The harmonic series — why a note never sounds alone
Six cases of a single series: octave, fifth, major third and the seventh partial that no key can reach — printed and calculated side by side.
A vibrating string does not only vibrate as a whole. It vibrates in halves, thirds and quarters at the same time, and every one of those parts sounds along with it. That is why a note never comes alone: what we take for a single sound is a whole tower.
This tower is not an opinion but a calculation. The nth partial has n times the frequency of the fundamental, so its distance is fixed before anyone writes it down. Notation can only come close — and the table further down says how far off it is.
The images are engraved and measured: the notes come from the sound, the partials from the calculation, and the trumpet’s natural notes additionally from the fingering table, which knows nothing about this page.
The series itself
Eight partials above a fundamental, written the way they are written. Wide leaps at the bottom, narrow steps at the top, and the gap shrinks with every step in between. This series is the same everywhere — it depends on the fundamental, not on the instrument and not on the key.
1:2:3:4:5:6:7:8 — The distances get smaller with every step from bottom to top
The octave
The second partial vibrates twice as fast as the first. It is the only one that comes out exactly even: printed and calculated land on the same spot, with nothing left over. That is why it carries the same name as the fundamental, and why two notes an octave apart are heard almost as one.
2:1 — Every step doubles the frequency and sounds like the same note
The fifth
The third partial lies an octave and a fifth above the fundamental. Calculated, it sits two cents higher than the key puts it — nobody hears that in a single note. Stack twelve such fifths on top of each other, though, and the remainder adds up until you no longer land where you started.
3:2 — The calculated note sits just above the printed one
The major third
The fifth partial is printed as a major third and lies almost fourteen cents below it. That is more than an eighth of a tone, and it is audible: a major chord sung in tune by a choir sounds calmer than the same chord on a piano. The key cannot do otherwise; it has one spot for every key signature.
5:4 — The calculated note sits audibly below the printed one
The seventh partial
The seventh partial is the first one notation clearly misses. It is printed as a minor seventh and calculated a good thirty cents below that — nearly a sixth of a tone. No key holds it. To hear it you need a voice, a brass instrument without valves, or a string you can stop freely.
7:4 — The calculated note sits so far below that no key reaches it
The trumpet without valves
Without valves a trumpet can play exactly the notes of this series and nothing in between. Here the series is not an explanation but an instrument: the same distances, the same narrowing towards the top. The three valves are then the answer to the gaps at the bottom — they lower a natural note far enough to close them.
2:3:4:5:6:7:8 — The same run of distances as in the series, from the second partial on
Printed and calculated
The table puts both sources side by side: on the left the note a partial is printed as, on the right where it really lies. The deviation is given in cents — a semitone has a hundred of them.
| Partial | Ratio | Deviation |
|---|---|---|
| 1 | 1:1 | +0 |
| 2 | 2:1 | +0 |
| 3 | 3:1 | +2 |
| 4 | 4:1 | +0 |
| 5 | 5:1 | -14 |
| 6 | 6:1 | +2 |
| 7 | 7:1 | -31 |
| 8 | 8:1 | +0 |
How the series is built
The first partial is the fundamental itself. The second vibrates twice as fast and sounds an octave higher, the third three times as fast, the fourth four times. It goes on like that, and because the gaps between neighbouring multiples keep shrinking, the notes move closer together as they rise: octave and fifth at the bottom, whole steps at the top.
That is exactly what decides what a brass instrument can play without keys or valves. Low in the series there are gaps, high up the notes sit close — which is why a high trumpet part can move by step and a low one cannot.
The series also explains timbre. Two instruments play the same fundamental and sound different because different partials are loud in each. The note is the same; the tower above it is another one.
What lies next to it
The distances in this series carry names of their own: Intervals. Why a trumpet needs three valves and what it can do without them: Trumpet fingering chart. And what follows from all this when an instrument has to be tuned: Guitar tunings.
Music symbols · Music notation glossary
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See AnotonaEvery note on this page is computed, not typed in — following the rule that also makes F# major spell its seventh degree E#, not F.