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www.harmonictheory.com© 2006 by Connie Achilles. All rights reserved.

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11-15-2005

The Lambda RectanglesEach is in the proportion of 3 units (width) to4 units (height). Musically the ratio of 3 : 4 is

the Perfect fourth.The cithara (from which the guitaris descended) was the instrument of Apollo.

It was tuned from the top string down, and the interval from the top string to the

2nd string was a Perfect fourth, as E’ to B.

l

The Pythagorean 3 - 4 - 5 TrianglesThe diagonal of the 3 x 4 rectangle is 5, (fromthe Pythagorean theorem , a^2 + b^2 = c^2. 3^2 = 9. 4^2 = 16. 5^2 = 25. 9 + 16 = 25.

Musically the ratio of 4 : 5 is a Major3rd B : G,2nd to 3rd string on the cithara.

Musically the ratio of 3 : 5 is a Major 6th, E : G1st (top) to 3rd string on the cithara.

4

3

apolloN APOLLON - The Pythagorean Definition of Godbased on an article by the lateAnne Macaulay, from the book“Homage to Pythagoras”. The samebook is also published with thetitle “Rediscovering Sacred Science”

a

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A - Alpha-1st letter of Greek alphabet.The Alpha circle, (the big encompassing circle)

diam = 10 units (10 x 6 = 60 ticks)circum = 10 x 22/7 = 31.43 units = 188.58 tickscircum =188.58 rounded = 189 ticks = 3^3 x 7

O - Omicron-16th letter of Greek alphabet.circum = 16 units (16 x 6 = 96 ticks)

diam = 16 x 7/22 = 5.09 unitsRounded diam = 5 units =30 ticks

(5 units is also the diagonal of the 3 by 4lambda rectangles.)

P - Pi-17th letter of Greek alphabet.diam = 17 x 7/22 = 5.41 units

diam = 5.41 x 6 = 32.46ticks rounded = 32 ticks

circum = 17 units = 102 ticks

L - Lambda the 12th letterof the Greek alphabet.

Symbolized by a 3 x 4 rectangle.(3 x 4 = 12)

Width = 3 x 6 = 18 ticks.Height = 4 x 6 = 24 ticks.

(Diagonal, shown at bottomleft = 5 x 6 = 30 ticks

= 5 units.)

nN - Nu-14th letter of Greek alphabet.

circum = 14 units = (14 x 6 = 84 ticks)diam = 14 x 7/22 = 4.45 units = 26.70 ticks

Rounded diam = 4.5 units = 27 ticks

11-15-2005

A B E

FCD

K P L

M NQ

R S

UT

O1 O2

O3

O4

x x

x

x

CD/DA=18/24=3/4=P4th

AD/AC=24/30=4/5=M3rd

CD/CA=18/30=3/5=M6th

AD/DF=24/36=2/3=P5th

RS/DC=12/18/2/3=P5th

DC/AE=18/36=1/2=P8ve

RS/BC=12/24=1/2=P8ve

radiusO4/AB=16/18=8/9=M2nd

CB/CP=24/27=8/9=M2nd

diameterO4/CT=32/45=A4th

diameterO1/DF=30/36=5/6=m3rd

CP/diameterO4=27/32=m3rd

radius O4/CP=16/27=M6th

BA/PK=18/20=9/10=M2nd

CD/diameterO4=18/32=9/16=m7th

QP/(DC/2)=5/9=m7th

QP/(radiusO4/2)5/8=m6th

radius04/diameterO1=16/30=8/15=M7th

radiusO4/radiusO1=15/16=m2nd

radiusO1/radiusO2=15/15=1/1=PU

not included

128/243

243/256

45/64

81/128

64/81

BA=BE=CD=CF=3 units (6x3=18 ticks)

CB=FE=DA=4 units (6x4=24 ticks)

CA=CE=5 units (6x5=30 ticks)

CT=7.5 units (6x7.5=45 ticks)

CP=4.5 units (6x4.5=27 ticks)

