Undertone scale

An undertone scale is a scale made from an octave-spanning segment of the subharmonic series, repeated at the octave. It is the inverse of an overtone scale. For example, the subharmonics 16 through 32 form the scale subharmonics 16–32. Undertone scales are also called modes of the subharmonic series, or aliquot scales.

Undertone scales are much less commonly used than overtone scales, and fewer of them have been explored.

Construction

The nth through 2nth subharmonics have the frequencies 1/n, 1/(n + 1), ..., 1/(2n), relative to the fundamental. Starting on the lowest of these and ascending, they form the ratios

2n/2n, 2n/(2n − 1), 2n/(2n − 2), ..., 2n/n

above the root, which spans one octave. For example, subharmonics 3–6 are 1/1, 6/5, 3/2 and 2/1 (the just minor triad).

The steps of an undertone scale are superparticular ratios, as in an overtone scale, but they get larger as the scale ascends (for example 10/9, 9/8, 8/7, 7/6, 6/5 for subharmonics 5–10), whereas the steps of an overtone scale get smaller. In terms of string length, the steps are equal increments of length (see EDL).

Relation to overtone scales

The undertone scale made of subharmonics n–2n is the inverse of the overtone scale made of harmonics n–2n: its intervals are those of the overtone scale in reverse order. So subharmonics 3–6 has steps 6/5, 5/4, 4/3, where harmonics 3–6 has steps 4/3, 5/4, 6/5.

Terminology

The terms subharmonic and undertone are used interchangeably here, although they are not exact synonyms; see subharmonic series. Modes are numbered by subharmonic: mode n of the subharmonic series is the segment n–2n. The label "under-n" (n/) can be used in the same sense as "over-n" (/n), see Overtone scale#Over-n scales.

See also

• Subharmonic series
• Overtone scale
• Otonality and utonality
• IFDO: An alternative framework for describing scales which overlaps substantially with undertone scales