MUSIC & GEOMETRY
A note to start this article with:
Perfect geometric shapes only occur when using Equal Temperament. Sonically though the intervals of Equal Tempered instruments/music aren't perfect (accept for the Octave). They are always a bit sharp or flat in comparison with the Harmonic Series (how sound occurs in nature) and Just Intonation.
The most harmonious sounding (Just) intervals do NOT create perfect symmetrical geometric shapes!
In this article all geometrically "perfect" shapes thus relate to 12Tone Equal Temperament as do any and all perfect geometrical shapes used as well as on this blog as on any other blog, website and book available.
"Music theory has no selfevident foundation in modern mathematics yet the basis of musical sound can be described mathematically (in acoustics) and exhibits "a remarkable array of number properties".^{}
In its form, rhythm and metre, the pitches of its notes (intervals) and the tempo of its pulse music can be related to the mathematical measurement of time and frequency, offering ready analogies in geometry."
Now, of course tones and chords are not the same "things" as for example polygons and polygrams. Units like Hertz and Degrees are not the same, they have their own function and use. But in the math behind many "things" there are formulas and ratios (relationships) that are very similar, if not identical. Would it thus be too 'farfetched' to say that the same "rules" apply for many  if not all  things in the universe?
This article will "zoomin" into geometry in particular.
Skip "INTRODUCTION (ABOUT 12TONE CIRCLES)" AND ...
↧JUMP TO "GEOMETRY OF INTERVALS"
↧JUMP TO "THE TETRACTYS & MUSICAL INTERVAL RATIOS"
↧JUMP TO "GEOMETRY OF SCALES"
↧JUMP TO "GEOMETRY OF THE GREEK MODES"
↧JUMP TO "SEQUENCE OF THE FALLING FIFTHS (OR FOURTHS)"
↧JUMP TO "GEOMETRY OF SYMMETRICAL SCALES"
↧JUMP TO "GEOMETRY OF CHORDS"
↧JUMP TO "GEOMETRY OF HARMONIC PROGRESSIONS"
↧JUMP TO "POLAR (NEGATIVE) HARMONY"
↧JUMP TO "TONNETZ GRID"
↧JUMP TO "GEOMETRY OF RHYTHM"
↧JUMP TO "THE COLTRANE CIRCLE"
↧JUMP TO "TORUS KNOT TONE CIRCLE"
↧JUMP TO "TONES, ASTROLOGY & GEOMETRY"
↧JUMP TO "COUNTING CORNERS (ANGLE DEGREE TO HERTZ)"
↧JUMP TO "3 DIMENSIONAL REPRESENTATION"
↧JUMP TO "PLATONIC SOLIDS"
↧JUMP TO "NUMBER SEQUENCES"
INTRODUCTION (ABOUT 12TONE CIRCLES)
In Western music theory there are 13 intervals from Tonic (unison) to Octave. These intervals are the: Unison, Minor Second, Major Second, Minor Third, Major Third, Fourth, Tritone, Fifth, Minor Sixth, Major Sixth, Minor Seventh, Major Seventh and Octave.
Note: click on the interval names for more basic info (Wikipedia). If you are interested in a more esotericphilosophical perspective on the intervals, then read the article: "The Function of the Intervals" on Roel's World.
When we look at these intervals and how they relate to one another in the musical tone circles, some geometric shapes appear. The most common tone circles in Western music are the "Chromatic Circle" and the "Circle of Fifths".
The Chromatic Circle is a visualization of the Chromatic Scale and the 12Tone Equal Temperament. Both clockwise and counterclockwise movements follow the Chromatic Scale (respectively up and down).
The Circle of Fifths is based on the stacking of (Perfect) Fifths, characteristic for the Pythagorean Temperament. If you move around the circle clockwise you go up a Fifth with every "step" you take, if you move around the circle counterclockwise you go up a Fourth with every "step" you take.
It is important to make not of though, is that the Circle of Fifths (on the right) is geometrically incorrectly drawn. In the case of the Circle of Fifths starting on C going clockwise around, the first 11 sections should be drawn a tiny bit bigger. The result of this is that the last section (F) would be relatively smaller then the other 11 sections, if you close the circle. When "Stacking" perfect Fifths the Circle isn't closed, but creates a spiral, the C after the F slightly overshoots (approx. about the size of a Quartertone.
For perfect geometric shapes, you should always use an "Equal Temperament". Any "Just Temperament" (based on the Harmonic Series) will not result in perfect geometric shapes.
RELATIVE MAJOR / MINOR TONALITIES
To give a "complete" picture in this introduction about tone circles you see on the right the Circle of ascending Fifths/Fourths with both Major and minor tonalities. The relationship between the Major and minor tonalities is of course the same in the Chromatic Circle.
For this article I have chosen to only us the circles with Major tonalities to have more space for drawing lines between the tones of the circle that form the geometric shapes (polygons and polygrams).
Even though the relative Major and minor are on the same side of the circle they do not automatically "replace" each other musically.
For example: the geometric relationship C Major and G♭ Major is the same as the geometric relationship between A minor and E♭ minor. The relationship between C Major and E♭ minor might look the same geometrically within the circle but is a different one! The tonic of C Major is a Tritone apart from G♭ Major, but only a minor 3rd apart from E♭ minor.
DIFFERENCES BETWEEN THE CHROMATIC CIRCLE AND THE CIRCLE OF FIFTHS:
You might have noticed that the tone circles above have different "shades of grey" for their tones. This is done for a reason. From a "philosophical" point of view you could say that  when you look at the "movement" around the circles  the Chromatic Circle represents 'Polarity' and/or 'Dualism' and the Circle of Fifths represents 'Unity' and/or 'Nondualism'.
Of course the Circle of Fifths can be read counterclockwise as well, but than it would not longer be the Circle of Fifths but the Circle of Fourths instead. In other words, the Chromatic Circle is bidirectional (same interval size in both directions) and the Circle of Fifths (clockwise) unidirectional, as is the Circle of Fourths (counterclockwise).
CHROMATIC CIRCLE
When we look at the Chromatic Circle, then we can see two "poles", C and F♯/G♭. From these "poles" we move around the circle in both directions. Now, imagine placing a "mirror" from C to F♯/G♭.
Then B is "mirrored" by C♯/D♭ (5♯/7♭  7♯\5♭), A♯/B♭ by D (10♯/2♭  2♯\10♭), A by D♯/E♭ (3♯/9♭  9♯\3♭), G by F (1♯/11♭  11♯\1♭), and G♯/E♭ by E (8♯/4♭  4♯\8♭). F♯/G♭ "mirrors" itself within (6♯  6♭) as does C (0♯  0♭). The numbers stay the same but the "polarity" has changed (sharp to flat, plus to minus, "male" to "female").
CIRCLE OF FIFTHS
When going around the Circle of Fifths the buildup and decrease of the number of sharp and flats is constant. The sharps going up 011 while the flats go down simultaneously 110. And another pattern to be seen:
0  1  2  3  4  5  6  5  4  3  2  1 ( 0).
As you can go around this circle endlessly An natural gradualness with no beginning, no end. Ouroboros, (the taildevouring snake). A unbroken circle, a crosscultural symbol often representing completeness which encompasses all space and Time, "infinity" or "unity".
GEOMETRY OF INTERVALS  'CONNECTING THE TONES'
