Diasem: Difference between revisions

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'''Diasem''' (also known as ~~'''Tas''', from Archytas, or~~ 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with step pattern 5L 2M 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed diasem'' LMLSLMLSL and ''left-handed diasem'' LSLMLSLML; these are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales. The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset a subminor third or a supermajor second. [[21edo]] is the smallest edo to support a non-degenerate diasem.

'''Diasem''' (also known as 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with step pattern 5L 2M 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed diasem'' LMLSLMLSL and ''left-handed diasem'' LSLMLSLML; these are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales.

Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[28/27]], and S represents [[64/63]].

Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[28/27]], and S represents [[64/63]]. This interpretation or more generally the series of [[generator sequence]] scales generated by GS(7/6, 8/7) or GS(8/7, 7/6), has been named '''Tas''' (from Archytas) by [[Scott Dakota]].

"Diasem" is a name given by [[groundfault]] (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "[[Semaphore]]") since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[semiquartal]] (5L 4s); it is also a pun based on the [[diesis]], which appears as the small step in the scale in the [[31edo]] and [[36edo]] tunings.

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The octave can be called the "perfect 9-step" in [[TAMNAMS]].

== ~~Important properties~~ ==

== Properties ==

Any diasem scale with positive step sizes has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:

* > 4\7 if L > m + s

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* < 4\7 if L < m + s

(This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.)

The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset L + m or L + s (respectively a flat minor third or a sharp major second in tunings of diasem with "reasonable" fifths and small s steps). [[21edo]] is the smallest edo to support a non-degenerate diasem.

<!--== Modes ==

Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. We have provided names based on the modes of the [[5L 4s]], [[5L 2s]] and [[7L 2s]] temperings of each mode.

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-'''Diasem''' (also known as '''Tas''', from Archytas, or 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with step pattern 5L 2M 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed diasem'' LMLSLMLSL and ''left-handed diasem'' LSLMLSLML; these are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales. The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset a subminor third or a supermajor second. [[21edo]] is the smallest edo to support a non-degenerate diasem.
+'''Diasem''' (also known as 2s in [[groundfault]]'s [[aberrismic theory]]) is a 9-note [[Maximum variety|max-variety-3]], [[generator-offset]] scale with step pattern 5L 2M 2s, equivalent to the [[semiquartal]] ([[5L 4s]]) mos with two of the small steps made larger and the other two made smaller. Diasem is [[chiral]], with two rotationally non-equivalent variants: ''right-handed diasem'' LMLSLMLSL and ''left-handed diasem'' LSLMLSLML; these are mirror images. The fact that the small step of diatonic is made smaller results in [[26edo]] and [[31edo]] diasem having better melodic properties than the respective diatonic scales.
-Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[28/27]], and S represents [[64/63]].
+Diasem can be tuned as a [[Just intonation subgroup|2.3.7 subgroup]] JI scale or a tempered version thereof, where L represents [[9/8]], M represents [[28/27]], and S represents [[64/63]]. This interpretation or more generally the series of [[generator sequence]] scales generated by GS(7/6, 8/7) or GS(8/7, 7/6), has been named '''Tas''' (from Archytas) by [[Scott Dakota]].
 "Diasem" is a name given by [[groundfault]] (though others have discussed the scale before her). The name is a portmanteau of "diatonic" and "semiquartal" (or "[[Semaphore]]") since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[semiquartal]] (5L 4s); it is also a pun based on the [[diesis]], which appears as the small step in the scale in the [[31edo]] and [[36edo]] tunings.
@@ Line 189: / Line 189: @@
 The octave can be called the "perfect 9-step" in [[TAMNAMS]].
-== Important properties ==
+== Properties ==
 Any diasem scale with positive step sizes has a fifth (large 5-step) between 4\9 (666.67¢) and 3\5 (720¢). The fifth is:
 * > 4\7 if L > m + s
@@ Line 195: / Line 195: @@
 * < 4\7 if L < m + s
 (This can be seen as follows: Let s' = m + s. Then the fifth generates the mos 5L 2s', which is either diatonic, 7edo or antidiatonic depending on the above conditions.)
+The scale has two chains of fifth generators (with 5 notes and 4 notes, respectively) with offset L + m or L + s (respectively a flat minor third or a sharp major second in tunings of diasem with "reasonable" fifths and small s steps). [[21edo]] is the smallest edo to support a non-degenerate diasem.
 <!--== Modes ==
 Diasem has 18 modes, 9 modes of LH diasem and 9 modes of RH diasem. We have provided names based on the modes of the [[5L 4s]], [[5L 2s]] and [[7L 2s]] temperings of each mode.