Luna and hemithirds
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The 7-limit hemithirds temperament functions as a strong extension of didacus, the 2.5.7 subgroup temperament, in the range between 25edo and 31edo tuning, defined by tempering out 3136/3125 such that two of its generators (hemithird, ~28/25, around 193.2 cents) reach ~5/4, three reach ~7/5, and therefore five reach ~7/4. Hemithirds extends didacus by tempering out 1029/1024, such that three intervals of ~8/7 reach 3/2, therefore finding 4/3 after fifteen generators in total.
Luna is a restriction of hemithirds to the 5-limit that is a microtemperament, supported by such high-precision tuning systems as 118edo and 441edo; another notable tuning of luna is 1000edo. It can further be re-extended to the 7-limit in the form of lunatic, but that extension is extremely complex (finding the 7th harmonic at 113 generators down).
See Hemimean clan #Hemithirds and Luna family #Luna for more information.
Intervals
Chords
Tuning spectrum
Gencom: [2 28/25; 196/195 352/351 385/384 625/624]
Gencom mapping: [⟨1 4 2 2 7 0], ⟨0 -15 2 5 -22 23]]
| Eigenmonzo (Unchanged-interval) |
Generator (¢) |
Comments |
|---|---|---|
| 14/13 | 192.872 | |
| 12/11 | 192.948 | |
| 15/11 | 192.995 | |
| 13/10 | 193.058 | |
| 16/13 | 193.066 | |
| 13/11 | 193.094 | |
| 15/13 | 193.118 | |
| 13/12 | 193.120 | |
| 11/8 | 193.122 | |
| 11/10 | 193.125 | |
| 18/13 | 193.144 | |
| 5/4 | 193.157 | |
| 6/5 | 193.198 | 5-odd-limit minimax |
| 10/9 | 193.200 | |
| 4/3 | 193.203 | |
| 16/15 | 193.210 | |
| 14/11 | 193.241 | 11-odd-limit minimax |
| 9/7 | 193.283 | 9-odd-limit minimax |
| 7/6 | 193.344 | 7-odd-limit minimax |
| 15/14 | 193.364 | |
| 11/9 | 193.426 | |
| 8/7 | 193.765 | |
| 7/5 | 194.171 |