other. This is true

for the 2/7 3/6 and 4/5. The number 9 does indeed behave in a similar fashion to the number zero.

32

1 = +1

2 = +2

3 = +3

4 = +4

5 = -4

6 = -3

7 = -2

8 = -1

36 = 9 = 0

All the number partners can be seen to emerge from the center, as well as the number 9 itself,

acting like some kind of axis point. Here is the second number sequence of the Vedic Square to

highlight this:

2 4 6 8 -1 3 5 7 9 2 4 6 8 -1 3 5 7 9

Around the 9 axis are the correct + and - numbers according to Buckminster-Fuller. In between the

8 and 1 is the “invisible” or uninvolved axis, also producing the right number partners, the same as

around the number 9. Around the uninvolved axis one will find the 1/8, 2/7, 3/6, 4/5 pairs, which

are the same relationships as around the number 9. This is really the first correlation that the

number sequences have with the mirroring of the musical Modal system, in that there is a visible

axis and an invisible/in-between axis.

The 2 and the 7 number sequence, for example, are what Buckminster-Fuller recognizes as +2

and -2. He states that the 2 has a plus twoness about it, and you can see that it does, in the light

of the sequence of numbers it creates. The 7 has a minus 2 about it, and you can see from the

above number 7 sequence that this is true. For it to remain true there has to be a hidden 9 in

between the 1 and 8 in the above sequence. This is where the visible and invisible axis really

make sense. The mean number between the 1 and 8 is going to be 4.5. This is true for the 2/7

sequence, the mean between those two numbers being 4.5. Here is a diagram to show the

relationship between number sequences of the Vedic Square, the 4.5 and the Indig numbers:

33

Another reoccurring theme within the mirroring of natural cycles will be seen to be the swap-over

factor.

4.5 4.5

1/8 – 2/7 – 3/6 – 4/5 - 5/4 – 6/3 – 7/2 – 8/1 - 9 - 1/8 – 2/7 – 3/6 – 4/5 – 5/4 - 6/3 – 7/2 – 8/1

Running numbers alone will obviously show them increasing or decreasing. But running them as

symmetrically related pairs, shows them crossing over the mirror point. They delve back into zero

point at the 4.5 axis, get swapped over the mirror axis and continue their flow. It should be noted

that around the 4.5 axis and the 9 axis everything is symmetrical. This is no different to the

previous set of numbers around the 9 and 4.5 axis, but in the last example it seems that there is a

four-way relationship functioning when the mirror flows are brought together. The 1/8 for example

occurs twice around the 9 axis. Does the 1 to the right of the 9 mirror to the 8 on the left, with the

remaining 1/8 being the remaining pair? The musical examples that follow also suggest this, and

place similar relationships in a tonal sense as is above in numerical cycles.

In summation - numbers 1 2 3 and 4 cycle in a clockwise fashion. Numbers 5 6 7 and 8 cycle

contrary to the first four numbers. The number 4.5 brings these two contrary flows to perfect

symmetry, and it is also a swap-over point. The 4.5 swap over tendencies will be followed

34

throughout the book, and each example shows that this is an obvious aspect of the axis found at

4.5. Both contrary cycles evident within the first eight numbers always add up to a 9 together. The

9 is also partnered with the 0, and one can see that the mean number between these two

extremes is the 4.5.

The whole pattern is established within the first 9 numbers. After this the number 10 begins the

contrary cycles all over again. The 4.5 is shifted and is represented as the number 13.5. This 13.5

is now perfect symmetry for the contrary cycles existing between the numbers 10 and 17. This

next diagram aims to show this simple expansion/contraction process through all number.

1 2 3 4 (4.5) 5 6 7 8 9

10 11 12 13 (13.5) 14 15 16 17 18

18 20 21 22 (22.5) 23 24 25 26 27

28 29 30 31 (31.5) 32 33 34 35 36

4.5, 13.5, 22.5, 31.5 are all 4.5s. 13.5 is 1+3 = 4, plus the .5, for example. The magnitudes are

forever changing, but the ratio between each mirror number pair is consistent. In the first example

we see the 1/8 on the outskirts, begin to compress inwards toward the 4.5, traveling through 2/7,

3/6 and 4/5. The 4.5 and the 9 are also axis points, both really equaling a 9. And this 9 is further

related to the number 0.

As the 1/8 begin to compress toward the 4.5, the 4/5 begins to expand outward from the 4.5,

traveling through the 6/3, 7/2 and 8/1. This flow meets the axis at 9, is switched over, and begins

to compress to the 13.5, whilst from the 13.5 there is an expansion through 13/14 to the outskirts

at 10/17.

The next 4.5 axis is then at 22.5, being perfect symmetry between the numbers 19 and 26. And so

on throughout number, into infinity.

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This is how clockwise and anti-clockwise cycles keep themselves in perfect symmetry around the

4.5 and 0 axis points. It is akin to the mystical expression that the Father is perfectly symmetrical,

not part of the duality (unchanging), but able to see all the creation from its vantage point, the still

point. There is nothing in existence that does not have a number or ratio.

Once the contrary cycling number pairs are ascertained, one can also witness their symmetrical

relationships amongst many divisions, like 10/17, for example. Instead of seeing the resulting

constant as 0.5882352941176470, one can see this constant as two sets of mirror contrary cycling

numbers that express their clockwise and anti-clockwise nature in symmetry that adds to a 9:

10/17 = 0.5882352941176470 5882352941176470 5882352941176470...........

58823529

41176470

One could say that the 4.5 acts as an axis:

58823529 4.5 -------------------------------------

41176470

The positive and negative aspects of each number are swapped over across the 4.5. For example,

the 5 (-4 indig) swaps to the 4 (+4 indig) across this 4.5 axis.

It is rather uncanny that this phenomena exists, and it is not limited to only a few divisions, but to

an infinite amount of divisions that display constants in the results. The prime numbers are the

most prolific. In a later chapter many more examples are given as an introduction to these contrary

number pairs within constants. If there is a formula that can be created in order to find these

examples, I certainly don't know what it is. However, whether these results appear by accident or

not, there is surely a case at least for their being known to exist in such a manner.

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Bases and their number sequences:

Taking each possible base, and the positions each number in any particular base occupies, a

simple single digit grid appears.

In base 2 the repeating single digit sequence is 1 2 4 8 7 5

In base 3 it is (2) 3 9 9 9 9 9

In base 4 it is (3) 4 7 1 4 7 1......

In base 5 it is (4) 5 7 8 4 2 1......

In base 6 it is (5) 6 9 9 9 9........

In base 7 it is (6) 7 4 1 7 4 1........

In base 8 it is -(7) 8 1 8 1 8.........

In base 9 it is - (8) 9 9 9..........

In base 10 it is - (9) 1 1 1 1 1.........

in base 11 it is - (1) 2 4 8 7 5 1...........

In base 12 it is - (2) 3 9 9 9 9 9

In base 13 it is - (3) 4 7 1 4 7 1

In base 14 it is - (4) 5 7 8 4 2 1

Take base 3 as an example. Reading from right to left, these are the numbers for the first few

position:

243 81 27 9 3 (2)

The far right is for the numbers 0-2. The next position is the amount of threes. After that the next

position is worked out by multiplying 3 by 3 = 9. The next position is 3 by 9 = 27. and so on. A

number like 371 in base 10, in base 3 would look like this – 111202. There will be 1*243, 1*81,

1*27, 2*9, 0*3, and 2 from the last.

If one wants to confirm that the grid below is correct, a quick study of bases would help one to do

so.

37

The grid:

1 2 4 8 7 5 1

(2) 3 9 9 9 9 9

(3) 4 7 1 4 7 1

(4) 5 7 8 4 2 1

(5) 6 9 9 9 9 9

(6) 7 4 1 7 4 1

(7) 8 1 8 1 8 1

(8) 9 9 9 9 9 9

(9) 1 1 1 1 1 1

This next table shows how the number sequences of the Vedic Square display the same four

triangles of the Mode boxes. In fact it will be seen that an infinity amount of numbers would do so,

but then the Vedic Square does represent that infinity of numbers. Added to that the above bases

grid, and one can also find the triangles of frequencies evolving in the way shown below. All in all,

the 45 degree angle plays a vital role in this kind of evolution. Like the 4.5, then 45 degree angle is

like a carrier for the circle of tones mirror structure.

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The Nine number sequences and the Triangles

1 2 3 4 5 6 7 8 9 = Bb

1 = C12 = G123 = B1234 = Eb12345 = G123456 = B1234567 = D12345678 = F#123456789 = Bb1234567891 = D12345678912 = F# etc

3 6 9 3 6 9 3 6 9 = F

3 = G36 = D369 = F#3693 = Bb36936 = D369369 = F#3693693 = A36936936 = C#369369369 = F3693693693 = A36936936936 = C#

etc

5 1 6 2 7 3 8 4 9 = B

5 = E51 = Ab516 = C5162 = E51627 = G516273 = B5162738 = Eb51627384 = G516273849 = B

2 4 6 8 1 3 5 7 9 = Bb

2 = C 24 = G246 = B2468 = Eb24681 = G246813 = B2468135 = D 24681357 = F#246813579 = Bb2468135792 = D24681357924 = F#

4 8 3 7 2 6 1 5 9 = Bb

4 = C48 = G483 = B4837 = D48372 = F#483726 = Bb4837261 = D48372615 = F#483726159 = Bb4837261594 = D48372615948 = F#483726159483 = A4837261594837 = C#48372615948372 = F

6 3 9 6 3 9 6 3 9 = Eb

6 = G63 = B639 = Eb6396 = G63963 = B639639 = Eb6396396 = G63963963 = B639639639 = Eb6396396396 = G63963963963 = Bb639639639639 = D6396396396396 = F#

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7 5 3 1 8 6 4 2 9 = F

7 = Bb75 = D753 = F#7531 = Bb75318 = D753186 = F#7531864 = Bb75318642 = D753186429 = F7531864297 = A75318642975 = C#

9 9 9 9 9 9 9 9 9 = Bb

9 = D99 = G999 = B9999 = Eb99999 = G999999 = B9999999 = Eb99999999 = G999999999 = Bb9999999999 = D99999999999 = F#

Each sequence can be extended and all four triangle of frequencies emerge.

8 7 6 5 4 3 2 1 9 = Ab

8 = C87 = F876 = A8765 = C#87654 = F876543 = A8765432 = C87654321 = E876543219 = Ab

40

Chapter four

The Lambdoma; verifying the mirroring of ratios

Below is a grid called the Lambdoma, which is said to have been constructed by

Pythagoras. It basically represents the natural overtone ratios that are produced when

things like strings are plucked or a brass instrument is blown through, or a bell is struck.

Each ratio is taken from the perspective of the C axis/fundamental. But with careful

inspection it will be seen how F# can also be considered an axis point.

41

Notice how at 11:1 and 1:11 the overtone is F#. In fact it can be seen that F# is F#

whenever two reciprocal ratios are represented, like the 10:7 and the 7:10, for example.

We will stay just with the 11:1 and 1:11.

It has been seen that at the 4.5/Tri-tone position of the major scale there is an perfectly

symmetrical in-between axis point. Could the same be true for this 11:1 and1:11 within

the Lambdoma? There is a similar arrangement of axis points here, the C and the F#.

This chapter will attempt to show how the Lambdoma, the C major scale and its mirror

show no conflict.

I am sure this kind of grid may not make sense to some, so here is, hopefully, a simple

explanation of how this grid describes sound within material objects. There are slight

differences between the idea of overtones and harmonics, but these are to do with

dissonance and consonance. Acoustic engineers prefer the word Harmonics, but the

ratios above are more or less the same..

When one bows and plays a violin string, or clangs on a big bell, one will hear what is

called the fundamental tone. Yet this tone that we all hear is actually composed of many

other tones as well, called Overtones (also called harmonics or partials). They are all

fainter than the fundamental tone, and can be represented as whole number ratios. The

grid above shows how C is the fundamental note. This is shown as a 1:1 ratio. Note also

that the note's frequency here, or Hertz, is seen as One Cycle Per Second. The next

ratio is that of 2:1, or Two cycles per second. This first overtone from the fundamental

produces the same note, except that the second note is one octave higher in pitch, and

more faint than the fundamental. Most musicians can hear this octave of the

fundamental when they pluck a string. In fact here is a diagram of, say, a violin string,

showing how octaves evolve when that string is progressively cut in half:

42

The whole string signifies the Fundamental pitch, 1:1. The 2:1, being the string cut

midway is the ratio for an Octave above the fundamental. The 4:1 is the half string being

cut in half again (or 4 of these slices would be required to complete the whole string

length), and signifies two octaves above the fundamental. All these, within the

Lambdoma will be shown as the note C. Where will this halving of the string end?

In between the main octave ratios are other ratios, such as 3:1. This ratio will produce

the interval of an octave plus a perfect 5th above the fundamental, which in this case will

be the note G. Here are the first few overtones plotted on the music stave:

The bottom note here is a low C. The next note is the C one octave above, then the G

above that, another C at 4:1, then the E above that, and so on.

Colour and other electro-magnetic forces are all related to frequency. Should the string

above be cut and cut and cut, it will enter the domain of colour frequencies, which have

cycles per second in the trillions. Using the law of octaves one may cut the fundamental

C1 a total of 2^26, and it will fall into the wavelength required to create the colour red, at

450 Terahertz. The question remains whether sound waves and electromagnetic waves

are the same. In simple terms as wavelength relationship they would both adhere to

similar type ratios, by which waves sum up or divide.

By using the law of octaves, scientists are able to determine a musical note that is being

emitted by the sun. This note is really too high for us to hear at its original frequency

43

(somewhere in the region of 50 to 100 Mhz) but a note will repeat itself by lowering it in

octaves, as well as raising it by octaves. All one has to do is halve or double the number

of the frequency continually, until our ears pick up a note.

Not all objects, however, produce all the overtones all the time. Brass instruments only

produce the odd numbered overtones, for example.

So far we are dealing with multiplications, 2:1, 3:1, 4:1 etc. What about the divisions?

Here is the fundamental note C set as an axis, with the multiplication ratios on the right

of it, and the division ratios on the left:

Overtone series of C with symmetrical reflection applied:

11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11

F# Ab Bb C Db F Ab C F C C C G C E G Bb C D E F#

Remember to view each position with an added :1 after it, so that the number 1 will be

1:1, the number 2 will be 2:1 etc.

Again we see that the 2:1 ratio to the right produces an octave, and this complies with

the law of doubling a number to produce the same note an octave higher. In symmetry to

this is the 1:2 ratio to the left. This too complies with the law of halving as number in

order to produce the same note one octave lower in pitch. If one played the 1:1 C note

on the violin, the wavelength would divide itself along that string, and one would hear a

faint G note, at a 3:1 ratio, the fundamental string divided into three parts/separate

wavelengths. In symmetry to this, the 1:3 ratio would produce the overtone F.

Studying the Lambdoma above again, hopefully it will begin to make more sense. The

F/G pair, for example, are the same mirror note partners as in the C major scale and its

mirror scale. This is what we see amongst the multiplication ratios and the divisions, the

same mirror note pairs as around the C axis of the Major scale, C/C, D/Bb, E/Ab, F/G,

G/F, etc. D and Bb, for example, are 9:1 and 1:9 respectively.

44

What is also confirmed in the Lambdoma is that at one point after the C there is another

axis point. That point occurs at the 11:1 and 1:11. As mentioned, this is the same note,

that of F#. Obviously the other notes do not qualify as axis points because they share

partners across the mirror that are not the same note.

Remember that this 11:1 and 1:11 axis is from the point of view of the C axis at 1:1.

There are other ratios for F# within the grid, but these are from the point of view of other

root notes. One axis point is the visible fundamental (the C axis), and the other is the

“invisible” or in between axis (the F# axis).

Here is the scale of C major and its mirror again:

C Db Eb F G Ab Bb C D E F G A B C

The note F# sits in the very center of both sides of the mirror, in between the F and the

G, four and a half moves to the left and right. This is the 4.5 axis, whereas in the

Lambdoma this same axis is at the F# 11:1 and F# 1:11. As seen in the Mode box

(chapter one) the C on the mirror side starts its cycle on the number 3, and on the

number 1 on the right hand side. The F#, therefore can also be seen to be positioned at

the numbers 4.5 and 6.5. This of course equals an 11.

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

C Db Eb F G Ab Bb C D E F G A B C

Around the C note axis appear mirror note pairs, C/C D/Bb E/Ab G/F etc. The F# is not

part of the scale, lying at the Tri-tone positions in both scales, between the notes F and

G. So F# = F# through the mirror (it symmetrically reflects itself). The note C is an axis

point because C = C in the mirror.

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If F# reflects to F#, and this functions as an 11 within the Lambdoma, then is the major

scale and its mirror also endowed with an 11 connection? The 4.5 + 6.5 of the F# axis

points did equal an 11.

Instead of using the modal mirror pair names, as in F-Lyd/G-Loc, we will show the

positions of the mirror pairs as numbers

3/1, 2/2, 1/3, 7/4, 6/5, 5/6, 4/7,

The 7/4 and 6/5 already shows the number 11 quality. The others do as well, if we shift

some of the relationships up by an octave. Here are two octaves of the C major scale:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

C D E F G A B C D E F G A B C

The 1 can also be an 8. Therefore the 3/1 mirror pair can also be designated as 3/8.

Then the 1/3 can be designated as 8/3. Together this equals an 11:11.

The 2/2 can be designated as 9/2, again an 11.

There are other reasons why the 11 should be so consistent through the pairing up of

notes that are symmetrical reflections of each other. This will be shown in the chapter

“tonal fountain”

It may be claimed, and in some ways it would be a fair point, that musical scales are

man made sequences of notes. Therefore the results within them, as claimed within this

book, cannot be taken as actually occurring within nature. In this chapter we have dealt

with the natural phenomena of sound, as held within the law of Overtones/Harmonics.

This law is perfectly expressed within a grid like the Lambdoma. If one studied the

overtone structure of two strings, one tuned to C and the other to G or F, the major scale

would naturally emerge within the first few overtones sounding from each string.

There is also a case that the first three prime numbers give rise to the major scale. A

chapter is dedicated to how this is achieved in the second volume of this book.

46

Here is the Major scale formula written as ratios relating to the Lambdoma. These ratios

apply to any major key, not just C Major:

1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:1 9:1 5:1 2:3 3:2 5:3 15:1 1:1

One can see that the top line of a mode box shown as ratios reflects perfectly around the

1:1 axis. If the above were the key of C Major then one would find, for example, that the

9:1 would represent the note D and the 1:9 would represent the note Bb along the top

line. One can check this against the Lambdoma, and then against the C major scale and

its mirror.

After the second line of a Mode Box there will be an asymmetrical picture. This is

because one Lambdoma is related only to one Fundamental, as in the Lambdoma

shown in this chapter, which is all related to the main note C. In a mode box of course,

one exposes seven major scales, that is, seven root fundamentals, two of which are

enharmonic. Therefore one would need to draw out at least six Lambdoma charts to

prove that the mode box does reflect in a natural way, and therefore the resulting

triangles of frequencies that emerge are part of the overall picture within a Mode Box.

The Mode Box behaves as one overall unit, not as a non-mirror side with the inclusion of

a mirror side.

47

Here is a rather complicated looking set of ratios. It is made up of two left hand sides of

a Mode Box. The only results one needs to focus on are those along the 45-degree

angles:

1:1 15:1 5:3 3:2 2:3 5:1 9:1 - 1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:9 1:5 3:2 2:3 3:5 1:15 1:1 - 1:9 1:5 3:2 2:3 3:5 1:15 1:1 1:5 1:11 5:1 3:5 1:15 15:1 5:3 - 1:5 1:11 5:1 3:5 1:15 15:1 5:3 3:2 1:11 5:1 9:1 1:15 15:1 5:3 - 3:2 1:11 5:1 9:1 1:15 15:1 5:3 2:3 3:5 9:1 1:1 1:9 1:5 3:2 - 2:3 3:5 9:1 1:1 1:9 1:5 3:2 3:5 1:15 15:1 1:9 1:5 1:11 2:3 - 3:5 1:15 15:1 1:9 1:5 1:11 2:3 1:15 15:1 5:3 3:2 1:11 5:1 9:1 - 1:15 15:1 5:3 3:2 1:11 5:1 9:1 1:1 15:1 5:3 3:2 2:3 5:1 9:1 - 1:1 15:1 5:3 3:2 2:3 5:1 9:1 1:9 1:5 3:2 2:3 3:5 1:15 1:1 - 1:9 1:5 3:2 2:3 3:5 1:15 1:1 1:5 1:11 5:1 3:5 1:15 15:1 5:3 - 1:5 1:11 5:1 3:5 1:15 15:1 5:3 3:2 1:11 5:1 9:1 1:15 15:1 5:3 - 3:2 1:11 5:1 9:1 1:15 15:1 5:3 2:3 3:5 9:1 1:1 1:9 1:5 3:2 - 2:3 3:5 9:1 1:1 1:9 1:5 3:2 3:5 1:15 15:1 1:9 1:5 1:11 2:3 - 3:5 1:15 15:1 1:9 1:5 1:11 2:3 1:15 15:1 5:3 3:2 1:11 5:1 9:1 - 1:15 15:1 5:3 3:2 1:11 5:1 9:1

Here is the 45-degree angle from bottom left to top right:

1:15 1:15 9:1 9:1 1:15 1:15 9:1 1:15 1:15 9:1 9:1 1:15 1:15 9:1

All patterns flowing from bottom left to top right along the 45-degree angles have

repeating sequences of ratios about them. As do all the patterns flowing from top left to

bottom right. Yet the patterns flowing along this other 45-degree angle hide even more

secrets. Here are the notes and ratios together along this 45-degree angle:

1:1 C, 1:5 Ab, 5:1 E, 9:1 D, 1:9 Bb, 1:11 F#, 9:1 D, 1:1 C

At this point the pattern repeats. These are the two triangles of the C major Mode box,

known as the Circle of Tones, C Ab E and D Bb F#. This should help those, who prefer

to see things in terms of ratios, and to see how the reversing of the C major formula

complies fully with the Lambdoma, which is a grid that represents natural sound and its

wavelength relationships expressed as whole number ratios.

48

Chapter five

Here the Fibonacci numbers will be treated in the same manner as the Mode Boxes.

In doing this one unearths the same triangle flows as within the mode box, and again

across a 45-degree angle.