There are many ways to visualize the connection between tones. One of the more "common" methods is to use geometric shapes, at least for the visual oriented persons and/or mathematical minds among us.
LINE
The simplest "shape" is a line. The "connected" tones in the Chromatic Circle and the Circle of Fifths are a "Tritone" apart from one another: CG♭, GD♭, DA♭, AE♭, EB♭ & BF.
The lines drawn represent Equal Tempered Tritones of 600 cents.
TRIGON (triangle)
The clockwise "connected" tones are a Major Third (400 cents) apart from one another: CE, EA♭ & A♭C / GB, BE♭ & E♭G / DG♭, G♭B♭ & B♭D / AD♭, D♭F & FA. Counterclockwise (and going up the scale) the "connected" tones are a Minor Sixth (800 cents) apart.
The triad formed by playing all 3 tones of the trigon together is called "Augmented Triad".
SQUARE
In the Chromatic Circle the clockwise "connected" tones are a Minor Third (300 cents) apart from one another: CE♭, E♭G♭, G♭A & AC/B♭D♭, D♭E, EG & GB♭ / FA♭, A♭B, BD & DF. Counterclockwise (going up the scale) the "connected" tones are a Major Sixth (900 cents) apart.
With the Circle of Fifths you have to swap the direction to get the same interval.
A Triad formed by two legs of the square is called the "Diminished Triad". It contains the "Devil's Interval" ("Diabolus in Musica") better known as "TriTone", found in between the Tonic and the Fifth (CA♭, EC, A♭E). All 4 tones (square) together forms a "Diminished 7th Chord", containing two simultaneous played TriTones a Minor Third apart.
HEXAGON
The "connected" signs form two Hexatonic or Whole Tone Scales when combined: C–D–E–G♭–A♭–B♭–C and B–D♭–E♭–F–GAB. The clockwise "connected" tones are a Major Second (Whole Tone, 200 cents) apart, counterclockwise (goung ip the scale) they are a Minor Seventh (1000 cents) apart.
DODECAGON
The Chromatic Circle visualize the Chromatic Scale: clockwise in Minor Second (100 cents) and counterclockwise (going up the scale) Major Seventh (1100 cents). The Circle of Fifths visualizes clockwise the Fifths (Equal Tempered 700 cents), counterclockwise the Fourths (Equal Tempered 500 cents).
DODECAGRAM
When you follow the lines of the Dodecagram across the Chromatic Circle, you find clockwise the Fifths (700 cents) and counterclockwise the Fourths (500 cents). The Circle of Fifths visualizes the Chromatic Scale: clockwise in Minor Second (Equal Tempered 100 cents) and counterclockwise (going up the scale) Major Seventh (Equal Tempered 1100 cents).
SUPERIMPOSED
With these geometric "shapes" combined, all possible toneconnections (intervals) of the 12Tone system can be made, as is clear from the combined graph.
NOTE: As you might have noticed, perfect Geometric shapes relate to the 12Tone Equal Temperament. Just Intonated intervals are larger or smaller then the Equal Tempered ones. This means that the geometric shapes would not "close" (end where they started) or result in shapes that ain't symmetrical.
When stacking Just Intervals a spiral appears, not a closed tonecircle such as the Chromatic Circle. So, even though Just Intonated intervals are more consonant, sound more natural (Harmonic Series), are more pleasing to the ear, the geometric shapes will be less then perfect. This by itself "proofs" (in my opinion) that a perfect geometric shape or mathematical formula does not per definition generate or represent the most perfect tone/interval/chord/scale/sound.
It is good to note though, that a spiral is considered mathematically symmetrical as well, the combination of a central dilation and a rotation whose centers are distinct is also "spiral symmetry". Geometrically (visually) though the spiral isn't symmetrical as the (Equal Tempered) shapes seen in the circles above that have "Reflection" (or "Mirror") Symmetry, Rotational Symmetry and Point Symmetry.
INTERVALS SUPERIMPOSED FROM ONE TONIC (IN COLOR)
This is what all intervals related to only one tonic (in this example the tone C) would look like. The colors used at the tonecolor combination (Concert Pitch 440Hz) as described in the Roel's World article about Sound (Music) & Light (Color).
THE TETRACTYS & MUSICAL INTERVAL RATIOS
The tetractys (Greek: τετρακτύς), or tetrad, or the tetractys of the decad is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number.
You can see the Tetractys as a geometric representation of many Just interval ratios (when relating a row to one or more above).
You might have noticed already the Octave 2:1, as well as the Just Fifths (3:2)  the Pythagorean scales are based on the stacking of the Just Fifth  and it's inversion, the Just Fourth (4:3).
When just looking at the single dot on top, you can "picture" the Tonic or Unison (1:1).
Now, picture the Tetractys to extend beyond the row with 4 dots, with more dots: 5, 6, 7, 8 ... et cetera, then we get: 5:4 (Just Major Third), 6:5 (Just Minor Third), 7:6 (Septimal Minor Third), 8:7 (Septimal Major Second), 9:8 (Pythagorean Major Second), 10:9 (Small Just Whole Tone), 11:10 (Neutral Second or "Ptolemy's Second"), 12:11 (Undecimal Neutral Second), et cetera ...
And if you would "skip" a row in the Tetractys, more intervals appear: 3:1 (Just Twelfth), 5:3 (Just Major Sixth), 7:5 (Septimal Tritone), 9:7 (Septimal Major Third), 11:9 (Undecimal Neutral Third), et cetera ...
Or "skip" 2 rows and even more intervals appear: 4:1 (Fifteenth or Two Octaves), 7:4 (Harmonic Seventh), 8:5 (Just Minor Sixth), 10:7 (Septimal Tritone), et cetera ...
And how about "skipping" 3 rows: 9:5 (Greater Just Minor Seventh), 11:7 (Undecimal Minor Sixth), 15:11 (Undecimal Augmented Fourth / 'Tritone'), et cetera ...
As you can see, many intervals can be formed by looking at the Tetractys as geometric representation of musical ratios. You can find an extensive list of intervals on Wikipedia where you can find additional ratios that related to the extended Tetractys displayed in this article.
GEOMETRY OF SCALES
A scale is a serie of tones at specific distances (intervals) from each other. The most common tone distances used in scales are semitones ("s" or "h" = half tone) and tones ("T" or "W" = whole tone / 2 semitones), but some scales do include larger intervals like the minor third (m3 = 3 semitones) and Major third (M3 = 4 semitones).
There are many different scales, too many to list them all in this article. Most scales (including the most common Major and Minor scales) are asymmetrical, only a small number of scales are symmetrical. I will only list a few of the most used scales, as well as several (less often used but) symmetrical scales.
When you connect the tones of scales (in sequence), Polygons (with Chromatic Circle) and Polygrams (with Circle of Fifths) appear. With all scales below the tone "C" is represented by the dot on the top):
C MAJOR AND C NATURAL MINOR SCALES
The Major and Minor polygons (or "Greek Modes Polygram) drawn (below) in the Chromatic Circle (CC) are the same shape just tuned, from Major to Minor with 90 degrees clockwise (CC).
Another interesting thing happens when you superimpose both polygons. You can place a "Mirror" where one half of the superimposed shape mirrors the other half.
C MAJOR (IONIAN)  C MINOR (AEOLIAN)  
TTsTTTs 
TsTTsTT 
C MAJOR & MINOR SUPERIMPOSED 
MIRROR 

