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, and the shorter that band the more consonant the interval.
One note is many: a string vibrates at its pitch and at every whole-number multiple of it. Every row here is one of those harmonics as a wave, and beneath them is what two of them add up to. Tap an interval and the text follows.
A fifth
The chosen pair is drawn in ink — the second and third harmonics, 2:3 — with their sum under the rule. The band down the drawing is one period of that sum: how long the two waves take to line up again. A fifth lines up every two cycles of the lower wave, and the brace joining the two rows on the left is the interval itself, its ratio beside it.
A major third
Harmonics 4 and 5. The band is four cycles long now: the longer the band, the rougher the interval sounds. A piano's third is fourteen cents off this one and still sweet — the band, not the cents, is what you hear.
The row
Every interval of the octave is in the row above the drawing, smoothest first — by how much the pair's sum wriggles, the same measure the app shapes its note halos by. It is nearly the order the series reaches them — the fourth is harmonics 3 and 4, the sixth 3 and 5 — and not quite a textbook's: the sixth comes before the thirds.
A minor second
Harmonics 15 and 16, and the band runs fifteen cycles before the waves meet again.
A chord
Three harmonics at once — 4, 5 and 6 — are a major triad, and the band is still four cycles: a triad is no rougher than its third. The degrees it holds stand in front of the sum, 1 3 5.
The whole note
The first eight harmonics together. Their sum is a saw-tooth — a bowed string — and one frequency alone, a sine, is what no instrument sounds like.
Hear it
♪ plucks the pair: struck, then each harmonic dying away at its own rate, the high ones first, as a string's do. What you hear is what is drawn.
Where you'll meet it
Pinned under the harmonic series essay, where the sections add harmonics one at a time and sound as you reach them; on the harmonic series page; and on each of the thirteen interval pages, with that interval's pair lit.