Fibonacci Mode Boxes

55:1 34:1 21:1 13:1 8:1 5:1 3:1 2:1 1 2:1 3:1 5:1 8:1 13:1 21:1 34:1 55:1Eb B G E C Ab F C C C G E C Ab F C# A 2:1 3:1 5 8 13 21 34 55 1C Eb B G E C Ab F C G E C Ab F C# A C

3:1 5 8 13 21 34 55 1 2D D F C# A F# D Bb G E C Ab F C# A C C

5:1 8 13 21 34 55 1 2 3C# G# G# B G D# C G# E C Ab F C# A C C G

8:1 13 21 34 55 1 2 3 5Ab F C C Eb B G E C Ab F C# A C C G E

13:1 21 34 55 1 2 3 5 8E C A E E G Eb B Ab F C# A C C G E C

21:1 34 55 1 2 3 5 8 13 D Bb Gb Eb Bb Bb C# A F C# A C C G E C Ab

34:1 55 1 2 3 5 8 13 21 A F# D Bb G D D F C# A C C G E C Ab F

55:1 1 2 3 5 8 13 21 34F C# Bb F# D B F# F# A C C G E C Ab F C#

All the ratios from the central red column travelling to the right are based on a

fundamental tone. To the left from this central red column, each interval created on

the right has been mirrored. Analysing this Fibonacci mode box is rather tricky, and a

full analysis is left for volume two. What is worth focusing on for now is that the same

triangle of frequencies are appearing on the mirror side along the 45 degree angle.

49

One will need to see that, in order to maintain the true mirror notes either side of the

red note at the central column, it is a question of bringing in other overtone series

built on different fundamental notes. Of course one could simply take my word for it!

But for those, like me, who require an explanation it will become apparent, with

enough study, how wonderful the tapestry is that Nature plots out, in terms of the

symmetrical picture.

After the initial Eb and F, the two triangles that have appeared are:

B Eb G

C# F A

These notes are in bold along the 45 degree angle on the left hand side. Plotting

these frequencies on a circle will produce one of the dual triangle symbols shown in

the first chapter. These particular two triangles will appear in six of the twelve

possible mode boxes, based on the major or natural minor scales. Obviously they

look like a star of David.

For it to occur within the Fibonacci series does strengthen the idea that they signify a

point of unity between the two sides of the mirror. After all, there are two sides

exposed within the above grid, and Nature is known to use the information on the

right hand side in order to regulate certain growth patterns, like spiralling galaxies,

conifer seeds, the nautilus shell, the distribution of leaves and branches on trees, and

many other things.

The very first thing that needs to be focused on is the different tonalities either side of

the mirror. As in the very first experiment, where we found C major mirroring to the C

Phrygian mode, we find that the overtones built on the C fundamental (first two lines)

adhere to the same principle. The 2:1 line is the same as the 1:1 line above it,

because the second line is merely an octave higher than the first. The mirror side

does not create the same overtone series that belongs to the non-mirror side. It is the

mirror side's undertones that reveal the overall twin triangle structure.

One cannot call the mirror side on the first line, the C Phrygian Overtones. Overtones

do not obey the Modal structure this way. On the one hand, each move on one side

of the mirror is reciprocated on the other side. In turn this creates tonality that really

belongs to other overtone series, built on different fundamental tones. In determining

50

these overtone series one is led into a mirror world that fits consistently with this

sides logic, yet also with its own mirror logic.

Any note can become the fundamental, and therefore each note can build a

Lambdoma grid of its own. In fact, building twelve Lambdoma charts would probably

help one understand what is going on. And this one reason the full analysis of this |

Fibonacci mode box is left for volume two!

The first nine movements of the Fibonacci numbers have been taken and cycled in

the same way the Mode Box of C Major was cycled. The cycles have been mirrored

according to the musical notes and how they are proportioned around the axis point

of each new line. The axis is always the red notes in the centre. As always the

diagonals on the left hand side display the Circle of Tones. This can be highlighted

with one or two examples. At the top left is the note Eb. The diagonal here runs from

the Eb all the way to the A, at the bottom centre:

Eb Eb F, B Eb G, C# F A,

After the initial three notes there are the two triangles of frequencies, B Eb G and

C# F A, and when combined they will produce the Circle of Tones – B C# Eb F G

A. It doesn’t matter which note one begins with; the result will be a circle of tones.

Below the top left Eb is the note C. Along this diagonal the other Circle of Tones is

produced:

C D, G# C E , Bb D F#

Firstly they are grouped up as two triangle of frequencies, and then when put in

sequential order produce the Circle of tones – C D E F# G# Bb. Also the top left

Eb and the centre A is a tri-tone interval. This is also true for the C and F#.

If we now continue the Fibonacci numbers for another nine movements, it is uncanny

that the thread will be picked up by the note F# on the number 89. The first Fibonacci

mode box begins with C, then after nine it begins with F#, establishing a strong link

with the results in the C major mode box, where C and F# are the invisible tri-tone

relationship. The 8:9 ratio would be that used for the movement of one tone, and six

of these are what produce the circle of tones. It’s strange how these things always

add up. A small fragment of the next Fibonacci mode box will be given here in order

to show this.

51

89 144 233 377 610 987 1597 2584 4181C Ab E C# A F D Bb F# D Bb G Eb B G# E C

Bb E C G# F C# A F# D Bb G Eb B G# E C F#

The triangles of frequencies will not only appear along the 45 degree angle. With

careful inspection one will notice them attempting to appear along the horizontal line

as well. No where more so than along the F# line. On the far left here one sees the

first triangle -C Ab E. It then gives way to the next triangle from the other circle of

tones , C# A F. After this, the first circle of tones appears with the second triangle

from it – Bb F# D. And lastly, there is swapping once more to the second circle of

tones with the appearance of the G Eb B triangle of frequencies.

The numbers on the left hand side are rather difficult to determine. The first move

from F# to D is that of a minor 6th. Therefore the reciprocal move to the left is also a

minor 6th. This resulting Bb may be determined by simply lowering the Bb 233 by

octaves.

233/2 = 116.5, 116.5/2 = 58.25

The resulting 58.25 would fit because it is lower than the F# 89. What then of the D

next to it? Halving 144 will give 72, and halving again will give 36. Even though 36

will fit one can see that the left hand side is not producing a Fibonacci sequence.

Well not quite anyway. If we add the 36 and 58.25 together we get 94.25. This is

quite close the F# 89, and given that we are dealing with approximate minor/major 6th

intervals it may be close enough to go along with. But then comes the question as to

whether 58.25 is a Bb pitch. The notes have been determined by their cycles per

second, so that 89 cps is the note F#, for example. Will this be true for the numbers

on the left? Certainly so! Because doubling or halving a number will produce an

octave of the same pitch. When we remember that a note can be spread across a

few numbers within its approximate range (Bb commences at 58, and is a Bb until 61

cycles per second. An octave higher and it commences at 116 until 122, with the gap

becoming wider.

52

The note F on the top left will probably be the F 21. This is because we can add it to

the D 36 and then add both numbers together and get close to the 58.25.

Moving on to the A we can assume it is an A at 13.75 (55 halved and halved again).

This 13.75 then adds to the 21 to make 34.75. As you can see this is nearly the total

for the note C# at 34. We can start deducing from all this that the left hand side is

really producing some in-between pitches to the ones written. This is not surprising

because the Phi ratio at 1.618 is only an approximation of a minor/major 6th, one

choosing either in order to keep the flow of notes going. Therefore it is more a

question of playing to the numbers, because as mentioned, a note can cover more

than the number that it is being represented with, just like the colour red has many

shades. In other words, the right hand notes are already approximations, so there is

little chance that things will add up perfectly when these notes are mirrored to the

other side. Then the Fibonacci numbers are equated with the Phi ratio even though

that too is an approximation, and also given he fact that the Phi ratio is not really a

finite number as far as we know.

Yet because the arrow is pointing right to left there is more of a subtraction system to

the left hand side of the mode boxes above. It is more 89 less 58.25 equals 30.75,

further confusing the issue, yet a workable structure nevertheless, depending on how

one may wish to use it.

Should the second Fibonacci mode box be completed, starting at the F#, one would

find the same Circle of Tones structure along the diagonals on the left hand side.

53

Here is the first Fibonacci mode box colour coded:

55

Eb34

B

21

G

13

E

8

C

5

Ab

3

F

2

C

1:1

C2

C

3

G

5

E

8

C

13

Ab

21

F

34

C#

55

A1

C

55

Eb34

B

21

G

13

E

8

C

5

Ab

3

F

2:1

C3

G

5

E

8

C

13

Ab

21

F

34

C#

55

A

1

C2

D

1

D

55

F34

C#

21

A

13

F#

8

D

5

Bb

3:1

G5

E

8

C

13

Ab

21

F

34

C#

55

A

1

C

2

C3

C#

2

G#

1

G#

55

B34

G

21

D#

13

C

8

G#

5:1

E8

C

13

Ab

21

F

34

C#

55

A

1

C

2

C

3

G5

Ab

3

F

2

C

1

C

55

Eb34

B

21

G

13

E

8:1

C13

Ab

21

F

34

C#

55

A

1

C

2

C

3

G

5

E8

E

5

C

3

A

2

E

1

E

55

G34

Eb

21

B

13:1

Ab21

F

34

C#

55

A

1

C

2

C

3

G

5

E

8

C13

D

8

Bb

5

Gb

3

Eb

2

Bb

1

Bb

55

C#34

A

21:1

F34

C#

55

A

1

C

2

C

3

G

5

E

8

C

13

Ab21

A

13

F#

8

D

5

Bb

3

G

2

D

1

D

55

F34:1

C#55

A

1

C

2

C

3

G

5

E

8

C

13

Ab

21

F34

F

21

C#

13

Bb

8

F#

5

D

3

B

2

F#

1

F#

55:1

A1

C

2

C

3

G

5

E

8

C

13

Ab

21

F

34

C#

This is by no means the end of the relationship that the Fibonacci numbers have with

the Circle of Tones. Next will be presented what are called Summation tones, within

the overtone series. This simply means that two pitches come together and between

them they add up and create a third tone from that union. The same is true for

Difference tones, where two pitches/numbers subtract and create a third tone from

that.

54

Summation Tones and the Fibonacci numbers

It is found that when the summation of any two pitches is taken into account the

overtones produced always equated closely to the triangles of frequencies that were

found to exist flowing through a mode box.

A summation tone is created when we add any of the overtone numbers together. An

example would be G3 (three cycles per second) and E5 (5 cps) add together to

produce the C8, and so the overtones’ frequencies speed up it seems. The Fibonacci

number series is what is produced in all three columns, and the summation tones are

those on the right. The first example, C1 and C2 produce G3. Then, just like the

Fibonacci number series, the previous two tones produce a third tone, in the shape of

C2+G3 producing the E5:

C 1 + C 2 = G 3

C 2 + G 3 = E 5

G 3 + E 5 = C 8

E 5 + C 8 = A 13 (nearly A)

C 8 + A 13 = F 21 (nearly F)

A 13 + F 21 = Db 34

F 21 + Db 34 = A 55

Db 34 + A 55 = F# 89

By 55 + F# 89 = D 144

F#89 + D 144 = Bb 233

D 144 + Bb 233 = G 377

B 233 + G 377 = Eb 610

G 377 + Eb 610 = B 987

Eb 610 + B 987 = Ab 1597

C 987 + Ab 1597 = E 2584

Ab 1597 + E 2584 = C 4181

E 2584 + C 4181 = A 6765

This way of combining the overtone series and the Fibonacci numbers leads straight

to the triangles of frequencies and keys evident within the mirror side of a mode box.

After G E C, which forms the grand triad, the proceeding triplets are those same

55

triangles that have been found on the mirror side of the Major scale, that go on to

define two Circles of Tones, each of which is two triangles combined. There is also

an in-and-out of the mirror flow occurring here. If we call the circle of tones a star of

David, then we can say that the triangles are swapping from one circle of tones/star

of David to the other. A F Db, which belongs to one star of David gives way to F# D

Bb, which belongs to the other star of David. This then gives way to G Eb B, which

re-establishes the first star of David, and finally this yields to the last triangle, Ab E C,

before commencing the whole journey again.

Summation tones lead to the same Circle of Tones/Star of David like structure.

Simply by summing up adjacent notes of one circle of tones the opposite circle of

tones will be created.

Here are one set frequencies relating to the overtone grid of the Lambdoma,

according to the pitch that would appear on any specific ratio:

C 128, D 144, E 160, F 168, G 192, A 216, B 240, C 256

Db 136, Eb 152, F# 176, Ab 208 , Bb 224

Here are the harmonics summed together:

C 128 and D 144 come together and create Db 272

D 144 and E 160 come together and create Eb 304

E 160 and F# 176 come together and create F 336

F# 176 and Ab 208 create G 384

Ab 208 and Bb 224 create A 432

Bb 224 and C 256 create B 480

One Circle of Tones has created the other. C D E F# Ab Bb (one circle of tones)

creates Db Eb F G A B (the other circle of tones).

56

Chapter six

The Invisible aspect of the Triangle of Keys

We have seen that for every Major type scale there is a symmetrical partner that is a

Minor type scale:

Lydian (major) mirrors to Locrian (minor)

Ionian (major) mirrors to Phrygian (minor)

Mixolydian (major) mirrors to Aeolian (minor)

Dorian mirrors to Dorian – the odd one out!

The Dorian is not really the odd one out here. It is actually the most important of all

positions, where the major/minor positions are able to swap-over and receive their

opposite tonal quality. It happens to carry the modal mirror relationship of 2/2, Dorian

mirrors to Dorian, which implies that for F# at the 4.5 of the C major Mode box, to

remain F# in the mirror, it too must be connected to the Dorian position. Is this F# a

2/2 from another major key, for example? After all, we have witnessed a Mode Box

made up of seven major key tonalities either side of the mirror. Finding the F#

connection would help to complete the investigation.

The answer is glaringly obvious when we remember that we are dealing with a series

of triangles of keys that are interrelated through mirroring. Mirroring a major scale in

effect produces another major scale (albeit commencing from its 3rd modal position) .

The keys used, for example, are C major and Ab major, leaving only E major as

some uninvolved aspect of the C Ab E triangle of keys. In the key of E major the F#

position is certainly Dorian.

Ion Dor Phr Lyd Mix Aeo Loc = Modal positions

E F# G# A B C# D# E = E major

It will be seen that the E major aspect of the triangle can be seen to join up with C

and Ab through the 4.5 tri-tone position.

57

Here is line one of the C major Mode Box, with the E major scale dissecting at the tri-

tone position either side of the mirror:

F# Dorian

E mirror axis D# C# B A G# C Db Eb F F# G Ab Bb C D E F F# G A B C E C Phrygian E major D# C major (Ab major) C# B A G#

F# (Dorian) E (major)

What we have here is the usual C major scale, and its mirror, with the E major scale

dissecting both at the 4.5/tri-tone positions either side of the mirror. The Dorian is

perfectly symmetrical, therefore it must be an F# Dorian on the other side too.

This would also make it logical that on one side the E major scale is ascending

through the tri-tone area, and on the other side it is descending through the tri-tone

area. Therefore at F#/F# the relationship has to be Dorian/Dorian.

The Dorian is the point of access to the other side because it is the only dual modal

partnership in a mode box that carries the same number either side, 2/2, establishing

itself as an axis/access point. Include to this this fact:; the number 9 is perfectly

symmetrical, and it comes to rest on the Dorian within a mode box:

1 2 3 4 5 6 7 8 9

C D E F G A B C D

The number 9 seems to be a catalyst to its perfect symmetry, and the two 4.5s are

related to the 9. The Dorian Mode seems to possess a special place amongst the

Positions.

58

By studying the two E major scales in red it will be seen that their opposite partners

still adhere to the correct dual modal partnerships. If the axis is placed horizontally at

the F# positions, the E from the right hand side scale mirrors the G# from the left

hand side scale, for example, and this will equal the usual Ionian/Phrygian mirror pair

as seen through the mode box of C major. Then the D# from the right hand side

(Locrian) will mirror to the A on the left hand side (Lydian) etc.

E (Ion) D# (Loc) C Ionian C# (Aeo) B (Mix) A (Lyd) G# (Phr) C Db Eb F F# (Dor) G Ab Bb C D E F F# (Dor) G A B C E (Ion) C Phrygian E Ionian (Major) D# (Loc) (Ab major) C# (Aeo) B (Mix) A (Lyd) G# (Phr)

Here is the F# Dorian mode mirrored in order to show this more clearly:

Dor Phr Lyd Mix Aeo Loc Ion Dor/Dor Phr Lyd Mix Aeo Loc Ion Dor

F# G# A B C# D# E F# G# A B C# D# E F#

Establishing this link at F# with the Dorian mode will explain the link with the number

9 at the tri-tone position. The 4.5 aspect is like a halving of the 9. The 4.5 adds up to

9 if the digits are simply added together. And halving the 4.5 would produce 2.25,

which again adds up to 9. It seems 9 cannot be broken into another number without

some form of asymmetry ensuing, although both sides of the apparent asymmetry

will always add up to a 9 (the 2/7, 3/6 4/5 1/8 number partnerships, for example). Yet,

this asymmetry is perhaps nature’s smokescreen, or even a cosmic joke, because in

reality, if both sides of the duality are focused on, there is never, in fact, a break from

perfect symmetry. This is due to the 4.5. The mean number between each of the

number pairs is always 4.5.

At arriving at this point we have treated the mirror as though it held an individual life

of its own. Yet this individuality unites with its opposite partner through a hidden tri-

59

tone position. Here the churning of clockwise and anti-clockwise cycling

superimposed over each step of the Major scale formula takes the Major scale

Modes on a `Tonality Journey' through seven other Modes/Major Keys. And having

gone through this perfectly symmetrical ride through the tri-tone, it appears on the

opposite side, once more unbalanced in terms of numerical modal partnership (1/3,

7/4, 6/5 etc), until it eventually arrives at a 2/2, finds unity, and carries on in this vein

continually, swapping over once more in an infinite dance between

expansion/contraction, Light/Dark, Positive/Negative, clockwise and anti-clockwise.

Every example given in this book always lead to this same result.

At the 4.5/tritone positions of each major scale will live this invisible aspect of each

triangle. The F# will be at the 4.5 of the C major scale. The Bb will be at the 4.5 of the

E major scale and the D will be at the 4.5 of the Ab major scale. Together this equals

– C E Ab, with F# Bb D flowing through the centres. All these keys are 4.5 partners of

each other. It would seem that these are the hidden symmetrical flows that unite two

sides of a system into one whole. At least in terms of the nature of number and

frequency cycles. And when these two triangles of frequencies are plotted onto a

circle, one inevitably stumbles upon a star of David symbol.

60

Chapter Seven

Where else?

In this chapter there will be shown small examples of how further to achieve the

triangle of frequencies. These triangles of frequency relationships are engrained in

fundamental ways , flowing through various number grids, music scales and overtone

structures.

We begin by what can be called number pyramids. How they are built is self

explanatory. The thing to focus on is the totals and the resulting frequency

relationships.

1

1 * 1 = 1 = 1 11 * 11 = 121 = 4 111 * 111 = 12321 = 9 1111 * 1111 = 1234321 = 7 11111 * 11111 = 123454321 = 7 111111 * 111111 = 12345654321 = 9 1111111 * 1111111 = 1234567654321 = 4

11111111 * 11111111 = 123456787654321 = 1 111111111* 111111111 = 12345678987654321 = 9

1 + 1 = 2 = 2 11 + 11 = 22 = 4 111 + 111 = 222 = 6 1111 + 1111 = 2222 = 8 11111 + 11111 = 22222 = 1 111111 + 111111 = 222222 = 3 1111111 + 1111111 = 2222222 = 5

11111111 + 11111111 = 22222222 = 7 111111111 + 111111111 = 222222222 = 9

Treating each total as a frequency number we have these notes appear; 1 * 1 = 1,

which is the note C, for example. This assumes the number one as one cycle per

second. Then 11*11 = 121 = the frequency for the note B, and so on.

Multiples pyramid = C, B G Dqt, Bb F# D, A F Db, etc

Additions pyramid = C, F A Db, F A Db, E G# C etc

61

The pattern soon settles down and notes, grouped in triplets, are seen to represent

the Triangles of frequencies that emerged from the Mode Box. After the initial C the

multiples are moving by the interval of a minor 6th, or approximately, after every three

minor 6th movements there is a descent by a quarter tone. If isolated in threes, these

notes all add up to one of the triangles of frequencies of the mode boxes. The

additions are moving by the inversion of the minor 6th, which is the major 3rd , F to A,

A to Db(C#), but after every three moves there the interval ascends by a quarter

tone.

Here is another number pyramid:

4 4 * 4 = 16 = 7

44 * 44 = 1936 = 1 444 * 444 = 197136 = 9 4444 * 4444 = 19749136 = 4 44444 * 44444 = 1975269136 = 4 444444 * 444444 = 197530469136 = 9 4444444 * 4444444 = 19753082469136 = 1

44444444 * 44444444 = 1975308602469136 = 7 444444444 * 444444444 = 197530863802469136 = 9

4 + 4 = 8 = 8 44 + 44 = 88 = 7 444 + 444 = 888 = 6 4444 + 4444 = 8888 = 5 44444 + 44444 = 88888 = 4 444444 + 444444 = 888888 = 3 4444444 + 4444444 = 8888888 = 2

44444444 + 44444444 = 88888888 = 1 444444444 + 444444444 = 888888888 = 9

Multiples of 4 = C, B G Dqt , Bb F# D, etc

Additions of 4 = C, F A Db, F A Db, E G# C etc

These pyramids can be extended further and all four triangles of frequencies will

emerge. These four triangles can be seen as two circle of tones structures, or two

star of David like symbols.

Any set of numbers will produce both the circle of tones structures. The following set

of numbers equal the note's frequency:

62

1 = C11 = F#111 = A1111 = Db11111 = F 111111 = A 1111111 = Db11111111 = E111111111 = G#1111111111 = C

11111111111 = E 111111111111 = G# 1111111111111 = C 11111111111111 = E 111111111111111 = G# 1111111111111111 = B 11111111111111111 = Eb 111111111111111111 = G

Another example:43.2 = F

432 = A 4320 = Db

43200 = E

432000 = G# 4320000 = C 43200000 = E

etcAnd another: 100 = G

1000 = B10000 = Eb100000 = G

1000000 =B

10000000 = Eb

100000000 = G

1000000000 = Bb10000000000 = D100000000000 = F#

63

Within fractions:

Related to divisions by the number 7:

.1 2 4 8 7 5 = C

1 . 2 4 8 7 5 = Eb 1 2 . 4 8 7 5 = G 1 2 4 . 8 7 5 = B 1 2 4 8 . 7 5 = Eb

1 2 4 8 7 . 5 = G

1 2 4 8 7 5 . = B

1 2 4 8 7 5 1 . = Eb

1 2 4 8 7 5 1 2 . = G

1 2 4 8 7 5 1 2 4 . = Bb

1 2 4 8 7 5 1 2 4 8 . = D

1 2 4 8 7 5 1 2 4 8 7 . = F#

1 2 4 8 7 5 1 2 4 8 7 5 . = Bb

Continuing in this fashion will see all four triangles of frequencies emerge. The

decimal point follows a 45 degree angle.