WWhWWWh  WhWWhWW  
2212221  2122122  
CC 

The Major and Minor polygrams drawn (below) in the Circle of Fifths (CoF) are turned (Major to Minor) with 90 degrees counterlockwise (CoF). 

CoF 
The two tones of the circles are not part of the superimposed C Minor and C Major polygons / polygrams are the C♯/D♭ and F♯/G♭. In both scales the intervals formed between those tones and the tonic (C in this case)  respectively a Minor Second and Tritone  are considered the most "dissonant" intervals in the scale.
The Minor Second though is part of the "tone connections" of the polygons / polygrams, as you can see listed below. Interestingly the Major Third does appear in the tone connections of the polygrams / polygons.
TONE    TONE  INTERVALS CC 
C 
 
G 
Fifth / Fourth 
D 
 
F 
Minor Third / Major Sixth 
D♯/E♭ 
 
E 
Minor Second / Major Seventh 
G♯/A♭ 
 
B 
Minor Third / Major Sixth 
A♯/B♭ 
 
A 
Minor Second / Major Seventh 
GEOMETRY OF THE GREEK MODES
All "Greek Modes" use the same shape polygon as the Major and Minor scales (Chromatic Circle). On the left an image of the IONIAN mode (or Major scale) polygon (with the tone C on top). The polygons of the other Greek Modes can be found by simply rotating the polygon:
 the DORIAN mode polygon is rotated 60 degrees (2 semitones) counterclockwise.
 the PHRYGIAN mode polygon is rotated 120 degrees (4 semitones) counterclockwise.
 the LYDIAN mode polygon is rotated 150 degrees (5 semitones) counterclockwise.
 the MIXOLYDIAN mode polygon is rotated 150 degrees (5 semitones) clockwise.
 the AEOLIAN (Minor) mode polygon is rotated 90 degrees (3 semitones) clockwise.
 the LOCRIAN mode polygon is rotated 30 degrees (1 semitone) clockwise.
You can also say the sequence of Whole (W) and Half (h) tones is "shifting":
 W  W  h  W  W  W  h = Ionian (the Major scale)
 W  h  W  W  W  h  W = Dorian
 h  W  W  W  h  W  W = Phrygian
 W  W  W  h  W  W  h = Lydian
 W  W  h  W  W  h  W = Mixolydian
 W  h  W  W  h  W  W = Aeolian (the Minor scale)
 h  W  W  h  W  W  W = Locrian
The fat W in the list above is the Whole Tone distance between the G and A in the polygon, opposite the D.
The shape of the "Greek Modes Polygon" might look a bit "odd" at first. This changes though, when you turn the Chromatic Circle and the Ionian Mode polygram itself 60 degrees counterclockwise:
SEPTAGON  SEPTAGRAM  SEPTAGRAM  SUPERIMPOSED 
2  1  2  2  2  1  2 D  C  B  A  G  F  E  D 
m3  M3  m3  M3  m3  M3  m3 D  F  A  C  E  G  B  D 
5th  5th  5th  TT  5th  5th  5th D  A  E  B  F  C  G  D 
All tone connections. 
When looking at this polygram now, you see how well "balanced" polygons and polygrams are formed with the Greek Modes.
 The "Gemstone" Septagon connects the 7 tones using Minor and Major 2nds.
 The red Septagram connects the 7 tones using Minor and Major 3rds.
 The purple Septagram connects the 7 tones using 5ths and one Tritone (TT).
The "connected" tones in the images above, are the "white keys" on the piano.
The shape of the red polygon might look familiar to you:
The nonconnected dots in the tone circles above  the "black keys" on the piano  form a pentagon and pentagram:
PENTAGON  PENTAGRAM  PENTA SUPERIMPOSED  ALL SUPERIMPOSED 
2  m3  2 m3  2 
4th <> 4th < > 5th > < 4th <> 4th 
White Keys (Greek Modes) Black Keys (penta) 
 The pentagon connects the 5 tones using Major 2nds and Minor 3rds.
 The pentagram connects the 5 tones using 4rds (descending) or 5ths (ascending).
THE "SEQUENCE OF FALLING FIFTHS"
The purple Septagram displayed earlier also visualizes another aspect in music as well: the "harmonic progression" or "chord progression", a succession of musical chords. This chord progression is the foundation of harmony in the Western musical tradition.
The numeral sequence of the falling (descending) Fifths goes as follows: (4 ) 7  3  6  2  5  1
Musicians and composers generally prefer using the Roman numerals instead though: (IV ) VII  III  VI  II  V  I
In C Major we get the following chord sequence:
(F Major ) B diminished  E minor  A minor  D minor  G Major  C Major.
All "steps" in the sequence "lead" to the tonic, the root of the tonality a composition is written in. I've put the (4) or (IV) in the sequence above between brackets, because the interval between F and B is not a Fifth but a Tritone.
"THE SEQUENCE OF FALLING FOURTHS"
