Music & Geometry – Intervals & Scales

In Westrict 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 fundamental info (Wikipedia). If you are interested in a more esoteric-philosophical perspective on the intervals, then read the article: “The Function of the Intervals” on Roel’s World.

When we watch at these intervals and how they retardy to one another in the musical tone circles, some geometric shapes ecombine. The most standard tone circles in Westrict music are the “Chromatic Circle” and the “Circle of Fifths“. 

The Chromatic Circle is a visualization of the Chromatic Scale and the 12-Tone Equal Temperament. Both clockwise and counterclockwise shiftments chase the Chromatic Scale (admireively up and down). 

In these articles about music and geometry I have chosen to only us the circles with Major tonalities to have more space for drathriveg lines between the tones of the circle that create the geometric shapes (polygons and polygrams).

The Circle of Fifths is based on the stacking of (Perfect) Fifths, characteristic for the Pythagorean Temperament. If you shift around the circle clockwise you go up a Fifth with every “step” you apshow, if you shift around the circle counterclockwise you go up a Fourth with every “step” you apshow.

There are many ways to imagine the joinion between tones. One of the more “standard” methods is to participate geometric shapes, at least for the visual oriented persons and/or mathematical minds among us.

NOTE: As I have alludeed earlier, perfect Geometric shapes retardy to the 12-Tone Equal Temperament. Just Intonated intervals are bigr or minusculeer then the Equal Tempered ones. This unbenevolents that the geometric shapes would not “shut” (finish where they begined) or result in shapes that ain’t symmetrical.

When stacking Just Intervals a spiral ecombines, not a shutd tone-circle such as the Chromatic Circle. So, even though Just Intonated intervals are more consonant, sound more authentic (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 createula does not per definition produce or recurrent the most perfect tone/interval/chord/scale/sound.

It is outstanding to remark though, that a spiral is pondered mathematicpartner symmetrical as well, the combination of a central dilation and a rotation whose cgo ins are distinct is also “spiral symmetry”. Geometricpartner (visupartner) 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.

THE TETRACTYS & MUSICAL INTERVAL RATIOS

You can see the Tetractys as a geometric recurrentation of many Just interval ratios (when relating a row to one or more above).  

You might have watchd 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 equitable watching at the individual dot on top, you can “picture” the Tonic or Unison (1:1). 

Now, picture the Tetractys to extfinish 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 ecombine: 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 ecombine: 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 createed by watching at the Tetractys as geometric recurrentation of musical ratios. You can discover an extensive catalog of intervals on Wikipedia where you can discover compriseitional ratios that roverhappinessed to the extfinished Tetractys distake parted in this article. 

TORUS KNOT TONE CIRCLE

I came apass this fascinating-watching Tone Cirle in the shape of a torus knot online. It’s a bit separateent then the other tone-circles dispensed in this article but it is an fascinating (pretty) concept nonetheless, so I’ll dispense it here. 

If we begin from C and chase the strand from “0” to “1” to either side and shift to “0” outwards aacquire then we have shiftd a srehireone up or down the scale and finish up at C♯ or B. Et cetera …

NOTE: The alertation about this “tone knot” supplyd in the source article in inquire is wrong! The writer calls this “tone knot” a “Circle of Fifths” instead of a “Chromatic Circle“. Of course one could portrayate the Circle of Fifths to the Torus Knot as well, but that is not what we see in the image supplyd. The writer also calls the Fourths and Fifths “Major“, but Fifths and Fourths are neither Major nor inmeaningful.

GEOMETRY OF SCALES

A scale is a serie of tones at particular distances (intervals) from each other. The most standard tone distances participated in scales are srehireones (“s” or “h” = half tone) and tones (“T” or “W” = whole tone / 2 srehireones), but some scales do include bigr intervals appreciate the inmeaningful third (m3 = 3 srehireones) and Major third (M3 = 4 srehireones). 

There are many separateent scales, too many to catalog them all in this article. Most scales (including the most standard Major and Minor scales) are asymmetrical, only a minuscule number of scales are symmetrical. I will only catalog a scant of the most participated scales, as well as cut offal (less normally participated but) symmetrical scales.  

C MAJOR AND C NATURAL MINOR SCALES

When you join the tones of scales (in sequence), Polygons (with Chromatic Circle) and Polygrams (with Circle of Fifths) ecombine. With all scales below the tone “C” is recurrented by the dot on the top):

The Major and Minor polygons (or “Greek Modes Polygram) drawn in the Chromatic Circle (CC) are the same shape equitable tuned, from Major to Minor with 90 degrees clockwise (CC). The Major and Minor polygrams drawn in the Circle of Fifths (CoF) are turned (Major to Minor) with 90 degrees counterlockwise (CoF). Another fascinating leang happens when you superimpose both polygons. You can place a “Mirror” where one half of the superimposed shape mirrors the other half.