1 2 4 8 7 5 can also be seen as the recurring sequence through binary, or the

sequence that emerges when numbers are doubled or halved (and the totals reduced

to single digit again)

10/17 = 0.5882352941176470

After the number 9 the sequence flips over and begins outlining the mirror numbers:

58823529

41176470

64

Inversions within music follow this rule, in that a 5th will invert to a 4th, for example, or

the 2nd inverts the a 7th. And of course these are the mirror numbers as evident

within a Vedic Square. The number 9, as always, is the sum of each mirror pair. The

triangle of frequencies emerge if we continually move the decimal point to the right,

like this:

0.5882352941176470 = D5.882352941176470 = F#58.82352941176470 = Bb588.2352941176470 = D

5882.352941176470 = F#

58823.52941176470 = Bb

588235.2941176470 = D

5882352.941176470 = F#

58823529.41176470 = Bb

588235294.1176470 = C#5882352941.176470 = F58823529411.76470 = A588235294117.6470 = C#

5882352941176.470 = F

58823529411764.70 = A

588235294117647.0 = C5882352941176470. = E5882352941176470.5 = G#

65

Phi:

1 . 6 1 8 0 3 3 9 8 8 7 4 9 8 9 4 = G#1 6 . 1 8 0 3 3 9 8 8 7 4 9 8 9 4 = C1 6 1 . 8 0 3 3 9 8 8 7 4 9 8 9 4 = E1 6 1 8 . 0 3 3 9 8 8 7 4 9 8 9 4 = G#

1 6 1 8 0 . 3 3 9 8 8 7 4 9 8 9 4 = B1 6 1 8 0 3 . 3 9 8 8 7 4 9 8 9 4 = Eb1 6 1 8 0 3 3 . 9 8 8 7 4 9 8 9 4 = G

1 6 1 8 0 3 3 9 . 8 8 7 4 9 8 9 4 = B

1 6 1 8 0 3 3 9 8 . 8 7 4 9 8 9 4 = Eb

1 6 1 8 0 3 3 9 8 8 . 7 4 9 8 9 4 = G

1 6 1 8 0 3 3 9 8 8 7 . 4 9 8 9 4 = B

1 6 1 8 0 3 3 9 8 8 7 4 . 9 8 9 4 = D1 6 1 8 0 3 3 9 8 8 7 4 9 . 8 9 4 = F#1 6 1 8 0 3 3 9 8 8 7 4 9 8 . 9 4 = Bb1 6 1 8 0 3 3 9 8 8 7 4 9 8 9 . 4 = D

Here is the grid presented in the chapter “Vedic Square” that shows the sequences

created by every number base system:

1 2 4 8 7 5 1

(2) 3 9 9 9 9 9

(3) 4 7 1 4 7 1

(4) 5 7 8 4 2 1

(5) 6 9 9 9 9 9

(6) 7 4 1 7 4 1

(7) 8 1 8 1 8 1

(8) 9 9 9 9 9 9

(9) 1 1 1 1 1 1

66

Having seen just how these triangles emerge through the whole of number, with the

involvement of the 45 degree angle as well, and the in/out mirror effect, it should be

no surprise that the above list of sequences will also expose them. These triangles

signify balance of a dual process. The sequences shown signify how much can come

about from just a small seed like arrangement. If experiments are the food of science,

we can see that a method for perhaps creating balance amongst frequency

relationships could be applicable.

67

Chapter eight

Fibonacci Numbers, number sequences and the Swap

Whilst practically every example given so far is indicative of information swapping between mirror

side and non-mirror side, this next section will concentrate solely on this phenomena.

The Fibonacci numbers are intrinsic to nature in various ways. They are a flow of numbers that

can represent such things as the breeding habits of rabbits, or cause the spiral effect we see as

galaxies. and conifer seeds, or nautilus shells. The Fibonacci numbers show up in the

arrangement of flower petals or leaves. Here are the first few Fibonacci numbers:

1 1 2 3 5 8 13 21 34 55 89 144

The preceding two numbers always become the next new number:

1+1 = 2 1+2 = 3 2 + 3 = 5 , 3 + 5 = 8 , 5 + 8 = 13 and so on.

Nature seems to obey this growth principle and it is also seen that the Fibonacci numbers are

closely associated with the PHI ratio, an infinitely recurring fraction whose first few digits are

1.618… This number is closely approximated within the Fibonacci series, for example:

21/13 = 1.615, 34/21 = 1.619 , 55/34 = 1.617, 89/55 = 1.618

As one includes more numbers from the series, this Phi ratio becomes more consistent. Phi is also

referred to as the Golden Mean or Golden Ratio.

The number sequence partners of the Vedic square will be teamed up with the flow of the

Fibonacci numbers (FN). Or to put it more simply, the FN series will be broken down to single digit.

However, if those single digits are viewed as numbers partners from the Vedic square, then it

allows us to include the mirror flow partners, 1/8, 2/7, 3/6, 4/5.

In doing this it will give the FN flows a clockwise or anti-clockwise characteristic. When a single

digit value for a FN is ascertained, the mirror number partner will be placed underneath, in order to

highlight how the overall flows are swapping sides of the mirror at a number 144. When we get to

68

the number 13 (8+5)) in the series, we cross add the digits and reduce this number down to the

number 4. This means that a number between 1 + 9 always represents the series. NS = Number

Sequence, and MNS = Mirror Number Sequence.

FN - 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946

NS - 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 MNS- 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2 3 5 8 4 3 7 etc.

one cycle

The NS – 1 1 2 3 5 8 4 3 7 1 8 9 becomes the MNS, and the swap-over is at the twelfth position, at

the number 144. As you can see, the number 13 is represented as a clockwise cycling number 4.

To say the 4 is clockwise cycling is referring to the number sequence that the number 4 is a root

of, 483726159, the 4th number sequence of the Vedic square. It’s also been seen how scales have

their own clockwise and anti-clockwise flows, with major tonality always mirroring to minor tonality

and vice versa. And above we see the number 9 allowing a number flow from this side of the

mirror to switch to the other side, and the switch is performed at the 4.5, the mean number running

through the mirror number pairs.

The mirror number sequences above still adhere to the FN. Here they are as if they were a FN series:

FN - 8 8 7(16) 6 (24) 13 (40) 19 32 51 83 134 217 351 568 919

MNS - 8 8 7 6 4 1 5 6 2 8 1 9 1 1

Here the 16+8= 24 (but broken down to a 6 in the MNS flow), the 16+24 = 40, etc.

It is known that the FN produce a repeating 24 single digit number sequence, whereas I have been

speaking of a 12 single digit number sequence along with a mirror 12 digit sequence that swaps

over across an axis point. There is no real conflict between this 12-number sequence and the 24-

number sequence. The only difference is that there is some case here to suggest that actually the

transmission of these numbers is being performed by symmetrical affinity that one number has

with another. There is a relationship here that is consistent with the results from other experiments,

and so the idea of cycles swapping continually from one side of the mirror to the other, may well

69

lead to the idea that the Fibonacci numbers are the builders of natural structures on two sides of

the mirror, or that information from both sides of the mirror acts as one whole.

To help show the above swap-over in the Fibonacci numbers clearer still, here it is again

according to the Indig number system as devised by Buckminster-Fuller. “IN” means Indig

Number, and “MIN” means Mirror Indig Number:

IN - +1 +1 +2 +3 -4 -1 +4 +3 -2 +1 -1 0 -1 -1 -2 -3 +4 +1 -4

FN - 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 etc

MIN- -1 -1 -2 -3 +4 +1 -4 -3 +2 -1 +1 0 +1 +1 +2 +3 -4 -1 +4

Remembering that the Phi ratio is associated with the Fibonacci numbers, as well as the fact that

there is swapping over the two sides of the mirror.

The 4.5 dominates the inner space between the number pairs. The mean number between all the

number pairs is always 4.5. It is also the zero in between the plus/minus indig numbers.

Dividing by the number seven, for example, also shows the same result of dual cycles swapping

over a mirror point:

1/7 = 142857142857

1 4 2 8 5 7 1 4 2 8 5 7

8 5 7 1 4 2 8 5 7 1 4 2

Here the 142 travels in and out if the mirror point as the sequence unfolds, with its mirror partner,

857 also cycling in and out of the mirror , tied to its partner by the symmetry of the number 9, with

its swap-over axis at the 4.5/0.

70

Here is the 3 6 9 sequence, as if it were the beginning of a Fibonacci number sequence. So far the

preceding two numbers (3 and 6), have equaled the next number, so this will be extended to show

the swapping over effect once more:

FN - 3 6 9 15 24 39 63 102 165 267 432 699 1131 1830 2961 4791

NS- 3 6 9 6 6 3 9 3 3 6 9 6 6 3 9 3

The overall sequence is – 3 6 9 6 6 3 9 3. When added together this sequence equals 45:

3 + 6 + 9 + 6 + 6 + 3 + 9 + 3 = 45

We will plot the 3 6 9 sequence once more together with each number's mirror partner, which in

this case is 6 3 9 . This will highlight yet another swapping over of sequences:

FN - 3 6 9 15 24 39 63 102 165 267 432 699

NS - 3 6 9 6 6 3 9 3 3 6 9 6 etc.

MNS - 6 3 9 3 3 6 9 6 6 3 9 3

Between the number 24 the 3 6 9 6 sequence begins to appear on the mirror side, whilst the 6 3

9 3 also manifests on its opposite side of the mirror. And so the journey continues infinitely through

this Fibonacci type flow.

Here we see that the swap-over hasn’t occurred on the number 9. This further strengthens the

idea of the swap-over point, at the 4.5. This of course is the mean number between 3 and 6.

Remember that these number partners (1/8, 2/7,3/6,4/5) are cycling in opposite directions to each

other, clockwise and anti-clockwise. Also that the Phi ratio is still maintained between the

Fibonacci-like flow of the 369 sequence ( for example, 267/165 = 1.618618...).

71

Mirror flow of the Fibonacci numbers

Here is another way to mirror the FN series. The very same result as previously ensues, but it may

help to visualize it this way, and also put the result within a different light. We can set an axis at the

0 and use the clockwise and anti-clockwise number partners of the Vedic Square to highlight the

evolution of the Fibonacci numbers on the mirror side.

Mirror point

(5) (4)135 82 62 29 24 14 1 4 6 7 8 8 0 1 1 2 3 5 8 13 21 34 55 89 144

One would be forgiven in thinking that the mirror side bares no relationship with the Fibonacci

numbers on the right. The first two are correct, but the number 7 totally kills off the sequence. Or

does it? What should have followed the 8 is the number 16 (8+8=16). Well, 16 breaks down to the

number 7 (1+6=7). The next number after the 7, on the mirror side, is a 6. This should have been

16+8=24. Obviously the 6 is the single digit value one obtains when cross adding the 24. It would

also work as 7+8 = 15 (1+5=6). The mirror sider is expressing a form of Fibonacci number growth

but expressing the result as mirror number value according to the clockwise and anti=clockwise

flowing number pairs of the Vedic Square. It’s a different kind of flow, but it does seem to relate

consistently.

On the other side of the mirror the number 13 partners the number 14. Broken down to single digit

this is 4 and 5, which are number sequence partners. This is true for all the symmetrical pairs.

72

Chapter nine

More Number sequence partners found within divisions

There are many constants that actually hide the flow of the number sequence partners

belonging to the Vedic Square. Here are some examples:

11/29 = 0.37931034482758620689655172413793

This recurring 28-figure constant is dominated by the number 9. Yet after the first fourteen

numbers there is a switch-over across the mirror point, and the fourteen mirror numbers of

the original fourteen emerge. Dissecting the equation in the centre where the swap-over

occurs, and then placing the next fourteen numbers under the first fourteen, can show this.

All vertical number pairs, as always, will be seen to add to 9, and be the correct number

pairs from the Vedic Square:

37931034482758

62068965517241

Here we see the 3 from the top sequence partner up with the 6 from the bottom sequence,

for example. All the pairs are mirror partners from the Vedic square. And the 4.5 is always

the mean number between each vertical number pair.

37931034482758

4.5 ------------------------------------

62068965517241

Why such a consistent mirror imprint of the original fourteen numbers is a mystery. But it is

a very common theme too.

11/19 = 0.57894736842105263157894736842105

If we take the first three numbers, 578, we will need to find a 421 in order to suspect that

this constant too is hiding a structure based on the Vedic Square number sequence flows.

There is indeed a 421 in the equation. The pairs as are follows:

73

578947368

421052631

11/38 = 0.28947368421052631578947368421053

Ignore the first number and then the pairs emerge:

894736842

105263157

The number 19 used as a divisor previously is a prime number. Doubling this 19 doesn’t

produce a prime number, but it still produces mirror flow number pairs of the Vedic

Square, after the initial number 2. Many prime numbers possess these flows.

11/28 = 0.39285714285714285714285714285714

After the initial 39 this sequence is based on numbers that are divisible by 7. Therefore we

know that the 285714 is a movement where the 2 and the 7 are partners, the 8 and the 1,

and the 5 and the 4. These pair together every third number.

285

714

11/14 = 0.78571428571428571428571428571429

Again this is a replication of numbers divisible by 7.

11/35 = 0.31428571428571428571428571428571

This one too shows the same set of numbers, after the initial 3. Again this is because

seven divides into thirty five.

11/13 = 0.846153846153846153846153846153

This one shows the mirror flows in the shape of 846 partnering 153 (8 with 1, 4 with 5, and

6 with 3):

74

846

153

11/26 = 0.42307692307692307692307692307692

After the initial 4 the sequence 230769 is another one hiding a number sequence pair, 2

with 7, 3 with 6 and 0 with 9.

307

692

11/23 = 0.47826086956521739130434782608695

Again the first three numbers are 478, so we look for a 521, which sure enough is in the

total, so the signs are that this constant too contains a mirror number sequence pair. The

two sets will be:

47826086956

52173913043

11/17 = 0.64705882352941176470588235294117

The first three numbers are 647, so we look for a 352, which indeed is in the total:

64705882

35294117

11/34 = 0.32352941176470588235294117647058

Here we have to ignore the first two numbers and then the pairs emerge:

35294117

64705882

11/47 = 0.23404255319148936170212765957446

75

Yet again there are mirror flows evident within this constant:

23404255319148936170212

76595744680851063829787

11/49 = 0.22448979591836734693877551020408

Again this will go on to produce number pairs relating to the Vedic square (or indig number system)

224489795918367346938

775510204081632653061

11/52 = 0.21153846153846153846153846153846

Ignore the 21 then:

153

846

There is no grand scientific claim to all this. It is merely interesting that certain fractions

show the swap-over effect that the nine cycles of the Vedic square highlight. Of course

one can also find the same nine single digit contrary flowing cycles in a 9*9 multiplication

table. This is in effect what a Vedic Square is.

The indig numbers again can be applied to these constants in order to represent them with

a clearer picture of the opposing flows prevalent within the number pairs. The first division

will be highlighted again in order to show this:

11/29 = 0.37931034482758620689655172413793

37931034482758

62068965517241

This can now be shown as follows:

+3 –2 0 +3 +1 0 +3 +4 +4 -1 +2 -2 -4 -1

-3 +2 0 -3 -1 0 -3 -4 -4 +1 -2 +2 +4 +1

Or as a graph:

76

One suspects that Prime numbers are playing a significant role in establishing these mirror

flows of Vedic Square number pairs

Dividing by the prime number 17 also gives a constant, which adds up to 72:

1/17= .0588235294117647

2/17= .1176470588235294

3/17= .1764705882352941

Yet, as seen, these constants hide the opposing flows of the Vedic Square mirror cycling number pairs:

05882352

94117647

11764705

88235294

17647058

82352941

These are only the first three examples. The number 19 also behaves similarly:

1/19= .052631578947368421

2/19= .105263157894736842

3/19= .157894736842105263 etc

Again this shows one overall cycle of numbers is emerging, beginning at different points.

This recurring number cycle adds up to 81.

77

052631578

947368421

105263157

894736842

157894736

842105263

The number 23 is another number with a recurring number cycle, which adds up to 99,

and a hidden mirror number sequence flow:

1/23= .0434782608695652173913

2/23= .0869565217391304347826

3/23= .1304347826086956521739

04347826086

95652173913

08695652173

91304347826

13043478260

86956521739

There are endless other examples of this process but these examples should be enough

to show that hidden mirror flows are evident within the division of numbers. This is the

process of things flowing in and out of the mirror at the 4.5 swap-over point, as seen in

music scales, Fibonacci number flows and now divisions. With the added insight that these

numbers also trace the two triangles from the circle of tones, as shown in the previous

section, one may conclude that there is more depth to the idea of mirror flows within every

day numbers.

The line between numerology and natural structure can be mighty fine. The fact that the

common number line has this inbuilt structure may be an open invitation by nature to

ponder its mirror side, including a pathway to the mirror universe.....for all we know.

78

Chapter ten

Both sides number flows – building a mirror universe

One will be entitled to wonder why the number 13, say, should mirror to the number 14.

Hopefully, it will be shown that this relationship is quite natural in the evolution of ‘both

sides’ number flows. But it is only the first way a number can be given a mirror number

partner. It all begins with the number 1 having the number 8 as a contrary cycling number

partner.

Just because the 1 is partnered with the 8 does not mean it is unbalanced. The 1 is a +1

flow and the 8 is a -1 flow, according to the indig numbers. In their positions the1/8 are

contrary flowing and they will cycle through asymmetry because they are sympathetic

toward each other. What is holding the balance is the 4.5 invisible axis running in between.

The case for mirror numbers can be seen as valid when one puts the number pairs

together and builds them up progressively.

1

What is the next total that breaks down to a 1?

10 = 1+ 0 = 1

The opposite cycling number partner of the 1 is the number 8. What is the next total that

breaks down to an 8?

17 = 1+7 = 8

Therefore 10 and 17 are mirror numbers. After the number 9 the whole square is born

again from 10 to 18 in effect:

The 9's being like junction points give rise to a double sine wave that can be traced

through each set of sequences. There must be something that completes cycles, and that

something is the 9, and a cycle consists of a "both sides" journey.

79

0=9 1=8 2=7 3=6 4=5 5=4 6=3 7=2 8=1

9=0

10=1711=1612=1513=14 14=1315=1216=1117=10

The idea of a mirror number table is really quite logical, and the following one can be

called the default one, because it is the steady parallel progress of the mirror number pairs

as they weave through the fabric of all number.

This table of number pairs can be seen as having a dual flow, clockwise and anti-

clockwise. The numbers coloured red can be seen as the negative numbers beginning

their infinite flow in the opposite direction. They will relate to the positive numbers in the

same way as always. For example, one would need to establish a plus/minus relationship,

as in the indig numbers.

And again the four-way mirror relationship is hinted at, because these plus/minus pairs are

on both sides of the mirror. There is the contrary flow this side, and there is the mirror

contrary flow on the other side.

80

Please do not attempt to memorize this list! There is an easier way to know any mirror

number. The rule is, if the first number is less than 4.5, then the mirror number is the next

greater number according to the mirror number pairs of the Vedic Square.

to infinity 3=6 4=5 5=4 6=3 7=2 8=1

axis 0=9 1=8 28=35 55=62 82 = 89 2=7 29=34 56=61 83 = 88 3=6 30=33 57=60 84 = 87 4=5 31=32 58=59 85 = 86 5=4 32=31 59=58 86 = 85 6=3 33=30 60=57 87 = 84 7=2 34=29 61=56 88 = 83 8=1 35=28 62=55 89 = 82 9=0 36=27 63=54 90 = 8110=17 37=44 64=71 91 = 9811=16 38=43 65=70 92 = 9712=15 39=42 66=69 93 = 9613=14 40=41 67=68 94 = 95 14=13 41=40 68=67 95 = 9415=12 42=39 69=66 96 = 9316=11 43=38 70=65 97 = 9217=10 44=37 71=64 98 = 9118=9 45=36 72=63 99 = 9019=26 46=53 73=80 100 =10720=25 47=52 74=79 101 =10621=24 48=51 75=78 102 =10522=23 49=50 76=77 103 =10423=22 50=49 77=76 104 =10324=21 51=48 78=75 105 =10225=20 52=47 79=74 106 =10126=19 53=46 80=73 107 =10027=18 54=45 81=72 108 = 99

109=116 112=113 115=110 110=115 113=112 116=109 etc 111=114 114=111 117=108

81

Here in the numbers one can also see a continual switch over across a mirror point at the

“4.5” positions. It begins in the first strip of numbers in between the 4 and 5. The 4.5

reappears in between the 13 (4) and 14 (5). The relationship 13/14, is swivelled round and

becomes 14/13 at this point.

The 4.5 re-appears in between 22 and 23 (22.5), and again in between the 31 and 32

(31.5) etc. What is established within the first nine numbers is the reoccurring dual number

partner flows of the Vedic Square.

82

Chapter eleven

Opposing forces

To show how opposing triangles affect each other one would need to draw out all

twelve mode boxes, one for each major scale. What follows is a more concise

procedure for understanding the way the mirror relates each triangle with the other

three triangles. This list shows all possible four triangles contained within the

mirroring of each major scale and its seven modes. Therefore all these triangle pairs

form the two circle of tones structure, each comprised of its two triangle relationships.

The (1) and (2) beside each pair of triangles designates which circle of tones they

belong to:

Major scale Related triangles C C E Ab – D F# Bb (1)

D D F# Bb – E G# C (1)

E E Ab C – F# Bb D (1)

F F A Db - G B Eb (2)

G G B Eb – A Db F (2)

A A Db F – B Eb G (2)

B B Eb G – Db F A (2)

Bb Bb D F# - Ab E C (1)

Eb Eb G B – F A Db (2)

Ab Ab C E – Bb D F# (1)

Db Db F A – Eb G B (2)

F#/Gb F# Bb D – Ab C E (1)

It must be remembered that a triangle is primarily three major keys, chords or notes

all threaded together through symmetry. Each major key can be mirrored three times

and will produce one of the triangles. The mode box then shows that, on the mirror

side, the triangles are the result of interrelated modal positions.

These triangles are now found to be each other’s mirror partners if the keys that

make up the remaining triangles are symmetrically reflected. What this means is that

two of the triangles forming one circle of tones or star of David like structure, can be

83

ascertained by mirroring the keys that contain the other two triangles. Let’s begin by

mirroring the key of F major:

S T T T S T T T T S T T T S F Gb Ab Bb C Db Eb F G A Bb C D E F

F Phrygian F Ionian

Here is how the triangles show opposing force with each other (the notes of one

triangle are mirrored across the F axis position):

CC = Bb

E = Gb Bb D

Ab = D (tri-tone)

Ab E

Gb

Here we can see that one triangle equals the other through the mirror. The tri-tone

relationship exists between Ab and D. In order to verify this for yourself, the first note

, C (from F Ionian), is seen to be in symmetry to the Bb note on the other side of the

mirror (F Phrygian scale). This procedure is followed by all the proceeding examples.