If you place the 4 on the other side of the reversed sequence then we get a perfect sequence of descending Fourths": 4  1  5  2  6  3  7:
F Major  C Major  G Major  D minor  A minor  E minor  B diminished.
This "Sequence of Falling Fourths" actually appears when you use the RADIAN measure of 57.29577951 degrees when going 'round the Circle of Fourths (or Circle of Fifths in clockwise movement), as Zachary G Covell demonstrates (see image on the right). Zack writes:
"After 7 times around the circle all four of the 90 degree lines are filled in. The 11th, 22nd, 33rd, and 44th step in the lines placing each one at a new radian of 57.29577951 degrees becomes the axes of the directions at 90 degrees to one another. Only a musician or someone who knows some music theory will see the 4152637 locations. The 4152637 are Harmonic Progression in Music."
Going around the Circle in 7 times isn't a "coincidence" either. When stacking (slightly tempered) Fourths / Fifths you will rise 7 octaves before you come across the "same" tone as you started with (but 7 octaves higher) and "close" the circle. I talk about "tempered" Fourths / Fifths in this case, because if you stack Just (perfect) Fourths / Fifths you would slightly 'overshoot' the circle with approx. a quartertone (this difference between 12 stacked Perfect 5ths and 7 stacked Octaves is also called the "Pythagorean comma") and would thus generate a spiral and not close the circle, as demonstrated by Zach.
The 11th, 22th and 44th are Tritones. The 33rd is not (it's a "Quartertone"), nor are the 55th, 66th, and 77th. But the 88th is, as is the 176th ... This is because there are some "doubling sequences" within the harmonic series as well. The most obvious is that of the 1st harmonic (fundamental): 1248163264et cetera. But, so does the "Tritone" in the harmonic series: 11224488176352et cetera. The 33rd harmonic (a "Quartertone") "doubles" as follows: 3366132et cetera, as does the 55th harmonic: 55110220et cetera ... you see the "system of doubles" within the harmonic series, right?
For the "number fetishists" and philosophers among us:
The numerologists among you might have noticed the 11, 22, 33 and 44 series, the so called "Master Numbers" (although not all numerologists use 44).
These numbers are also the first 4 double digit palindromic numbers, numbers that remains the same when its digits are reversed. The Circle of Fifths can be used both ways, the interval stays the same only the direction changes, ascending (clockwise) vs. descending (counterclockwise) 5th.
In the Bible the numbers 11 and 22 symbolize disorder, chaos and judgment. The Tritone is commonly experienced as the most "dissonant" or "tensionful" in the 12Tone system. 33 represents "God's Jugement".
If you count from the 1st line drawn in the circle, the 33rd Harmonic becomes the last "leg" of the cross and the beginning of the last "sequence" (of 4) to close the circle/cycle.
In the Bible a 44 day period began on the day Jesus was crucified and ended with his resurrection. When you look at the circle, the 44th harmonic completes / ends the circle/cycle. It "ends" the "period" between "crucifixion" (33rd Harmonic) and 44th Harmonic.
I know these things don't have much to do with math or geometry, but are fun nonetheless.
GEOMETRY OF SYMMETRICAL SCALES
In music, a symmetric scale is a music scale which equally divides the octave. The concept and term appears to have been introduced by Joseph Schillinger^{} and further developed by Nicolas Slonimsky as part of his famous "Repository of Scales and Melodic Patterns". In 12tone equal temperament, the octave can only be equally divided into 2, 3, 4, 6 or 12 tonespaces, which consequently may be filled in by adding the same exact interval or sequence of intervals to each resulting note (called "interpolation of notes").
EQUALLY SPACED SCALES
CHROMATIC SCALE at CC  CHROMATIC SCALE at CoF  HEXATONiC SCALE at CC & CoF 
sssssssssss 
sssssssssss 
TTTTT 
11111111111  11111111111  22222 
MIRRORED SCALES
These 7tone scales  when split in the middle  are mirrored on both sides of the center interval:
C NEOPOLITAN  C DORIAN  C MAJORMINOR  C DOUBLE HARMONIC  
sTT  T  TTs  TsT  T  TsT  TTs  T  sTT  sm3s  T  sm3s  
122  2  221  212  2  212  221  2  122  131  2  131  
CC 

