Luna and hemithirds: Difference between revisions
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{{Infobox Regtemp | |||
| Title = Hemithirds | |||
| Subgroups = 2.3.5.7 | |||
| Comma basis = [[1029/1024]], [[3136/3125]] (2.3.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11) | |||
| Edo join 1 = 25 | Edo join 2 = 31 | |||
| Generator = 28/25 | Generator tuning = 193.239 | Optimization method = CWE | |||
| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]], [[25L 6s]] | |||
| Mapping = 1; -15 2 5 | |||
| Odd limit 1 = 7 | Mistuning 1 = ??? | Complexity 1 = ??? | |||
| Odd limit 2 = 11 | Mistuning 2 = ??? | Complexity 2 = ??? | |||
}} | |||
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 [[cent]]s) 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. | 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 [[cent]]s) 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. | ||
Revision as of 18:06, 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. |
Lua error in Module:Infobox_regtemp at line 131: attempt to perform arithmetic on local 'generator_size' (a nil value).
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 |
|---|---|---|
| 7-limit hemithirds | ||
| 0 | 0.0 | 1/1 |
| 1 | 193.2 | 28/25, 125/112 |
| 2 | 386.5 | 5/4 |
| 3 | 579.7 | 7/5 |
| 4 | 773.0 | 25/16 |
| 5 | 966.2 | 7/4 |
| 6 | 1159.4 | 49/25, 125/64 |
| 7 | 152.7 | 35/32 |
| 8 | 345.9 | 49/40, 128/105 |
| 9 | 539.2 | 175/128 |
| 10 | 732.4 | 32/21, 49/32 |
| 11 | 925.6 | 128/75 |
| 12 | 1118.9 | 40/21 |
| 13 | 112.1 | 16/15 |
| 14 | 305.3 | 25/21 |
| 15 | 498.6 | 4/3 |
| 16 | 691.8 | 112/75 |
| 17 | 885.1 | 5/3 |
| 18 | 1078.3 | 28/15 |
| 19 | 71.5 | 25/24 |
* In CWE 7-limit hemithirds tuning
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 |