Luna and hemithirds: Difference between revisions
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== Intervals == | == Intervals == | ||
In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. | In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. | ||
{| class="wikitable sortable center-all right-2" | {| class="wikitable sortable center-all right-2" | ||
|- | |- | ||
! rowspan="3" | # !! rowspan="3" | Cents* !! colspan=" | ! rowspan="3" | # !! rowspan="3" | Cents* !! colspan="1" | Approximate ratios | ||
|- | |- | ||
! colspan="1" | Intervals of extensions | |||
|- | |- | ||
! Hemithirds | ! Hemithirds | ||
| Line 22: | Line 21: | ||
| 0.0 | | 0.0 | ||
| '''1/1''' | | '''1/1''' | ||
|- | |- | ||
| 1 | | 1 | ||
| 194.4 | | 194.4 | ||
| 28/25 | | 28/25 | ||
|- | |- | ||
| 2 | | 2 | ||
| 388.9 | | 388.9 | ||
| '''5/4''' | | '''5/4''' | ||
|- | |- | ||
| 3 | | 3 | ||
| 583.3 | | 583.3 | ||
| 7/5 | | 7/5 | ||
|- | |- | ||
| 4 | | 4 | ||
| 777.7 | | 777.7 | ||
| '''25/16''' | | '''25/16''' | ||
|- | |- | ||
| 5 | | 5 | ||
| 972.1 | | 972.1 | ||
| '''7/4''' | | '''7/4''' | ||
|- | |- | ||
| 6 | | 6 | ||
| 1166.6 | | 1166.6 | ||
| 49/25 | | 49/25 | ||
|- | |- | ||
| 7 | | 7 | ||
| 161.0 | | 161.0 | ||
| '''35/32''' | | '''35/32''' | ||
|- | |- | ||
| 8 | | 8 | ||
| 355.4 | | 355.4 | ||
| 49/40 | | 49/40, 128/105 | ||
|- | |- | ||
| 9 | | 9 | ||
| 549.9 | | 549.9 | ||
| 175/128 | | 175/128 | ||
|- | |- | ||
| 10 | | 10 | ||
| 744.3 | | 744.3 | ||
| '''32/21''', 49/32 | |||
| '''32/21''' | |||
|- | |- | ||
| 11 | | 11 | ||
| 938.7 | | 938.7 | ||
| 128/75 | | 128/75 | ||
|- | |- | ||
| 12 | | 12 | ||
| 1133.1 | | 1133.1 | ||
| 40/21 | | 40/21 | ||
|- | |- | ||
| 13 | | 13 | ||
| 127.6 | | 127.6 | ||
| '''16/15''' | | '''16/15''' | ||
|- | |- | ||
| 14 | | 14 | ||
| 322.0 | | 322.0 | ||
| 25/21 | | 25/21 | ||
|- | |- | ||
| 15 | | 15 | ||
| 516.4 | | 516.4 | ||
| '''4/3''' | | '''4/3''' | ||
|- | |- | ||
| 16 | | 16 | ||
| 710.8 | | 710.8 | ||
| 112/75 | | 112/75 | ||
|- | |- | ||
| 17 | | 17 | ||
| 905.3 | | 905.3 | ||
| 5/3 | | 5/3 | ||
|- | |- | ||
| 18 | | 18 | ||
| 1099.7 | | 1099.7 | ||
| 28/15 | | 28/15 | ||
|- | |- | ||
| 19 | | 19 | ||
| 94.1 | | 94.1 | ||
| 25/24 | | 25/24 | ||
|} | |} | ||
Revision as of 17:15, 1 April 2025
|
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 |
| 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 |
| 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 |