Diaschismic: Difference between revisions

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[[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]

Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.

Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14 steps. 58 is the largest number of tones for a [[mos]] where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region. In general, ratios closer to intervals of 12edo, such as [[17/16]], are simpler in terms of generator steps, while ratios further from 12edo intervals, such as [[11/9]], are more complex. The 13th harmonic is just beyond the specified generator range, so it does not appear in the diagram.

The 13th harmonic is just beyond the specified generator range, so the diagram does not show it. It is found as a comma step below [[28/17]], or alternatively [[16/13]] is found as the [[fifth complement]] of [[11/9]].

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise, though it requires using the sharp, second-best approximation of [[5/4]] (aka the 104c [[val]]).

Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise.

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 [[File: Diaschismic 12et Detempering.png|thumb|Diaschismic as a 58-tone 12et detempering]]
-Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14. 58 is the largest number of tones for a mos where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region.
+Diaschismic is naturally considered as a [[detemperament]] of the [[12edo|12 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -14 to +14 steps. 58 is the largest number of tones for a [[mos]] where intervals in the 12 categories do not overlap. Each category is divided into four or five qualities separated by an interval reached by 6 fifths minus a semioctave, which represents the generic comma step, representing the ratios [[51/50]], [[55/54]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], and many more. Combining this division with the minor and major diatonic qualities of the 12 equal temperament, diaschismic gives us nine or ten qualities for each diatonic category in addition to the five qualities in the tritone region. In general, ratios closer to intervals of 12edo, such as [[17/16]], are simpler in terms of generator steps, while ratios further from 12edo intervals, such as [[11/9]], are more complex. The 13th harmonic is just beyond the specified generator range, so it does not appear in the diagram.
-The 13th harmonic is just beyond the specified generator range, so the diagram does not show it. It is found as a comma step below [[28/17]], or alternatively [[16/13]] is found as the [[fifth complement]] of [[11/9]].
+Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise, though it requires using the sharp, second-best approximation of [[5/4]] (aka the 104c [[val]]).
-Notice also the little interval between the largest of a category and the smallest of the next. This interval spans 46 generator steps, so it vanishes in 46edo, but is tuned to the same size as the comma step in 58edo. [[104edo]] tunes it to one half the size of the comma step, which may be seen as a good compromise.
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