CoF 
A few notes about the scales above:
 The (modern) Dorian scale isn't the only Greek mode with perfect symmetry (all Greek modes use the same shape polygon), but it's "turned" so that the top of the "gemstone" polygon and the C on top of the Tone Circles align. The Dorian scale was traditionally the "1st mode" till the Ionian scale was added.
 The "MajorMinor" scale (a Heptatonic scale) is actually the natural minor (Aeolian) scale but with a Major 3rd instead of a minor 3rd.
 The Double Harmonic scale is also know as the "Arabic", "Gypsy Major" or "Byzantine" scale. This scale contains a minor 2nd, major 3rd, perfect 4th and 5th, minor 6th, major 7th.
SCALES WITH REPEATING SEQUENCES
These scales  that "split" the octave in 2  display the same sequence of intervals on both sides of the middle, except for the Augmented Scale that splits the octave in 3:
C TRITONE  C 2SEMITONE TRITONE  C DIMINISHED *  C AUGMENTED *  
sm3T  sm3T  ssM3  ssM3  Ts  Ts  Ts  Ts  m3s  m3s  m3s  
132  132  114  114  21  21  21  21  31  31  31  
CC 

CoF 
* The intervals of the Diminished and Augmented Scale pitchpairs (21) and (31) can also be reversed: (12) and (13) to create the Auxiliary Diminished and Augmented Scales. The Polygons / Polygrams created will be the same, just turned 30 degrees clockwise.
A few notes about the scales above:
 The Tritone scales are Hexatonic (6tone) scales.
 The Diminished and scale (also known as "Korsakovian" and "Pijper" scale) are Octatonic (8tone) scales. The "Auxiliary" Diminished scale is the enharmonic equivalent of the "Petrushka Chord".
 The Augmented scale (used frequently by Jazz saxophonists such as John Coltrane, Oliver Nelson and Michael Brecker) is a Hexatonic scale containing an interlocking combination of two augmented triads a minor third apart.
GEOMETRY OF CHORDS
Just like with scales also chords form geometric shapes. For this article I will only share the shapes and Polygons for C Major and C Minor. These shapes are the same though for all 12 tonalities.
CHROMATIC CIRCLE
C MAJOR (CM)  C MINOR (Cm)  SUPERIMPOSED  MIRROR  12 TONALITIES 
CIRCLE OF FIFTHS
C MAJOR (CM)  C MINOR (Cm)  SUPERIMPOSED  MIRROR  12 TONALITIES 
Other common chords in all 12 tonalities superimposed (the colored polygons are Cbased):
C dim  C Aug  Cm7  C7  CM7  
CC  
CoF 
As you might have noticed, there are various chords that "generate" the same polygon when the 12 tonalities have been superimposed, like the Cm7 and C7 in both the Circle of Fifths and the Chromatic Circle.
There are of course many more chords variations with additional notes such as for example 4, 6, 9, 11 and more. Perhaps when I have some free time I'll add some more to this article.
HARMONIC PROGRESSIONS
When you draw a square around the Circle of Fifths and draw lines between the corners of the square, 4 triangles appear. In each triangle we see the chord progressions that belong to the Major tonality in the center of the outer ring, highlighted by the green triangle. Diminished chords are usually not drawn into the tone circle, but to complete the picture I have added it into a 3rd ring in the center.
This concept only works with the Circle of Fifths. It does not work this way with the Chromatic Circle.
The example below is based on C Major, but you could turn the Circle of Fifths in any direction to see the chord progressions for any other Major tonality.
DEGREE  TONE  FUNCTION  CHORD (TRIAD) 
1  C  TONIC  C MAJOR 
2  D  SUPERTONIC  D MINOR 
3  E  MEDIANT  E MINOR 
4  F  SUBDOMINANT  F MAJOR 
5  G  DOMINANT  G MAJOR 
6  A  SUBMEDIANT  A MINOR 
7  B  LEADING TONE  B DIMINISHED 
A few examples below of the 'movement' created by chord progressions:
BLUES PROGRESSION
A standard Blues uses the 1st, 4rd and 5th degrees:
I → IV → I → V → (IV) → I
See how the movement between the chords of a standard Blues takes place in the outer ring.
TURNAROUND
A standard turnaround uses the 1st, 2nd, 3rd, 5th and 6th degrees:
III → VI → II → V → I
A turnaround creates a "zlike" patern, from the inner ring to the outer ring.
"AUTUMN LEAVES" PROGRESSION & THE GEOMETRY OF THE AUTUMN EQUINOX
When you "map" the chord progressions of some compositions using the Circle of Fifths, geometric shapes may appear as well. Robert Aguirre discovered a very interesting one, relating the Jazz standard "Autumn Leaves" with the Autumn Equinox:
"Since you went away the days grow long ... And soon I'll hear old winter's song ..."
POLAR HARMONY  CHANGING "POLARITY" ("NEGATIVE HARMONY")
The following images are based on a concept shared by Ernest Levy (writer of "A Theory of Harmony") where every chord has a "reflection", a "polar opposite" chord within that key center, created by rotating every single tone around the axes of that key center. With other words, with this concept you can substitute a chord by changing it's "polarity". Some musicians refer to this method as "Negative Harmony". The mirror "reflection" is representative of the reflective qualities between the harmonic and subharmonic series, ("otonality & utonality").
In this concept the axes between the tonal center (root) and Fifth (the interval between the second and third harmonics, generally considered as the most important and consonant interval) is used to "mirror" chords and scales.
A simple way to visualize it is to "mirror" the chord in the Tone Circle.
In the first image you see the polygon of G7 and the the tonic of C. In order to create the "mirror" in the Chromatic Circle you need to draw a line between the tonics of the G7 and C chords and add perpendicular to this G  C axis the "mirror" (image 2). In image 3 you see the mirrored polygram of G7, that turns out to be a F Minor 6 (as noted in the 4th image).
G7  C  ADD AXIS & MIRROR  MIRROR G7  Fm6  C  
CC 

