Spatial Mapping of Just Intonation in Overtone Dimensions

This record explores a geometric approach to understanding pitch relationships in just intonation.
It visualizes three overtone series—3rd (perfect fifths), 5th (major thirds), and 11th (undecimal intervals)—as spatial axes, representing each pitch as a hand-drawn map.
Particular attention is given to the 11/8 series, where the numerator of the frequency ratio appears to increase similarly to powers of 11, suggesting an intriguing alignment between harmonic structure and mathematical patterning.
This attempt is part of a broader exploration into understanding just intonation as a multidimensional space using spatial cognition.

Reiji's Observations

• This is a configuration of the circle of fifths, circle of thirds, and circle of elevenths based on just intonation, arranged in a space that is both 3D and 4D.
• The blue spiral line, the green vertical lines, and the yellow diagonals represent the fifth, third, and eleventh intervals respectively.
• The blue letters indicate fifth-based notes, pink letters over the green lines indicate third-based notes, and brown letters aligned with yellow diagonals indicate eleventh-based notes. The blue "C" represents the tonic.
• Repeating the fifth cycle 12 times in just intonation results in a pitch approximately 23 cents sharper.
• The third cycle repeated 3 times ends up about 41 cents flatter.
• The eleventh cycle repeated twice leads to about 98 cents flatter.

Observing the Calculation Method

• Fifths: Multiply by 3/2, and divide by 2 if it exceeds one octave: 1 → 3/2 → 9/4 → 9/8 → 81/32 → 81/64 → 243/128…
• Thirds: Multiply by 5/4, divide by 2 when above the octave: 1 → 5/4 → 25/16 → 125/64 → 625/256 → 625/512…
• Elevenths: Multiply by 11/8 and divide by 2 if it goes above: 1 → 11/8 → 121/64 → 1331/512 → 1331/1024…

At this point, I noticed that the numerators in the eleventh cycle resemble Pascal’s Triangle.
That’s an amazing discovery!

Advanced Observations

(While testing this at the piano...)

• Normally, the denominators of interval ratios are powers of 2.
• Basic intervals include the octave, perfect fifth, major third, natural seventh, major second, augmented fourth, minor sixth, major seventh, and minor second.
• To construct an interval between a major and minor third, something like the 11th root of 9 might be used.
• However, using it directly could cause harmonics to clash and create instability.
• Shifting the entire overtone structure downward can resolve this and make it more stable.

Overtone Geometry Map (hand-drawn sketch)

A geometric diagram mapping the 3rd, 5th, and 11th overtone series into spatial dimensions.
Blue: Perfect fifth series (3rd partial), Pink: Major third series (5th partial), Brown: 11th partial series.

Ratio Expansion Log (manual calculation)

Fractional expansions of overtone series, with emphasis on how the numerators in the 11/8 series resemble exponential growth with base 11.