Ed4: Difference between revisions

'''Equal divisions of the fourth harmonic''' ([[4/1]], tetratave) or '''equal divisions of the double octave''' are [[equal-step tuning]]s 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 in[[consistency]] in any non-trivial [[integer limit]]. For example, given [[5edo]], two octaves of which, in cents are:

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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 [[EDO]]s (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]].)

The same approach can also be taken with respect to [[regular temperament]]s, and this would suggest using only square ratios, meaning that [[subgroup]]s are composed of prime squares like 4.9.25 rather than [[prime]]s 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 [[generator]]s 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 ([[81/80]])² and generated by ~[[4/1]] and ~[[9/4]] instead of ~[[2/1]] and ~[[3/2]]

 

 

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