Diachrome: Difference between revisions

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~~In [[aberrismic theory]],~~ '''~~diachrome~~''' (~~also~~ denoted 5s) is a set of three ~~5L2m5s~~ scale ~~patterns~~:

'''Diachrome''' or '''chromedye''' (denoted dia5s or 5s in [[groundfault]]'s [[aberrismic theory]] systematic naming) is a set of three 5L 2m 5s [[scale pattern]]s:

* 5sL: LsLsLsmLsLsm

* 5sR: LmsLsLsLmsLs

* 5sC: LsLsLmsLsLsm

* 5sC: LsLsLmsLsLsm (''interchroid'' structure)

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]] and [[7L 5s]] children of [[5L 2s]] that the L = m and m = s tunings reflect.

~~''Diachrome'' is named from ''diatonic'' and ''chromatic'' for the [[5L 7s|5L7s]] and [[7L 5s|7L5s]] children of [[5L2s]] that the L=m and m=s tunings reflect.~~

== Structure ==

5sC has two chains of "fifths" 3L + m + 3s both having 6 notes and offset by 3L + m + 2s. 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.

=== 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.

=== 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.)

=== Balance ===

The 5sC pattern, LsLsLmsLsLsm, is an even-regular 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.

== Diachrome in edos ==

Diachrome is available in good RTT edos that have quasi-just or Parapyth fifths.

The first edos with a diachrome tuning are {{EDOs|24, 29, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 48}}.

== Temperament interpretations ==

Diachrome is ~~most accurate to~~ JI ~~in tunings that have quasi-just or Parapyth fifths~~. In ~~both~~ 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] ===

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:

=== 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 5.8{{c}} 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]].

* The 2.3.5 tuning has L = 10/9, m = 256/243, s = 81/80.

* The 2.3.7 tuning has L = 567/512, m = 256/243, s = 64/63.

The tempered tuning thus has the mappings

* 3/2 = 3L + m + 3s,

* 5/4 = 2L + s,

* 7/4 = 4L + 2m + 3s.

~~The property of tempering out 5120/5103 thus lends 41edo, 46edo, 53edo, and 58edo some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.~~

=== 2.3.7.11.13 ~~Parapyth~~ ===

The property of tempering out 5120/5103 thus lends 41edo (6:3:1), 46edo (7:3:1), 53edo (8:4:1), and 58edo (9:4:1) some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.

Diachrome can be given a [[~~Parapyth~~]] (2.3.7.11.13[29 & 41 & 46]) tempering:

* The L step becomes 12/11

=== 2.3.7.11.13 parapyth ===

* The m step becomes ~~256/243~~~22/21~104/99

Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:

* The s step becomes 28/27~33/32

* The L step becomes 12/11;

* The m step becomes 22/21~104/99~256/243;

* The s step becomes 28/27~33/32~1053/1024.

The tempered tuning thus has the mappings

* 3/2 = 3L + m + 3s,

Line 30:

Line 49:

* 11/8 = 2L + m + 3s,

* 13/8 = 3L + 2m + 4s.

~~By not tempering out 144/143, Parapyth distinguishes '''ms''' from '''L''' by tuning the former to 13/12 and the latter to 12/11.~~

The 5sL version of diachrome tempered to ~~Parapyth~~ is known by [[Margo Schulter]] under the name "Penthesilia[12]".

By not tempering out 144/143, parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11.

The 5sL version of diachrome tempered to parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]".

=== 2.3.11.19.23/5.31 [17 & 24] ===

There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral | neutral]] (2.3.11.19.23/5.31[17 & 24]):

* The L step is 12/11;

* The m step is 256/243~128/121~93/88~19/18;

* The s step is 33/32~32/31~95/92.

The tempered tuning thus has the mappings

* 3/2 = 3L + m + 3s,

* 11/8 = 3L + 2s = 2L + m + 3s,

* 19/16 = L + m + s,

* 23/20 = L + m,

* 31/16 = 5L + 2m + 4s.