The process is pretty much the same with the Circle of Fifths, except that you add the "mirror" directly in between the tonic and fifth of the tone center:  
CoF 
Substituting G7 with Fm6 is changing the direction of approach to resolve at C. G to C is a Descending Fifth, F to C an Ascending Fifth. The F Minor 6 chord could also be seen / notated as a D7^{b5} chord, but it will not "accentuate" the dominant function as traditionally with G7 or alternatively it's "reflection" Fm6.
Famous musicians like saxophonist Steve Coleman (interesting read: Symmetrical Movement Concept) and Herbie Hancock have used "Negative Harmony" in their music. This subject gained most "exposure" and "popularity" due to Youtube videos by/with Jacob Collier.
TONNETZ GRID
Another way to visualize (and perhaps easier way to "map" the "polar conversion" of) a chord, is by using the Tonnetz Grid. In musical tuning and harmony, the Tonnetz Grid (German for tonenetwork) is a conceptual latticediagram representing tonal space first described by Leonhard Euler in 1739.
This way of visualizing "negative harmony" can be found in detail at this imgur post. For this Roel's World article I will only show you the example as described above with G7 and C. I'll use a Tonnetz Grid image by Jan Ott for it:
As you might have noticed, the in the Tonnetz Grid "connected" tones on the horizontal lines are a 5th apart. The tones connected diagonally from lefttop to rightbottom are a Minor 3rd apart, the tones connected diagonally from bottomleft to topright are a Major 3rd apart.
NEORIEMANNIAN TONNETZ TORUS
An interesting variation of the Tonnetz Grid shown above, is the Torus (or Donut) shaped NeoRiemannian Tonnetz Torus.
The red lines connect notes that are a Major Third (and Minor Sixth) apart.
The green lines connect notes that are a Minor Third (and Major Sixth) apart.
The blue lines connect notes that are a Perfect Fifth / Perfect Fourth apart.
With tose 3 lines (one of each color) triangles can be formed. These triangles represent either Major (redgreenblue / RGB) or Minor (greenredblue) Triad (3tone) Chords.
GEOMETRY OF RHYTHM
So far in this article we have looked at the geometric relationships between tone (pitch) and geometric shapes (polygons and polygrams). Music though is more then tones alone. An important aspect of music is rhythm and just as with tones, polygons can be used to visualize rhythm.
How does this work?
Well a circle could be divided in any kind of time signatures. You could divide a circle into 4 (standard '4/4 time') or into 3 ('3/4 time'  Waltz), into 5 (or '5/4 time'  best known from Dave Brubeck's "Take Five"), into 7 ('7/8 time'  Balkan rhythm), or what ever division you wish to use.
On the left you see a division of a circle into 32 parts (481632). This can be used for '4/4 time' musical pieces.
You could draw various standard geometric shapes like for example squares, triangles, lines, hexagons, hexagrams, et cetera.
That though will "limit" the rhythmic options you have. Small adjustments in standard shapes could turn a "static" groove into something more irregular or a groove that swings or shuffles ...
STANDARD '4/4 TIME' MEASURE  EXAMPLE 1:  
In this example on the left you see 3 geometric shapes:
1 5sided uneven polygon, 1 dodecagon and 1 line.
The blue uneven 5sided polygon visualized the kick drum, the yellow dodecagon visualizes the closed hihat and the red line visualizes the snare on the 2 and 4.


1  2  3  4  
Kick drum 
♪ 

♪ 

𝄽 



♪ 

♪ 

𝄾 

𝄿 
𝅘𝅥𝅯 
Snare 
𝄽 
♩ 
𝄽 
♩ 

Closed Hat 
♬ 

♬ 

♬ 

♬ 

♬ 

♬ 

♬ 

♬ 

STANDARD '4/4 TIME' MEASURE  EXAMPLE 2:  
In the example on the left you see 4 geometric shapes: 1 line, 1 blue square, 1 green square and 1 polygon with 12 uneven sides.
The blue square visualizes the kick drum in a so called "Four on the Floor" pattern (common for Disco and House music), the green square visualizes the open hihat, the yellow unevensided polygon visualized the closed hihat and the red line visualizes the snare on the 2 and 4. 

