Harmonic Proportions

Harmonic proportions — ratios of small integers — are the natural language of acoustics, tuning, and resonance. They are also where the mismatch between the real number system and physical reality shows up most clearly.

The Error Accumulation Problem

When harmonic ratios are approximated in base 10 (or in binary floating-point), the denominators often contain prime factors that the base cannot represent exactly. A fifth ($3/2$), a major third ($5/4$), a seventh harmonic ($7/4$) — these are all exact ratios, but their decimal representations are either terminating by luck ($5/4 = 1.25$) or repeating by necessity ($7/4 = 1.75$, fine, but $7/6 = 1.1\overline{6}$).

The problem compounds when ratios are chained. Tuning a scale means stacking fifths, thirds, and sevenths. Each step introduces a small representational error; the errors accumulate; and by the time you've gone around the circle of fifths you've drifted from where you started. This is the comma problem in music theory — the Pythagorean comma, the syntonic comma, the septimal comma — each a measure of how much error has accumulated from chaining exact ratios in an inexact base.

MMP's Reading

MMP frames this as a physicalization problem. The ratios themselves are physical — they terminate in the right base. The error is not inherent to the ratios; it is a consequence of describing them in a base whose prime factors don't match the harmonic series.

The harmonic series has prime factors $\{2, 3, 5, 7, 11, \ldots\}$. Base 10 has only $\{2, 5\}$. Base 12 adds $3$. Base 60 adds $\{2, 3, 5\}$. No finite base covers the full harmonic series — but some bases are dramatically better than others for specific domains.

The practical implication: for any domain defined by harmonic ratios up to a given prime limit, there is an optimal base. Computing in that base eliminates representational error entirely for all ratios within the limit. This is not an approximation strategy — it is the observation that the error was never in the numbers, only in the choice of base.

TODO: work through specific examples — 5-limit, 7-limit, 11-limit — and their optimal bases