The harmonic series refers to the infinite sequence of natural harmonics, consisting of all natural number multiples of a base frequency: 1,2,3,... So, for example, the harmonic series starting on the base note of A 440hz would be 440hz, 880hz, 1320hz, and so on. The individual pitches in the harmonic series are called harmonics or overtones, terms which are used interchangeably. In general, the lower harmonics are more important.
All musical intervals in just intonation are ratios of harmonics, which means that the ratio of the frequency of the notes are ratios of whole numbers.
Mathematically, the use of equally-tempered scales to approximate whole numbers amounts to using roots of two to approximate rational numbers. Because roots of two are irrational numbers, these approximations are always off by a certain amount. The fact that some tunings match the harmonic series better than others can be seen as random or happenstance, although it is also a fundamental result of immutable mathematical truths.
12-ET, the tuning used in most Western Music, matches the harmonics up through 6, and also matches 8,9,10,12,15, and 16 well. However, the match to 5 (and multiples of 5, like 10) is audibly out-of-tune, which is why major and minor thirds are audibly out-of-tune in 12-ET. And this tuning does not match 7,11,13, or 14 in any meaningful way. The poor match to 7 and 11 is familiar to all experienced players of brass instruments, as brass instruments can easily produce high harmonics. Starting on a C, the 7th harmonic appears as a note that is about 1/3rd of a step flatter than a B-flat, and the 11th harmonic manifests as a note almost exactly halfway between an F and F#.
31-ET, by contrast, matches all of the harmonics up through 16, with the exception of 13. This means that, for the most part, the just intervals involving the 7th and 11th harmonics are well-matched in 31-ET but not 12-ET. A brass instrument player might be pleased to learned that all the harmonics up through the 12th are now more-or-less useful.
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