Harmonic series and overtones

Definition#

Every pitched sound is a stack of sine waves at whole-number multiples of one frequency. The lowest is the fundamental — the note you hear — and the ones above it are its harmonics (or overtones, or partials). Their relative loudness is what makes a saw sound different from a square at the same pitch, and on a chip they are not an abstraction: in a 32-sample wavetable, a partial is literally how many times a shape repeats inside the frame.

In tracker terms#

The multiples map onto intervals, which is what makes them usable as musical material rather than as timbre alone. In lsdj note names, for a base of A3 220 Hz (source: s-intense-tech-01-wave-synth-deep-dive-part-1):

Read the interval column, not the note column. The names are reproduced as the source gives them, and they are internally inconsistent: with A4 = 440 Hz and octave numbers changing at C, as they do in every tracker, 660 Hz is E5, 1100 Hz C#6, 1320 Hz E6 and 1540 Hz ≈ G6 — each of the source's names is an octave low, as though the octave number changed at A rather than at C (arithmetic done here, unverified against the video's on-screen text). The A's and the interval column are right either way. A second reason not to lean on the names: LSDj's own octave display shifted by one between versions — the source states "for version 4, add 1 octave", and abrasive's lsdj_tune for LSDJ 3.9.9 takes its reference as --base A5 440 where version 6 shows the same pitch as A4 (sources: s-intense-tech-01-wave-synth-deep-dive-part-1, s-little-scale-abrasive-lsdj-tune; see rom-retuning).

Multiples of 2 are octaves, of 3 fifths (9 is a major tenth), of 5 major thirds; 7 is "kind of an out-of-tune minor 7th" and 11 an out-of-tune flat fifth (source: s-intense-tech-01-wave-synth-deep-dive-part-1). The same list, written as intervals rather than notes, is nitro2k01's in the ghost-channel thread (source: s-chipmusic-lsdj-ghost-channel).

A naming clash worth knowing: sources in this wiki use "overtone" both ways. Defense Mechanism counts ×2 as "the 1st overtone" (the classical convention: overtone n = partial n+1); nitro2k01's table indexes from the fundamental, so his 3 is the octave + fifth. Where a number matters, this wiki writes the multiplier (×3 = octave + fifth) rather than an overtone ordinal.

These are just intervals — small whole-number ratios — and equal temperament is not. A tempered fifth is about 2 cents narrower than a 3:2, a tempered major third about 14 cents wider than 5:4. Below the fifth nobody notices; from ×5 up the mistuning is audible against tempered notes played on other channels, which is why chip techniques that harvest harmonics stop at the fifth (sources: s-chipmusic-lsdj-ghost-channel, s-intense-tech-01-wave-synth-deep-dive-part-1).

Examples#

• Reading a waveform as harmonics. A saw contains all multiples, a square only the odd ones (which is also why a 50 % pulse and a square sound alike), a triangle the odd ones falling off faster; a sine has none at all — "a pure tone … If you filter a pure sine, nothing happens until the cut off frequency gets to the frequency of the tone" (sources: s-intense-tech-01-wave-synth-deep-dive-part-1, s-chipmusic-lsdj-faq).
• A filter cutoff calibrated in harmonics. In LSDj's synth screen every multiple of 10 in CUTOFF is one partial: bandpass at 20 sounds the octave, 30 the octave + fifth, 40 two octaves — so the filter selects an interval, not just a brightness (source: s-intense-tech-01-wave-synth-deep-dive-part-1). Raising Q on an allpass filter promotes one partial to near-melodic prominence without losing the root (source: s-defensemech-ghost-channel-allpass).
• Hearing the series on a chip. Walking a bandpass with high Q up LSDj's CUTOFF sounds the partials one by one: 10 fundamental, 20 octave, 30 fifth, 40 octave, 50 major third, 60 fifth, 70 an out-of-tune seventh, 80 octave, 90 a major second, A0 a major third, B0 a sharp fourth, C0 a fifth — and then it runs out, "it probably only generates up to here really". On a square signal the even partials are simply not there to find: 20 does nothing audible and the first thing that answers is the fifth at 30 (source: s-defensemech-wave-channel-ramble).
• A harmonic as a second voice. The ghost-channel is this concept used structurally: draw a second wave at ×2 or ×3 inside the same single cycle and the channel plays two notes, at the price of being able to play only the intervals in the table above.
• The series is just intonation. Tuning a chip to whole-number ratios instead of 2^(k/12) means tuning it to the harmonic series — which is why little-scale's koto retune is written as the fractions 9/8 · 6/5 · 3/2 · 8/5 · 2/1 and needs no logarithms (source: s-little-scale-koto-tuning; see alternative-tunings, rom-retuning). nitro2k01, having done the same to LSDj: "it gives each scale it's own character" (source: s-little-scale-nanoloop-18-tet).
• Harmonics as measurement. asger-alstrup's 1995 register dumps read the sid's waveforms as their partial content, which is how the triangle's and noise's internals were reconstructed (source: s-sidmusic-alstrup-waveform-examinations).

Related#

ghost-channel · wavetable-programming · filter-programming · semitone-math · chord-arpeggio · chip-samples · sid · alternative-tunings · rom-retuning

Sources#

s-intense-tech-01-wave-synth-deep-dive-part-1 · s-chipmusic-lsdj-ghost-channel · s-defensemech-ghost-channel-allpass · s-defensemech-wave-channel-ramble · s-chipmusic-lsdj-faq · s-sidmusic-alstrup-waveform-examinations · s-little-scale-koto-tuning · s-little-scale-nanoloop-18-tet · s-little-scale-abrasive-lsdj-tune

To verify#

• The cent figures for the tempered fifth (≈2 ¢ narrow) and major third (≈14 ¢ wide) are standard tuning arithmetic added here for scale, not taken from a wiki source (unverified).
• The octave numbers in the table above: the source names ×3 as E4, ×5 as C#5, ×6 as E5 and ×7 as ≈ G5, which the frequencies contradict by an octave (see the note under the table). What would settle it is the video's on-screen text or an LSDj screenshot of the same notes, neither of which the transcript preserves (unverified).