Harmonic series
The series of integer partials.
What it means
A vibrating body does not vibrate only as a whole but simultaneously in halves, thirds, quarters, and further fractions. The notes that thereby arise are called partials; their frequencies are integer multiples of the fundamental frequency. Together they form the harmonic series, which lies above every fundamental in the same order.
Musical use
The distances decrease going upward: above the fundamental follow first the octave, then the fifth, then another octave, after that the major third, the fifth, and further up the notes draw together into stepwise seconds. From this several phenomena are explained.
First, tone colour: which partials stand out and how strongly determines the character of a sound — many high partials sound bright and cutting, few sound soft.
Second, consonance: intervals that appear low down in the series rest on simple numerical ratios and sound calm. The octave, the fifth, and the third are exactly these.
Third, natural instruments: the natural horn and natural trumpet have only the notes of this series available. From this their peculiarities follow — large gaps in the low register, a playable scale in the high register. The horn fifths are a direct consequence of this.
Fourth, the limits of tuning: individual partials, for instance the seventh, deviate noticeably from the notes of our tone system and cannot be represented within it.
The frequencies of the overtones are integer multiples of the fundamental. The intervals between neighbouring partials become ever smaller going upward: octave, fifth, fourth, major third, minor third. Which overtones are how strongly present determines the tone colour of an instrument. Clarinets, for instance, emphasize the odd partials, which accounts for their hollow sound.
Performance practice
Use the series when intonating: when a combined sound is pure, partials coincide and the beats stop. String players can play harmonics as an audible demonstration; on the piano the effect can be shown by silently depressing a key and striking a lower related note forcefully. And work on tone colour through the embouchure or attack — it decides which partials respond.