The F major scale is not associated with these two triangles, as seen in the table

above. It will become clear that two triangles from each major key are able to swap to

the other two unrelated triangles of the other circle of tones structure. This will

further underline the in-and-out of the mirror type relationship that seems to exist.

The next key to be symmetrically reflected will be A major (the second aspect of the

F A Db/C# triangle).

A Bb C D E F G A B C# D E F# G# A

A Phrygian A Ionian

C = F# (tri-tone)

E = D

Ab = Bb

84

This has yielded a similar result, except that the tri-tone relationship is now between

C and F#. Here is the third key from the triangle, C#/Db major and its mirror:

C# D E F# G# A B C#/Db Eb F Gb Ab Bb C Db

C# Phrygian Db Ionian

C = D

E = Bb (tri-tone)

Ab = F#

Now for the second triangle associated with F A C#, which is G B Eb. Together,

these two triangles of frequencies/keys make up a circle of tones:

G Ab Bb C D Eb F G A B C D E F# G

G Phrygian G Ionian

C = D

E = Bb (tritone)

Ab = F#

There is another interesting result here. The key of G above produces the same

results as the key of C#/Db. This is therefore another tri-tone relationship – G to C# is

a tri-tone.

B C D E F# G A B C# D# E F# G# A# B

B Phrygian B Ionian

C = A#(Bb)

E = F#

Ab(G#) = D (tritone)

Again the result is consistent with the opposing triangles. Also the result for B Major

is the same as for F major, which is yet another tri-tone relationship – B to F is a tri-

tone.

85

Eb Fb Gb Ab Bb Cb Db Eb F G A Bb C D Eb

Eb Phrygian Eb Ionian

C = Gb (tritone)

E = D

Ab = Bb

Lastly one sees that the result in Eb major is the same as in A major – Eb to A is a

tri-tone.

One would expect that the triangles of C E Ab and D F# Bb will show a similar

opposing force with the triangles just used to expose them. Let us only mirror C

major to show this is the case, as we know that the other five major keys show

similar results:

C Db Eb F G Ab Bb C D E F G A B C

C Phrygian C Ionian

F = G

A = Eb

Db = B

As you can see this produces the other two triangles that do not belong within the C

major key and its inbuilt circle of tones structure. Therefore:

C Db

Bb D B Eb

= Ab E A F

F# G

Pick any root note of any major key, and, in symmetry, it will create triangles, which

then oppose each other through the mirror. The triangles are mirrors of each other,

and this is an insight into how vibration can be transformed from one side of the

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mirror to the other. The gateway is through the triangles, at the tri-tone, and related to

the access point through the Dorian Modes. Chapters fourteen and fifteen will

highlight the Dorian connection even further.

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Chapter twelve

Cycling a Chord

In this chapter the “in and out of the mirror” effect is shown through the cycling of

chords belonging to a major scale.

A chord/triad can be built on all seven different notes of the major scale. These triads

are also associated with the relative modal position that any of the notes are

occupying. The root chords of a scale are reckoned quite simply:

1 2 3 4 5 6 7 8 C D E F G A B C = C Major

135, CEG, forms the primary triad built on the note C. So the triad of C is called C

major, but written simply as C, with its major quality implied. This triad is also

occupying the Ionian Mode position.

The next primary triad will be built from the D note, and it will be a 2 4 6, D F A. This

triad is a minor triad and is shown as Dm. This triad is also occupying the Dorian

Mode Position, Dm being the root triad of D Dorian. All the seven root triads are

worked out this way. Next will be the E minor triad on numbers 3 5 7, E G B, and this

triad will be root triad of the Phrygian mode, and so on.

After the root triads it is customary to continue adding notes to these triads. This is

called Extending the chord. To the C triad, CEG, one can add the note B, making the

spelling 1 3 5 7. This chord is then named C-maj7, because it is C major with an

added 7th. The triad of Dm, DFA, can also be embellished with another note, the C,

and this will give the chord the name Dm7 (D minor 7th). This is known as building

chords in thirds. The easy way to visualize it is that there is a skipping of a note every

time a new note is added to a chord. In the case of the triad built on D, one begins

with D, skips the next note, adds the F, skips the next, adds the A, skips the next,

and adds the C.

These chords are usually numbered according to roman numerals. So the chord built

at the first position will be known as the I chord, the chord built on the second

position will be known as the II chord etc.

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I II III IV V VI VII I

C Dm Em F G Am B-dim C

Knowing this will make the ensuing examples easier to understand.

This next list of chords is to be read from right to left:

Dm7 Cm7 A#m7 G#m7 F#m7 Em7 Dm7

D F A C Eb G A# C# F G# B D# F# A C# E G B D F A C

This is an unbroken chain of mirrored chords where the axis shifts to new roots in a

continual cycle. The idea is to start at the far right of the listed chords and

systematically go about reversing the formula of the minor-7 chord. The formula for a

minor-7 chord is 1 b3 5 b7. This simply means that, from the root 1 (red D) move

up a minor 3rd (b3) to the note F, then from the root 1 again move up a 5th to the note

A (D to A is a 5th interval), and from the root 1 again move up a minor 7th (b7) to the

note C (D to C is a minor 7th interval). This formula always describes a minor 7th

chord, beginning on any root note.

The red note D on the right is the central point where we start. By reflecting those

same intervals around that red D we produce the chord of Em7. For example, the

note D to the note F is an interval of a minor third (b3) so we move by this amount in

the opposite direction, D down to B. Do this with the other intervals and the reflected

chord emerges.

b7 5 b3 1/1 b3 5 b7

E G B D F A C

The mirror chord becomes that of Em7. The red note E is now the new root, and we

reflect the chord Em7 in the same way we did previously (Em7 is made up of the

notes E G B D). The result is F#m7. The F# is then made axis point again and the

same procedure follows until the Dm7 chord arrives again in order to commence a

new cycle. That’s a daisy chain of mirror chords!

The flow of chords is always up one tone and of course this will end up as the Circle

of Tones type structure - Dm7 Em7 F#m7 G#m7 A#m7 Cm7. It also flows right

to left as indicated by the big black arrow.

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One question is, does this structure apply to any type of chord, and will this structure

appear in less obvious ways? After all it is quite obvious that the above chords flow

upwards one tone at a time and will add up to a Circle of Tones.

The above list of mirrored chords shows, however, that the six chords still relate to

the original Triangle of major keys (C E and Ab) that has been discussed. With study

it can be seen that Dm7 and Em7 are the II and III chords of the C Major scale, F#m7

and G#m7 are the II and III chords of E Major and A#m7 (Bbm7) and Cm7 are the II and III chords of Ab Major. So far we have uncovered the II and III chords of the

Triangle of Keys, C E Ab.

There are also overlaps. Em7 and F#m7 are the II and III chords of D Major, the

G#m7 and A#m7 are the II and III chords of F# Major and Cm7 and Dm7 are the II

and III chords of Bb Major. Again this points to the Circle of Tones type structure. Not

only do we have represented here two triangles of keys in the shape of C E Ab,

and D F# Bb, but the two joined together are C D E F# Ab Bb which can be

seen as the Circle of Tones. Remember that notes like G# and Ab are the same note

in Equal Temperament.

In the above list of mirrored chords, this is how the Major Keys and the II and III chords flow:

Major Keys - Bb G# F# E D C

Dm7 (III)(II) Cm7 (III)(II) A#m7 (III)(II) G#m7 (III)(II) F#m7 (III)(II) Em7 (III)(II) Dm7

D F A C Eb G A# C# F G# B D# F# A C# E G B D F A C The way to read this is – Dm7 is the II and Em7 is the III of the C Major scale. Then

Em7 is the II and F#m7 is the III of the D Major scale. Then F#m7 is the II and G#m7

is the III of the E Major scale and so on (which produces a circle of tones). All this

flow is moving firstly from one side of the mirror through to the other side and back

again continually.

This next chord shows this in and out of the mirror effect more clearly. The C7 chord

is mirrored this time. Its formula is 1 3 5 b7, and this is the formula that is continually

mirrored, in the same way as in the list of mirrored chords above.

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C7 A#m7b5 G#7 F#m7b5 E7 Dm7b5 C7

C E G A# C# E G# C D# F# A C E G# B D F Ab C E G Bb

Triangle 1

The circle of tones connection is still here yet the chords change between Dominant

and minor-7b5. This is because of Double Reflection. The dominant 7 chord mirrors

to the minor-7b5 and vice versa, for example, C7 = Dm7b5 and Dm7b5 = E7 when

the formulas are mirrored, and as you can see by the flows of mirrored chords above.

Dm7b5 C7

D F Ab C E G Bb

E7 Dm7b5

E G# B D F Ab C

Only in this case this double reflection is happening as we move through the circle of

tones structure, a whole tone step at a time, and the two chords are swapping sides

of the mirror together. C7, the dominant chord, moves across the mirror point to

Dm7b5, This in turn swaps back to a dominant chord on its opposite side of the

mirror, and so on. This is a symmetrical ripple, like skimming a C7 pebble into a lake

and with each succeeding bounce it really traverses two lakes of tonality.

The Dominant chords define one Triangle, C E G#, whilst the m7b5 chord defines

the other Triangle, D F# A#. Therefore it is the triangles that are guiding the flow of in

and out of the mirror.

This last example is the 1 3 5 triad continually mirrored. The triad of C is chosen and

the 1 3 5 formula is continually applied:

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C Gm D Am E Bm Gb Dbm Ab Ebm Bb Fm C

C E G Bb D Gb A C E G# B D Gb Bb Db E Ab C Eb Gb Bb D F Ab C E G

Here you have the circle of tones descending from both the majors and the minors.

Remember to follow the arrow and that the red notes are axis points. Begin at C at

the far right and plot down the 1 3 5 triad. This is made up of a major third (C to E)

and a minor third (E to G). These proportions are then mirrored from the C axis. This

produces the next triad. The major triad C has become minor, Fm. The root of that

minor triad is then made an axis point and the intervals are mirrored once more. This

time it will be a minor third followed by a major third that is mirrored, producing the

next triad of the cycle, which is Bb. The Bb note then becomes the new axis, and the

procedure is repeated until the cycle is complete.

All in all the groups of major and minor triads that emerge are separated by a tone as

they travel in and out of the mirror:

Major triads - C Bb Ab Gb E D = circle of tones (or two triangles)

Minor triads - Fm Ebm Dbm Bm Am Gm = circle of tones

This time the circle has defined itself according to major and minor triads. The circle

of tones structure itself will emerge in a variety of ways, sometimes in a major role

and sometimes in a minor role. If we remember that the triad of C is out of the mirror

and the triad of Fm is in the mirror we can imagine an overall journey here of triads

swapping across the axis/mirror point.

Fm C

Mirror side F Ab C E G

Bb Fm

Mirror side Bb D F Ab C

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Chapter thirteen

Swings four ways

Another phenomena existing because of symmetry also adds further witness that the

two sides of the mirror are well structured together.

After the single notes of a scale, the next step is usually to create the seven primary

triads. The root triad will represent the home chord of the Key.

A major scale is also known to possess a Relative Minor. The minor scale, the

Relative Minor is a scale closely associated with the major scale, through its 6th

position. Music from both of these scales are written in the same key signature. C

Major and A minor are two such scales that are said to be relative to each other.

The major triad and the relative minor triad swap over across the mirror axis point.

Here is the major scale again, and its mirror:

3 4 5 6 7 1 2 3/1 2 3 4 5 6 7 1

C Db Eb F G Ab Bb C D E F G A B C

This was also the very first major scale mirrored. The left hand scale has already

been seen to be a Phrygian. But here we are concerned with how the seven primary

triads relate to each other through the mirror. Here is a list of the seven primary triads

again:

I II III IV V VI VII

C major, D minor, E minor, F major, G major, A minor, B-diminished

Here they are as they appear on a music stave:

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Chords are normally numbered according to the Roman numeral system. Individual

notes are numbered normally, as seen in the mirror scales above. The C major triad

is composed of the notes CEG. If a non-musician study the C major scale, it should

be quite clear that there is a sequence within the choice of the notes used to make

the C major triad; the root note, the 3rd and the 5th. The pattern established here is to

set one note as root, miss another, choose the next, miss the next, and choose the

next. And this pattern is extended greatly within the construction of chords with more

than three notes in them. One simply misses the next note after the 5th, and adds the

7th, then the 9th (same as the 2nd) then the 11th (same as the 4th ) etc. Here is an

example of a C major 9th chord

C E G B D = 1 3 5 7 9

So this is the basic way of harmonizing scales. The mirror triads exist and are built by

reversing the flow of the intervals that make up each individual triad.

The mirror triad of the C major triad would be calculated this way:

C E G = C Ab F

The note C is the axis, so is the same both sides of the mirror. The note E reflects

around the C axis and becomes the note Ab in the mirror (both a major 3rd away from

the axis). Likewise the note G reflects around the C axis and becomes the note F in

the mirror (both a perfect 5th away from the axis). This triad is rather well known of

course; it is that of F Minor. Therefore, what has happened is that a major triad has

mirrored to a minor triad. This is consistent with the fact that a major scale mirrors to

a minor scale.

This C-triad and F minor triad connection is even subtle, because this minor triad is

the relative minor of the Mirror major Key, that of the Ab major we have already seen.

F minor is at the 6th position of the Ab major scale, making it the relative minor of that

scale.

As already explained, and hopefully making more sense, the major triad and the

relative minor triad swap over across the mirror axis point, C major triad for mirror F

minor triad.

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The 6th position of the C major scale is the position for the A minor triad, which is the

Relative Minor. The mirror triad of the A minor triad is calculated:

A C E = Eb C Ab

We are still in the home key of C Major, and that is the visible axis that holds the

symmetry around itself. Therefore the A note reflects to the Eb note, the C note is still

C, and the E note mirrors to the Ab note. This is the triad of Ab major, the 135 of its

own key. And therefore, relative minor swaps for mirror relative major, or A minor

triad swaps for Ab major triad.

The resulting mirror triads of Ab major and F minor, in the mirror key, function as the

root chord (Ab major) and its relative Minor chord (F minor). This is what creates the

four-way relationship. It is found that the root chord from one side of the mirror

reflects to the relative minor chord from the other side of the mirror. And the root

chord from the other side of the mirror reflects to the relative minor chord from this

side. That is some coincidence, given the fact that the modes move in a reverse flow

on the mirror side, and even begin that flow disjointed, with the 3rd mode reflecting to

the 1st mode on the other side of the mirror. And here amongst that disjointedness we

find a perfect relationship:

Root Relative Minor

C major triad A minor

Ab major triad F minor

Things crossover at specific axis points in order to gain access to the other side.

Earlier in history the major and minor tonalities were associated with feminine and

masculine. Here we would see the masculine having its counterpart on the other side

of the mirror, and vice versa. Alternatively, it should be obvious that both sides of the

mirror would require their own duality.

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Chapter fourteen

Dorian Symmetry- Explanation of the Keys

In this chapter we will come to see the perfect symmetry held around the Dorian

Mode axis, and how it is a secret catalyst to the creation of the major and minor scale

system, as used in the West.

It will be shown how the Dorian mode, the second mode of the major scale, becomes

‘pregnant’ with the sharps and flats, and gives birth to the key signatures. I realize

that sounds rather “out there”, but it is justified in terms of what is to follow. It is true

that the circle of 5ths, as we generally use it today, establishes keys whose contents

of sharps and flats increase by one at a time, rising to a maximum of six sharps and

six flats. Below is a circle of 5ths, showing how many sharps or flats are in any

particular Key Signature.

Major keys in a circle of 5ths

C

(b) F G (#)

(bb) Bb D (##)

(bbb) Eb A (###)

(bbbb) Ab E (####)

(bbbbb) Db B (#####)

Gb/F# (bbbbbb/######)

Sharps and Flats

0 1 2 3 4 5 6

C - G D A E B F# - sharps

C- F Bb Eb Ab Db Gb - flats

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Interestingly, one can run the F#/Gb notes along the center in-between the two music

staves. This shows how C and F# reflect the same mirror note positions. More

importantly, if you follow the arrows, you will see that in reaching the F#/Gb axis the

process happens by contrary motion. One would intuit that it is a changing from

clockwise to anti-clockwise flow.

Notice something rather uncanny here, as we plot the scale of C major and its mirror

again:

C Db Eb F G Ab Bb C/C D E F G A B C

The note pairs around the C axis are the same as the major keys that have identical

sharps and flats within their signature. D on one side, for example, is in reflection with

Bb on the other side. The D major key has two sharps and the Bb major key has two

flats. Again E/Ab are mirror note partners. The E major key has four sharps, and the

Ab major key has four flats. And so on through the mirror pairs. This consistency is

rather uncanny, and it will be seen that the idea of a “Dorian tonal pregnancy” is not

as far fetched as first may seem, and is actually rather empirical.

We have already seen how the Dorian dissects the 4.5, at the hidden F# position

either side of the mirror. Does the D Dorian mode hold any symmetrical secrets of its

own? You will see that each new flats and sharp needed to build a new major key

are all born in symmetry around the D Dorian axis.

When building major scales, each new major scale in turn has only one note that is

different to the proceeding major scale. The only difference in notes between C major

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F#/Gb---------------------------------------------------------------------------------

and G major, for example, is the note F#. C major is the right hand scale as shown

above. Here is G major:

G A B C D E F# G = G Major

Think of this major scale as any doh-reh-meh-fah-sol-lah-the-doh sounding scale.

The only note that is different to the C major scale's notes is the F# in place of the

natural F.

The Dorian Mode takes into account both sides of the Duality and deals with both

types of cycles, clockwise and anti-clockwise. It is at the Dorian where the journey

begins into more and more subtle structures and inner mirror patterns.

All twelve major keys have a special link with the D Dorian Mode.

The F# and Gb major keys are at full expansion and full contraction in the circle of

5ths. The F# major scale contains six sharps, and the Gb major scale contains six

flats. These two places are known as the poles within music, and they will be at

180% to C major, the key with no sharps or flats. As each new key progresses and

gains more sharps and flats, this distribution is reflected around the Dorian axis point,

as will be seen. This type of symmetry happens nowhere else but within the D Dorian

Mode.

The Dorian Mode which belongs to the scale of C Major possess no sharps or flats,

in other words, it is all the white notes of the piano beginning at D and ending on D

an octave up or down from that point.

T S T T T S T D E F G A B C D = D Dorian

The intervals that make up this Mode (the Tones and Semi-tones) are then 'mirrored'.

The note D is set as an Axis point and the notes are written as mirror partners around

this axis.

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T S T T T S T T S T T T S T 2 - D E F G A B C D E F G A B C D - 2

The red D is the axis point. See the formula on the right of D reverse itself to the left

of D. The same mode emerges either side of the axis point, but the note partners will

be different. In effect, this is the second line of the C Major Mode box.

Mirror note partners are D/D, E/C, F/B, G/A, A/G, B/F, C/E, which are visible

mirror pairs, and D#/Db, Gb/A#, G#/Ab, A#/Gb, C#/Eb, which are the 'in between'

mirror note pairs. Some of the in-between note partners are shown in parenthesis

next, so you will see how they are in symmetry around the D axis:

center center

D E F G (Ab) A (A#) B C (Db) D (D#) E F (Gb) G (G#) A B C D

Now we focus on each increase of sharp or flat in Key signature that is connected to

one of the two poles, F# and Gb.

G Major contains one sharp and F Major has one flat:

G A B C D E F# G

F G A Bb C D E F

The sharp and the flat appear at the F and the B notes respectively. These two notes

are mirror partners around the Dorian Mode. You can add the sharp and the flat and

see that they mirror perfectly around the D axis point. Here is the D Dorian Mode and

its mirror partner again:

b* #*D E F G A B C D E F G A B C D

One note is a sharp one side of the mirror and the other note is a flat the other side of

the mirror.

The Key of D major contains two sharps and the Key of Bb major contains two flats:

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D E F# G A B C# D

Bb C D Eb F G A Bb

The sharp and flat notes are F and C, and B and E respectively. We know that F and

B are mirror partners; so are C and E.

b* # b #*D E F G A B C D E F G A B C D

This is the pattern as we add more sharps and more flats. The altered notes (the flats

and sharps) are always mirror partners in the perfectly symmetrical Dorian Mode.

The Major scale of A has three sharps and the scale of Eb has three flats.

A B C# D E F# G# A

Eb F G Ab Bb C D Eb

The new altered notes are Ab and G#. Again these are reflected around the axis note

D. This is also halfway point, and the very center of the D Dorian Mode, which is like

an axis point too. G#/Ab is the same when symmetrically reflected around the D axis.

b* # b #* D E F G (Ab) A B C D E F G (G#) A B C D

E Major has four sharps and Ab major has four flats:

E F# G# A B C# D# E

Ab Bb C Db Eb F G Ab

The new altered notes are Db and D#. This is again the result within the D Dorian

mode too where D# mirrors to Db.

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b* # b #* D E F G (Ab) A B C (Db) D (D#) E F G (G#) A B C D

Two more sharps and flats and we will arrive at the poles of F# and Gb.

B Major has five sharps and Db major five flats:

B C# D# E F# G# A# B

Db Eb F Gb Ab Bb C Db

The new altered notes are the Gb and the A#. These two notes are also mirror pairs

around the D Dorian Mode.

# b* # b #* b D E F G (Ab) A B C (Db) D (D#) E F G (G#) A B C D

Lastly F# Major has six sharps and Gb Major has six flats:

F# G# A# B C# D# E# F#

Gb Ab Bb Cb Db Eb F Gb

The new altered notes are E# and Cb. This last result too is similar to that within the

D Dorian Mode and its mirror partner.

# # b* # b #* b b D E F G (Ab) A B C (Db) D (D#) E F G (G#) A B C D

These are the way the sharps and flats progress. The note C is the beginning of the

journey but has no sharps or flats. Every addition of a sharp and a flat within the key

signatures, which develop toward the musical poles, is seen to reflect perfectly

around the D Dorian mode axis.

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To verify the growth of sharps and flats there is then a second witness, the Major

scale of C itself, as seen earlier. This is the home key which gives rise to the D

Dorian Mode.

Look at the C Major scale, its mirror, and the note pairs either side of the axis,

together with the amount of sharps and flats required to build the major scale from

each root note:

0 5 3 1 1 4 2 0 2 4 1 1 3 5 0 C Db Eb F G Ab Bb C D E F G A B C

Gb (6) F# (6)

Mirror partners - C/C D/Bb E/Ab F/G A/Eb B/Db. The center of both scales being

F#/Gb means that this is another axis point. It is interesting that the two poles live at

the center of this natural of Major scales. Was this an accident of Nature? Or is it an

intended seed set in symmetry in order for the duality to contain the potential of self

unity, or in a kind of neutral state, swap over at the poles and then continue infinitely

to higher or lower circles of twelve keys? In fact the answer is yes, because were we

to create the next twelve cycles of keys, this indeed would be the evident structure.

