Subharmonic series
The subharmonic series (or undertone series) is a sequence of tones generated by unit-fraction frequency ratios under a fundamental: 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ... ad infinitum. Each member of this series is a subharmonic. The term undertone, which is not an exact synonym, is also used in the same sense by many people, especially because undertones do not occur as partials of a sound's timbre.
Relation to the harmonic series
The subharmonic series is the inverse of the harmonic series. The nth subharmonic lies below the fundamental by the same interval that the nth harmonic lies above it, so every interval of one series has a mirror image in the other.
Each octave-reduced subharmonic is the octave complement of the octave-reduced harmonic with the same number.
| n | Harmonic n | Cents | Subharmonic n | Cents |
|---|---|---|---|---|
| 3 | 3/2 | 702.0 | 4/3 | 498.0 |
| 5 | 5/4 | 386.3 | 8/5 | 813.7 |
| 7 | 7/4 | 968.8 | 8/7 | 231.2 |
| 9 | 9/8 | 203.9 | 16/9 | 996.1 |
| 11 | 11/8 | 551.3 | 16/11 | 648.7 |
| 13 | 13/8 | 840.5 | 16/13 | 359.5 |
As in the harmonic series, the steps between adjacent members are superparticular ratios. The interval from the (n + 1)th subharmonic up to the nth is (n + 1)/n, so the steps get smaller the further down the series goes.
Chords and scales
Any material taken from the subharmonic series is a utonality in Harry Partch's terminology (see otonality and utonality). For example, the subharmonics 4, 5 and 6, from highest to lowest, are in the ratio 1/6 : 1/5 : 1/4, which is 10:12:15, the just minor triad. It is the utonal counterpart of the major triad 4:5:6 built from the harmonics 4, 5 and 6.
A segment of the series that spans one octave, repeated at the octave, forms an undertone scale, the inverse of an overtone scale.