CQ=3.67 units (6x3.67=22 ticks)

PK=PL=3.33 units (6x3.33=20 ticks)

QP=.83 units (6x.83= 5 ticks)

RS=2 units (6x2=12 ticks)

TU=9 units (6x9=54 ticks)

diameter O1 and O2=5 units (6x5=30 ticks

radius O1 andO2=2.5 units (6x2.5=15 ticks)

diameter O3=10 units (6x10=60 ticks)

radius O3=5 units (6x5=30 ticks)

diameter O4=5.33 units (6x5.33=32 ticks)

radius O4=2.67 units (6x2.67=16 ticks)

Dimensions and Musical Ratios from those Dimensions in Apollo Geometry

6

810

B

G

E

Eb

Bb

G

6

108

5/5

(G)

5

30

16/15

(Ab)

5.33

32

6/5

(Bb)

6

36

4/3

(C)

6.66

40

8/5

(Eb)

8

48

9/5

(F)

9

54

10/5

(G)

10

60

n nu circle

a alpha circles

6-5-2005

Eb48(5 x8/5)

C 40 (5 x 4/3)

G 3

0 (5

x 1

0/10

)G

30 (2.5 x 10/5)

Ni F# 32 (10 x 8/15)

Db 32 (5 x16/15)

D 40 (10 x 2/3)

B48

(10 x4/5)

A 54 (10 x 9/10)

E 36 (10 x 3/5)

F 54 (5 x 9/5)

Bb 36 (5 x 6/5)

G 30 (2.5 x 10/5)

G 3

0 (5

x 1

0/10

)

A

G

E

FCD

K PL

O1 O2

O3

O4

x x

x

x

O5

H

diam O1 and O2 = 5 units = 30 ticks

diam O4 = 5.33 units = 32 ticks

DF = AE = GH = 6 units = 36 ticks

KL = 6.66 units = 40 ticks

po

(I have called this the feminineoctave because it contains thekomal or soft intervals. Like the

Moon, which is reflected light,these might be called reflectivesounds. They are the “reflected”

intervals of the masculine overtonalintervals which are heard directly.)

DG = FH = 8 units = 48 ticks

diam 05 = 9 units = 54 ticks

diagonals DH = FG = 10 units = 60 ticks

diameter O3 = 10 units = 60 ticks

omicron circles

pi circle

The Masculine Octave

The right side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 5 - 10 units or 30 - 60 ticks

string length ratios(denominator larger than numerator)

ex. 10 x 8/15 = 5.33 units5.33 x 6 = 32

(I have called this the masculineoctave because it contains the

overtonal or direct intervals. Like theSun, whose light is direct, these arethe overtonal intervals which can be

heard directly above the fundamental tone.)

The Feminine Octave

The left side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 5 - 10 units or 30 - 60 ticks

vibrational ratios(numerator larger than denominator)

ex. 5 x 16/15 = 5.33 units5.33 x 6 = 32

5/5

(G)

5

30

8/15

(F#)

5.33

32

3/5

(E)

6

36

2/3

(D)

6.66

40

4/5

(B)

8

48

9/10

(A)

9

54

10/10

(G)

10

60

Dimensions And Musical Ratios In Apollon Geometry - Octave 5 - 10 units, 30 - 60 ticks

3

45

B

G

E

Eb

Bb

G

3

54

5/5

(G)

2.5

15

16/15

(Ab)

2.67

16

6/5

(Bb)

3

18

4/3

(C)

3.33

20

8/5

(Eb)

4

24

9/5

(F)

4.5

27

10/5

(G)

5

30

6-5-2005

Eb24

(2.5 x8/5)

C 20 (2.5 x 4/3)

G 3

0 (5

x 1

0/10

)G

30 (2.5 x 10/5)

F# 16 (5 x 8/15)Db 16 (2.5 x16/15)

D 20 (5 x 2/3)

G 1

5 (2

.5 x

5/5

)

B24(5 x4/5)

A 27 (5 x 9/10)

E 18 (5 x 3/5)

