Ed4

Equal Divisions of the Double Octave -- frequency ratio 4/1, aka "Quadruple" -- are closely related to Equal Divisions of the Octave -- frequency ratio 2/1, aka "Duple" -- in other words, ED2 or EDO scales. Given any odd-numbered ED2, an ED4 can be generated by taking every other tone of the ED2. Such a tuning shows the pathological trait of inconsistency in any non-trivial integer limit. For example, given 5ED2 (aka 5edo), two octaves of which, in cents are:

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

...taking every other tone yields:

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0 480 960 1440 1920 2400...

The resultant scale we can call 5ED4.

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 37ED2 (aka 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 37ED2 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 ED2 systems. 37 tones of 37ED2 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.

See: Equal Temperaments

Individual pages for ED4s

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