The harmonic series figure
Every harmonic of one note as a wave, highest on top and fainter as they multiply, and their sum beneath — the chosen pair in ink, one period banded down the stack so the band's length is the consonance.
One note is many: a string vibrates at its pitch and at every whole-number multiple of it, and the intervals between those multiples are where consonance comes from. This figure draws that — every harmonic as a wave, and what two of them add up to. It is the figure of the harmonic series essay and of every interval's page.
What it shows
A row for every harmonic of one note, the highest on top, labeled f, 2f, 3f … 16f. The note is the root of the instrument figure on the same page, or C3 when there isn't one; the figure never names it, because the lesson is not about a note. Each row is that harmonic's sine wave at the strength the series gives it, one over its number — so row by row, reading down from 16f to f, the waves get slower and stronger.
The chosen pair — or a stack of three or more — is drawn in ink, zoomed so the lower one fills its row, at their relative strengths. Under a rule beneath the rows is their sum: the two waves added, over sixteen cycles of the lowest.
One period is banded. A stack of harmonics repeats every so many cycles of the lowest one, and the band down the whole figure is that period, with each chosen row's cycles inside it counted. The band's length is the consonance: a fifth (2:3) repeats every two cycles, a major third (4:5) every four, a minor second (15:16) every fifteen. A period longer than sixteen cycles widens the drawing and the frame scrolls it. That is how the interval pages rank their intervals without saying a word.
On the leading edge a brace joins the chosen rows — the interval, literally the distance between two waves — with the numbers before it: the ratio and, when equal temperament bends it, by how many cents; a stack's ratio sits there the same way. The interval itself is in front of the sum's row, since the sum is the interval as you hear it: its degree, in the degree's color — or, for a stack, the degrees it holds from its lowest note: 4:5:6 is 1 3 5, a major triad. Hover any of the numbers, or a cycle count, for what it means.
Everything the figure plays is plucked: struck, then each harmonic dying away at its own rate, the high ones first, as a string's do — a held sum of sines is a buzzer, and the envelope is what makes it a note. In the essay, a link marked ♪ sounds the state it sets, and the sections that add harmonics one at a time sound as you reach them (after your first tap on the page — a browser plays nothing before one). The timbre section goes further: it hands the figure a recipe, a strength for each lit harmonic, and the rows are drawn and played at those strengths — a string plucked over the neck, then by the bridge — the one place the figure departs from the series' own 1/n; and blows the pipe instead of plucking it. Pinned in the essay the figure draws only the lit rows, without the row of intervals: the text is the control there. One harmonic alone draws no sum: there is nothing to add.
There are no note names, degrees or colors on the harmonics themselves. A first draft had them, and it read as a scale; the point is how early an interval appears in the series, and that is a number.
What you can do
- The row of intervals above the drawing is the control, smoothest first — octave, fifth, fourth, major sixth, major third, minor third, tritone, minor sixth, major second, major seventh, minor seventh, minor second. Pick one and its pair lights. The order is the roughness measure NoSheet shapes its note blobs by: how much the two waves' sum wriggles (the mean of its second derivative over a cycle, on a log scale, a unison 0 and a semitone 1). It is nearly the order the series reaches them, and not a textbook's: the sixth is smoother than the thirds, and the figure says so.
- ♪ sounds the pair — pure sines, so what you hear is what is drawn.
- A stack (1-2-3-4-5-6-7-8, the triad 4-5-6, the minor triad 10-12-15) is a link state the essay uses; it lights no chip.
The state is in the address — #series=2-3 is the fifth, #series=4-5-6 the major triad — so any pair is a link.
Reading the cents
The number under a partial is how far that harmonic sits from the nearest note a piano has. The number in the caption is how far the interval is from its equal-tempered size. Neither is what makes an interval rough: the major third is fourteen cents off and sweet. Roughness is the ratio's complexity — the length of the band.
Where it appears
On the harmonic series page, on each of the thirteen interval pages with that interval's pair lit, and pinned under the essay, where every section lights a pair or a stack as you scroll to it.