F 27 (2.5 x 9/5)

Bb 18 (2.5 x 6/5)

G 15 (5 x 5/5)

G 30 (2.5 x 10/5)

G 3

0 (5

x 1

0/10

)

A27

(5 x9/10)

F27

(2.5 x9/5)

AB

E

FCD

K PL

O1 O2

O3

O4

x x

x

x

O5

radius O1 and O2 = 2.5 units = 15 ticks

radius O4 = 2.67 units = 16 ticks

AB = BE = DC = CF = 3 units = 18 ticks

KP = PL= 3.33 units = 20 ticks

DA = CB = FE = 4 units = 24 ticks

Dimensions And Musical Ratios In Apollon Geometry - Octave 2.5 - 5 units, 15 - 30 ticks

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(I have called this the feminineoctave because it contains thekomal or soft intervals. Like the

Moon, which is reflected light,these might be called reflectivesounds. They are the “reflected”

intervals of the masculine overtonalintervals which are heard directly.)

CP = 4.5 units = 27 ticks

radius 05 = 4.5 units = 27 ticks

diagonals CA = CE = 5 units = 30 ticks

diameter O1 and O2 = 5 units = 30 ticks

radius 03 = 5 units = 30 ticks

omicron circles

pi circle

omicron circles

alpha circle

lambda rectangles

lambda rectangles

nu circle

lambda rectangles

The Masculine Octave

The right side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 2.5 - 5 units or 15 - 30 ticks

string length ratios(denominator larger than numerator)

ex. 5 x 8/15 = 2.67 units2.67 x 6 = 16

(I have called this the masculineoctave because it contains the

overtonal or direct intervals. Like theSun, whose light is direct, these arethe overtonal intervals which can be

heard directly above the fundamental tone.)

The Feminine Octave

The left side of the largest circle1 unit = 6 ticks, 10 units = 60 ticks

10 units is the diameter of the largest (Alpha) circleOctave covers 2.5 - 5 units or 15 - 30 ticks

vibrational ratios(numerator larger than denominator)

ex. 2.5 x 16/15 = 2.67 units2.67 x 6 = 16

5/5

(G)

2.5

15

8/15

(F#)

2.67

16

3/5

(E)

3

18

2/3

(D)

3.33

20

4/5

(B)

4

24

9/10

(A)

4.5

27

10/10

(G)

5

30

2000 = The Perfected Work + The Gnosis2/1 = 2.000

729 Euergesia + 1271 h Gnwsiı

C

B

RS/BC=12/24=1/2 P8ve

R S1224

x

xx

BC/Rs=24/12=2/1 P8ve

D C

A E

DC/AE=18/36=1/2 P8ve

18

36

x

xx

AE/DC=36/18=2/1 P8ve

The Octave

1/2 - .500

500 = the number

o ariqmoı

1/2 - .500500 = Thou art + IEOU (Gnostic name of God)

15 Ei + 485 IEOU

11-15-2005

500 = Goddess + The all-holy1/2 - .500

15 Qea+ 485 o panagioı

x x

x

30 30

60

O1 O2

O3

diamO1 (or O2)/diamO3=30/60=1/2=P8vediamO3/diamO1 (or O2)=60/30=2/1=P8ve

The Perfect Fifth and Perfect Fourth

D F

A E

AD/DF=24/36=2/3 P5th

24

36

x

xx

DF/DA=36/24=3/2 P5th

D C

R S

RS/DC=12/18=2/3 P5th

12

18

x

xx

DC/RS=18/12=3/2 P5th

3/2 = 1.5001500 = Light, life

fwı

3/2 = 1.5001500 = To know the God Apollo

85 oida + o Qeoı Apollwn

3/2 = 1.5001500 = Beloved lyre

1150 Eufileı + 350 kiqariı

4/3 = 1.3331333 = Beloved Hestia (var)810 filiı +523 Estih

4/3 = 1.3331333 = Apollo + Harmony1061 Apollwn + 272 Armonia

3/4 = .750750 = Helios(Sun) + Foundation (radius of Sun)