1  2  3  4  
Kick drum 
♩

♩

♩

♩


Snare 
𝄽 
♩ 
𝄽 
♩


Open Hat 
𝄾 

𝅘𝅥𝅯 
𝄿 
𝄾 

𝅘𝅥𝅯 
𝄿 
𝄾 

𝅘𝅥𝅯 
𝄿 
𝄾 

𝅘𝅥𝅯 
𝄿 
Closed Hat 
♬ 
𝄿 
𝅘𝅥𝅯 
♬ 
𝄿 
𝅘𝅥𝅯 
♬ 

𝄿 
𝅘𝅥𝅯 
♬ 

𝄿 
𝅘𝅥𝅯 
These were just two very simple examples, but you could make these patterns as complicated as you wish.
XRONOMORPH
The absolute coolest free software you can use for turning geometric shapes into rhythm is called "XronoMorph".
"XronoMorph is a free OS X and Windows app for creating multilayered rhythmic and melodic loops (hockets). Each rhythmic layer is visualized as a polygon inscribed in a circle, and each polygon can be constructed according to two different mathematical principles: perfect balance and wellformedness (aka MOS). These principles generalize polyrhythms, additive, and Euclidean rhythms. Furthermore, rhythms can be smoothly morphed between, and irrational rhythms with no regular pulse can also be easily constructed.
Each polygon can play an independent sound, and XronoMorph comes with a useful selection of samples to play the rhythms. Alternatively, you can load your own VST or AU plugins, or send MIDI to an external software or hardware synth. The rhythmic loops can be saved as presets within XronoMorph; they can also be saved as Scala scale tuning files, which means XronoMorph can be used as a tool for designing wellformed (MOS) and perfectly balanced microtonal scales."
Watch the following videos to get a better idea about what you can do with this amazing software:
You can download XronoMorph for free from: www.dynamictonality.com
THE COLTRANE CIRCLE
An interesting variant to the "Circle of Fifths" is the "Coltrane Circle", created by saxophonist John Coltrane (perhaps based on Nicolas Slominksy's Thesaurus of scales and musical patterns?) and was used by Yusef Lateef for his work "Repository of Scales and Melodic Patterns".
Being a saxophonist myself and a fan of Coltrane's work I had no choice then to write something about this tone circle in this article as well. ;)
Below on the left you see a scanned copy of an original drawing of the "Coltrane Circle".
In have modified an image made Corey Mwamba from his article "Coltrane's Way Of Seeing" that clarifies the scanned image:
In the drawing (on the left) there are a couple of sharps notated, they have been replaced by Corey Mwamba with their enharmonic equivalents (C♯ = D♭ and F♯ = G♭) in his drawings.
Note: if the lines, numbers and the Pentagram in the Coltrane Circle (on the left) were drawn by John Coltrane himself is not clear. 
There is yet another interesting polygon to be drawn. 
NOTE: There is a lot more that can be said about the Coltrane Circle as well as the geometric relationships between chords and chord progressions in some of his music, in particular the album Giant Steps. If you like to know more about this, then read this Roel's World article: "The Coltrane Tone Circle".
TORUS KNOT TONE CIRCLE
I came across this interestinglooking Tone Cirle in the shape of a torus knot online. It's a bit different then the other tonecircles shared in this article but it is an interesting (pretty) concept nonetheless, so I'll share it here.
If we start from C and follow the strand from "0" to "1" to either side and move to "0" outwards again then we have moved a semitone up or down the scale and end up at C♯ or B. Et cetera ...
⟲  STEPS ON STRANDS  ⟳ 
Major 7nd 
0 ⇠ 1 ⇠ 0 ⇢ 1 ⇢ 0  Minor 2nd 
Minor 7nd  0 ⇠ 2 ⇠ 0 ⇢ 2 ⇢ 0  Major 2nd 
Major 6th  0 ⇠ 3 ⇠ 0 ⇢ 3 ⇢ 0  Minor 3rd 
Minor 6th  0 ⇠ 4 ⇠ 0 ⇢ 4 ⇢ 0  Major 3rd 
5th  0 ⇠ 5 ⇠ 0 ⇢ 5 ⇢ 0  4th 
Tritone  0 ⇠ 6 ⇠ 0 ⇢ 6 ⇢ 0  Tritone 
4th  0 ⇠ 7 ⇠ 0 ⇢ 7 ⇢ 0  5th 
NOTE: The information about this "tone knot" provided in the source article in question is wrong! The writer calls this "tone knot" a "Circle of Fifths" instead of a "Chromatic Circle". Of course one could assign the Circle of Fifths to the Torus Knot as well, but that is not what we see in the image provided. The writer also calls the Fourths and Fifths "Major", but Fifths and Fourths are neither Major nor minor.
TONES, ASTROLOGY & GEOMETRY
If you know a little about astrology, then you must have recognized some of the geometric shapes used in the tone circles earlier in this article, in particular those of the intervals. The tone circles might have also reminded you of the way the Zodiac Circle is draw.
Note: The Zodiac Signs in this article are placed clockwise and with Aries "on top". Traditionally the Zodiac Signs are drawn counterclockwise with Aries on the left. Another difference is that the "cusps" (the line in between the signs) traditionally align with North, South, East and West.
Both tone circles though are drawn as traditionally used, thus turning the Zodiac Circle to match both tone circles (music is the main subject of this article after all) seemed to be more convenient reading and comparing the tone circles with the Zodiac Circle.

LINE  TRIGON  SQUARE  HEXAGON  DODECAGON  DODECAGRAM  SUPERIMPOSED 

6{2}  4{3}  3{4}  2{6}  t{6}  t{6/5}={12/5} 

CC  TRITONES  MAJOR THIRDS MINOR SIXTHS 
MINOR THIRDS MAJOR SIXTHS 
MAJOR SECOND MINOR SEVENTH 
MINOR SECOND MAJOR SEVENTH 
FIFTHS FOURTHS 
ALL INTERVALS SUPERIMPOSED 
CoF  FIFTHS FOURTHS 
MINOR SECOND MAJOR SEVENTH 


Opposition  Trine  Square  Sextile  Semi Sextile 
Quincunx 




180°  120°  90°  60°  30°  150°  

"POLARITIES" Opposition 
"TRIPLICITIES" Fire 
"QUADRUPLICITIES" Cardinal Fixed Mutable 
"DUALITIES" Masculine Feminine

