Luna and hemithirds: Difference between revisions
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Revision as of 17:17, 1 April 2025
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This page on a regular temperament, temperament collection, or aspect of regular temperament theory is being revised for clarity as part of WikiProject TempClean. |
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. The canonical extension to the 13-limit tempers out 385/384 and 441/440 to reach ~55/32 at four ~8/7s and therefore ~11/8 at 22 generators down, and then 1001/1000 to interpret the generator as ~143/128 and find ~13/8 at 23 generators up.
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 by adding 4375/4374 to the comma list, 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
In the following table, odd harmonics and subharmonics 1–35 are labeled in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| Intervals of extensions | ||
| Hemithirds | ||
| 0 | 0.0 | 1/1 |
| 1 | 194.4 | 28/25, 125/112 |
| 2 | 388.9 | 5/4 |
| 3 | 583.3 | 7/5 |
| 4 | 777.7 | 25/16 |
| 5 | 972.1 | 7/4 |
| 6 | 1166.6 | 49/25, 125/64 |
| 7 | 161.0 | 35/32 |
| 8 | 355.4 | 49/40, 128/105 |
| 9 | 549.9 | 175/128 |
| 10 | 744.3 | 32/21, 49/32 |
| 11 | 938.7 | 128/75 |
| 12 | 1133.1 | 40/21 |
| 13 | 127.6 | 16/15 |
| 14 | 322.0 | 25/21 |
| 15 | 516.4 | 4/3 |
| 16 | 710.8 | 112/75 |
| 17 | 905.3 | 5/3 |
| 18 | 1099.7 | 28/15 |
| 19 | 94.1 | 25/24 |
* In CWE undecimal didacus
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 |