Diachrome

In aberrismic theory, diachrome (also denoted 5s) is a set of three 5L2m5s 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 5L7s and 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.

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.

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:

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

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 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]".