Music

Overtones and the Harmonic Series

When a string vibrates at frequency $f$, it also vibrates simultaneously at $2f$, $3f$, $4f$, $5f$, $\ldots$ — the overtone series. These are not separate sounds; they are the resonant structure of the fundamental. The ratios between them are small integers: the octave is $2/1$, the fifth is $3/2$, the major third is $5/4$, the harmonic seventh is $7/4$.

This is not a convention. It is a physical fact about how standing waves work. The ear evolved in this acoustic environment and recognizes these ratios as consonant because they are — their waveforms align periodically, their beat frequencies are zero or near-zero, and their combination tones reinforce rather than clash.

12-TET and Disphony

Twelve-tone equal temperament (12-TET) divides the octave into 12 equal semitones, each a ratio of $2^{1/12}$. This is an irrational number. Every interval except the octave is a compromise — close to a just ratio but not exact. The fifth in 12-TET is $2^{7/12} \approx 1.4983$ instead of $1.5000$. The major third is $2^{4/12} \approx 1.2599$ instead of $1.2500$.

These errors are small but not zero, and they accumulate in chords. A 12-TET major chord beats — the waveforms never fully align. The system was adopted not for musical perfection but for practical orchestration: instruments in different keys needed to play together, and a fixed-interval system allowed a single piano to accompany them all. The cost was intonation. The benefit was logistics.

Limit Tuning

Just intonation uses exact integer ratios throughout. A $5$-limit tuning uses only primes up to $5$ — octaves ($2$), fifths ($3$), major thirds ($5$). An $11$-limit tuning adds the seventh harmonic ($7$) and the eleventh ($11$), giving access to intervals that 12-TET cannot approximate well.

In MMP terms, limit tuning is the recognition that musical intervals are physical numbers — they terminate exactly in the right base — and that the disphony of 12-TET is a precision error caused by representing exact ratios in an irrational approximation.

TODO: demonstrate specific comma errors in 12-TET vs just intonation numerically; connect to HarmonicProportions