318 Hlioı + 432 katabolh

D

A

AD/diam O4=24/32=3/4 P4th

24

32x

xx

damO4/AD=32/24=4/3 P4th

11-15-2005

O4

2/3 = .6666666 = the heart, soul, mind

h frhn

D C

A

DC/DA=18/24=3/4 P4th

24

18

x

xx

AD/DC=24/18=4/3 P4th

2/3 = .6666666 = Whole; perfect + wisdom

108 olh + 558 mhtiı

D C

A

DA/AC=24/30=4/5 M3rd

24

30

x

xx

CA/AD=30/24=5/4 M3rd

The Major Third and Major Sixth4/5 = .800800 = Faith

pistiı

4/5 = .800800 = Of the Universe

Kosmon

800 = Lord; God4/5 = .800

Kurioı

5/4 = 1.2501250 = Divine (divine = 25. 25^2 x 2 = 1250)

qeia

D C

A

DC/CA=18/30=3/5 M6th

30

18

x

xx

AC/CD=30/18=5/3 M6th

3/5 = .600600 = Thou

su

h qeothı

600 = The Godhead3/5 = .600

5/3 = 1.6661666 =Beloved Demeter

1198 prosfilhı + 468 Dhmhthr

1666 =Beloved Hestia5/3 = 1.666

1150 eufuleı + 516 Estia

16/27 = .5925592 = Divinity

agaqothı

27/16 = 1.68751687 = Consciousness + Kosmos

1087 Suneidhsiı + 600 Kosmoı

1687 = Consciousness + the Godhead27/16 = 1.6875

1087 Suneidhsiı + 600 h Qeothı

1687 = Consciousness + Thou27/16 = 1.6875

1087 Suneidhsiı + 600 Su

C

P

radius 04/PC=16/27= 16/27 M6th

27

16x

xx

CP/radius 04=27/16=27/16 M6th

11-15-2005

O4

600 = Kosmoskosmoı

3/5 = .600

The Minor Third and Minor Sixth 5/8 = .625625 = to learn; to know friendship; amity

74 daenai + 551 filia

5/8 = .625625 = divine + cosmos25 qeia + 600 kosmoı

5/8 = .625625 = divine + Thou

25 qeia + 600 Su

8/5 = 1.6001601 = Lights; wisdom

fwta

P

QP/(radius O4/2)=5/8= 5/8 m6th

5

8x

xx

(radius 04/2)/PQ=8/5=8/6 m6th

Q

O4

D F

A

AC/DF=30/36=5/6 m6th

30

36

x

xx

DF/AC=36/30=6/5 m6th

P

PC/diamO4=27/32= 27/32m6th

27

32x

xx

diamO4/PC=32/27=32/27 m6th

C

O4

5/6 = .833832 = beloved by God

qeofiloı

6/5 = 1.2001200 = Light, lamp

lucnon

27/32 = .8437843 = joy + kithara (lyre)702 cara + 141kiqara

32/27 = 1.1851185 God + Mind; understanding

284 Qeoı + 901 gnwmh

32/27 = 1.1851185 flute + path

701 Auloı + 484 dromoı

C

11-15-2005

The 8/9 9/8 Major Second and the 9/16 16/9 Minor Seventh

P

AD/PC=24/27=8/9M2nd

2724

x

xx

PC/AD=27/24=9/8M2nd

C

A

D

8/9 = .888888 = Olen (Apollo’s first prophet)

