140edo: Difference between revisions
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The '''140 equal divisions of the octave''' divides the [[octave]] into 140 parts of 8.57 [[cent]]s each. | The '''140 equal divisions of the octave''' ('''140edo'''), or the '''140(-tone) equal temperament''' ('''140tet''', '''140et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 140 parts of 8.57 [[cent]]s each. | ||
== Theory == | == Theory == | ||
In the 5-limit, | In the 5-limit, 140et tempers out [[15625/15552]], making it a kleismic system, and the kwazy comma, {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It supports the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]]. | ||
If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]]. | If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]]. | ||
Revision as of 10:24, 29 July 2021
The 140 equal divisions of the octave (140edo), or the 140(-tone) equal temperament (140tet, 140et) when viewed from a regular temperament perspective, divides the octave into 140 parts of 8.57 cents each.
Theory
In the 5-limit, 140et tempers out 15625/15552, making it a kleismic system, and the kwazy comma, [-53 10 16⟩. It is most notable, however, in the 7-limit, where it tempers out 2401/2400, 5120/5103, 10976/10935 and 65625/65536. It supports the 7-limit rank-2 temperaments tertiaseptal, hemififths, countercata and bisupermajor, and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the optimal patent val for 13-limit countercata. In the 11-limit it tempers out 385/384, 1331/1323, 1375/1372, 5632/5625, 6250/6237 and 9801/9800, and in the 13-limit 325/324, 352/351, 625/624, 676/675, 847/845, 1001/1000, 1716/1715 and 2080/2079.
If we use the val ⟨140 223 325 394] (140bbd) we obtain a tuning for porcupine temperament; the generator 19\140 is 0.023 cents flat of the POTE generator.
Prime harmonics
Script error: No such module "primes_in_edo".
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 15625/15552, [35 -25 2⟩ | [⟨140 222 325]] | -0.104 | 0.346 | 4.03 |
| 2.3.5.7 | 2401/2400, 5120/5103, 15625/15552 | [⟨140 222 325 393]] | -0.055 | 0.311 | 3.63 |
| 2.3.5.7.11 | 385/384, 1331/1323, 1375/1372, 2200/2187 | [⟨140 222 325 393 484]] | +0.115 | 0.439 | 5.12 |
| 2.3.5.7.11.13 | 325/324, 352/351, 385/384, 625/624, 1331/1323 | [⟨140 222 325 393 484 518]] | +0.119 | 0.401 | 4.68 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 9\140 | 77.14 | 22/21 | Tertiaseptal / tertia |
| 1 | 13\140 | 111.43 | 16/15 | Stockhausenic |
| 1 | 37\140 | 317.14 | 6/5 | Hanson / countercata |
| 1 | 41\140 | 351.43 | 49/40 | Hemififths |
| 1 | 53\140 | 454.29 | 13/10 | Fibo |
| 1 | 59\140 | 505.71 | 75/56 | Marfifths |
| 2 | 3\140 | 25.71 | 64/63 | Ketchup |
| 2 | 19\140 | 162.86 | 11/10 | Kwazy / bisupermajor |
| 2 | 41\140 (29\140) |
351.43 (248.57) |
49/40 (15/13) |
Semihemi |
| 4 | 37\140 (2\140) |
317.14 (17.14) |
6/5 (126/125) |
Quadritikleismic |
| 4 | 58\140 (12\140) |
497.14 (102.86) |
4/3 (35/33) |
Undim |
| 5 | 43\140 (13\140) |
368.57 (111.43) |
10125/8192 (16/15) |
Qintosec |
| 10 | 29\140 (1\140) |
248.57 (8.57) |
15/13 (176/175) |
Decoid |
| 20 | 54\140 (2\140) |
497.14 (17.14) |
4/3 (126/125) |
Degrees |
| 28 | 54\140 (2\140) |
497.14 (17.14) |
4/3 (126/125) |
Oquatonic |