Ed4

Equal divisions of the double octave – frequency ratio 4/1, aka "quadruple" – are equal-step tunings closely related to equal divisions of the octave – frequency ratio 2/1, aka "duple" – in other words, ed2's or edos. Given any odd-numbered edo, an ed4 can be generated by taking every other tone of the edo. Such a tuning shows the pathological trait of inconsistency in any non-trivial integer limit. For example, given 5edo, two octaves of which, in cents are:

0 240 480 720 960 1200 1440 1680 1920 2160 2400 …

Taking every other tone yields:

0 240 480 720 960 1200 1440 1680 1920 2160 2400 …

0 480 960 1440 1920 2400 …

The resultant scale we can call 5ed4.

This approach yields more useful scales starting with edo systems which are larger, where a composer might decide a single degree is too small to be useful. As one example, consider 37edo, which is well known to be an excellent temperament in the 2.5.7.11.13.27 subgroup, but whose single degree, approximately 32.4¢, might be "too small" in some context (e.g. guitar frets). Taking every other step of 37edo produces 37ed4, an equal-stepped scale which repeats at 4/1, the double octave, and has a single step of 64.9¢. (See also 65cET.)

Ed4 scales also have the feature that they ascend the pitch continuum twice as fast as edo systems. 37 tones of 37edo is one octave, while 37 tones of 37ed4 is 2 octaves. Thus, fewer bars would be needed on a metallophone, fewer keys on a keyboard, etc.

Individual pages for ED4's

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1	3	5	7	9	11	13	15	17	19
21	23	25	27	29	31	33	35	37	39
41	43	45	47	49	51	53	55	57	59
61	63	65	67	69	71	73	75	77	79
81	83	85	87	89	91	93	95	97	99