Note that L = m + s in this tuning, and it makes the scales nonstrictly [[Rothenberg propriety | proper]].

=== 2.3.7.13.19.23 [17 & 19 & 41] ===

A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering:

* The L step is 13/12;

* The m step is 256/243~96/91~19/18;

* The s step is 28/27~27/26.

The tempered tuning has the mappings

* 3/2 = 3L + m + 3s,

* 7/4 = 4L + m + 5s,

* 13/8 = 4L + m + 3s,

* 19/16 = L + m + s,

* 23/14 = 3L + 2m + 4s.

Unlike the above parapyth tempering, this one sharpens s~28/27 slightly towards 27/26 instead of flattening it towards 33/32, sharpens 32/27 slightly towards 19/16 instead of flattening it to 13/11.

It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper.

=== Pele temperament ===

In the [[pele]] temperament ([41 & 46 & 58]), following interpretations are available:

* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80;

* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49;

* L = 12/11, m = 21/20~22/21~256/243, s = 28/27~33/32~65/63;

* L = 13/12, m = 21/20~22/21~104/99~256/243, s = 27/26.

== External links ==

* [https://sw3.lumipakkanen.com/scale/XvmXDTFp0 5sRA Aeolian (46edo 7:3:1 Aberschismic chromedye)]

* [https://sw3.lumipakkanen.com/scale/Xvmsw1KBx 5sRA Aeolian (46edo 6:3:2 Parapyth chromedye)]

[[Category:Aberrismic theory]][[Category:Rank-3 scales]]

[[Category:Aberrismic theory]]

[[Category:Rank-3 scales]]

