Ed8
Equal divisions of the triple octave (frequency ratio 8/1, aka "octuple") are closely related to equal divisions of the octave (frequency ratio 2/1, aka "duple"), in other words, ED2 or EDO scales. Given any number which is coprime to 3 for ED2, an ED8 can be generated by taking every third tone of the ED2. For example, given 5ED2 (aka 5edo), three octaves of which, in cents are:
- 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...
...taking every third tone yields:
- 0 240 480 720 960 1200 1440 1680 1920 2160 2400 2640 2880 3120 3360 3600...
- 0 720 1440 2160 2880 3600...
The resultant scale we can call 5ED8.
This approach yields more useful scales starting with ED2 systems which are larger, where a composer might decide a single degree is too small to be useful. As one example, consider 31ED2 (aka 31edo), which is well known to be a consistent temperament in the 11-limit, but whose single degree, approximately 38.7¢, might be "too small" in some contexts (e.g. guitar frets). Taking every third step of 31ED2 produces 31ED8, an equal-stepped scale which repeats at 8/1, the triple octave, and has a single step of 116.1¢.
ED8 scales also have the feature that they ascend the pitch continuum three times as fast as ED2 systems. 31 tones of 31ED2 is one octave, while 31 tones of 31ED8 is three octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.
See: Equal Temperaments