320edo: Difference between revisions
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+rank-2 temperaments |
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=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per octave | |||
! Generator<br>(reduced) | |||
! Cents<br>(reduced) | |||
! Associated<br>ratio | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 7\320 | |||
| 26.25 | |||
| {{monzo| -2 13 -8 }} | |||
| [[Sfourth]] (5-limit) | |||
|- | |||
| 1 | |||
| 131\320 | |||
| 491.25 | |||
| 3645/2744 | |||
| [[Fifthplus]] | |||
|- | |||
| 1 | |||
| 157\320 | |||
| 588.75 | |||
| 45/32 | |||
| [[Untriton]] (5-limit) | |||
|- | |||
| 2 | |||
| 19\320 | |||
| 71.25 | |||
| 25/24 | |||
| [[Vishnu]] / [[narayana]] | |||
|- | |||
| 5 | |||
| 133\320<br>(5\320) | |||
| 498.75<br>(18.75) | |||
| 4/3<br>(81/80) | |||
| [[Pental]] | |||
|- | |||
| 8 | |||
| 133\320<br>(9\320) | |||
| 566.25<br>(33.75) | |||
| 104/75<br>(55/54) | |||
| [[Octowerck]] | |||
|- | |||
| 10 | |||
| 19\320<br>(13\320) | |||
| 71.25<br>(48.75) | |||
| 25/24<br>(36/35) | |||
| [[Decavish]] | |||
|- | |||
| 10 | |||
| 133\320<br>(5\320) | |||
| 498.75<br>(18.75) | |||
| 4/3<br>(81/80) | |||
| [[Decal]] | |||
|} | |} | ||
Revision as of 14:25, 28 December 2021
The 320 equal division divides the octave into 320 equal parts of precisely 3.75 cents each.
It tempers out 65625/65536 (horwell) and 420175/419904 (wizma) in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports the varuna temperament, the rank-3 temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the optimal patent val. It also provides the optimal patent val for the rank-4 werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to a rank-3 temperament.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-507 320⟩ | [⟨320 507]] | +0.2224 | 0.2224 | 5.93 |
| 2.3.5 | [23 6 -14⟩, [-28 25 -5⟩ | [⟨320 507 743]] | +0.1574 | 0.2036 | 5.43 |
| 2.3.5.7 | 65625/65536, 235298/234375, 321489/320000 | [⟨320 507 743 898]] | +0.2361 | 0.2229 | 5.94 |
| 2.3.5.7.11 | 441/440, 8019/8000, 41503/41472, 65625/65536 | [⟨320 507 743 898 1107]] | +0.1928 | 0.2173 | 5.80 |
| 2.3.5.7.11.13 | 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655 | [⟨320 507 743 898 1107 1184]] | +0.1845 | 0.1993 | 5.31 |
| 2.3.5.7.11.13.17 | 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224 | [⟨320 507 743 898 1107 1184 1308]] | +0.1565 | 0.1968 | 5.25 |
| 2.3.5.7.11.13.17.19 | 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520 | [⟨320 507 743 898 1107 1184 1308 1359]] | +0.1741 | 0.1899 | 5.06 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 7\320 | 26.25 | [-2 13 -8⟩ | Sfourth (5-limit) |
| 1 | 131\320 | 491.25 | 3645/2744 | Fifthplus |
| 1 | 157\320 | 588.75 | 45/32 | Untriton (5-limit) |
| 2 | 19\320 | 71.25 | 25/24 | Vishnu / narayana |
| 5 | 133\320 (5\320) |
498.75 (18.75) |
4/3 (81/80) |
Pental |
| 8 | 133\320 (9\320) |
566.25 (33.75) |
104/75 (55/54) |
Octowerck |
| 10 | 19\320 (13\320) |
71.25 (48.75) |
25/24 (36/35) |
Decavish |
| 10 | 133\320 (5\320) |
498.75 (18.75) |
4/3 (81/80) |
Decal |