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.

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]".