Wlhn

888 = JesusIhsouı

8/9 = .888

9/8 = 1.125

1125 = to lead to a placekatagw

9/8 = 1.1251125 = Apollo + Truth

1061 Apollwn + 64 alhqeia

9/16 = .5625562 = verily

dhpou

16/9 = 1.7771777 = the treasure + soul; understanding

1058 o qhsauroı + 719 qumoı

1777 = the treasure + purity1058 o qhsauroı + 719 kaqariothı

16/9 = 1.777

radius O4/CF=16/18=8/9M2nd

18

16x

xx

CF/radiusO4=18/16=9/8M2nd

C F

O4

P

ADx2/PC=48/27=16/9m7th

2748

x

xx

PC/ADx2=27/48=9/16m7th

C

A

D

diamO4/CF=32/18=16/9m7th

18

32x

xx

CF/diamO4=18/32=9/16m7th

C F

O4

The 32/45 45/32 Tritone, the 9/10 10/9 Major Second and 5/9 9/5 Minor Seventh

9/10 = .900901 = mind, understanding

gnwmh

10/9 = 1.1111110 = the microscosm

o Mikroı Kosmoı

1110 = the macrocosmo Kosmoı o oloı

10/9 = 1.111

5/9 = .5555555 = foundation; image

idruma

9/5 = 1.8001800 = breath, spirit, inspiration +

the creation of God576 pneuma + 1224 ktisiı Qeou

9/5 = 1.800720 = mind; understanding +

the Holy Spirit720 nouı + 1080 to pneuma agion

P

(DC/2)/PQ=9/5=9/5m7th

5

9

x

xx

PQ/(DC/2)=5/9=5/9m7th

Q

CD

P

AB/KP=18/20=9/10M2nd

20

18

x

xx

KP/AB=20/18=10/9M2nd

BK

A

32/45 = .7111711 = Muse

Mousa

45/32 = 1.4061407 = Vital, full of life

zwtikoı

diamO4/CT=32/45=32/45tritone

32

45

x

xx

CT/diamO4=45/32=45/32tritone

T

C

O4

The 15/16 16/15 Minor Second, and the 8/15 15/8 Major Seventh

15/16 = .9375938 = compassion

sumpaqhı

16/15 = 1.0661065 = understanding, knowledge

sunesiı

8/15 = .5333533 = Sun + Jupiter318 Hlioı + 215 Diaı

15/8 = 1.8751875 = Apollo + Pythagoras1011 Apollw + 864 Puqagoraı

15/8 = 1.8751875 = Apollo + the divine wisdom1061 Apollwn + 814 h qeia sofia

radiusO1/radiusO4 =15/16=15/16m2nd

16

15

x

xx

radiusO4/radiusO1=16/15=16/15m2nd

O4

O1

AC/diamO4 =30/32=15/16m2nd

32

30

x

xx

diamO4/AC=32/30=16/15m2nd

A

C

O4

radiusO4/AC =16/30=8/15M7th

16

30

x

xx

AC/radiusO4=16/30=15/8m2nd

A

C

O4

diamO4/diamO3 =32/60=8/15M7th

32

60xx

diamO3/diamo4=60/32=15/8m2nd

x

x

O4

O3

8/15 = .5333533 = the love + foundation101 h agaph + 432 katabolh

#

There is an excellent explanation of the mathematical geometry of the vesica piscis and the fishes in the net in bothJohn Michell’s Dimensions of Paradise and David Fideler’s Jesus Christ, Sun of God. The same story is is related about

both Pythagoras and Jesus knowing the exact number of fishes in the net. In the bible, the exact number of fishes is given as 153. In the Pythagorean version, the actual number is not given—only that Pythagoras knew exactly howmany fish were in the net, and when he was found to be correct, asked that the fishes be freed back into the water.

To add musically to this incredibly deep symbol, if the note at A432Hz is used as the basis (and it is a good basisbecause the radius of the Sun is 432,000 miles and the gematria for 432 is “foundation” katabolh), then the geometric

mean of octave A216 and A108 is Eb153 (152.735 exact)! Also, although A432 is the hertz basis, I am using the note D as the ratio of 1/1, the ratio from which all other ratios are generated. The octave D144 is in relation to Eb153 in the ratio of 17/16, which is again in Greek gematria o logoı iquı, “the true ratio,” when 17/16 is used in it’s decimal form 1.062.

