According to recorded history, Pythagoras was the first individual to study the relationships between music and mathematics. He established that the perfect consonances of music were simple, whole-number ratios — octave (1:2), perfect fifth (2:3), and perfect fourth (3:4) — and that all musical relationships could be described mathematically. Both the tonal pitches of a given scale, and the string lengths used to produce them, could be described through whole-number ratios.

In mapping the spaces of mathematics and music, I use color as a key to indicate relations between tone, as well as shapes to map the different types of intervals.

Pure Pythagorean tuning uses only ratios or divisors of 2 and 3. This is used as a starting point, numbers further up in the overtone series are introduced to generate the ratios of what is often called “Just tuning”: for example, 6:5, the just minor third (D–F).

The diagrams available in the music section present a complete analysis of the musical ratios 2-3-5, and information on the modes, the pattern of whole and half steps based on those ratios. Links to audio recordings of the modes appear at the bottom of this page. — Connie Achilles

> Download entire 28-page Music section in PDF format (1.1 MB)
Selections from Music section provided below

Harmonic Lattice from D

In describing the musical scale, modern discussions usually illustrate the octave using the span from C–C, which is easily demonstrated using only the white keys on a piano keyboard.

In my approach, I use the starting note of D 288 Hz to represent unison. The reason for this is that D, if placed in the center of an ascending and descending octave, is the only note that offers the same symmetry whether ascending or descending.

This diagram showing the harmonic latice derived from D is an important one that underlies my work and also illustrates the symbols I use to denote pitches. Circles are used for perfect intervals: unison (1/1), octave (2/1), perfect fifth (3/2), and perfect fourth (4/3). Ovals are used for other intervals that appear in the cycle of fourths and fifths. Finally, diamonds are used to indicate intervals with divisors of 2, 3, and 5, rather than just 2 and 3.

Overview of the Arithmetic, Harmonic, and Geometric Means

This illustration shows that perfect symmetry of the ratios in a double-octave with D as a starting point rather than C.

As the diagram shows, starting from the center D, the rising and falling pitch ratios are a perfect mirror image of one another.

This chart also gives the formulas of the Arthimetic, Harmonic, and Geometric Means from classical Greek mathematics and music theory. These means are used in tuning to find the intermediate intervals that exist between two extremes.

Color Tunings in Hue, Saturation, and Brightness

In this tone mandala, the innermost circle shows 12 equal divisions of the octave. This circle of squares thus corresponds to 12-tone equal temperament tuning, in which Gb and F#, for example, represent the same pitch.

In the middle circle, the enharmonic notes of 12-tone ET represent different pitches. For example, Gb and F# are now slightly different tones.

The outermost circle shows the 2-3-5 limit harmonic ratios, which divides the octave into 22 separate notes.

The colors associated with each tone value are taken from the 360-degree circular color model of hue, saturation, and brightness on the Macintosh computer.

How A B C D E F G are Generated

Using D as a starting point, this diagram shows how the diatonic scale can be generated using only the ratios of 3/2 and 4/3 — the arithmetic and harmonic means of the octave.

Defining D as 288 hertz, the Greek gematria value of stibos (path), the corresponding gematria values of 288 and the other notes are given. This section deals extensively with the relations between hertz and corresponding gematria values.

The Arithmetic and Harmonic Mean of the Perfect Fifth

Applying the arithmetic and harmonic means to the perfect fifth — G:D and D:A — describes the location of the intermediate intervals of major and minor thirds.

In this diagram, the arithmetic means are shown above, the harmonic means are shown below. The gematria values of the summed triads are also given.

The Triads of Pythagoras and Jesus

Using the means generated in the previous diagram, we see that the arithmetic-generated triad of B-D-F# (above) gives the gematria value of 888 — Jesus — teacher of the Christian mysteries.

Similarly, the harmonic-generated triad of Bb-D-F (below) gives the sum of 864 — Pythagoras — the teacher of the Pythagorean mysteries.

Arithmetic and Harmonic Means of the Major Third

In the complete collection of music diagrams that is available for download, I have conducted an extensive analysis of the arithmetic and harmonic means of the octave, fifth, and third.

This particular example shows the arithmetic and harmonic means of the major third, which produces the large 9/8 major second and the small 10/9 major second. The hertz sum of the harmonic mean of C-D-E in this case is equivalent again to 864, the gematria value of Pythagoras.

The Pattern of Whole and Half Steps in Musical Modes where the Pattern is the Same in Both Upper and Lower Tetrachords

The seven classical modes formed the basis of musical performance in ancient Greece.

Each Greek mode consists of an upper and lower tetrachord (group of four adjacent notes) joined together to form the mode. In this diagram, I have shown the three modes where the pattern of steps is the same in both the upper and lower tetrachords.

Recordings of the modes in mp3 format are also provided on this website via the links below. (Note: the names of the modes given are the names used in modern music theory.)

Audio Files of the Musical Modes (mp3 format) © by Connie Achilles. All rights reserved.