Diachrome

Diachrome (also denoted 5s in groundfault's aberrismic theory) is a set of three 5L 2m 5s scale patterns:

• 5sL: LsLsLsmLsLsm
• 5sR: LmsLsLsLmsLs
• 5sC: LsLsLmsLsLsm

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.

Structure

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 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 a diregular scale according to the 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 24, 29, 31, 34, 36, 38, 39, 41, 43, 44, 45, 46, 48.

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.

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.

• 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

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

The tempered tuning thus has the mappings

• 3/2 = 3L + m + 3s,
• 7/4 = 4L + m + 5s,
• 11/8 = 2L + m + 3s,
• 13/8 = 3L + 2m + 4s.

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 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 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.