# 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

### 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 two notable JI interpretations. 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. 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 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]".