This next diagram shows all the above information in a graphical way and may be

easier to imagine when it comes to the wonderful symmetry that Nature displays

around the Dorian Mode.

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The Dorian Distribution of Sharps and Flats

An example is the C# note pointing rightwards with its arrow to the D note. This

means that C# is the new sharp required to construct the D major scale, and it is

born in symmetry as the Eb note points itself toward the left, in order to construct the

Bb Major scale. The number 2 that joins them up signifies the amounts of flats or

sharps within their respective key signatures. The major scales are reckoned

vertically at these points and the notes colored blue are not usually within the C

major (on the right), or C Phrygian (on the left) scales.

There was a time when major tonality was called masculine and minor tonality was

called feminine. This has grown out of favour nowadays, probably for good reason

because it perhaps sounds a little belittling to women. Although I am sure this was

not the intended purpose. The reason for placing this sexual identity on tonality was

because of the music produced. The major sounding music was quite lively (imagine

marching band music), and considered dispersive. Whereas the minor sounding

music (like Greensleeves) was considered an aspect of sub-consciousness, the

hidden depths of the sea, feminine traits. And this must also have been a reason why

the major key always had a relative minor key.

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If C were in the masculine role and D the feminine, it would also explain why D is the

note 'pregnant' with the six sharps and six flats. The thing about the key of C too is

that it displays 'home' keys, the roots of all twelve major scales. The story being told

when the key of C is mirrored is that of symmetrically linked major scale Key

signatures either side of its axis point.

In the D Dorian (not the key of D but the key of C in its second modal position) is the

story of how the sharps and flats are born and distributed amongst the two poles (of

six flats and six sharps) at opposite ends, and in doing so the journey creates

naturally occurring Keys around a circle of 5ths. This creation is a working system

that complies with the law of symmetry, and it is done within the Home key of 'home'

Keys, C Major. The distribution within the symmetrical Mode of D Dorian is musical

expansion and contraction reaching to the two poles. Here is a circular diagram

showing this symmetrical picture. It may help to imagine this as a circular mode box:

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And here is the same diagram having been warped slightly. Many thanks to Dale

Pond for this very good idea:

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Chapter fifteen

Shades of Dark and Light

Every scale is different. For every scale there is a blending of colours, each

contributing to one overall flavor that contributes to the overall sound of the scale.

Some scales take a trained ear in order to distinguish them apart but other scales are

easily recognized by the masses. In the western culture the major scale and the

relative minor scale are heard within almost all the music we listen to. One can

usually tell the difference between a bright sounding scale and a dark sounding

scale. Some scales sound very “country” and with a happy “major” sounding feel

whilst other scales sound mysterious and mellow, which is the trait of the minor

scale. So what gives a scale this quality, where each shade of colour held within the

scale induces an appropriate response within the listener, and produces such

powerful emotions as to make an audience weep unashamedly, or a person alone in

their room reflect and relive memories that the music is stirring within, as if it

somehow contained the power to make this happen? The answer lies within the

sometimes over-feared field of music theory and is based on the scale’s make-up,

that is, its formula. This is the “colouring” aspect of tonality.

Imagine a bow striking a low note on a violin. Then the player moves the first finger of

his left hand down the violin string, and the note is heard rising higher and higher in

pitch (until it sounds like a cat squeal!). The player can simply slide his finger up and

down the string and will always produce a note. This is true for bicycle pump playing

as well, and highlights the fact that Nature’s Sound is a continuous tone that is there

to be sliced up to our liking. There are no gaps between pitches within natural

sound. This slicing up technique is what is meant by a scale’s formula. Start off with a

low tone and then, instead of sliding with the finger, leave definite gap between each

step that is played. These steps then are like musical slices of the continuous tone.

Each step is given a letter between A and G. There is a numbering system too that is

used to describe each step of the overall formula.

Continuous tone rising and falling

Above is continuous pitch traveling higher and higher, or lower and lower. Below, the

sound is sliced up into a scale.

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C D E F G A B C

The note is struck between the gaps in the line, at a pleasing ratio (mostly) between

the beginning note and this new slice. The brightness and darkness of a scale

depend on the arrangement of these steps. Almost everyone can hum the doh reh

meh fah sol lah teh doh scale. Each of these seven steps that exist within this scale

is predetermined and not accidental. Each step is based on a leap upwards or

downwards in pitch by a certain ratio.

Slicing the continuous tone creates steps and in turn steps create formulas, that is,

collections of steps. When this is understood one can imagine how many different

sounding scales there are. Musical scales are not made up of exactly the same

arrangements of steps and so not only sound different to each other, but also give

rise to different effects on the listener.

It is the formula that reigns within the building of scales and not necessarily the actual

pitches of the notes themselves. The formula is the provider of colour, and therefore

the scale’s place on the musical light/dark spectrum.

There is a system within the major scale known as the modes. Further to this the

modes are known by their light or dark qualities. Here are the different shades

relative to the Modes of any major scale, beginning with the brightest sounding:

Lydian brightest

Ionian

Mixolydian

Dorian axis

Aeolian

Phrygian

Locrian darkest

It should be no surprise that the perfectly symmetrical mode, the Dorian, contains the

axis point between light and dark tonality. In this experiment we will start with what is

regarded as the darkest mode of a Major scale, the Locrian. This mode actually

mirrors to the Lydian mode, which is regarded as the brightest mode of the Major

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scale. The Lydian mode is the brightest because of the formula it is built on. There is

what musicians call a sharp 4th within its make-up so it is even brighter than the Major

scale (Ionian Mode) itself. This is because the Major scale contains what is called a

Perfect 4th, which is one semi-tone lower than a sharp 4th, and it is that extra raising

of one semitone that has made the Lydian Mode even brighter than the Major scale

(you will see this in the next diagram where the “#” shows expansion/light, and the “b”

shows contraction/dark). What makes the Modes darker in quality within this major

scale is the addition of flatted notes (shown as a “b” sign). The last Mode in the

diagram below contains a b2 b3 4 b5 b6 b7 within its formula and one can see why it

is the darkest Mode. You can see this shift from light to dark if we show the formulas

required to make up each mode.

To establish such a table one must use the same root note for every one of the

seven formulas, for that is how the shades of colours can be seen to be flowing

through the circle of modes. The first line’s information can be interpreted as – T, T ,

T, S, T, T, S, and is placed in the gaps in between the numbers. This next list uses

the note C as the root of all seven modes:

1 2 3 #4 5 6 7 8 (C Lydian mode formula) - Brightest

1 2 3 4 5 6 7 8 (C Ionian mode formula)

1 2 3 4 5 6 b7 8 (C Mixolydian)

1 2 b3 4 5 6 b7 8 (C Dorian) Axis

1 2 b3 4 5 b6 b7 8 (C Aeolian)

1 b2 b3 4 5 b6 b7 8 (C Phrygian)

1 b2 b3 4 b5 b6 b7 8 (C Locrian) Darkest

The Ionian mode is actually the very root of the parent major scale. The scale above

it, the Lydian has its raised 4th and is the brightest because every scale within the

structure of the seven modes is darker in quality than the Lydian mode. The

Mixolydian contains one flat, and the Dorian contains two etc.

To really explain this one would again need to go back to music basics. I do realize

all this may sound rather confusing for non-musicians, and study will be required in

order to understand why these particular formulas mean what they do. When the

above list is represented on the music stave, as shown shortly, it should become

clearer as to why these formulas are used. So if some are rather unsure as to the

use of these flats and sharps, remember that we are dealing with the circle of 5ths,

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which contains twelve major scale key signatures. Therefore, if C were root note of

all the modes above, then each mode would reside within a different major scale.

Knowing how many flats or sharps each major scale contains within its key will help

to establish the right understanding as to how musical light and dark is distributed.

Basically , these C based scales in the above list are getting darker and darker

sounding, started from the Lydian mode, which is the brightest.

The bottom mode in the above diagram, the Locrian, contains five flats. Therefore it

resides within a major scale that contains these five flats as part of its Key signature.

We know that Db major is that major scale. The C Locrian will be the seventh mode

of that major scale, because the Locrian mode is always the seventh mode. Here is

the Db major scale, with its modal positions written above the notes:

Ion Dor Phr Lyd Mix Aeo Loc Ion

Db Eb F Gb Ab Bb C Db = Db Major

Therefore the previous list based on modal colour is still related to the circle of 5ths,

as the major scales evolve, and the shades of light and dark that it creates are a

natural progression, with the Dorian mode position being its central axis point.

Here is another way of seeing the above information. All the modes are generated

from the C root position (C Ionian, C Dorian, C Phrygian, C Lydian etc):

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It is no surprise now that the Dorian Mode is the Constant (symmetrically reflects as

Dorian both sides of the mirror) as this is where light and dark meet, and it will be

directly over the F# point within the C Dorian Mode. It affirms the Dorian as having an

axis quality of its own as seen previously. The tri-tone position demands it and it

demands it at the dissection of 9 into two equal halves either side of the mirror, at the

4.5 positions. The Dorian can gain access to the mirror side, through its perfect

symmetry.

Also you will find that the above Modes, all built on the root note C, belong to the

following Major scales:

Db Ab Eb Bb F C G

This means that the first mode of C Locrian resides within the Db Major scale, the C

Phrygian within the Ab Major scale, and so on. If we put these Major scales in

sequence starting from the note C we have this list of scales:

C Db Eb F G Ab Bb

And these are the notes required to build the C Phrygian mode, which is the mirror

scale of C Major, as shown in the mode box. This C Phrygian Mode brings us back to

the very first step of the mirror process, a cycle of light to dark and dark to light

complete, perhaps only spiraling faster and faster and slower and slower.

This next diagram takes the C major modes through a circle of 5ths beginning with

the brightest mode, F Lydian. These shades are then mirrored, and in effect, this is a

mode box arranged in the right sequence of light and dark, that is, through the cycle

of 5ths. If you have made it this far into the book you should probably know how to

mirror scales by now.

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Mode Box in shades of light to dark through cycle of 5ths

7 LOC F Gb Ab Bb Cb Db Eb F G A B C D E F LYD 4

3 PHR C Db Eb F G Ab Bb C D E F G A B C ION 1

6 AEO G A Bb C D Eb F G A B C D E F G MIX 5

2 DOR D E F G . A B C D E F G A B C D DOR 2

5 MIX A B C# D E F# G A B C D E F G A AEO 6

1 ION E F# G# A B C# D# E F G A B C D E PHR 3

4 LYD B C# D# E# F# G# A# B C D E F G A B LOC 7

7 LOC F Gb Ab Bb Cb Db Eb F G A B C D E F LYD 4

The right hand side of this mode box shows the progressive shades of colour, with F

Lydian (4) being the brightest sounding scale, then the C Ionian being the next

brightest, all the way down to the B Locrian (7), which is the darkest sounding scale.

This then correlates to an opposite journey on the mirror side. The thing to look for

here is that the top left scale starts with maximum contraction, in the form of six flats,

at F Locrian (7). The next line has only four flats, and the line after that has two flats,

until the Dorian, which is where light and dark meet, balances out and switches over

the contraction for expansion. The journey continues with two sharps, four sharps

and six sharps. At this point, if the cycle is repeated, we see a point where maximum

contraction becomes maximum expansion, between B and F. This B to F is the

visible tri-tone of the C major scale.

This diagram says a lot more than at first seems to be the case. Here we have

tonality distributing expansion and contraction as the circle of 5ths evolves. There is

no faltering, or creation of gibberish on the mirror side, but actually a well laid out

structure, showing how the Dorian is perfect point of symmetry, and swap-over axis.

As it has been shown that the 4.5 is also the swap-over axis, then it should be quite

clear that there is a Dorian aspect to the 4.5, as seen in the chapter “the invisible

aspect of the triangle of keys””. The Dorian and the 4.5 are literally partners in the

tonal fountain, where light and dark have their axis, and the swapping over of

qualities is performed. Swapping from major type to minor type, from expansion to

contraction, from tonal light to tonal dark. Positive to negative, as in Buckminster-

Fuller's indig number (where -4, for example, switches to +4, as in full contraction

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becoming full expansion), clockwise to anti-clockwise. The Dorian/4.5 is where

nature has a unity, between both sides of the mirror. It is perfectly symmetrical at this

point, and can witness the whole of duality.

Here is a simple chart showing the evolution of light/dark within the above left hand

side of the mode box in true shades:

bbbbbb

bbbb

bb

DORIAN

##

####

######

bbbbbb

bbbb

bb

DORIAN

##

####

######

This compares to the way the light/dark quality of the indig numbers expand and

contract across the 4.5 axis point, shown in the “Vedic square” chapter, where it is

seen how maximum expansion immediately becomes maximum contraction at the

swap-over point, and the 4.5 is in effect the zero point. Here it is shown in music and

number fashion, as nature would inevitably leave its imprint on such structures.

Could this map, showing the symmetry of expansion and contraction, be of any use

to anyone other than a musician?

It just so happens that F Lydian , the brightest mode, mirrors to F Locrian, the

darkest. They exist at the 7th and 4th position, or the 4th and 7th position of their

respective major scales. The light/dark relationship between the two F's is echoed in

a similar light/dark relationship through the two B's, also a 7/4.

4 F Lydian/Locrian 7

7 B Locrian/Lydian 4

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dark light

F Locrian - F Gb Ab Bb Cb Db Eb F G A B C D E F - F Lydian

light dark

B Lydian - B C# D# E# F# G# A# B C D E F G A B - B Locrian

One journeys in through the F and comes out through the B. Both B/F and F/B are

the visible tri-tone positions of the C Major scale and its mirror. Notice that F Locrian

and B Lydian are in effect the same scale.

Gb = F#

Ab = G#

Bb = A#

Cb = B

Db = C#

Eb = D#

F = E#

In one manifestation the scale is full expansion, and in the other manifestation it is full

contraction. These two scales are modes and therefore they belong to a parent major

scale. The scale that these two modes belong to is F# or Gb Major, the two poles.

This is what ties in the visible tri-tone aspect with the invisible. C to F# is one tri-tone,

and F to B is the other. The F# is not visible in the C major scale, nor the mirror of the

C major scale. The F and B are visible in that they are two notes that belong to C

Major and are a tri-tone apart (F - G -A - B = three tones). When thinking of a pole

shift, we can see how it is possible musically.

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Shades from the Center

Again, there is a hidden structure behind the distribution of these shades of tonal

colors, and it is emerging from the center of each formula. Observe this diagram:

Darkest mode

Phr. Lyd 4

7 Loc.

S T T S T T T

Brightest mode

The formula from left to right is that of the Locrian Mode (STTSTTT). Reversing the

flow of the formula gives us the Lydian mode (TTTSTTS). So the brightest sounding

mirrors to the darkest. Here is a Lydian scale starting on the note C

T T T S T T S

C D E F# G A B C

This is the brightest mode of the major scale family. Now if we use this formula in

reverse, the result is the C Locrian mode, the darkest.

S T T S T T T

C Db Eb F Gb Ab Bb C

The flows meet in the middle. Commencing the next scale from the central semi-tone

in the formula above would create the Phrygian mode (S T T T + S T T). If we

continue our experiment using the Phrygian mode as our new starting point we will

see that it is the next darkest mode after the Locrian that emerges, and it is paired

with the next less brightest mode. And in the middle waiting to start the next formula

is the Aeolian mode, which is the correct shade that develops next.

Aeo 3 Phr. Ion. 1

S T T T S T T

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The Phrygian mode is one shade lighter than the Locrian mode, and the Ionian mode

is one shade darker than the Lydian mode. As you can see the two respective modes

flow left to right and right to left, creating the respective shade; S T T T S T T creates

the Phrygian mode, and T T S T T T S creates the Ionian mode. Therefore this is

simply the Ionian/Phrygian relationship we have uncovered before. The formulas

meet in the middle, and if we make this Aeolian point (built using TSTT+STT) the

new starting point of the next scale we get even brighter on one side whilst getting

darker on the other. The Aeolian will mirror to the Mixolydian mode.

Dor

6 Aeo Mix 5

T S T T S T T

As well as the dual looking formula, which creates shades of tonal colour, the next

mode is waiting in the center in order to evolve the next pair of shades between light

and dark tonality. Let’s continue in this vein with the other modal partners. Dorian is

the next starting point in the dual journeys:

2 Dor Mix Dor 2

T S T T T S T

5 Mix Ion Aeo 6

T T S T T S T

1 Ion Lyd Phr 3

T T S T T T S

4 Lyd Loc Loc 7

T T T S T T S

So, the Modal structure of the Major scale is seen as evolving in sequential shades of

brightness or darkness from a central point within each formula, which can be read

either left to right, or right to left. The above information is related to the Mode Boxes

in a subtle way. The modal partners are consistent with those found in the C Major

Mode Box, like Ionian/Phrygian, Dorian/Dorian etc, and if anything we are seeing

how the inner qualities given to the different shades of the Modes springs from a

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hidden axis at the central tri-tone position. This center, 4.5 position, is in effect a

Tonal Fountain.

The last example in the above list throws up a rather interesting oddity. The Locrian

mode is mentioned twice. It’s as if the journey is about to commence again through

the mirror at this point, with the Locrian mode waiting in the middle to start the

process all over again. The Locrian Mode on the mirror side of the C Major Mode Box

is that of F Locrian which resides within the Key of F#/Gb Major. This interplay

between the invisible axis and visible axis, swapped over at the tri-tone position, will

be seen in the chapter “In and Out of the Mirror”. It is the place where things are

made manifest on the opposite side of the mirror. The way in which these

relationships flow is not necessarily confined to musical scales. One must surely ask

in what other fields of study viewing information as swapping over from mirror/non-

manifest to non-mirror/manifest may be of benefit.

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Chapter sixteen

In and Out of the Mirror

If the non-musician can at least understand the principle behind this next series of

diagrams, they will get an insight into how one may view musical information as

existing firstly on one side of the mirror and then on the other side, in a continual

sequence, which is held together by the function of the Circle of Tones structure, that

is, the two triangles of frequencies. There have been other musical examples that

have shown this process, and there are of course also numerical ways of seeing this

information (as seen in the flow of the Fibonacci numbers, for example). There is a

continual swapping between what I have termed the invisible axis to the visible axis.

One moment the information is part of the visible scale or sequence, and then next it

is emitting from the in-between axis, or rather, the uninvolved axis, from the other

side of the mirror.

To help weed through the jungle of music theory, the non-musician may like to bear

in mind that they merely need to be satisfied that the 45-degree angle of these music

scale boxes show the flow of this circle of tones structure. It isn’t only the 45-degree

angle that carries this structure. It in fact occurs vertically along the scale box, and

then transfers itself to the 45-degree angle. Cyclic events discussed so far always

have this structure flowing through them.

The other two Mode Boxes

The whole mode box of C Major can be transposed using the other keys that made

up one Triangle (C Ab E). You will see all the modal relationships that occurred in C

major’s mode box remain intact in the next two mode boxes, even though other notes

are being used; the formulas that build the modes being exactly the same. Therefore

any musical relationships established in the C major mode box will remain true for

the other two Mode boxes. Plus we are also able to observe how the triangles of

keys continue their flow at the 45-degree angles.

The same two triangles are flowing at this angle within the left hand side of the Ab

Major Mode Box, yet the arrangement of them and the axis pitch have changed. In

the C major mode box, the axis pitch that held the two triangles around it was the

note Bb. Even though there has been a change, the axis point at Gb still maintains

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two triangles of keys either side of it. This circle of tones structure is alive at the 45-

degree angle.

Mode Box Two - Ab Major

PHR Ab Bbb Cb Db Eb Fb Gb Ab Bb C Db Eb F G Ab ION

DOR Bb C Db Eb F G Ab Bb C Db Eb F G Ab Bb DOR ION C D E F G A B C Db Eb F G Ab Bb C PHR

LOC Db Ebb Fb Gb Abb Bbb Cb Db Eb F G Ab Bb C Db LYD

AEO Eb F Gb Ab Bb Cb Db Eb F G Ab Bb C Db Eb MIX

MIX F G A Bb C D Eb F G Ab Bb C Db Eb F AEO LYD G A B C# D E F# G Ab Bb C Db Eb F G LOC

This looks rather complicated but it is the same type of information as in the first

mode box. The only difference is that other notes are being used, but all the formulas

and relationships are exactly the same as the C Major mode box.

The circle of tones is now grouped up as – Ab C E – Gb - Bb D F# - with the Gb

acting as an axis between them. This was actually the flow of notes along the vertical

line in the C major mode box, and the implication is that the 45-degree angle has

become the swap-over point, as the journey through the triangle of keys continues.

The triangles that appeared vertically along the C major mode box, have now

reached the 45-degree angle and will swap over to the other side of the mirror..

The central axis position between the two triangles is always made up of the

enharmonic pitch held at the Lydian position on the mirror side (Gb in this case is an

enharmonic of F#), the last relationship shown as F# in the above mode box. There

is a swap between expansion and contraction, musically speaking. The axis appears

at the Locrian mode position, the Gb sitting there in the center of the two triangles

above.

Remember that the flow of these triangles is cyclic. If one drew out a 32 by 32 mode

box, one would see the Gb and F# continually change the contraction/expansion

aspect.

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This 4-Lydian/7-Locrian relationship occupies the interval that has been shown to

being the visible tri-tone within a major scale. Earlier this was established as the

notes F and B. In the above major scale it is between the notes Db and G. The

distance between these two notes, as always, is that of a tri-tone (Db – Eb – F – G, is

three whole tones). This is now the visible tri-tone of the Ab major scale, always at

the 4/7 positions:

Visible tri-tone

Ab Bb C Db (D) Eb F G Ab

Invisible tri-tone

It is at the visible tri-tone position of a scale that the invisible tri-tone meets. When

understood, one will find this overall set of relationships quite uncanny. The invisible

hands over to the visible, in a continual swapping-over effect, which occurs at a

special point within the cycles. You will see this occurring within all Major scales

when mirrored. This process shown to be evident within natural cyclic phenomena is

too re-occurring for it to be an accident of Nature. The invisible tri-tone in the scale

above would be between the root note, Ab, and the note D (in between the Db and

Eb, at the 4.5 position). This handing over from visible to invisible will be witnessed

again when the mode box for the scale of E is drawn.

The Lydian is known as the brightest mode of the major scale, whilst the Locrian is

known as the darkest mode. So here we also have a combination of Light and Dark,

wherein a hidden symmetrical link with the tri-tone axis positions is seen to swap

from contraction to expansion. There have been many examples of this in the book.

We have plotted two of the mode boxes that relate to one triangle of Keys. The E

Major Mode box is the third such mode box that exists in an unbroken thread within

the C E Ab triangle of keys. This unbroken thread could also be viewed as a kind of

seed, or three things in one.