153

153

153

153

153

153

153

153

153

153

153

153

153

153

153

153

153 Hz306 Hz

216 Hz

EbEb-1

108 Hz

GM

GM

153 as the Measure of “the Fish” in Archimedes The whole number ratio 153:265, which Archimedes uses for √3 in his treatise On the Measurement of the Circle, corresponds to the height and length of each rhombus in the diagram of the 153 fishes in the net. Since 153:265 was known as the “measure of the fish” or vesica piscis in pre-Christian times, those skilled in mathematics would have immediately recognized the story of the 153 fish in the net as a geometrical “story problem.” In tems of accuracy, the ratio 153:265 differs rom the real value of √ by about one part in

10,000.

A-2 A-1

11-15-2005

b

#

#

180°(0)

209°31(30)

216°(36)

247°19(67)

280°(100)

315°(135)

322°40(143)

0°(180)

19°41(200)

24°(204)

45°(225)

66°36(247)

90°(270)

95°06(275)

120°(300)

146°09(326)

151°55(331)

Total degrees = 2527°57’ . 2527=7x19^2

864 Pythagoras Puqagoraı

1275 Orpheus Wrfeuı

388 The Sun o Hlioı

Total=2527

If the enharmonic notes used in equal temperament (like the black keys on a keyboard)are separated out and treated as different notes, the total number of notes is 17 rather than the

12 notes of 100 cents each in equal temperament. This is just the right number of notes needed tofill in the diamond net of the vesica piscis. There are 16 small diamonds and the one big diamond

which is the outer frame. The ratio of 17/16 in it’s decimal form is 1.062. What makes this remarkable is that 1062, in Greek gematria, is o logoı iquı, “the true ratio”

A

A#

Bb

B

C

C#

Db

D

D#

Eb

E

F

F#

Gb

G

G#

Ab

A

A#

Bb

B

C

C#

Db

D

D#

F#

F

E

E

Gb

G

G#

Ab

11-15-2005

☀

✩

✩

✩

✩

✩

✩

Light, though really threefold, has traditionally also a sevenfold nature. In the scheme of geometric representation, the first emanation from, or manifestation of, the Absolute, in the series we are now considering, is figured as a Cube developed from an original point, which is one of its angles, and this is portrayed in its visible aspect of symmetry as a Hexagon with six internal lines radiating from the center.

The . . . New Creation is symbolized by the Cube, for a true knowledge of which a sense beyond that of mere physical vision is required. By this the presence of the seventh ray is revealed and the Eight points are manifest in their true relation. Thus from One are seen to proceed Seven and the Seven are connected with the parent One by rays or lines of Three several lengths (or which are now seen to be of three several lengths), wheras in the flat representation only six were seen, and they were apparently of equal length.

☀✩

✩

✩

✩

✩

✩

✩

Theoretically there is a seventh radius but never more than six are visible because the seventh approaches the eye and unites the central visible point with the eighth point lying exactly behind it.

The six outward points, with their connecting lines represent the Kosmos, or manifestation of the Divine Power, and the seventh, the central point, is the Divine Source, the revelation of God to man.

But the eighth or invisible original is required to complete the figure of the cube, which is only apprehended by the higher reason.

Now we are in the presence of three geometrical entities which by their joint operation determine the form of the Cube, and these are first, the length of its side which is figured as 100—the basic number in the Denary system (far older than decimal notation), on which the whole of the Gematria may be said to be built.

From Gematria, A Preliminary Investigation of The Cabalaby Bligh Bond and Simcox Lea

11-15-2005

☀✩

✩

✩

✩

✩

✩

✩

There are, as will readily be seen, three rays of this first order emanating from any one point in the cube, and connecting that point with the three lying nearest to it in the cube. Secondly there are to be seen three more connecting lines which are the diagonals on the square sides of the cube, linking the first point with the three next in order of remoteness, and these lines, on the scale now determined, measure each 100 x √ 2, or 141.42—their total, 424.26, being integrally expressed as 424 or 425 by substituting unity for the fraction. Finally, there is the One ray leading diametrically across the cube to the opposite point, and the measure of this is 100 x √ 3, or 173.205. Call this 173. thus we have the total of the Seven rays