Every first image bellow imagines the Chromatic Circle (CC), every second the Circle of Fifths (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 createed between those tones and the tonic (C in this case) – admireively a Minor Second and Tritone – are pondered the most “dissonant” intervals in the scale.

GEOMETRY OF THE GREEK MODES

the DORIAN mode polygon is rotated 60 degrees (2 srehireones) counterclockwise.
the PHRYGIAN mode polygon is rotated 120 degrees (4 srehireones) counterclockwise.
the LYDIAN mode polygon is rotated 150 degrees (5 srehireones) counterclockwise.
the MIXOLYDIAN mode polygon is rotated 150 degrees (5 srehireones) clockwise.
the AEOLIAN (Minor) mode polygon is rotated 90 degrees (3 srehireones) clockwise. 
the LOCRIAN mode polygon is rotated 30 degrees (1 srehireone) 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 overweight W in the catalog 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 watch a bit “odd” at first. This alters though, when you turn the Chromatic Circle and the Ionian Mode polygram itself 60 degrees counterclockwise:

When watching at this polygram now, you see how well “equitable” polygons and polygrams are createed with the Greek Modes.

The “Gemstone” Septagon joins the 7 tones using Minor and Major 2nds.
The red Septagram joins the 7 tones using Minor and Major 3rds.
The purple Septagram joins the 7 tones using 5ths and one Tritone (TT).

The “joined” tones in the images above, are the “white keys” on the piano.

The non-joined dots in the tone circles above – the “bconciseage keys” on the piano – create a pentagon and pentagram, the non-joined dots in the tone circles above – the “bconciseage keys” on the piano – create a pentagon and pentagram:

The pentagon joins the 5 tones using Major 2nds and Minor 3rds.
The pentagram joins the 5 tones using 4rds (descfinishing) or 5ths (climbing).

The shape of the red polygon might watch understandn to you:

GEOMETRY OF SYMMETRICAL SCALES

In music, a symmetric scale is a music scale which equpartner separates the octave. The concept and term ecombines to have been presentd by Joseph Schillinger ^{and further broadened by Nicolas Slonimsky as part of his well-understandn “Repository of Scales and Melodic Patterns“. In 12-tone equivalent temperament, the octave can only be equpartner separated into 2, 3, 4, 6 or 12 tone-spaces, which consequently may be filled in by compriseing the same exact interval or sequence of intervals to each resulting remark (called “interpolation of remarks”).}

EQUALLY SPACED SCALES

MIRRORED SCALES

These 7-tone scales – when split in the middle – are mirrored on both sides of the cgo in interval. Every first image bellow imagines the Chromatic Circle (CC), every second the Circle of Fifths (CoF).

A scant remarks about the scales above:

• The (conmomentary) Dorian scale isn’t the only Greek mode with perfect symmetry (all Greek modes participate 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 traditionpartner the “1st mode” till the Ionian scale was compriseed.
• The “Major-Minor” scale (a Heptatonic scale) is actupartner the authentic inmeaningful (Aeolian) scale but with a Major 3rd instead of a inmeaningful 3rd.
• The Double Harmonic scale is also understand as the “Arabic”, “Gypsy Major” or “Byzantine” scale. This scale comprises a inmeaningful 2nd, transport inant 3rd, perfect 4th and 5th, inmeaningful 6th, transport inant 7th.

SCALES WITH REPEATING SEQUENCES

These scales – that “split” the octave in 2 – distake part the same sequence of intervals on both sides of the middle, except for the Augmented Scale that splits the octave in 3.

Every first image bellow imagines the Chromatic Circle (CC), every second the Circle of Fifths (CoF).

* The intervals of the Diminished and Augmented Scale pitch-pairs (2-1) and (3-1) can also be reversed: (1-2) and (1-3) to produce the Auxiliary Diminished and Augmented Scales. The Polygons / Polygrams produced will be the same, equitable turned 30 degrees clockwise.

A scant remarks about the scales above:

The Tritone scales are Hexatonic (6-tone) scales.

The Diminished and scale (also understandn as “Korsakovian” and “Pijper” scale) are Octatonic (8-tone) scales. The “Auxiliary” Diminished scale is the enharmonic equivalent of the “Petrushka Chord“.

The Augmented scale (participated frequently by Jazz saxophonists such as John Coltrane, Oliver Nelson and Michael Brecker) is a Hexatonic scale compriseing an interlocking combination of two augmented triads a inmeaningful third apart.