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BOX THREE E MAJOR

PHR E F G A B C D E F# G# A B C# D# E ION

DOR F# G# A B C# D# E F# G# A B C# D# E F# DOR

ION G# A# B# C# D# E# Fx G# A B C# D# E F# G# PHR

LOC A Bb C D Eb F G A B C# D# E F# G# A LYD

AEO B C# D E F# G A B C# D# E F# G# A B MIX

MIX C# D# E# F# G# A# B C# D# E F# G# A B C# AEO

LYD D# E# Fx Gx A# B# Cx D# E F# G# A B C# D# LOC

Here the same circle of tones exists at the 45-degree angle, albeit mostly as

enharmonic equivalent notes. The Cx is really a D note (the x means double sharp),

and the B# is the enharmonic of C. So here we still have the E, Ab/G#, E, and the D,

F#, Bb/A# triangles.

The middle axis at the 45-degree angle has shifted again. This time it is D. It should

be noted that the same circle of tones is also occurring vertically in all three mode

boxes, but is disguised as seven modes, each belonging to different major scales.

The vertical flips to the 45-degree angle after every cycle of seven modes, using

three major third moves that defines one triangular relationship. It does this for as

long as vibration and number can be counted and plotted, that is, infinitely. More

uncanny is the fact that the Cx key (C double sharp) is really a gateway to another

circle of 5ths. This will be shown in the next chapter.

As mentioned, the note D, which was the invisible tri-tone note of the previous scale,

becomes the visible note that is axis to the two triangles. More examples should

make this quite clear. Here is the scale of E major with the visible and invisible tri-

tone relationships:

Visible tri-tone

E F# G# A (Bb) B C# D# E

Invisible tri-tone

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The invisible tri-tone notes of all three modes boxes have been F#, D and Bb, which

itself is one of the two triangles that make up the circle of tones. The three roots are

of course the C Ab E triangle.

Below is a visual of the three mode boxes and their 45-degree angle activity. In effect

they are a series of Ionian/Phrygian relationships traveling along the diagonal of the

mode box.

Helter skelter mode boxes

C Major Ab Major E Major

C C Ab Ab E E

E C C Ab G# E

G# C E Ab B# E

Bb C Gb/F# Ab D E D C Bb Ab F# E

F# C D Ab A# E

A#/C F#/Ab D/E

This arrangement between the triangles occurs in two ways within a Mode Box.

Across the 45-degree angle of the left hand side of the whole C Major Mode Box

emerges the same notes as above and they will be seen to occupy the same type of

modal quality/position throughout the other two mode boxes. In other words the first

triangle starts as C Phrygian, then across the 45-degree angle, becomes E Phrygian

and G# Phrygian, with the Bb Phrygian acting as axis point to other triangle, then D

Phrygian, F# Phrygian, and A# Phrygian. This modal relationship covers both

triangles, so the 45-degree angle is describing a set of seven Phrygian Modes. Two

of the Phrygian modes are enharmonic of each other, for example, A# and Bb. And,

as mentioned, these enharmonic equivalent pitches are necessary for the

transference from visible axis to invisible axis.

We know that these triangles of keys that keep emerging are connected to clockwise

and anti-clockwise cycles. They represent a system where a major/clockwise cycle is

always mirrored to the opposite minor/anti-clockwise cycles. To be ongoing this

procedure would require a traveling from one side of the mirror to the other, back

again and so forth. And it may be that this is what the zigzagging motions of these

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three mode boxes laid out next to each other are showing, one 45-degree angle

becoming the swap-over point into the other 45-degree angle. This would cover all

three mirrors. It could be seen as three generations of matter, according to their

frequencies, but each generation swapping from one side of the mirror to the other.

Further evidence of this two-way mirror relationship that continually swaps over can

be seen musically in this next experiment, where the three mode boxes that

comprised one triangle of keys relationship will be looked at more closely.

Let's start with the Ionian mode. We move down the C Major mode box and find the

Ionian on the mirror side (left hand side of mode box). It is at the note E. We then

move to the Ab Major mode box and find the Ionian residing on the mirror side there

too, and it falls on the note C. Lastly we repeat this procedure with the E Major mode

box. The Ionian modes appeared on these notes:

E C G#

As you can see this equates to an Augmented Triangle of Keys. No surprise there

really. Yet if we follow this procedure with all the Modes we will see how this triangle

remains constant and how it combines with the other triangle found on the right hand

side of the mode boxes to form the Circle of Tones. In fact both possible circles of

tones are involved.

To ascertain the other triangular modal relationships we follow a similar procedure to

the first example by focusing on which notes within the three mode boxes each

particular mode is found. This is also the first clue that the augmented triangles flow

from one side of the mirror to the other, and the main reason for focusing on the

results. The same numerical relationships are also involved, and may explain the

natural double reflection principles between the relationships. Imagine the first line

as a 1/3, then the second line as a 2/2, just like in the mode boxes. Here is the full

list:

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Modes occurring on the mirror side of the three mode boxes non-mirror side

Ionian (on the notes) E C G# C Ab(G#) E

Dorian D Bb F# D Bb F#

Phrygian C G# E E C G#

Mode's Major keys -(F#) (D) (A#) (C) (G#) (E)

Lydian B G Eb F Db A

Mixolydian A F C# G Eb B

Aeolian G Eb B A F C#

Locrian F Db A B G D#(Eb)

Take the Phrygian line as another example. Here we find that the Phrygian modes

appear on the mirror side of the three mode boxes at the notes C G# and E. On the

non-mirror side they appear at the notes E C and G#. Both sets have appeared on

either side of the mirror.

The other lines are also replicating. All except the Dorian, which shows that it has

direct access to the other side, by virtue of its perfectly symmetrical quality. The

Dorian is a special position, and it was shown how, at the 4.5 position in the center of

the Major scale, there is an invisible “Dorian” quality, which is why F# equals F# on

the other side of the mirror, for example.

In the diagram above, from the Lydian mode onwards the notes, when put in

sequence, form a Circle of Tones – Db Eb F G A B. This is the other possible circle

of tones (which is also the other two possible triangles of keys). The red notes in

brackets above are the Major scales that house the particular modes creating this

circle of six Lydian keys. By doing this the other circle of tones emerges. The B

Lydian resides as a mode within F# Major, for example, and the G Lydian mode

resides within D major, and so on.

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What is quickly apparent here is the Triangle of Keys relationship existing between all

the modes. Each set of modes is one of the four triangles that make up the circle of

tones. Each note is separated by a minor 6th interval (the inversion of a Major 3rd),

which eventually defines each series of notes as an Augmented type Chord/Triangle.

The triangle of notes, chords and Keys are always defined by the intervals 1 3 #5 , in

reverse or some inversion of them which is still the formula for an Augmented chord.

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Swapping the invisible tritone for the visible tritone

The next set of diagrams further highlights the visible and invisible axis relationships

that exist through the symmetrical reflections of a plain C major scale and its modal

structure. Here is a stripped down version of the mode box of C major:

Mirror point

C - PHR C C C C - ION

D - DOR E D C D - DOR

E - ION G# E C E - PHR

F - LOC Bb F C F - LYD

G - AEO D G C G - MIX

A - MIX F# A C A - AEO

B - LYD A# B C B - LOC

C

We can see that the note C is in symmetrical reflection to a series of notes

throughout the mode box. In analyzing these relationships one gains an insight into

the cyclic events that occur either side of the mirror, and how axis points interact

within the whole picture. Here is a list of these relationships:

C = C in C Phr/Ion

C = E in D Dor/Dor

C = G# in E Ion/Phr

C = Bb in F Loc/Lyd

C = D in G Aeo/Mix

C = F# in A Mix/Aeo

C = A# in B Lyd/Loc

One can see that every enharmonic pair (the Bb and A# in the above example)

actually relates to what can be called the visible tri-tone interval, F to B in this case,

or F Lyd/Loc – B Loc/Lyd. What is interesting here is that the Bb and A# are positions

occurring within the Gb and F# major keys, which is the invisible tri-tone interval from

C. The Bb resides at the F Lydian (which comes from C major), and F Locrian (which

125

comes from Gb major). The A# resides at the B Locrian (which is C Major), and the B

Lydian (which is F# major). Here is the scale of C once more to show the two Tri-tone

points:

Invisible Tri-tone axis

C D E F (F#/Gb) G A B C

Visible Tri-tone

1. F Lydian - C Major

F Locrian - Gb Major

2. B Locrian – C major

B Lydian – F# major

In the mode box diagram above it is seen that on one side of the mirror is the root

note along the 45-degree angle, that is, the note C. In reflection to the root is the

Circle of Tones, flowing as two triangles of Major thirds either side of a central

position. Bb acts as centre to C E G# one side and D F# A# on the other. In fact

every note on the left hand side creates triangles along the 45-degree angles. If one

were to focus on the note B occurring along the 45-degree angle on the right hand

side of the mode box, there too in symmetrical reflection to it would be a circle of

tones.

Db B

F B

A B

Cb(B) B

Eb B

G B

B B

Along this particular 45-degree angle we see clearer evidence of the two tri-tone axis

points playing the visible/invisible role. The Cb(B) position on the mirror side

Is occurring at the F Lyd/Loc position, and in reflection is another B on the non-mirror

side. This is one tri-tone axis (the Cb/B is at the tri-tone position in either F scale both

126

sides of the mirror). The B/B also occurs at the bottom, at the Loc/Lyd position. It was

already seen how this tri-tone is a hidden Dorian aspect too, and it is this ability to

have direct access to the other side of the mirror that allows the Dorian to swap the

full Light/Expansion for the full Dark/Contraction. This means, musically, that the

brightest Mode (the Lydian) swaps for the darkest mode (the Locrian). It is also a

four-way swapping, Lydian for Locrian, and Locrian for Lydian.

What we see is the information around the root position axis being replicated at the

tri-tone axis. For example, if one look at the original Mode box for C major, they will

notice that at the B Locrian/Lydian position the notes C and A# are mirror partners.

This is also true for the F Lydian/Locrian position in the same mode box, where C

and Bb are mirror partners. Sharps and flats have swapped, which signifies a swap

from light to dark, tonally speaking.

C = A# in B Loc/Lyd

C = Bb in F Lyd/Loc

This means that a Mode Box is a complete unit unto itself displaying all twelve notes

of the chromatic scale. In fact if one count notes like Db and C# as two separate

notes there are displayed in the Mode Box in all twenty different notes, including a Cb

and E#.

Each unit is replicated twelve times and is representative of all twelve Major and

Minor keys that evolves through the circle of 5ths. The next diagrams again shows

how every note of C Major along its 45-degree angle creates a Circle of Tones on the

mirror side, made up of two triangles separated by the major 3rd interval.

F G

A G

C# G

Eb G

G

127

Following each notes individual 45-degree angle exposes how triangles are always

formed on the mirror side at the corresponding 45-degree angle. F A C# is one

triangle of keys, which belongs to the second circle of tones. The Eb is probably a

central axis here for the second augmented triangle that would go on to produce the

other possible triangle. If the flow were uninterrupted by the mirror point we would

see the notes G B D# go on to produce the second triangle of keys. In fact those

notes appear at the Aeolian modal position on the mirror side (along the 45-degree

angle). Then the Eb and D# would be the enharmonic pair involved, where the

swapping between contraction and expansion would occur, due to the replication of

information at the visible and invisible tri-tone positions.

The Major 3rd intervals will meet in the center of the Mode Box at the 45-degree

angle, and this will define the most perfectly symmetrical point between the two

triangles:

C

E

G#

A

Bb

D

F#

A#

If the axis point around the triangles were to be the note A then the other notes would

be seen to mirror comfortably around this axis point. Here is the key of A Major

mirrored in order to highlight this:

A Bb C D E F G A B C# D E F# G# A

A Phrygian A Ionian

It is in this key/axis point that the note C equals F#. If this experiment is carried out

on the other two Mode Boxes the three axis points will be A C# F. If you observe the

Major scale above again you will see that these Augmented triangles or triad

relationships naturally occur by moving the first Major 3rd interval away from the root

axis, A to C# left to right and A to F right to left.

128

F A C#

This is further evidence of relationships existing in and out of the mirror, turning both

halves of the mirror into one whole unit.

The Swap-over point

It has been seen that the Lydian/Locrian positions are the visible tri-tone, whilst the

invisible tri-tone axis is at the 4.5 position. There is a swap-over point between the

two sides of the mirror that is rather subtle, but the logic is there, and it will require

some study to fully understand how this occurs.

The tri-tone position is a place of replication. What this means here is that the

symmetry around the note F, for example, is exactly the same as around the note B.

Likewise, the symmetry around C is the same as the symmetry around the F#/Gb

poles. And in a sense it will mean that both F or B can be either Lydian or Locrian in

role.

To show how the Lydian/Locrian play each other’s roles, here are other examples. If

we take a closer look at the two Circles of Tones we will find another intriguing result:

F

C D E F# G# A#

The notes F Lydian and B Locrian display symmetrical characteristics that other note

pairs/modal partnerships do not display. One of the reasons for this is their Tri-tone

interval relationship. The Lydian/Locrian position is really a partnership of perfect

light/dark, the Lydian being the brightest Mode of a Major scale whilst the Locrian is

the darkest sounding. In C Major the Lydian mode is generated from the note F whilst

the Locrian is generated from the note B. These two notes perform an interesting

function within the above Circle of Tones, at the point where the note F would

normally be (even though it is not used within the scale). The notes of the Circle of

Tones are in symmetry around the note F.

129

C D E F F# G# A# min 2nd min 2nd

min 3rd min 3rd

4th 4th

The note B also proves to be an axis point when it swaps itself over with the note F

that should be lying between E and F#. All the notes around B show perfect

symmetry with one another:

C D E B F# G# A# 5th 5th

Maj 6th Maj 6th

Maj 7th Maj 7th

The intervals reflected are the inversions of the original intervals (although one will

need to start at the bottom to tie together the min 2nd and maj 7th, and then work

upwards for the rest).

This experiment can be carried out on the other Circle of Tones.

F#/Gb

Db Eb F G A B

This will produce the same set of intervals a sin the first example. The note C can be

used between F and G instead. The C takes the place of the axis F#/Gb, its Tri-tone

partners, and it will then be seen that the notes either side of C are in symmetrical

reflection, a sin the previous examples:

Db Eb F C G A B 5th 5th

Maj 6th Maj 6th

Maj 7th Maj 7th

130

It is the tri-tone that really brings this ability to swap-over about. Any musical

information established at one root, can be replicated a tri-tone interval away.

These next two charts will show that the information at the note C is being replicated

at the note Gb, which is a tri-tone interval away from the note C. This first chart is

showing the major and minor triads as they mirror around the C axis.

Chords and Mirror chords for the key of C Major

C Db Eb F G Ab Bb C D E F G A B CC Phrygian C Ionian

One example within this chart is the chord C, which is comprised of the notes C E G.

These reflect to the notes C Ab F, which form the triad of F minor.

131

Non- Mirror Mirror

Non- Mirror Mirror

Non- Mirror Mirror

Non- Mirror Mirror

CCm

DbDbm

DDm

FmF

EmE

EbmEb

EbEbm

EEm

FFm

DmD

DbmDb

CmC

GbGbm

GGm

AbAbm

BmB

BbmBb

AmA

AAm

BbBbm

BBm

G#mG#

GmG

F#mF#

Chords and Mirror chords for the key of Gb Major

Gb G A B Db D E Gb Ab Bb Cb Db Eb F Gb

The C major grid begins at the opposite end of the Gb major grid, yet all the

relationships are the same (F minor equals C, for example). A similar chart may be

drawn for F#, and it will give similar results. In fact , here is the D major grid to show

that the relationships between chords and mirror chords will be different to the above

two grids.

Chords and Mirror chords for the key of D Major

D Eb F G A Bb C D E F# G A B C# D

D Phrygian D Ionian

132

Non- Mirror Mirror

Non- Mirror Mirror

Non- Mirror Mirror

Non- Mirror Mirror

GbGbm

GGm

AbAbm

Bm B BbmBb

AmA

AAm

BbBbm

BBm

AbmAb

GmG

GbmGb

CCm

DbDbm

DDm

FmF

EmE

EbmEb

EbEbm

EEm

FFm

DmD

DbmDb

CmC

Non- Mirror Mirror

Non- Mirror Mirror

Non- Mirror Mirror

Non- Mirror Mirror

DDm

EbEbm

EEm

GmG

F#mF#

FmF

FFm

F#F#m

GGm

EmE

EbmEb

DmD

G#G#m

AAm

BbBbm

C#mC#

CmC

BmB

BBm

CCm

C#C#m

BbmBb

AmA

AbmAb

In both the C and Gb grids, the chord C equalled the chord of F minor. Whereas in

the above grid, the chord C equals the chord of A minor. It is only at the tri-tone

intervals where information replicates.

We will now see how the swap-over to the mirror side is achieved because of this

inherent symmetrical ability for the Lydian/Locrian to replicate each other’s

information around their axis positions. One can become the other, just like poles can

switch form positive to negative. They contain within them both the visible and

invisible tri-tone relationships – F and B Lyd/Loc coming from C and F#/Gb.

Lyd Loc

C D E F (F#/Gb) G A B C - C major

Be aware of the arrangement on the mirror side of the C Major Mode box. At the note

F on that side is the Locrian Mode position. This is the F Locrian mode, and it

belongs to the Gb major scale. The note F on the right hand side is on the Lydian

mode position, as shown above. Yet it is also partnered with the B Locrian on the

same side, and can take on its qualities in terms of reflection. It does this, and the

swap-over to the mirror side occurs, by the F note “slipping” into the invisible axis at

Gb, as shown above. From here, the mirror side is entered, and this Gb becomes the

axis that supports the two triangles of keys around it.

Fragment of C Major Mode box:

Lyd/ Loc/

Loc Lyd

C D E F (Gb) G A B C

Loc/ Lyd/

Lyd Loc

F Gb Ab Bb Cb Db Eb F = (Gb major)

Cb = B

The F Loc is from the Gb major scale. Here we see the roles of Lyd/Loc swap-over,

and access to Gb from one side of the mirror to the other accomplished. And it is

133

accomplished by the visible F Lyd/Loc entering the invisible tri-tone area at F#/Gb,

the poles) on the right hand side, and re-establishing itself at the F Loc/Lyd on the

mirror side.

Deeper than this is the fact that Cb will really be taking the whole process through a

tonal spiral, and will do this because of the slight difference between itself and the

note B. For our purposes, it is just a question of spotting that certain positions take on

each other's qualities and so are able to swap these qualities to the mirror side,

through the tri-tone gateways.

This whole scenario is then repeated in the Ab major mode box.

Fragment of Ab Major mode box:

Lyd/ Loc/

Loc Lyd

Ab Bb C Db (D) Eb F G Ab

Loc/ Lyd/

Lyd Loc

C# D E F# G A B C#

Again the Db Lyd/Loc enters the mirror side by the invisible tri-tone route of D.

This access to the mirror side isn’t only between the C and Gb, and the

Lydian/Locrian along those scales. The note F# is also involved, as it is the other

pole within the major scale system

Lyd/ Loc/

Loc Lyd

C D E F (F#) G A B C

Lyd/ Loc/

Loc Lyd

B C# D# E# F# G# A# B

(F)

134

Here the F Lyd/Loc of the right hand side enters the mirror side by the invisible tri-

tone route of F#. The mirror side is the F# major scale, beginning from the B Lydian

position at the seventh line of the mode box.

The fact that musical data is seen to appear in and out of the mirror at certain

strategic points is obviously what one would consider a musical exercise. Yet, with

the added insights regarding the Fibonacci numbers, and the Phi ratio, as well as

many other number based examples, one can seriously begin to entertain the idea

that there is a system in place where there are swap-over points between the visible

and invisible phenomena in the universe. However unlikely that may seem, there is at

least some justification for experimenting along those lines. After all, if one has a

definite experiment they wish to conduct, in order to research whether one can make

an object disappear this side and reappear on the other side, surely it would be

worthwhile giving that kind of experiment the go ahead!

Whatever the scientific implications, one cannot dismiss the fact that the circle of

tones is playing a vital cyclic role in connecting both sides of the mirror.

135

Chapter seventeen

Chords through the Triangle

In these examples triads will be symmetrically reflected through the Keys that comprise each

individual triangle. This will further highlight the transference of information from one side of the

mirror to the other. The first example starts with the root triad of C Major, which is comprised of the

notes C E G, and it will be cycled through the triangles of keys, with which it shares a mirror link –

C Ab E, D Bb F#. Firstly we will mirror the C triad through the key of C Major and its mirror scale:

* * * * * * C Db Eb F G Ab Bb C D E F G A B C Phr Ion

The chord C equals Fm (F Ab C). This has already been seen to be the case in the chapter

“swings four ways”

Now we mirror the chord C through the key of Ab Major, the major scale that houses C Phrygian

above:

* * * * * * G# A B C# D# E F# G#/Ab Bb C Db Eb F G Ab

Phr (C) (E) Ion

Here the chord C equals Am. The E note on the right hand side does not appear within the scale

as such, but is in between the Eb and F. Likewise, it mirrors to the C note on the left hand side, in

between the B and C# (as the asterisks show). Think of it as an equal distance either side of the

mirror point, falling on a chromatic note not part of either scale.

We then take the chord C through the key of E Major which is the key that houses G# Phrygian

above:

136

* * ** * *

E F G A B C D E F# G# A B C# D# E Phr (G#) (C#) (G) (C) Ion

The chord C equals C#m (made up of the notes E C# and G#). Again the G and C notes are in

between the pitches that make up E major. And on the mirror side the C# and G# are not notes

contained within the E Phrygian mode, but in between the C/D and the G/A.

The above triangular relationship consisting of the keys C E and G#/Ab is only one of the

possible triangles here, the other triangle that makes up the Circle of Tones being D F# Bb. By

mirroring the chord C through the Keys of the other Triangle an interesting result emerges. We

start by mirroring D Major and trace the C triad over it and the mirror triad through the mirror scale:

* * * * * * D Eb F G A Bb C D E F# G A B C# D Phr Ion

The chord C equals Am. This result is similar to that when the chord C is mirrored through the key

of G#/Ab Major. Interestingly, D and Ab are a Tri-Tone apart. So again we see the same result

displayed in two keys that are a tri-tone apart.

The keys of F# and Bb Major also yield similar replication of the first triangle’s results. Through F#

major the chord C equals Fm establishing a tri-tone link with C Major (C/F# are a tri-tone interval

apart), and through Bb Major the chord C equals C#m, establishing a tri-tone link with E Major.

Triangle’s Keys - C/F# Ab/D E/Bb

Chord C equals - Fm Am C#m

137

Notice how the minor chords are three relative minors to the above major scales of C Ab and E,

one of the triangles. The major/relative minors are C/Am, Ab/Fm and E/C#m. Notice also the tri-

tone relationship each Key pair shares.