☀✩

✩

✩

✩

✩

✩

✩

☀✩

✩

✩

✩

✩

✩

✩

side of square = 100100 x 3 = 300

diagonal of square side of cube100 x √ 2 = 141.42141.42 x 3 = 424.26

(424 or 425)

diagonal across cube100 x √ 3 = 173.205

Thus we have for the total of the Seven rays the sum of 300 + 424.5 + 173, which is 897 or 898 and may with equal propriety be taken as either, as the actual figure is halfway between. Hence we find in Gematria that whereas 897 gives AIZ EN THASIA EKKLHSIAI, 898 is the number by Gematria of OI Z ASTERES, the seven stars or rays which are the angels of the seven Churches.

In the three incommensurable orders of magnitude derived from the measures of the cube we have clearly a very suggestive parallel to the description in the Pistis Sophia of the three kinds of Light in the Lord’s enduma, which we are told vary in the quality of their beams (variae lucis) and in the imcommensurable nature of their proportions (vario tupw aliis aliis praestantibus infinito modo).

What are the authors of this old book trying to tell us? Let us assume that they really mean something—that they have a rational idea which they are trying to express. With this assumption let us then endeavor to see what sort of parallel modern science can offer to the interpretation we have placed upon the ‘seven rays’ and the three cardinal measures of the cube.

Those who have studied the physics of Light will know that the reputed Seven colours of the spectrum are really Three, and that the blending of these three produce the effect of Seven. There are three primary sensations of colour, and three only, the Red, the Green and the Violet-blue. And these each occupy a certain part of the field, indefinite as to its boundaries, but having an acme of intensity at or near the centre of each section. Near the centre of the spectrum, at a point where the purest green is observed, is the position of one of the fixed lines of colour known as Frauenhofer’s E line. There are other such lines, lettered A to H, distributed over the field, and these are measured according to their wave-length, in inverse order to their frequency, and the lengths are given in what are called tenth-metres (see Ganot’s Physics).

The whole series of light-rays comprises those whose wave-length varies from a little more than 7600 tenth-metres in the extreme Red to a little less than 3900 in the extreme Violet—altogether about an octave of light. Now if we take the Green ray at about line E or 5270 in measure then our series must be as follow:—

Red 7453 . . . . or 5270 x √ 2

Green 5270

Violet . . . . or 7453 x (1/√ 3)

and this is remarkable, because 7453 brings us to the intense Red close to Frauenhofer’s A line (7604) and between it and B, whilst 4302 for the Violet brings us to the point of the most intense deep Blue, and practically on to Frauenhofer’s line G which is 4307.

We must be content to leave our instructed readers to judge as to the real nature of so striking an approximation. If, on further investigation, the parallel appears based on undeniable physical fact, then a few more such instances may go far to rehabilitate the lost wisdom of Antiquity. And if it be true that the data of the colour-scale—which are derived from molecular motions—are indeed founded upon harmonics whose source is not, as in the case of sound and other physical measure, based upon the interaction of forces having whole-number proportionals, but upon those mysterious entities, the Roots of Two and Three, then we are face to face with a condition pointing to a genesis of motion in a region of space unknown to us, and wherein the dynamic laws operate in a relation quite unfamiliar. It looks as if a fourth and interior dimension must be assumed, and is this not precisely what some physicists say of the Atom, that it must possess some substance in a fourth dimension?

From GEMATRIA, A Preliminary Investigation of the Cabala, by Frederick Bligh Bond and Thomas Simcox Leapublished by Research into Lost Knowledge Organization Trust

Looking at Red, Green, and Blue from the hexagonal cube viewpoint

Looking at Cyan, Magenta, and Yellow from the hexagonal cube viewpoint

Blue 43027453 x (1/√ 3)Frauenhoferline G (4307)

Red 74535270 x √ 2Frauenhoferline A (7604)

Green 5270Frauenhoferline E (5270)