Ed4: Difference between revisions

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'''Equal divisions of the ~~double octave~~''' ~~– frequency ratio~~ [[4/1]], ~~aka "quadruple" –~~ are [[equal-step tuning]]s 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:

'''Equal divisions of the fourth harmonic''' ([[4/1]], tetratave) or '''equal divisions of the double octave''' are [[equal-step tuning]]s 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.

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]].)

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.

== Individual pages for ED4's ==

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 quality compared to the original temperament (see also [[macrodiatonic and microdiatonic]]). For example, [[meansquared]] is the counterpart of [[meantone]], tempering out [[81/80]]<sup>2</sup> and generated by ~[[4/1]] and ~[[9/4]] instead of ~[[2/1]] and ~[[3/2]]

== Individual pages for ED4's =

{| class="wikitable center-all"

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-'''Equal divisions of the double octave''' – frequency ratio [[4/1]], aka "quadruple" – are [[equal-step tuning]]s 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:
+'''Equal divisions of the fourth harmonic''' ([[4/1]], tetratave) or '''equal divisions of the double octave''' are [[equal-step tuning]]s 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:
 240 480 720 960 1200 1440 1680 1920 2160 2400 …
@@ Line 11: / Line 11: @@
 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]].)
+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.
-== Individual pages for ED4's ==
+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 quality compared to the original temperament (see also [[macrodiatonic and microdiatonic]]). For example, [[meansquared]] is the counterpart of [[meantone]], tempering out [[81/80]]<sup>2</sup> and generated by ~[[4/1]] and ~[[9/4]] instead of ~[[2/1]] and ~[[3/2]]
+== Individual pages for ED4's =
 {| class="wikitable center-all"