The chord C then becomes Fm Am and C#m through both triangles. F A and C# are also

another triangle of the other possible Circle of Tones.

We can also try this experiment on the triads of Ab ( the notes for this triad are Ab C Eb), and E

(E G# B), as they too are root chords of one of the triangles.

Triangle of Keys in Ab/D in E/Bb in C/F#

Ab triad mirrors to C#m Fm Am

in E/Bb in C/F# in Ab/D

E triad = Am C#m Fm

The root triads belonging to this triangle all mirror to the same three minor chords in different

orders. We also know that if we put the three root chords through the second possible triangle of

that particular circle of tones the three minor chords will still emerge as Fm Am and C#m. We

must also bear in mind that one triangle from one circle of tones combines with the other two

triangles from the other possible circle of tones. So:

Triangles C/F# Ab/D E/Bb (both triangles of one circle of tones)

chord C = Fm Am C#m

chord Ab = Am C#m Fm one triangle of other circle of tones

chord E = C#m Fm Am

At this stage, we will take the seven possible primary triads through each of the triangle of keys.

This experiment will be to show how the two individual circle of tones run into each other and

swap-over across the mirror.

Let’s now turn our attention to the second triad of the Major scale, D minor, and mirror it in a

similar fashion through the two triangles of keys, C E G# and D Bb F#:

138

C/F# Ab/D E/Bb

Dm triad = Eb G B

As you can see we have discovered the missing triangle, Eb G B. Together with the F A and C#

these two triangles make up the other circle of tones.

F A C#, Eb G B = second possible circle of tones

Let’s proceed by applying this process to the third triad of the Major scale, E minor, through the

triangles of C E Ab(G#) and D F# Bb:

C/F# Ab /D E/Bb

Em = C# F A

Here we have returned to the first triangle from the other possible circle of tones, but the chords

are major instead of minor (because a major triad always mirrors to a minor triad). Now for the

fourth triad:

C/F# Ab /D E/Bb

F triad = Cm Em G#m

This is the first time the resulting triangle has not ventured into the other possible circle of tones.

This is a very subtle clue that another swap-over of some kind has occurred. The circle of tones,

does not possess a perfect 4th in the shape of F. Its next destination note is F#. As it is, the Major

scale employs a semi-tone movement at this point, from E to F so the flow of one circle of tones is

broken.

We find that from the triads F, G and Am, that the same mirror triads emerge, Cm Em and G#m

(this reverts to a C E G# major triad when the A-minor triad is mirrored through the circle of tones).

When we put the seventh triad of the major scale through the triangles, that of B diminished, the

result is G dim. Eb dim and B dim. This has re-established the link with a Triangle from the first

circle of tones again, because again there is only a semi-tone between the B and C notes of the

major scale here. Here the second circle of tones that had occurred through the F G and A, has its

next destination, after B, to the note C#, so again the swap-over has occurred.

139

COT 1 COT 2 COT 1

C D E F G A B C

Gap Gap

The first circle of tones will move to F#, and the second circle of tones will move to C#. The cycle

eventually becomes that of a movement of three places within one circle of tones followed by a

movement of four places within the other. The above can also be seen to be occurring on the 4/7

relationship of the F Lydian/B Locrian, which is the ‘visible’ tri-tone relationship within a major

scale.

Here is another view of how the circle of tones movements flow and swap-over at the semi-tone

points of the major scale.

All movements of a tone

COT 1

F# G# A#

C major - C D E F G A B C D etc

B Db Eb Db(C#)

COT 2 All movements of a tone

Imagine that there is an invisible in-between journey in the flows above. The circle of tones flowing

through the notes C D E is in visible relationship with this major scale. When that particular circle

of tones dives out of sight at the note F, it flows through the F# major key instead, which is a tri-

140

tone away from C major. Hence the notes F# G# A# are the next moves of its journey. The circle

of tones has switched at the F#, due to the disturbance of the semi-tone movement of the Major

scale formula. This is one of its functions with the invisible tri-tone relationship. It will reappear

when the next semi-tone movement of the major scale happens, between the B and C (it's other

function with the visible tri-tone relationship).

The second circle of tones was flowing on the other side of the mirror, but the switch over at the tri-

tone was access point to this side. Again a hidden Dorian aspect has caused this to happen,

because the F# is a hidden Dorian modal position.

To recap all this, what has been seen is that the triads of C major opened up one consistent

triangle of chords, F A C#, until the triads reached the F note of the scale. At this point there was a

swap into the second circle of tones, which had been flowing through the mirror of C major until

that point. The actual F A C# went onto the note F#, disappeared from C Major, and became

visible again at the next semi-tone area, between B and C. Whilst this was happening to one circle

of tones flow, the other circle of tones made its appearance in C major, from the triads built on the

root notes of F G A and B.

The interaction between these two circle of tones structures is by flowing in and out of each other's

mirror side.

141

Chapter eighteen

Triangles in and out

The Mode Box of C major is shown once more, including the names of the Modes that occupy

each individual position.

1. ION = IONIAN MODE 2. DOR = DORIAN 3. PHR = PHRYGIAN 4. LYD = LYDIAN 5. MIX = MIXOLYDIAN 6. AEO = AEOLIAN 7. LOC = LOCRIAN

142

3

Phr

C

4

Lyd

Db

5

Mix

Eb

6

Aeo

F

7

Loc

G

1

Ion

Ab

2

Dor

Bb

3/1Phr/Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C 2

Dor

D

3

Phr E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2/2Dor/Dr

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D 1

Ion

E

2

Dor

F#

3

Phr G#

4

Lyd

A

5

Mix

B

6

Aeo

C#

7

Loc

D#

1/3Ion/Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E 7

Loc

F

1

Ion

Gb

2

Dor

Ab

3

Phr Bb

4

Lyd

Cb

5

Mix

Db

6

Aeo

Eb

7/4Loc/Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F 6

Aeo

G

7

Loc

A

1

Ion

Bb

2

Dor

C

3

Phr D

4

Lyd

Eb

5

Mix

F

6/5Aeo/Mix

G

6

Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G5

Mix

A

6

Aeo

B

7

Loc

C#

1

Ion

D

2

Dor

E

3

Phr

F#

4

Lyd

G

5/6Mix/Aeo

A

7

Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A4

Lyd

B

5

Mix

C#

6

Aeo

D#

7

Loc

E#

1

Ion

F#

2

Dor

G#

3

Phr

A#

4/7Lyd/Loc

B

1

Ion

C

2

Dor

D

3

Phr

E

4

Lyd

F

5

Mix

G

6

Aeo

A

7

Loc

B

By studying the names of the modes above each note you will see how the original dual modal

relationships are kept intact throughout this Mode Box. Everything is systematic and symmetrical ,

and it is worth focusing on some of the results. For example, the Ion\Phr relationship is maintained

throughout. Every time Ionian is represented on one side of the mode box there is a Phrygian

mode in symmetrical reflection to it (a 3/1 or a 1/3 relationship). Here are the Phrygian Modes

found on the left hand side (mirror side) of the whole Mode box. For example, E Phr is found on

the Dorian mode line (second line down in the mode box). G# Phr is found within E Ionian (third

line down). Bb Phr is found on the F Locrian line etc. As mentioned they also flow at a 45-degree

angle across the mirror side of the mode box.

C Phr. E Triangle of Keys G# (the first triangle found from the mirroring of C Major)BbDF# Triangle of KeysA#

The result here is similar to the Circle of Tones structure. Also exposed are the same two Triangle

of Keys when the major scale modes were mirrored.

The next table shows the Modes as they occur throughout the left hand side of the Mode box

along the 45-degree angles. Follow the individual numbers across this angle, and it should be

easy to see them:

Ion. Dor. Phr. Lyd. Mix. Aeo. Loc.

1 Ab 2 Bb 3 C 4 Db 5 Eb 6 F 7 G 1 C 2 D 3 E 4 F 5 G 6 A 7 B 1 E 2 F# 3 G# 4 A 5 B 6 C# 7 D# 1 Gb 2 Ab 3 Bb 4 Cb(B) 5 Db 6 Eb 7 F 1 Bb 2 C 3 D 4 Eb 5 F 6 G 7 A

1 D 2 E 3 F# 4 G 5 A 6 B 7 C# 1 F# 2 G# 3 A# 4 B 5 C# 6 D# 7 E#(F)

These notes are, as always, all representations of the triangle of keys that find their home on the

mirror side of a mode box. Bear in mind that notes such as Bb/A# are the same note in equal

temperament. But in the mirror environment they also imply tonal expansion and contraction, and

also swapping over across a mirror point.

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The above example throws up an interesting result because at the Lydian mode point there is a

change from one Circle of Tones (COT) to the other, due to the fact that the COT contains a #4,

therefore at the F note position of the Major scale (the perfect 4th) the directional flow changes to

meet that of the other COT (as seen in the previous chapter “Chords through the Triangle). In

other words, the semitone movement within the major scale flips the two COT flows over. Within

each list there are two triangles of keys rotating around a central axis point.

Each triangle of notes/keys flips either side of the axis in and out of the mirror commences,

looking rather like a tonal double helix:

i) Ab ii) Bb iii) C iv) D v) E vi) F# 2

C D E F# Ab Bb 2 E F# Ab Bb C D 1 * F# G# A# C D E 2 ii) Bb iii) C iv) D v) E vi) F# i) Ab 2 D E F# Ab Bb C 2

F# Ab Bb C D E *

This same experiment can also be shown on the remaining modes, 2 to 7. Each strip is a left hand

side of a Mode Box relating to one of the mirror Major keys.

In all, the central notes are F# Ab Bb C D E, a Circle of Tones. The Triangle of Keys always

swap over either side of the mirror point and weave themselves back to the beginning. As you

have seen this is also a characteristic shared within numbers, including the Fibonacci numbers.

Again, the question is can these structures be useful? Have they been tested? And , as they

signify balance and unity of the dual cyclic flows in frequency (and number), can this be an initial

basis for experimentation using resonators or even lasers? If one treats the mode box as a whole

unit of information, made up of symmetrically related partnerships, then by presenting a real

structure made up of the same frequencies, would in effect create the flow of the twin triangle

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structure. This means that one of the experiments would be looking for a way to get information to

appear on the mirror side. This can come to mean many things, including the fact that it will not

produce the desired effect. Even so, some of the things it can come to mean, such as

teleportation, makes it intriguing approach in my mind.

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Chapter nineteen

A link between a triangle of frequencies and the body's response

It had already occurred to me that there may be a link between the information in

these mirrored grids and the human body, and even consciousness itself. Whether

that link be purely physical, or even spiritual in nature is besides the point here. By

uniting natural cycles, exposing the mirror side and realizing that the system

becomes one Whole, rather than two disjointed sides, does lead one to wonder if the

brain or mind itself has a relationship with this kind of result. What is the relationship

between the firing of neurons at specific frequencies, and this overall Whole system

that is exposed through mirroring? Is there a unity aspect to mind or consciousness?

The next picture is taken from a book by David Gibson, called “The art of mixing”

It is a guide to sound engineers who produce music for us to listen to. This picture

below is a sound engineer's guide to the various parts of the body that respond to

specific frequency ranges. I was rather surprised by the result.

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The numbers given here relate directly to one of the triangle of Keys, namely, the G

Eb B.

40Hz = Eb

800 Hz = G

1000 Hz = B

5000 Hz = Eb

10000 Hz = Eb

Therefore, there is an aspect of the human response toward certain frequencies, and

it does relate to one of the triangles of this circle of tones, the G Eb B triangle..

Obviously some of the pitches involved are slightly sharp or slightly flat, but they do

fall within this G E Bb range. The human body seems to be built with the triangle of

frequencies in mind. Are there any other areas of the physical body that also

translate to these triangles? In order to answer this there would need to be a lot of

thinking, use of imagination, and hopefully experiments performed. At least there is

data available that can perhaps be a reasonable choice to make when one is

deliberating a possible experiment. There are ideas presented throughout this book

on how to set up various experiments.

Volume two delves much deeper into presenting a case for consciousness, built from

the idea of this recurring mirror structure made up of its triangles of frequencies.

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Visible and invisible axis points within numbers

The following number sets present one idea on how to set up clusters of frequencies

that may provide insights into the make up of the body and mind.

A number like 101 has its axis point visible, the middle 0 being the axis, with the

number 1 either side of it. Then will come 111 with the axis again being visible at the

central number 1, and this time the number 1 merging from both sides to its centre.

There are nine three-digit numbers, with a 1 either side, with this potential, as will be

seen.

A number like 1001 has its axis invisible, in between the 00, with a 01 either side of

this axis. Again there are nine four-digit numbers with this potential. In fact,

regardless of the amount of digits there are always nine such sets of numbers with

such axis points.

What will be consistently seen when these mirror numbers are added together from

their respective groups is the second number sequence of the Vedic Square, 2 4 6 8

1 3 5 7 9.

Firstly, here are the three digit numbers with the visible axis. The totals in red are

further broken down to single digit, in order to expose the second number sequence

of the Vedic Square. Note also that with each set of nine the totals increase by a

value of 1010.

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Visible axis

101 202 303 404 505 606 111 212 313 414 515 616 121 222 323 424 525 626 131 232 333 434 535 636 141 242 343 444 545 646 151 252 353 454 555 656 161 262 363 464 565 666 171 272 373 474 575 676 181 282 383 484 585 686 191 292 393 494 595 6961460 2470 3480 4490 5500 6510 2 4 6 8 1 3

707 808 909717 818 919727 828 929737 838 939747 848 949757 858 959767 868 969777 878 979787 888 989797 898 999

7520 8530 9540 5 7 9

Invisible axis

1001 2002 3003 4004 5005 6006 1111 2112 3113 4114 5115 6116 1221 2222 3223 4224 5225 6226 1331 2332 3333 4334 5335 6336 1441 2442 3443 4444 5445 6446 1551 2552 3553 4554 5555 6556 1661 2662 3663 4664 5665 6666 1771 2772 3773 4774 5775 6776 1881 2882 3883 4884 5885 6886 1991 2992 3993 4994 5995 699614960 24970 34980 44990 55000 65010 2 4 6 8 1 3

7007 8008 9009

149

7117 8118 91197227 8228 92297337 8338 93397447 8448 94497557 8558 95597667 8668 96697777 8778 97797887 8888 98897997 8998 9999

75020 85030 95040 5 7 9

Here the totals increase by a value of 10010. After numbers that contain four digits

there are extra mirror combinations possible, other than the ones listed next. After

dealing with the lists that contain similar type patterns to the above the other

possibilities will be dealt with. The next examples have the visible axis at the central

zero.

Visible axis

10001 20002 30003 40004 50005 11111 21112 31113 41114 51115 12221 22222 32223 42224 52225 13331 23332 33333 43334 53335 14441 24442 34443 44444 54445 15551 25552 35553 45554 55555 16661 26662 36663 46664 56665 17771 27772 37773 47774 57775 18881 28882 38883 48884 58885 19991 29992 39993 49994 59995 149960 249970 349980 449990 550000 2 4 6 8 1

60006 70007 80008 9000961116 71117 81118 9111962226 72227 82228 9222963336 73337 83338 9333964446 74447 84448 9444965556 75557 85558 9555966666 76667 86668 9666967776 77777 87778 9777968886 78887 88888 9888969996 79997 89998 99999

650010 750020 850030 950040 3 5 7 9

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These next sets have the axis in between the central zeros:

Invisible axis

100001 200002 300003 400004 500005111111 211112 311113 411114 511115122221 222222 322223 422224 522225133331 233332 333333 433334 533335144441 244442 344443 444444 544445155551 255552 355553 455554 555555166661 266662 366663 466664 566665177771 277772 377773 477774 577775188881 288882 388883 488884 588885199991 299992 399993 499994 599995

1499960 2499970 3499980 4499990 5500000 2 4 6 8 1

600006 700007 800008 900009611116 711117 811118 911119622226 722227 822228 922229633336 733337 833338 933339644446 744447 844448 944449655556 755557 855558 955559666666 766667 866668 966669677776 777777 877778 977779688886 788887 888888 988889

` 699996 799997 899998 999999 6500010 7500020 8500030 9500040 3 5 7 9

The last two examples produce individual totals increasing by a value of 100010 and

1000010 respectively. One can see from these few examples that the pattern is set.

The second number sequence of the Vedic square will repeatedly appear. The last

example is only one of the possibilities within the set of six numbers evolving from

either a visible or invisible axis. The number 101101 is also a mirror number, and

other lists are possible. The next number in this particular list would be 102201 and

so on. Here is a list covering these numbers:

101101102201103301104401105501106601107701108801109901

1049510 2

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The number 2 is still the single digit number that this sequence breaks down to.

Knowing that from now on the same number sequence of the Vedic Square will

emerge we can leave this particular list and move on to a different mirror number

combination.

110011 – this will produce the same number sequence.

We leave the mirror number lists at this point to concentrate on any possible musical

applications for such numbers. Because these numbers are related in some respects

to the visible/invisible aspects of the mode box, and to the number sequence of the

Vedic Square, and to Buckminster-Fuller’s indig number system, it is worth viewing

these numbers as cycles-per-second and listing the vibrations that would emerge.

We begin with the first mirror number total 1460, and then move on to the next total

2470 etc.

1460 cps = F#2470 cps = Eb3480 cps = A4490 cps = C#5500 cps = F6510 cps = G#7520 cps = A#8530 cps = C9540 cps = D

F# G# A A# C C# D Eb F

One can see the tri-tone relationship here between F# and C

The overall scale also has a very Phrygian type sound about it. One could view it as

F Phrygian with the A and D as added notes.

Doubling or halving a number produces octaves so it would be possible to play this

scale all within one octave.

Notes can also be shown as belonging to each individual list:

101 = G-qt 111 = A 121 = B-qt 131 = C 141 = C#

151 = D-qt 161 = Eb-qt 171 = E-qt 181 = F-qt 191 = F#-qt

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This hexagonal cluster resonator consists of the frequencies for the numbers

101 1001 10001 100001 1000001 100001 10001 1001 101:

G G G

B B B B Eb Eb Eb Eb Eb G G G G G G

B B B B B B B

G G G G G G

Eb Eb Eb Eb Eb

B B B B G G G

The cluster consisting of three G notes are all tuned to 101 cycles per second. The

next cluster consisting of B notes are all tuned to 1001, and so on. Observe that we

have uncovered one of the Augmented triangles of keys, discovered in the Mode Box

– G B Eb, yet remember that the notes are all a quarter tone flat. It would be a

question of tuning a mode box down by this interval (not to mention the myriads of

other tunings, as not all the notes of any mode box would be strictly a quarter tone).

Therefore we can join this particular cluster resonator with the other triangle that

goes on to make the Circle of Tones. So where is this other triangle, namely F A C#?

It will be found within the numbers 707 7007 70007 700007 7000007 700007 70007

7007 707. To get the image of this simply replace the above hexagram with the

appropriate notes, which will be tuned to the new frequencies.

The hexagonal cluster resonators are placed at 45% angles from their respective

partners. Here are the four prevalent relationships:

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G B Eb C E G#

D F# Bb F A C#

The other two possible triangles that go on to make the other circle of tones

structure, C E G# and D F# Bb, will be found within the numbers 131 1331 13331

133331 1333331 etc for C E G#, and 181 1881 18881 188881 etc for D F# Bb. It

must be held in mind some of the notes are close approximations.

Here is a rough sketch of hexagonal resonator formations that are the triangular

frequencies of the circle of tones, following in/out of mirror movements. The format

would be designed to be looped continuously.

G# C E Bb D F# C E G# D F# Bb E G# C F# Bb D

F# G# Bb C D E

Bb D F# C E G# D F# Bb E G# C F# D Bb G# C E

The notes in red are the axis points that exist between the triangles of frequencies.

Together these axis points form yet another circle of tones, showing how the

triangular frequencies shift through every point of the scale. The size of this ‘machine’

would depend on the individual resonators that are packed to form each hexagram

cluster.

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The above diagram bares a close resemblance to the ‘in and out of the mirror’

triangles we saw in chapter 18.

In the present form the largest resonator would be vibrating at 131 cps (for the C).

The smallest resonator would be resonating at 1888881 cps (for the D). This may be

a hybrid Nano-resonator! Here perhaps one needs to use the doubling and halving

rule in music, as this produces octaves. One can experiment with relative sizes of the

resonators until something that has the potential of being built emerges.

To be noted about these hexagonal structures is that all the notes in reality fall within

the quarter tone region.

These visible axis and invisible axis number lists have exposed once again the Circle

of tones. One cannot call this structure a phenomena of no importance, as it is the

only structure that signifies unity of the mirror sides. These cycles exposing the Circle

of Tones are based on number sequences that are intrinsic to all of Number itself.

There was correlation in the Fibonacci number flows and every single experiment

carried out so far.

Possible machine constructions using the triangle of frequencies

Nano scale resonators.

Double up the 101, 1001, 10001, 100001 etc numbers, until they are on a nano

scale. Each resonator will be a different size, the proportions of which will adhere to

the circle of tones flow within these particular numbers. After the proportions are set,

these resonators can be stacked up , one over the other, according to the next

possible set of numbers that will emerge:

101 = G#/A

202 = G#/A

404 =G#A

also, 808, 1616, 3232, 6464, 12928, 25856, 51712, 103424, 206848

A typical nano scale resonating structure would look a little like this

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These differently proportioned resonators will adhere to the hexagonal resonator

numbers. The octaves of the 101 frequency will be obtained as many times as is

necessary in order to make each resonator of nano scale proportion. All the

resonators will need to be nano scale at the beginning of the stack, therefore the

biggest resonator is the one that is required to be the right nano scale size

Is a resonator built to the frequency of 206848 cps small enough to be considered

nano scale?

If the 206848 resonator is the largest of the cluster, then the next number needed to

fit into the same octave range is the 1001, which will be doubled until it reaches the

same octave range.

1001, 2002, 4004, 8008, 16016, 32032, 64064, 128128, 256256, 512512,

The 256256 frequency is the closest, and will therefore become the second largest

resonator.

The number 10001 is the next to be doubled until the same octave range is found.

10001, 20002, 40004, 80008, 160016, 320032, 640064,

The number 320032 is the closest frequency and will therefore become the next

largest nano resonator.

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two resonators tuned to F# bolted to the ceiling holding up the pyramid of resonators

Nano sized resonators

FLOOR

157

Chapter twenty

Working Conclusion

After so many experiments regarding cycles and tonality, the conclusion is rather simple

actually. It is the Dorian mode axis that is responsible for the access points to the other

side of the mirror. To physically gain entrance through this access point may not be as

easy as verifying that it exists on paper. Most humans with an imagination can start

thinking on setting up experiments that may prove to be a way to penetrate through

these access points. And to aid in the experiment it will be the frequencies of the Dorian

mode, or a set of symmetrical like numbers, represented as frequencies, that would be

first on the list when thinking in terms of swapping something from one side of the mirror

to the other. Equations that have proved to contain mirror flows would also be high on

the list. And the all important 45-degree angle must also be seen as the carrier of this

access point. How do we know that in order to teleport an object the mirror information

as set out in this book may not be relevant?