@@ Line 1: / Line 1: @@
-In [[aberrismic theory]], '''diachrome''' (also denoted 5s) is a set of three 5L2m5s scale patterns:
+'''Diachrome''' or '''chromedye''' (denoted dia5s or 5s in [[groundfault]]'s [[aberrismic theory]] systematic naming) is a set of three 5L 2m 5s [[scale pattern]]s:
 * 5sL: LsLsLsmLsLsm
 * 5sR: LmsLsLsLmsLs
-* 5sC: LsLsLmsLsLsm
+* 5sC: LsLsLmsLsLsm (''interchroid'' structure)
-sL 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.
+sL 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]] and [[7L 5s]] children of [[5L 2s]] that the L = m and m = s tunings reflect.
-''Diachrome'' is named from ''diatonic'' and ''chromatic'' for the [[5L 7s|5L7s]] and [[7L 5s|7L5s]] children of [[5L2s]] that the L=m and m=s tunings reflect.
 == Structure ==
-sC has two chains of "fifths" 3L + m + 3s both having 6 notes and offset by 3L + m + 2s. 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.
+=== Lattice ===
+sC 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 ===
+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 ===
+The 5sC pattern, LsLsLmsLsLsm, is an even-regular 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.
+== Diachrome in edos ==
+Diachrome is available in good RTT edos that have quasi-just or Parapyth fifths.
+The first edos with a diachrome tuning are {{EDOs|24, 29, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 48}}.
 == Temperament interpretations ==
-Diachrome is most accurate to JI in tunings that have quasi-just or Parapyth fifths. In both 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] ===
-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:
+=== 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 5.8{{c}} 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]].
 * The 2.3.5 tuning has L = 10/9, m = 256/243, s = 81/80.
 * The 2.3.7 tuning has L = 567/512, m = 256/243, s = 64/63.
 The tempered tuning thus has the mappings
 * 3/2 = 3L + m + 3s,
 * 5/4 = 2L + s,
 * 7/4 = 4L + 2m + 3s.
-The property of tempering out 5120/5103 thus lends 41edo, 46edo, 53edo, and 58edo some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.
-=== 2.3.7.11.13 Parapyth ===
+The property of tempering out 5120/5103 thus lends 41edo (6:3:1), 46edo (7:3:1), 53edo (8:4:1), and 58edo (9:4:1) some importance in aberrismic theory; 5120/5103 has been named the ''Aberschisma'' for this reason.
-Diachrome can be given a [[Parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:
-* The L step becomes 12/11
+=== 2.3.7.11.13 parapyth ===
-* The m step becomes 256/243~22/21~104/99
+Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:
-* The s step becomes 28/27~33/32
+* The L step becomes 12/11;
+* The m step becomes 22/21~104/99~256/243;
+* The s step becomes 28/27~33/32~1053/1024.
 The tempered tuning thus has the mappings
 * 3/2 = 3L + m + 3s,
@@ Line 30: / Line 49: @@
 * 11/8 = 2L + m + 3s,
 * 13/8 = 3L + 2m + 4s.
-By not tempering out 144/143, Parapyth distinguishes '''ms''' from '''L''' by tuning the former to 13/12 and the latter to 12/11.
-The 5sL version of diachrome tempered to Parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]".
+By not tempering out 144/143, parapyth distinguishes m + s from L by tuning the former to 13/12 and the latter to 12/11.
+The 5sL version of diachrome tempered to parapyth is known by [[Margo Schulter]] under the name "Penthesilia[12]".
+=== 2.3.11.19.23/5.31 [17 & 24] ===
+There's also a diachrome tempered by an extension of [[Rastmic_clan#Neutral | neutral]] (2.3.11.19.23/5.31[17 & 24]):
+* The L step is 12/11;
+* The m step is 256/243~128/121~93/88~19/18;
+* The s step is 33/32~32/31~95/92.
+The tempered tuning thus has the mappings
+* 3/2 = 3L + m + 3s,
+* 11/8 = 3L + 2s = 2L + m + 3s,
+* 19/16 = L + m + s,
+* 23/20 = L + m,
+* 31/16 = 5L + 2m + 4s.
+Note that L = m + s in this tuning, and it makes the scales nonstrictly [[Rothenberg propriety | proper]].
+=== 2.3.7.13.19.23 [17 & 19 & 41] ===
+A diachrome is also available in the 2.3.7.13.19.23[17 & 19 & 41] tempering:
+* The L step is 13/12;
+* The m step is 256/243~96/91~19/18;
+* The s step is 28/27~27/26.
+The tempered tuning has the mappings
+* 3/2 = 3L + m + 3s,
+* 7/4 = 4L + m + 5s,
+* 13/8 = 4L + m + 3s,
+* 19/16 = L + m + s,
+* 23/14 = 3L + 2m + 4s.
+Unlike the above parapyth tempering, this one sharpens s~28/27 slightly towards 27/26 instead of flattening it towards 33/32, sharpens 32/27 slightly towards 19/16 instead of flattening it to 13/11.
+It tunes m + s to 23/21, therefore larger than L~13/12, and makes the scales strictly proper.
+=== Pele temperament ===
+In the [[pele]] temperament ([41 & 46 & 58]), following interpretations are available:
+* L = 10/9, m = 21/20~22/21~256/243, s = 56/55~64/63~66/65~81/80;
+* L = 11/10, m = 21/20~22/21~256/243, s = 40/39~45/44~50/49;
+* L = 12/11, m = 21/20~22/21~256/243, s = 28/27~33/32~65/63;
+* L = 13/12, m = 21/20~22/21~104/99~256/243, s = 27/26.
+== External links ==
+* [https://sw3.lumipakkanen.com/scale/XvmXDTFp0 5sRA Aeolian (46edo 7:3:1 Aberschismic chromedye)]
+* [https://sw3.lumipakkanen.com/scale/Xvmsw1KBx 5sRA Aeolian (46edo 6:3:2 Parapyth chromedye)]
-[[Category:Aberrismic theory]][[Category:Rank-3 scales]]
+[[Category:Aberrismic theory]]
+[[Category:Rank-3 scales]]