MUSIC & GEOMETRY
"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  and that includes music and geometry/math  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 "INTERVALS & POLYGONS"
↧JUMP TO "SCALES & POLYGONS / POLYGRAMS"
↧JUMP TO "CHORDS & POLYGONS / POLYGRAMS"
↧JUMP TO "THE COLTRANE CIRCLE"
↧JUMP TO "TONES, ASTROLOGY & POLYGONS / POLYGRAMS"
↧JUMP TO "3 DIMENSIONAL REPRESENTATION"
↧JUMP TO "TORUS KNOT TONE CIRCLE"
↧JUMP TO "RHYTHM AND GEOMETRIC SHAPES"
↧JUMP TO "COUNTING CORNERS (ANGLE DEGREE TO HERTZ)"
↧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.
RELATIVE MAJOR / MINOR TONALITIES
To give a "complete" picture in this introduction about tone circles you see on the right the Circle of 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".
INTERVALS & POLYGONS / POLYGRAMS  '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.
TRIGON (triangle)
The clockwise "connected" tones are a Major Third 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 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 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 apart.
With the Circle of Fifths you have to swap the direction!
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) apart, counterclockwise (goung ip the scale) they are a Minor Seventh apart.
DODECAGON
The Chromatic Circle visualize the Chromatic Scale: clockwise in Minor Second and counterclockwise (going up the scale) Major Seventh. The Circle of Fifths visualizes clockwise the Fifths, counterclockwise the Fourths.
DODECAGRAM
When you follow the lines of the Dodecagram across the
Chromatic Circle, you find clockwise the Fifths and counterclockwise the Fourths. The Circle of Fifths visualizes
the Chromatic Scale: clockwise in Minor Second and
counterclockwise (going up the scale) Major Seventh.
SUPERIMPOSED
With these geometric "shapes" combined, all possible toneconnections (intervals) of the 12Tone system can be made, as is clear from the combined graph.
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).
SCALES & POLYGONS / POLYGRAMS
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
C MINOR (AEOLIAN)  C MAJOR (IONIAN)  
TsTTsTT 
TTsTTTs 
C MAJOR & MINOR SUPERIMPOSED 
MIRROR 

WhWWhWW  WWhWWWh  
2122122  2212221  
CC 

CoF 
The two tones that are not part of the polygons in both the Chromatic Circle (CC) and Circle of Fifths (CoF) are the C♯/D♭ and F♯/G♭. In both scales, (Minor and Major) 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.
CHROMATIC CIRCLE (CC)
The Major and Minor polygons drawn in the Chromatic Circle (CC) above might seem different at first, but the shapes are the same, just tuned, from Minor to Major with 90 degrees counterclockwise. Another interesting thing happens when you superimpose both polygons. You can place a "Mirror" where one half of the superimposed structure mirrors the other half.
TONE  TONE  INTERVALS  
C 
 
G 
Fifth / Fourth 
D 
 
F 
Minor Third / Major Sixth 
D♯/E♭ 
 
E 
Minor Second / Major Seventh 
G♯/A♭ 
 
A♯/B♭ 
Major Second / Minor Seventh 
B 
 
G 
Major Third / Minor Sixth 
CIRCLE OF FIFTHS (CoF)
With the Circle of Fifths when you superimpose both Major and Minor polygrams you get an even prettier geometric shape. And also here you can place a "Mirror" where one half of the superimposed structure mirrors the other half.
TONE  TONE  INTERVALS  
C 
 
G 
Fifth / Fourth 
F 
 
D 
Minor Third / Major Sixth 
A♯/B♭ 
 
A 
Minor Second / Major Seventh 
D♯/E♭ 
 
E 
Minor Second / Major Seventh 
G♯/A♭ 
 
B 
Minor Second / Major Seventh 
Another difference with the polygons of the Chromatic Circle is the intervals listed on the right in the table above. The Major Second / Minor Seventh and the Major Third / Minor Sixth are not listed.
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 is the only Greek mode with perfect symmetry. 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.
CHORDS & POLYGONS
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 MINOR (Cm)  C MAJOR (CM)  SUPERIMPOSED  MIRROR  12 TONALITIES 
CIRCLE OF FIFTHS
C MINOR (Cm)  C MAJOR (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.
CHORD 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 
8  C  OCTAVE  C MAJOR 
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.
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".
TONES, ASTROLOGY & POLYGONS
If you know a little about astrology, then you must have recognized some of the shapes used in the tone circles above. 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) would traditionally align with North, South, East and West.
Both tone circles though are drawn as traditionally used, thus "adjusting" 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 
MINOR SECOND MAJOR SECOND MINOR THIRDS MAJOR THIRDS FOURTHS TRITONES FIFTHS MINOR SIXTHS MAJOR SIXTHS MINOR SEVENTH MAJOR SEVENTH 
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)".
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 containing the TriTones 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:
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.
RHYTHM AND GEOMETRIC SHAPES
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'  most known from "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
"COUNTING CORNERS"
Another way to look at geometry and sound is the angles and the tones you could generate with them if you change the unit from degrees to Hertz:
POLYGON  ANGLES  ANGLE DEGREE  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, nor is a geometric shape and a series of tones. But there does seem to be a lot of similarities between them, in fact between many things in the universe. Some "laws" seem to apply for a lot of things of not everything.
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 
Could the Polyhedrons represent some of the intervals in 3dimensional space? This all might be a bit "farfetched" for some of you (visitors of my blog), but it's pretty cool nonetheless, I think.
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