Regardless of future idea for experimentation, it is enough to be aware that this mirroring

phenomena does carry a justified result . And that it exists is a simple matter of logic and

reason, because such things as Mode Boxes work as a whole unit, and so verify the

mirror side as an essential aspect.

4.5

1) 4.5 swaps sharps for flats in circle of 5th

2) 4.5 swaps minor for major in mode box

3) 4.5 is tri-tone position

4) 4.5 is a Dorian position. Dorian is a 2/2 position mirror pair.

5) Dorian is seen as center of the light/dark tonality distribution in circle of 5ths

6) subsequent circles of 5ths are linked by the tri-tone/4.5

7) 4.5 dominates the number line, in terms of mirror number pairs

8) 4.5 is swap-over point of indig numbers and Vedic square/9*9 maths table sequences

9) number pyramid examples show 4.5 to be symmetry point amongst single digit totals.

158

10) dual flows of Fibonacci sequences shows 4.5 as swap over point.

11) 4.5 is linked to the 45 degree angle

As an example of 4.5 being a swap-over point, the single digit totals of one pyramid the

numbers will be shown as indig numbers. Here we will see how positive and negative

always swap-over around the 4.5. Here is pyramid number 1's additions:

2 = +2

4 = +4

6 = -3

8 = -1

(4.5 axis)1 = +1

3 = +3

5 = -4

7 = -2

9 axis

9 acts as a zero, as does 4.5.

The multiplications show the symmetry around the 4.5 in a different manner. Here's is

pyramid number 1's multiplication totals as single digit:

1

4

9

7

(4.5) 7

9

4

1

9This next magic square is also an example of 4.5 and 9 axis symmetry:

159

Another witness to the C/F# axis is the colour spectrum:

160

We are used to hearing that it is symmetry breaking that leads to manifest structures. In

other words, the universe functions through asymmetry. The question is whether perfect

symmetry rules asymmetry. If the answer is the former, then such a system maintained

by an axis such as 4.5 is somehow relative to reality. And also bearing in mind that this

4.5 was also shown to equate as a zero point. This may lead to an alternative way of

setting up various experiments. For example, one may set up sound chambers that

contain resonators set to specific frequencies, as maintained around the 4.5 axis within a

mode box (shown in chapter one). Their relative interactions can be investigated by

placing objects whose frequency is already ascertained, and one can log the effects the

sound chamber has on any object. Under investigation will be the idea of balance, which

is seen in a Mode Box, as well as the symmetrically placed mirror numbers around both

the 9 and 4.5 axis points. This type of experiment is by no means limited to only the use

of resonators. One may use lasers, coils, wires, always set up to mimic the arrangement

in a Mode Box, or a Lambdoma grid or a Fibonacci grid. With both sides of these grids

now evident, all these experiments will take on a new type of investigative quality.

The fact that 4.5 switches over the positive and negative aspects of cycles, can also be

experimented with, as a way to investigate the exploitation of the in and out of the mirror

phenomena that occurs.

161

The starting point for a universe is surely that of perfect symmetry. And, once

commenced, each type of clockwise and anti-clockwise cycle can be traced and seen to

be consistent with each other, meaning, plotting one coordinate will give a mirror cycle

coordinate.

The picture will not complete without the mirror side and its structural components being

observed. That is what gives one the focus of the twin cycles and their relationship one

with the other. Because all is number, and all is wavelength, then all evolves out of

primary contrary cycling relationships.

Focusing only on the one side of the mirror isn't exposing the full unit and how it

expresses itself. This is shown through musical cycles, and through the basic number

cycles that exist. The truth about these things are held in a simple seed. And what can

be built through this can become the extremely complicated formulas we have

developed.

There is potential in mimicking these natural flows, in that we can build things that are

focused on this dual process and so utilize it. In terms of space travel, there need not be

only a focus on traveling this side, but a traveling based on in-and-out of the mirror.

There are specific points either side of the mirror that a thing can be seen to swap to,

due to the contrary fashion of spin. Uncannily, there was an old Egyptian belief based on

all of us possessing a Siamese twin on the other side. This and the fact that the

Egyptians revered the Dorian mode scale ratios is even more uncanny, because the

results unearthed in this book correspond perfectly with such a system. So, from this

idea of a Siamese twin, we could access our Siamese counter-part by establishing the

axis where information is switched over to the mirror side. Also, all these apparent

dualities exist at some point as merged/united/married. Therefore, there is either a third

force possible, or a neutralizing force that allows the mind to access other dimensions.

The most potent place is where these two contrary spinning forces unite, because it is

the neutral point decides what goes where and how much of it is expressed. If the

'desire' is for a strong manifestation, then the oscillations are vigorous and wide. If the

desire is for a weak manifestation, then oscillation will be mild and narrow. We can love

passionately or mildly. Desire does not allow that which is manifest to remain in the rest

position (although it originates from there), but will continually make the manifestation

switch from clockwise to anti-clockwise, positive to negative, this side of mirror and then

162

other side etc, at the "invisible" axis, at its 4.5 moment. This is what is generally meant

by swapping of qualities across the mirror point, and the 4.5 performs this duty.

It isn't quite enough to ask where along a wavelength is this switching over to the mirror

performed. One has to imagine that there are two mirror wavelengths that are contrary to

each other. This is really a four-way relationship; mirror positive to the negative on this

side, and mirror negative to the positive on this side. Mind can be at either side of the

mirror, once it is aware of where to swap sides. This is done at the neutral, and it is the

4.5 access/axis point to the other side, that sits at this gateway.

This way of thinking inevitably conjures up the work and thinking of Carl Jung, who

believed that mind had a marriage point, where the ego is unseated and the whole

masculine/feminine aspect are united. This is also the meaning assigned to a symbol

called the Merkaba. Perhaps not a scientific meaning, but nevertheless it has been

equated with contrary cycles of light in unity, the symbol of the seat of consciousness

itself. And, as Amit Goswami, has predicted, the universe is a sea of consciousness.

And as Carl Hogan has also predicted, the universe is a hologram in nature. Putting

these things together, one is also tempted to predict that the 4.5 is the tonal fountain

point itself, where mirror cycling pairs manifest, or where the hologram has its central

source.

Entanglement is another phenomena that may be related to this swap over effect at the

4.5.

163

Merkaba – Is there a case?

Within a few months of giving the mirroring of formula idea some thought now and again,

a certain experiment led to the conclusion that a simple set of cycles , as in the major

scale's seven cycles and mirrored seven cycles, were capable of offering an insight into

the duality of forces themselves; the great expansion and contraction principles that

govern the motions of a wave. There is also a moment where nature sets herself aside

from those oscillations brought about by activities between the positive and the negative

aspects, as in electric charge. We can call that place rest. Or simply the unity of that

duality, neither positive nor negative, or simply both in full potential; even if that period of

rest were an infinitesimal moment when compared to that thing's active interest in the

universe.

When the Indig numbers pertaining to Buckminster-Fuller are compared to the number

sequences of the Vedic square, one sees at the 4.5 a swapping from full positive to full

negative. The plus-4 instantly switches to minus-4. This is in keeping with the musical

examples too, where within the C major mode box the modes at the tri-tone (Lydian and

Locrian) are also a representation of full sharps and full flats. The Lydian contains the full

quota of sharps of the key system, and the Locrian contains the full quota of flats. The

tri-tone sits at the 4.5 of the scales. It is this axis that creates an in and out of the mirror

system within the keys. The tri-tone is also a hidden Dorian relationship, and point of

perfect symmetry. There is much consistency about this system, as it appears

throughout the other experiments. All in all it is evidence of whole unit function, between

two sides of the mirror.

It embraces the ways of clockwise and anti-clockwise cycles, as well as tonal shades, a

sequential journey from light to dark, within musical keys.

When the whole unit is exposed, it then shows that it ties in with another consistent

unifying structure. The structure resembles the symbol known as a star of David. One

can justifiably argue that the result may look like a star of David but is merely two

triangles of frequencies, and amongst other things it obviously looks like a star of David.

Yet there are reasons for calling the result a star of David , which I will attempt to convey

in a logical manner as we go on. Please remember that this is now conjecture, and does

not affect the actual results exposed.

164

On further pondering, the symbol can also be equated with what is known as a Merkaba.

The reason one can tie the results in with this symbol are several. The dimensional

aspects are taken into account. The symbol itself has been equated with the unity of the

contrary fields of light. It is this that makes the association of this symbol together with

the results of mirroring formulas not seem arbitrary. The Merkaba has also been

associated as the symbol for consciousness in unity. There is no more apt place for this

symbol to appear than a duality of contrary cycles in unity. And that is where it seems to

appear, as shown by the results in this book.

I originally discovered the Merkaba symbol in the work of Carl Jung. It seemed to be

used as a symbol for consciousness, which in turn was equated with spirit. Made up of

contrary rotating tetrahedra, it was associated with unity and light.

Having then discovered that clockwise and anti-clockwise cycles, such as musical scales

and fundamental number sequences also unified and provided a similar result, that was

in essence a star of David, the connection became intriguing to me.

Natural cycles are what produce this one consistent mirror structure, and it is related in a

subtle way to the patterns and growth principles we see in nature. The Fibonacci

numbers, for example, are seen to express themselves in many ways within nature.

Frequencies too are an intrinsic part.

The star of David is also a 3D representation of a Merkaba symbol. Other results

showed that there were two of these symbols that related to each other, one either side

of the mirror. The mirror sides became united, as expressed through two sets of

triangles of frequencies. These triangles also swapped information across to the other

side of the mirror, and this could be clearly seen in the music scale examples. When the

Fibonacci numbers are represented as a sequence of frequencies, the same symbol

would emerge, behaving behaving the same way as seen in the mode box.

Having seen this consistency, with only this symbol emerging and uniting the contrary

flow of cycles, many more examples began to emerge. Some of these examples are not

included in this volume, but are saved for volume two.

165

There is now definite data available and it can be tested to see if it does relate to

consciousness, because of this possible connection to the star of David/Merkaba

symbol.

There is now a way to predict an experiment's outcome, and it can be falsified. All these

things now place this symbol into a scientific arena. Why, because if one is dealing with

the unity of consciousness, the data is presented in order to experiment with such a

prediction. The theory can easily be verified or falsified. If one merely said that a

Merkaba is the unity of consciousness, one could easily say ”well prove it then”. Now at

least if someone wants proof, there is a means to prove it. So the claim is that by

mimicking the lay out of certain grids, bringing together two sides of the mirror, and the

grids being based on natural cyclic phenomena, tests on the data can reveal if

consciousness reacts to the frequencies involved. Will what looks like a star of

David/Merkaba, running through these grids (uniting both sides of the mirror) also induce

consciousness to react in a similar way, enduce balance and unity?

Our DNA vibrates, and can be assigned specific frequencies. Our bodies have organs

that are sensitive to and resonate most strongly to given frequencies. These frequencies

equate to one of the triangles of notes. They are our portal to manifestation. When we

decide on the general experience we wish in our present manifestation, we emerge into

the duality according to our sign and progress according to the number pairs. We are

susceptible to certain characteristics. Eventually we are ready for a life that will bring

together many other lifetimes to completion. This is not the end of the matter, but the

beginning of another matter. And so on it goes, infinity, just as it is written in the number

cycles. If nature has such a constant system, who are we to say that we are aloof from

this? This is our system.

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Flower of Life

One can take each cycle of the major scale (one of the modes) and show it this way:

This shape is known as the Seed of Life, and it is a derivative of the Flower of Life.

There is also a mirror circle of modes. It will look no different to the above, but it links in

through the Phrygian mode being in the center, interfacing with the Ionian, then rotating

counter-clockwise, and creating the Ion/Phr, Dor/Dor/ Phr/Ion, Lyd/Loc etc modal pairs.

Each mode does individually cycle, and will display a tonal colour when brought into a

composition. These Modes conjure emotions, from country pub floral dancing light

activity to dark shred metal, they create the mood. Music is also related to the

transcendental, beyond emotional pull. For humans it delivers on many levels.

That the Mode Box should contain a flow that comes to resemble a Merkaba, then the

above diagram only goes to add to this notion.

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The flower of life is also equated with the Egg of Life:

The shape of the Egg of Life is said to be the shape of a multi-cellular embryo in its first

hours of creation

A basic one dimensional depiction of the “Tube Torus shape is formed by ratcheting the

Seed of Life and duplicating the lines in its design. In Physics, the Tube Torus is

considered a basic structure in the study of Vortex forms. Some say the Tube Torus

contains a code of vortex energy that describes light and language in a unique way,

perhaps as something of an Akashic Record

Taken from a wikipedia explanation of the flower of life.

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This tube torus is in many ways closer to the result of the twin Merkaba symbol behind

many processes. A diagram shown in part one was broken up into two Merkaba symbols

as such, because of the fact that each pair of interlocking triangles happened once

across a Mode box of the Major scale. However, when showing frequencies like this:

1 = C

12 = G123 = B1234 = Eb12345 = G

123456 = B

1234567 = D12345678 = F#123456789 = Bb1234567891 = D

12345678912 = F#

123456789123 = Bb

1234567891234 = D

12345678912345 = F123456789123456 = A1234567891234567 = Db12345678912345678 = F

123456789123456789 = A

1234567891234567891 = Db

12345678912345678912 = F

123456789123456789123 = A

1234567891234567891234 = C12345678912345678912345 = E123456789123456789123456 = Ab

Here one sees that the evolution of the triplets are 1) G B Eb, 2) D F# Bb, 3) F A Db, 4)

C E Ab. Numbers 1 and 3 constitute one of the interlocking triangle symbols, as well as

2 and 4.

Taken as is, then these four triangles of frequencies can be made to resemble a tube

torus. And more importantly, it also shows how the triangles are moving in and out of the

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mirror, with one triangle on one side of the mirror yielding to another triangle on the other

side at the time. The chapter on opposing forces showed this.

All in all Balance is what these triangles of keys, that look like a Merkaba symbol, are

signifying. It is not too unreasonable to include the duality of masculine and feminine into

these mix. There are a few reasons why this should be considered. The major and minor

connection, the clockwise and anti-clockwise connection, the positive and negative,

expansion and contraction connections. All these have long been associated with

masculine and feminine. And in that light they are seen to evolve from and back to a

marriage point within natural cycles.

The Merkaba is seen as a dual spinning force of light, of the masculine and feminine

forces in unity. It may be too unscientific in some ways to associate these forces with

sexuality, but it is also quite plain that sexuality does play a role within nature, and here

we see the symmetry that such a role possesses.

What exactly would these dual spinning forces of light be? Are they simply the positive

and negative forces? Are they dispersive and attractive? We know that light has a

clockwise and anti-clockwise nature. We also see that these dual flows have their mirror

motions traced and kept in perfect symmetry by a symbol that does look just like a

Merkaba, as if all the mirror pairs evolve from and back toward its perfect point of unity.

A non dual symbol is quite an uncanny one to unearth using simple number and music

cycles, and focusing on the position aspect around an axis.

Once the connection between the mirror structure that looks like a Merkaba and the idea

of a Merkaba within ancient writings, we see that there is little difference between the

“story” both methods tell of. It is a story where the masculine will unite with the feminine.

Carl Jung helped make this view palatable for the western world. The view here is that a

time will come when the ego is unseated, and the conscious and sub-conscious aspects

within a human will unite once more. This is the marriage that spiritual people can talk

about. They can not only talk about it, but they can choose to see the very same “story”

being told within nature's cyclic forms.

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In chapter eleven the various frequency responses of individual areas of the body were

shown. These frequencies were seen to be one of the triangles of the circle of tones, G

Eb B. It is reasonable to suggest that this relationship is like an interface between what

happens outwardly within nature, inwardly within the human form, and further inwardly to

the point of unity within the human form, access way to the Spirit form perhaps.

Conjecture:

In order for the most possible information to emerge with the least effort, nature employs

cycles, sequences which can bee thought of as seeds. Theses seeds then allow bigger

systems to grow out from them. Such is the nature of the 4.5, and the nine number

sequences of the Vedic Square. We have also seen that every other base system

possible boils down to only nine separate sequences. This compactness can give rise to

infinite variety.

With the information shown in this book one may wish to get a little “sci-fi” and imagine

the creation of a teleportation machine. Because the circle of tones structures swap-over

through either side of the mirror a machine can be built that brings together the right

frequencies in order to cause, through summation tones, the opposite circle of tones

structure to emerge. Blending all six tonal combinations together and placing an object in

the chamber that is oscillating these frequencies, causes the object to be transferred to

the mirror side. The Dorian mode and the 4.5 would naturally be access points to the

other side. This is because each of the notes comprising a circle of tones are 4.5

intervals of each other (the tri-tone interval), and are also Dorian mode partners of each

other (this will become quite apparent in the following chapters) . This would be a sci-fi

experiment that can be performed in reality.

The Mode Box, or Fibonacci mode box is the sound chamber, so to speak. The reason

for thinking this is the fact that the whole box is needed in order to see the mirror

structure of the triangles emerge. Some of the reasons for wishing to create a kind of

living Mode box, for example, is to analyse the effect it would have on consciousness.

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The Mode box does not need to be created in the audible range. One can experiment

with octaves and also temperament.

The dual triangle structure, in terms of the results exposed through mirroring cyclic

formulas, is a means to travel between the dimensions. It is like the spaceship we are

hoping to build. It is like a discarded vehicle, hidden under a pile of leaves, waiting to be

discovered and used.

A few scientists I have come across recently have generally dismissed this result that

leads to the recognition of the Merkaba vehicle. Their logic is simple "what objective

definition of the Merkaba exists?". They then assume it is mystical mambo jumbo and

forget to even peruse the results that inspired me to approach them in the first place.

The defining qualities of a Merkaba is what is known as the unification of the contrary

spinning fields of light. This, apparently, is the so called non-objective description of it,

given by many mystics in the past. The problem that certain scientists have had for now

is that one cannot bring about any objective results in order to confirm this "belief". This

is where I have to differ in opinion, because I argue that those results are readily

available.

Perhaps the myths from the past, where a Merkaba could be activated, was done purely

from mind power. Yet mind cannot conjure up a reality that cannot be verified in some

mathematical sense. Nature doesn't act by disobeying its own rules.

It is only when a whole unit is represented that one can take in the overall picture and

seek to analyze the results. And it is maintained that only one result emerges throughout

the many different approaches and natural cycles used. The result is very simple in the

end, and almost obvious. But there is the prize of viewing this same contrary spinning

field of light structure that is the Merkaba - dual tetrahedron, star of David symbol. It is

no mean feat to actually find two corresponding contrary field of light symbols that look

exactly like each other, based on natural cycles including light, and one being objective

in result and the other being a statement made by so called mystics when talking of the

inter-dimensional vehicle.

A theory of Merkaba can be put forward based on the application of symmetrically

reflecting natural formulas, that is, formulas that we know exist within nature. We know

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that forces exist in ratio to each other. We know that, within themselves, there is contrary

spin. We know of the wave structure of matter. We know that the Phi ratio and Fibonacci

numbers appear in nature, in the form of the spiral of galaxies. if this spiral is related to

gravity, we know that Fibonacci numbers somehow express these lines of forces.

Light waves obey the same principles inherent within sound waves, that is, they both

combine to create summation waves and divide to create difference waves. The

relationships found with music cycles adhere to the behavior of light waves. The use of

the octave is the common sense behind this relationship. By the 40th octave the light

spectrum has reached the colour range. A few octaves higher is the infra red light

domain, and on and on toward the x-ray waves and gamma ray waves. What is

established within one octave will be true for the proceeding octaves. The same is true

for simple numbers.

Double helix of DNA, right handed clockwise structure. The thought that occurs is linked

to the Egyptians believing they had a Siamese twin on the other side. Could this

Siamese twin have a DNA double helix that is left handed and anti-clockwise in make-

up? If the results of mirroring natural cycles show a unification of clockwise/anti-

clockwise mirror pairs, then it may stand to reason that the mirror side is the function of

this so called Siamese twin entity, and in marrying the two sides of the mirror at the 4.5

is to bring about a merging of the two entities. What would emerge if DNA were merged,

one clockwise and the other anti-clockwise?

Another Experiment:

A volunteer walks into a sound proofed room. There is a chair in the center of the room.

Near the ceiling of the chamber is three sets of speakers, in front of which is a set of

resonators, which have as many resonators as one mode box. Through the speaker is a

set of wires which connects to all the resonators, and through the wires is a high

frequency tri-tone interval pitch relative toe ahcvo9 f the three mode boxes , which are in

front of the speakers. Mode box one will have the sound form the speaker that is

emitting the F#/Gb frequency. Mode box two will have the sound of the tri-tone interval D

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(relative to the Ab major mode box). The third mode box will receive the sound Bb

(relative to the E major mode box).

The volunteer reflects on the fact that each mode box consists of a twin triangle

structure. He will imagine this symbol and he will focus on stillness and meditate.

After a period of meditation, the volunteer can proceed to conduct various other

experiments, normally termed paranormal. This would embrace the many studies in

consciousness conducted at Princeton, for example.

Symmetry:

Some of the more wacky claims associated with the mirror process is that universes

come in twins, that is, there is a mirror universe to our own. Black holes are meeting

points of these two universes, and that information is exchanged at these places

between them.

The results in this book are in accord with counter-spinning phenomena, and they have

been logged without mention of anything spiritual. That is because, firstly, it is a wise

thing to do to present the results as is. Yet the possibility of a connection to this Merkaba

has had to be discussed.

This Merkaba has been associated with consciousness, as the duality of forces

becoming unified. Throughout this book we have seen how this is possible by using

simple number and music systems including the nature of sound itself, which extends by

association through the light spectrum, as the wavelength decreases and frequency

increases.

It has taken many years of thought to finally realize that my own journey was to come to

the conclusion that all along I had been in the process of unearthing an interface that

has the potential for dimension travel. I would have laughed at such a conclusion

seventeen years ago. I would have strongly disagreed with it some fourteen years ago,

and explained that it was purely a new musical adventure, and I was going to provide an

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alternative musical structure for musicians to improvise with. Twelve years ago I would

have been slightly more open to the idea of an interface, but still needed some

convincing. In fact the final pieces of the puzzle for me started coming into place after

meeting a few like minded individuals some seven years ago, on the Internet. And

through our own sharing of information, I am now convinced that there does exist a

matrix of frequencies and patterns that will elevate the mind to a form of non-dual

existence. It is not a new thing. One is not creating a new frame of mind. One is

awakening the married mind, to conscious awareness and control of its desire and

wishes.

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