The degrees listed above relate to the angles of 2 dots relatively to each other from the center point of the geometric shapes..
Read more about music, astrology and "Tone Zodiacs" in the article "Zodiac Signs & Tonality (Music)".
"COUNTING CORNERS"
Another way to look at geometry and sound is angles (arc minutes) and the tones you could generate with them if you change the unit from degrees to Hertz:
POLYGON  ANGLES  ANGLE DEGREE  DEGREE TO HERTZ  HARMONIC  
TRIGON  3  x  60°  =  180  1st 
QUADRAGON  4  x  90°  =  360  2nd 
HEXAGON  6  x  120°  =  720  4rd 
DODECAGON  12  x  150°  =  1800  10th 
The sum of the angles of these polygons could create the tone of F♯ at 180Hz, 360Hz, 720Hz and 1800Hz.
Other polygons  not found in the standard tone circles  would also create interesting tones. For example, the Pentagram (5 angles of 108°) would generate a C♯ a harmonic 5th at above 180Hz, a Septagon (7 angles of 128 4/7°) would generate an A♯ at 900Hz, the Octagon (8 angles of 135°) generating a C♯ at 1080Hz, et cetera ...
With other words, the "series" of Polygons with increasing number of angles relates closely to the Harmonic Series in music.
Of course Hertz and Degrees is not the same thing. Most sound frequencies are 'directional' and/or 'omnidirectional'. No 'sharp edges' as with a triangle or square, but spherical and/or coneshaped.
You can't just "swap" units and say it's all the same. But there does seem to be a lot of similarities between the math behind the units, as between many things in the universe. Some 'laws' seem to apply for a lot of things if not everything.
3 DIMENSIONAL REPRESENTATION
Some authors have taking mapping tone relationships one step further, turning the more "traditional" 2D Polygons into 3D shapes. At the website www.cosmometry.net for example Marshall Lefferts converted the 2D Circle of Fifths into a 3D Vector Equilibrium.
Marshall Lefferts writes:
"This illustration shows that the tritone intervals are exactly opposite each other in the circle of fifths". [left]
"In this way we can easily see the six pairs of opposite notes that comprise the basic 12tone system of music."
"Extending this into the 3dimensional space of the vector equilibrium, one of the primary components of cosmometry, we can map these pairs like this (notice that the circle of fifths is maintained in the sequence of notes as they are arrayed visually around the center)." [image right]
"Buckminster Fuller stated that the vector equilibrium is composed of sixpairs of equal and opposite vectors radiating from its center point. The music system aligns with that model conceptually, though of course the frequency differences of the actual tritone notes are not equal."
Instead of blogging more about the 3 dimensional representations of musical harmony and tonerelationships it might be more useful to share some very interesting videos made by others with you:
PLATONIC SOLIDS
We can't call this geometry article "complete" without mentioning the Platonic Solids.
The reason to mention these is because of their obvious relation with a few of the shapes found earlier in this article, in particular the Trigon (triangle) and Quadragon (Square).
The Platonic Solid that is a direct "oneonone" match with musical geometry is the Tetrahedron. You can create a Tetrahedron by connecting 4 Trigons (the Major Thirds / Minor Sixths relationships within one octave within the tone circle).
For the Octahedron you would need 2 2/3 octaves, and 5 octaves for the Icosahedron.
The 3 Squares that represent the Minor Thirds and Major Sixths relationship within the tone circle only forms half a Hexahedron. You would need to cover 2 octaves to generate 6 Squares to complete the Hexahedron.
The only Platonic Solid that does not relate to the polygons created within the standard tone circle is the Dodecahedron, only in the Coltrane circle we could draw a Pentagon (and Pentagram).
Also interesting are the Schäfli Symbols {corners of the polygon shape of a face, edges connected to the corners}. Their recursive definition (or inductive definition) look similar to several tone interval ratios:
POLYHEDRON 
SCHLÄFLI 
AS RATIO 
INTERVAL  
{3, 3}  3:3 = 1:1  UNISON  
(Cube) 
{4, 3}  4:3  PERFECT 4TH  
{3, 4}  3:4 (3:2·2)  PERFECT 5TH  
{5, 3}  5:3  JUST MAJOR SIXT  
{3, 5}  3:5 (2·3:5)  JUST MINOR THIRD 
"For the Pythagoreans and later Greek philosophers, tetrachords were considered to be the auditory geometry of a “perfect” 4sided, 4pointed tetrahedron solid. As a geometrical model for music, the two tetrachords in a Greek mode could have been intended to represent two opposing and interlocking tetrahedrons balanced around a shared center, thereby creating the 8 vertices of a cube or hexahedron. Alternatively, the vertices (or points) of this cube could have also been intended to pinpoint the center of 8 triangular faces that form its geometric opposite, the octahedron.
In either case, the geometric perfection represented by two balanced tetrachords is the founding principle behind the Greek modes and thus even our present major and minor system of scales.
Greek geometers found that tetrahedrons could be used to construct more complex perfect shapes. For example, a pair of octahedrons can be balanced to form either the 12 vertices of an icosahedron or 12 pentagonal faces of a dodecahedron. Because of this, tetrachords, octaves and a stack of twelve perfect 5ths were associated with the geometrical shapes of the five perfect solids, which the Greeks considered integral to the structure of the universe."
From: "Interference  A grand scientific Musical Theory" by Rich Merrick (www.interferencetheory.com)
12TONE TETRAHEDRON
For creating a 12Tone Tetrahedron you need to combine 4 (2D) Trigons.
1 Trigon is formed by the (3) tones of an Augmented Triad.
Each trigon shifts a semitone up or down at the corners with the neighboring trigons.
MAPPING NUMBER SEQUENCES
You could also generate interesting looking polygons/polygrams by mapping mathematical sequences onto the Chromatic Circle and the Circle of Fifths. You could map practically any series of numbers, you could use your own birthday, telephone number, numerological character values of a word or what ever you feel like.
In the examples below I use semitones, every number in the sequence represents the number of semitones I go up the chromatic scale. The number "0" does represent nil movement around the circle.
You could of course also chose to use whole tones instead (the amount of semitones up the scale then becomes the number from the sequence x 2) or any other progression if intervals if you wish.
In the examples below I stop mapping when I doubled a tone. Of course you could also continue mapping until you have connected all 12 tones ...
PI 3.14159265 from C
For this example we use the first 9 digits of Pi. We start at C and go the number of semitones up as follows in the 8tone (plus octave) sequence:
C +3 D♯ +1 E +4 G♯ +1 A +5 D +9 B +2 C♯ +6 G +5 C
PHI 1.61803 from C
For this example we use the first 6 digits of Phi. We start at C and go the number of semitones up as follows in the 5tone / (pentatonic) scale (plus octave) sequence:
C +1 C♯ +6 G +1 G♯ +8 E +0, +5 C
Fibonacci Sequence: 0, 1, 1, 2, 3, 5 from C
For this example we use the first 6 digits of the Fibonacci Sequence. We start at C and go the number of semitones up as follows in the 5tone / (pentatonic) scale (plus octave) sequence:
C +0, +1 C♯ +1 D +2 E +3 G +5 C
If you like to read more about music and number sequences, then check "Music Composition & Math (numbers)".
REFERENCES MENTIONED IN THIS ARTICLE:
 Wikipedia (articles about: Pi, Phi, Fibonacci, and other).
 www.cosmometry.net by Marshall Lefferts
 "Coltrane's Way Of Seeing" by Corey Mwamba