Diachrome: Difference between revisions

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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}}.

== Temperament interpretations ==

Diachrome is interesting for having at least ~~two~~ notable JI interpretations. 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. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]].

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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~~ ===

=== 2.3.7.11.13 parapyth ===

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

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

* The L step becomes 12/11

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

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* 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.

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]".

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 towards 27/26 instead of flattening it towards 33/32. In so doing, it supplies more accurate tunings of 7/4 and 7/6.

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]]

@@ Line 19: / Line 19: @@
 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 interesting for having at least two notable JI interpretations. 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. These commas are notable for being the two most common interpretations for aberrisma scale steps in [[aberrismic theory]].
@@ Line 30: / Line 30: @@
 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 ===
+=== 2.3.7.11.13 parapyth ===
-Diachrome can be given a [[Parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:
+Diachrome can be given a [[parapyth]] (2.3.7.11.13[29 & 41 & 46]) tempering:
 * The L step becomes 12/11
 * The m step becomes 256/243~22/21~104/99
@@ Line 40: / Line 40: @@
 * 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.
+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]".
+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 towards 27/26 instead of flattening it towards 33/32. In so doing, it supplies more accurate tunings of 7/4 and 7/6.
+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]]