Harmonic series

Play the lowest C on a piano and listen closely as the sound dies away. What seems to be a single note is in fact a bundle of notes. A musical sound is not simple but compound: along with the main sound, secondary sounds ring out, very faint and almost imperceptible, that play an important part in music, above all in forming timbre (Lacerda 1966, ch. XLIV).

The main sound is the fundamental, and the ones that accompany it are its harmonics (Lacerda 1966, ch. XLIV).

Where the harmonics come from

The explanation lies in a vibrating string. A string fixed at both ends vibrates as a whole, from end to end, and that vibration gives the fundamental. But while it vibrates as a whole, it also vibrates divided into two halves, and that secondary vibration produces the note an octave above. And also in three thirds, four quarters, five fifths, and so on; each of those secondary vibrations produces a harmonic (Lacerda 1966, ch. XLIV).

Blatter describes the same thing through physics, and it holds for the tube of a wind instrument too: besides the wave that gives the fundamental, waves form at twice the frequency, three times, four times, and so on, in theory without end. Those frequencies are the partials, or harmonics, and together they form the harmonic series (Blatter 2007, part 6, appendix II).

The series on C

Starting from a low C, the first harmonics are these. The first four, in bass clef:

Hear the passage

The fifth to the tenth, in treble clef:

Hear the passage

Numbering from the fundamental, which is harmonic 1:

Harmonic	Note	Interval from the one before	Frequency ratio
1	C	—	—
2	C	octave	2:1
3	G	perfect fifth	3:2
4	C	perfect fourth	4:3
5	E	major third	5:4
6	G	minor third	6:5
7	B flat, a little low	minor third, narrow	7:6
8	C	major second, wide	8:7
9	D	major second	9:8
10	E	major second	10:9

The rule for the ratios is simple: each harmonic's frequency is the fundamental's multiplied by its number. The second harmonic vibrates twice as fast as the fundamental, the third three times as fast. So the ratio between two neighbouring harmonics is the ratio of their numbers: the third is to the second as 3 is to 2, and that is the interval of a perfect fifth (Blatter 2007, part 6, appendix II).

Every note has its series. To find the harmonics of another note, transpose the series on C, keeping the same intervals (Lacerda 1966, ch. XLIV).

The intervals keep shrinking

Looking down the interval column, a pattern appears: octave, fifth, fourth, major third, minor third, minor third, major seconds, then minor seconds. The intervals between neighbouring harmonics start large and keep getting smaller. Med notes that none of them is exactly the same size as another, even when they share a name: of the two minor thirds, the one from the fifth to the sixth harmonic is larger than the one from the sixth to the seventh (Med 1996, ch. XVI).

That is why some harmonics do not match the piano's notes. The seventh, B flat, is a little low; the eleventh falls between F and F sharp; the thirteenth between A flat and A; the fourteenth, another B flat, is also low (Med 1996, ch. XVI). The series goes on without limit, but in practice harmonics are followed only up to the fifteenth: above it, the intervals become smaller than a semitone.

Blatter puts numbers on the problem. The minor third between the sixth and seventh harmonics, at 7:6, is so small that it has never been of use; the major second between the seventh and eighth, 8:7, is too wide for Western taste. And there are two major seconds of different sizes, 9:8 and 10:9, and they cannot both be "the" right major second (Blatter 2007, part 6, appendix II).

Harmonics make timbre

A flute and a violin playing the same A have the same fundamental. What makes one sound like a flute and the other like a violin is the harmonics. Timbre depends on how far each sound's series extends and on which harmonics stand out in it (Lacerda 1966, ch. XLIV).

Med says the same in other words: every note has, proportionally, the same series, but the intensity and quality of the harmonics vary from one instrument to another, and not every instrument produces the whole series. His example is the clarinet, which would produce only odd-numbered harmonics (Med 1996, ch. XVI). That is a simplification: in the clarinet's low register the odd harmonics dominate strongly, which is what gives it its hollow sound, but the even ones do not vanish entirely.

An instrument made only of harmonics

Brass instruments without valves, such as the military bugle, can only play harmonics. The player does not change the length of the tube; they change the lip pressure and choose which harmonic of the tube's series sounds. That is why bugle calls always use the same few notes, those of a major triad, from the third to the sixth harmonic:

Hear the passage

With valves, the trumpet and the horn change the length of the tube, and each valve combination gives a new harmonic series on another fundamental. That is how they reach the notes that lie between the harmonics of a single series.

What the series explains, and what it does not

The series is tempting as an explanation for almost everything in music. Blatter separates carefully what it explains well from what it explains only in part.

The most important intervals. The octave, fifth and fourth are the intervals that matter in Western music, and in most of the world's music, and they are exactly the first in the series, the easiest to hear (Blatter 2007, part 6, appendix II). The series also explains which note of an interval sounds like its base. In a fifth, the upper note belongs to the lower note's series, and so the lower note sounds like the root (Blatter 2007, part 3, art. 1).

The history of harmony. Some use the series to explain why harmony supposedly evolved from fifths and fourths to thirds, since the fourth is followed by the major and minor thirds. Blatter grants there is most likely some truth to it, but notes that nobody has shown the relationship beyond doubt, and that going up the series one soon meets intervals Western music does not use. If there is an influence, it is limited (Blatter 2007, part 6, appendix II).

Tuning. The series is also the basis of just intonation, which tunes intervals by the harmonics' ratios. Blatter calls it an attractive idea, with some validity, but not the universal solution many of its advocates believed. The 5:4 major third sounds a little small to many ears, and the 6:5 minor third a little large. In the end, intonation too is a matter of taste (Blatter 2007, part 6, appendix II).

What comes next

The harmonic series is the starting point for the question of tuning. If the harmonics' ratios give pure intervals, why is the piano not tuned that way? The answer runs through tuning systems and temperament, and the choice of which imperfection to spread around.