Diamech
Diamech (also denoted 4s in groundfault's aberrismic theory) is a family of quasi-diatonic ternary scale patterns with scale signature 5L2m4s, so named by groundfault who considers 33edo's L:m:s = 5:2:1 a good tuning for it. The name diamech is a blend of dia- and the TAMNAMS prefix mech- for machinoid. 23edo is the only edo without a diatonic fifth that has a 5L2mks ternary scale pattern, namely 5L2m4s with L:m:s = 3:2:1.
Structure
The three standard diamech scale patterns, denoted 4sL (LsLsLmLsLsm), 4sC (LmLsLsLmLss), and 4sR (LsLmLsLsLms), may be constructed via MOS substitution, giving generator sequences for the three scales:
| chirality | filling MOS | UDP for filling MOS | step pattern | generator sequence | ||
|---|---|---|---|---|---|---|
| template MOS: | LXLXLXLXLXX
|
intvl. class of gen.: | 2-steps | |||
| 4sC | mssmss |
4|0(2) | LmLsLsLmLss
|
GS(L+m, L+s, L+s) | ||
| 4sR | smssms |
2|2(2) | LsLmLsLsLms
|
GS(L+s, L+m, L+s) | ||
| 4sL | ssmssm |
0|4(2) | LsLsLmLsLsm
|
GS(L+s, L+s, L+m) | ||
Tunings and interpretations
Untempered 2.3.7
The standard untempered 2.3.7 interpretation for diamech has L = 9/8, m = 49/48, s = 64/63.
Huntmic (2.3.7.13/5[640/637])
Huntmic is an important temperament for diamech offering an elegant combination of interordinals and making 7/4 sharper in systems exaggerating 49/48 and splitting it into two equal parts. Assuming the 2.3.7 interpretation above and the constraint m:s = 2:1 makes L + m = 147/128 exactly half of 2L + m + 2s = 4/3, making it natural to set 147/128 equal to 15/13, tempering out 640/637. 53edo (9:2:1) and 58edo (10:2:1) offer particularly good tunings for this interpretation; while the 58edo tuning's s step is smaller than that of the 53edo tuning, 58edo has better 7/6, 9/7, 15/13, and 13/10 as well as giving you a free 11:13:15.