Acoustics & Music Theory
why it sounds like that
Concept

Undertone series — where it appears

The mirror of the harmonic series: frequencies at a fundamental divided by one, two, three and onward, which no vibrating body produces. Read downwards, the ratios of a series still describe intervals, and a half-diminished seventh tuned as 1/7 : 1/6 : 1/5 : 1/4 keeps its coincidences as long as a series-tuned dominant seventh.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

a major triad, C–E–G. The partials are at 261.6, 329.6, 392.0 Hz. Each row is a harmonic series they are consistent with to within 30 cents: the filled dots are the partials in their assigned slots and the open dots are slots the series predicts that nothing occupies. There are 2 such series with harmonic numbers up to 16, the best fitting them to 10.1 cents with a fundamental of 65.54 Hz and no unoccupied slot. The rest sit at one half of it and their arithmetic is exactly as exact; what separates them is the count of empty slots, which is why a residue pitch built from few partials is reported an octave out by a minority of listeners and not by the rest.

The root an ear supplies

A major triad's notes fit 4:5:6 with nothing missing and a fundamental two octaves below the bass. A minor triad's fit two different series with two different answers a major sixth apart, and the model cannot choose between them. The ambiguity theorists argued about for two centuries is a computable quantity, and the spectrum invented to remove it is one no object produces.

intervals · The triad Of every four-note chord a harmonic series holds, one lasts longest, and it is the one with a seventh. Every chord of four different pitch classes whose notes are partials of one series (odd parts up to 15), tuned exactly and read both upwards and downwards, 140 chords in all over C3: the longest any spacing within 2 octaves keeps every pair's partial coincidence, grouped by the largest odd part the chord uses. Largest part 7: 2 chords, best 0.87 s (1 : 3 : 5 : 7); Largest part 9: 8 chords, best 0.73 s (1 : 3 : 7 : 9); Largest part 11: 20 chords, best 0.57 s (1/1 : 1/3 : 1/5 : 1/11); Largest part 13: 40 chords, best 0.54 s (1 : 5 : 9 : 13); Largest part 15: 70 chords, best 0.69 s (3 : 5 : 9 : 15). The best is odd parts 1, 3, 5 and 7 — the dominant seventh as 4 : 5 : 6 : 7 — and its mirror, 1/1 : 1/3 : 1/5 : 1/7, which is a half-diminished seventh; both last 0.87 s, the tempered diminished seventh's 0.86 s. Nothing a series holds passes it.

Four notes cannot get below the seventh partial

Tune a dominant seventh as 4 : 5 : 6 : 7 — every note moved, not only the seventh — and its coincidences last 0.868 seconds, which is the one-note retuning's figure to the millisecond. Tuning the third and fifth buys nothing at the top and costs seven voicings a fifth of their life. Every other chord of four notes a harmonic series holds does worse, and the reason is not temperament: no four pitch classes avoid a semitone, a tone and a tritone together, and none of those three can be tuned to coincide below the seventh partial. The half-diminished seventh, which the one-note retuning ruined, reaches the same ceiling read down the series.

timbre · Harmonic series

Named alongside it

The objects these essays reach for when they reach for this one.

Harmonic seriesJust intonationChord qualityConsonanceDominant seventhFrequency ratioResidue pitchSeptimalTriadVirtual pitchVoicing

All concepts