Ed4

Equal divisions of the fourth harmonic (4/1, tetratave) or equal divisions of the double octave are equal-step tunings obtained by dividing the octave in a certain number of equal steps. They are 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.

Odd-numbered ed4s are nonoctave systems, in fact ones with the highest attainable relative error for the octave, but even-numbered ed4s are not and exactly correspond to EDOs (2ed4 to 1edo, 4ed4 to 2edo, 6ed4 to 3edo and so on).

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 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.

The same approach can also be taken with respect to regular temperaments, and this would suggest using only square ratios, meaning that subgroups are composed of prime squares like 4.9.25 rather than primes themselves. There is a tetratave-repeating "squared counterpart" to every octave-repeating temperament in a prime subgroup of any rank, created by squaring both the basis elements and the commas. The generators of the temperament are also squared, and the general structure and any MOS scales will be the same, but with extreme stretching in place, creating a very distinct harmonic and melodic quality compared to the original temperament (see also macrodiatonic and microdiatonic scales). For example, meansquared is the counterpart of meantone, tempering out 81/80² and generated by ~4/1 and ~9/4 instead of ~2/1 and ~3/2

Individual pages for ED4's

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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

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• 321ed4