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Fun with the Circle Of 5ths & Identifying Chords
In the process of teaching myself music theory, I have become very
impressed with the power of the Circle of 5ths (here I am talking about
the standard circle in 12-equal temperament). I have cataloged someinteresting properties, and I am always on the lookout for more.
1. A line through the center identifies notes which are a tri-tone
apart, and the line slices the circle into 2 keys, which are named by
moving one step clockwise from the line (the end points are in
both keys). This also sheds light on how keys are related to each
other, by being close or far apart around the circle. If they are
close, then they share a lot of notes.
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2. There is no need to write on the diagram where the relative minor
keys are, or draw a separate diagram for minor keys, because they
can easily be found (more on this later):
3. The pattern of the chromatic scale (notes most closely related in
pitch) is obtained by drawing a line through the center, say d#to a,
and then looking counter-clockwise (descending through the
chain of 5ths) to find the note below d#, and clockwise
(ascendinga 5th) to the note above d#. The pattern of the diatonic
scale is obtained by slicing the circle to isolate the particular key
like before, and then hopping around the half circle by 2 steps
each time, never straying to the other side of the tri-tone line.
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4. Likewise, if you can only remember part of the circ le and you want
to construct the rest, you need only draw a line through the center
from the note that you remember, and you can locate its sharp (by
going clockwise one step) or its flat (by going counter-clockwise):
5. Chord Resolution
It is easy to remember what dominant 7th chord resolves to what
tonic chord (major or minor) by going counter-clockwise around
the circ le of 5ths.
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6. Symmetries
The circle and the positions of the dots are invariant under the
following operations:
o 12 clockwise rotations and 12 counter-clockwise rotations
(rotate c to g, c to d, etc...) But rotating clockwise by n/12 of a
full turn has the same result as rotating (12-n)/12
counterclockwise, so there are only 12 distinct rotations in
all, not 24.
o There are 12 lines through the center for reflections (axes of
symmetry). 6 are lines through pairs of notes (tri-tones) and 6
are through the mid points between notes.
o There is one reflection through the center point, but it is
equivalent to a rotation of 180 degrees, so we won't discuss
this one further.
o
7. Rotating
an interval line or a chord shape corresponds to transposing to a
different key. Here we rotate the C major triad shape 4 positions
clockwise to get E major:
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8. If we antic ipate a litt le the chord diagrams that occur later on this
page, it is even possible to determine what are the chords within a
given key. In the key of C major, we can fit in the shape for the F
major triad, we shift it once clockwise and get the C major and
then the G major. When we try to shift it again, the 3rd hits the tri -
tone line and has nowhere else to go but up to the note f giving us
D minornow. Continuing to shift, we get Am and Em. Then we hit
the tri-tone line again, and this time the chord shape is deformed
into B-diminished. The main idea is that you can determine the
chords by what chord shape will fit. If I would have realized this
earlier, maybe I wouldn't have had to make these chord charts !
9. Intervals
o If we regard c as the lowest note in the octave, we see the
right half of the circle is comprised of major intervals (capital
M) and the left side is composed of the complementary minor
intervals (baby m). Complementary means that the number of
semi-tones of a segment and its mirror image always adds to
12. And of course we can rotate this arrangement of arrows
to eminate from any other note on the circ le, which is what is
really important.
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o If we now let the left half circle be notes in the octave
below c and the right half be notes above, we get mirror
images of exclusively major intervals.
o We can play the same game by letting the right side of the
circ le be in the octave below c and the left side above and
get mirror images of minor intervals, but since it is late at
night I won't draw that.
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10. Reflections
Reflections are very cool because they always have the property
of turning major triads into minor ones and vice versa! First
observe the symmetry between C and Cm:
o We can reflect through any symmetry line; the rule is
that the 5th of the major triad is interchanged with the root of
the minor triad . Thus we can easily find the name of any of
these reflected chords.
o This implies an equal importance of major and minor triads.
See Theory of Chordspage.
o Not only is the root triad reflected, but the entire key is
reflected. In the picture you can check that all the notes of C
major are mapped to the notes of C (natural) minor.
o As an example, consider reflecting the C major triad through
the line joining abto d. The note g is mapped to a which
means that C is mapped to Am(NOTE: this reflection
is not what is in the picture above or the table shown below).
The tri-tone line of C is b-f and that is the perpendicular
bisector of the ab to d line. All the notes in the key of C are
preserved, but C is interchanged with Am. This perhaps
sheds some light on why Am. is called therelativeminor to C!
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Reflection between C major and C minor (the parallel minor)
Roman
numberNotes
Chord
nameChord name Notes
Roman
number
I c-e-g C Cm g-eb-c i
ii d-f-a Dm Bb f-b-d VII
iii e-g-b Em A e -c-a VI
IV f-a-c F Gm d-b -g v
V g-b-d G Fm c-a -f iv
vi a-c-e Am Eb
bb
-g-eb
III
viio b-d-f B dim D dim a
b-f-d ii
o
11. Why does any major triad always reflect to a minor triad? It is
enough to show this for C major, since we can rotate to any other
major chord. From the root note c, the 5th is 1 step clockwise (we
always measure distances clockwise, so from now on I'll just say
the distance is "+1 steps") and the major 3rd is +4 steps and the
minor 3rd is -3 steps. From rotational symmetry, this is true for all
the chords.
12. Now we make an arbitrary reflection of the circ le, and so c is
mapped to c', and likewise e e', and g g'. We have to show
that this new triad is a minor chord. c' and g' are still adjacent to
each other, since reflections can't change distances, but now g' is
-1 steps from c', so g' is the root and c' is the 5th. Since in fact all
intervals are reversed in reflections (we'll prove this at the end),
the distance from c to e is +4 and thus the distance from c' to e' is
-4. This means that the distance from g' to e' is -3 and we have all
the intervals for a minor chord.
13. Lemma: In any reflection, the direction of any interval is
reversed.
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14. If a point x on the circle is counter clockwise from a
point y then the distance (measured clockwise) satisfies d(p,x)
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