Pentatonic Scale - Tuning

Tuning

Ben Johnston gives the following Pythagorean tuning for the minor pentatonic scale:

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Proportions (composed of naturals) in the above table may be derived by proceeding with the principle that historically gives the Pythagorean diatonic and chromatic scales, stacking perfect fifths with 3:2 frequency proportions (C-G-D-A-E). Considering the anhemitonic scale as a subset of a just diatonic scale, it is tuned thus; 20:24:27:30:36 (A-C-D-E-G = 5/3-1/1-9/8-5/4-3/2). Assigning precise frequency proportions to the pentatonic scales of most cultures is problematic as tuning may be variable.

For example, the slendro anhemitonic scale and its modes of Java and Bali are said to approach, very roughly, an equally-tempered five note scale, but, in fact, their tunings vary dramatically from gamelan to gamelan.

Specially trained musicians among the Gogo people of Tanzania sing the fourth through ninth (and occasionally tenth) harmonics above a fundamental, which corresponds to the frequency proportions 4:5:6:7:8:9. Up to eight, this is an octaval scale of five notes (8 is the same note an octave higher as 4), while nine is a major second above eight, and a major ninth above four. The 6:7:8 bit includes two septimal ratios rarely found on western staves (septimal minor third & septimal whole tone).

Composer Lou Harrison has been one of the most recent proponents and developers of new pentatonic scales based on historical models. Harrison and William Colvig tuned the slendro scale of the gamelan Si Betty to overtones 16:19:21:24:28. (1/1-19/16-21/16-3/2-7/4) They tuned the Mills gamelan so that the intervals between scale steps are 8:7-7:6-9:8-8:7-7:6. (1/1-8/7-4/3-3/2-12/7-2/1 = 42:48:56:63:72)