Diachrome: Difference between revisions
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'''Diachrome''' (also denoted 5s in [[groundfault]]'s [[aberrismic theory]]) is a set of three 5L 2m 5s scale patterns: | |||
* 5sL: LsLsLsmLsLsm | * 5sL: LsLsLsmLsLsm | ||
* 5sR: LmsLsLsLmsLs | * 5sR: LmsLsLsLmsLs | ||
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5sL and 5sR are [[chiral]] pairs, and 5sC is achiral. The three chiralities are also determined by the number of ms and sm substrings they have. | 5sL and 5sR are [[chiral]] pairs, and 5sC is achiral. The three chiralities are also determined by the number of ms and sm substrings they have. | ||
''Diachrome'' is named from ''diatonic'' and ''chromatic'' for the [[5L 7s | ''Diachrome'' is named from ''diatonic'' and ''chromatic'' for the [[5L 7s]] and [[7L 5s]] children of [[5L 2s]] that the L = m and m = s tunings reflect. | ||
== Structure == | == Structure == | ||
=== Lattice === | === Lattice === | ||
5sC has two chains of "fifths" (the 7-step 3L + m + 3s) both having 6 notes and offset by 3L + m + 2s. The interval L + s is thus reached by stacking two fifths and reducing, and m is reached by stacking -5 fifths and reducing. The three leftmost notes of the lower chain (the one not offset by 3L + m + 2s) are joined to the three rightmost notes of the upper chain by 2L + m. 5sL (resp. 5sR) similarly has two chains of fifths, but the lower chain has 7 (resp. 5) notes and the upper chain 5 (resp. 7) notes. | 5sC has two chains of "fifths" (the 7-step 3L + m + 3s) both having 6 notes and offset by 3L + m + 2s. The interval L + s is thus reached by stacking two fifths and reducing, and m is reached by stacking -5 fifths and reducing. The three leftmost notes of the lower chain (the one not offset by 3L + m + 2s) are joined to the three rightmost notes of the upper chain by 2L + m. 5sL (resp. 5sR) similarly has two chains of fifths, but the lower chain has 7 (resp. 5) notes and the upper chain 5 (resp. 7) notes. | ||
=== As substitution scale === | === As substitution scale === | ||
In terms of [[MOS substitution|substitution]], 5sL = subst('''XXXXXXmXXXXm''', '''X''', '''Ls'''), 5sR = subst('''XXXXXXmXXXXm''', '''X''', '''sL'''), and 5sC = subst('''XXXXXmXXXXXm''', '''X''', '''Ls'''). (See [[User:Inthar/Notation]] for the explanation of this notation.) | In terms of [[MOS substitution|substitution]], 5sL = subst('''XXXXXXmXXXXm''', '''X''', '''Ls'''), 5sR = subst('''XXXXXXmXXXXm''', '''X''', '''sL'''), and 5sC = subst('''XXXXXmXXXXXm''', '''X''', '''Ls'''). (See [[User:Inthar/Notation]] for the explanation of this notation.) | ||
=== Balance === | === Balance === | ||
The 5sC pattern, LsLsLmsLsLsm, is a diregular scale according to the [[Ternary scale theorems#|classification of ternary balanced scales]]. In particular, it is (as an abstract scale word) MV3 but not SV3. | The 5sC pattern, LsLsLmsLsLsm, is a diregular scale according to the [[Ternary scale theorems#|classification of ternary balanced scales]]. In particular, it is (as an abstract scale word) MV3 but not SV3. | ||
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The first edos with a diachrome tuning are {{EDOs|24, 29, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 48}}. | The first edos with a diachrome tuning are {{EDOs|24, 29, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 48}}. | ||
== Temperament interpretations == | == Temperament interpretations == | ||
Diachrome is interesting for having at least four notable JI interpretations. In all the interpretations below, L + s = 9/8, and m = 256/243. | Diachrome is interesting for having at least four notable JI interpretations. In all the interpretations below, L + s = 9/8, and m = 256/243. | ||
=== 7-limit[5120/5103] === | === 7-limit[5120/5103] === | ||
In the 7-limit, diachrome has two JI tunings which are very similar and can be identified by tempering out [[5120/5103]], the 5.8c gap between 81/80 and 64/63. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]]. | In the 7-limit, diachrome has two JI tunings which are very similar and can be identified by tempering out [[5120/5103]], the 5.8c gap between 81/80 and 64/63. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]]. | ||
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It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper. | It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper. | ||
[[Category:Aberrismic theory]][[Category:Rank-3 scales]] | [[Category:Aberrismic theory]] | ||
[[Category:Rank-3 scales]] | |||