171edo: Difference between revisions
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| Line 54: | Line 54: | ||
| 0.401 | | 0.401 | ||
| 5.71 | | 5.71 | ||
|- | |||
| 2.3.5.7.11.13 | |||
| 243/242, 364/363, 441/440, 625/624, 3584/3575 | |||
| [{{val| 171 271 397 480 592 633 }}] (171) | |||
| -0.149 | |||
| 0.386 | |||
| 5.50 | |||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| Line 61: | Line 68: | ||
| 0.418 | | 0.418 | ||
| 5.96 | | 5.96 | ||
|- | |||
| 2.3.5.7.11.13 | |||
| 352/351, 385/384, 1001/1000, 1188/1183, 1331/1323 | |||
| [{{val| 171 271 397 480 591 633 }}] (171e) | |||
| +0.189 | |||
| 0.471 | |||
| 6.71 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 169/168, 325/324, 385/384, 1331/1323, 1375/1372 | |||
| [{{val| 171 271 397 480 591 632 }}] (171ef) | |||
| +0.505 | |||
| 0.576 | |||
| 8.21 | |||
|} | |} | ||
| Line 82: | Line 103: | ||
| 77.19 | | 77.19 | ||
| 256/245 | | 256/245 | ||
| [[Tertiaseptal]] / | | [[Tertiaseptal]] / tertia (171e) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 88: | Line 109: | ||
| 119.30 | | 119.30 | ||
| 15/14 | | 15/14 | ||
| [[Septidiasemi]] / | | [[Septidiasemi]] / sedia | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 100: | Line 121: | ||
| 175.44 | | 175.44 | ||
| 448/405 | | 448/405 | ||
| [[Sesquiquartififths]] / | | [[Sesquiquartififths]] / sesquart | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 106: | Line 127: | ||
| 182.46 | | 182.46 | ||
| 10/9 | | 10/9 | ||
| [[Minortone]] / [[mitonic]] / | | [[Minortone]] / [[mitonic]] / mineral (171) / ore (171e) / goldmine (171ef) | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 196: | Line 217: | ||
| 498.25<br>(98.25) | | 498.25<br>(98.25) | ||
| 4/3<br>(200/189) | | 4/3<br>(200/189) | ||
| [[Term]] / | | [[Term]] / terminal / terminator | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 202: | Line 223: | ||
| 49.12 | | 49.12 | ||
| 36/35 | | 36/35 | ||
| [[Ennealimmal]] (171e) / | | [[Ennealimmal]] (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171e) | ||
|- | |- | ||
| 19 | | 19 | ||
Revision as of 13:25, 7 September 2021
The 171 equal divisions of the octave (171EDO), or the 171(-tone) equal temperament (171TET, 171ET) when viewed from a regular temperament perspective, is the tuning system derived from dividing the octave into 171 parts of 7.01754 cents each.
Theory
171EDO is a remarkable edo which serves as a microtemperament for the 7-limit, approximating the 9-odd-limit tonality diamond within about 2/5 of a cent. The excellence of its 7-limit approximations is good enough to make it the eleventh zeta integral EDO but not enough to make it a gap EDO.
Remarkable 5-limit commas 171EDO tempers out are 32805/32768 (schisma), 7629394531250/7625597484987 (ennealimma), 19073486328125/19042491875328 (enneadeca), and 95367431640625/95105071448064 (gammic comma), and remarkable 7-limit commas 171EDO tempers out are 2401/2400 (breedsma), 4375/4374 (ragisma), 65625/65536 (horwell comma), 250047/250000 (landscape comma), 420175/419904 (wizma), and 703125/702464 (meter comma). So, 171EDO supports a number of 7-limit rank-2 temperaments: pontiac, sesquiquartififths, term, ennealimmal, tertiaseptal, supermajor (tempering out 4375/4374 and 52734275/52706752), enneadecal, neptune (tempering out 2401/2400 and 48828125/488771072), mitonic (tempering out 4375/4374 and 2100875/2097152), and mutt. It is also an excellent tuning for the 5-limit schismatic microtemperament, tempering out 32805/32768, and the no-fives temperament tempering out [-59 39 0 -1⟩ (nanisma).
171 factors into primes as 32 × 19, and it shares the nearly pure 7/6 of 9EDO and the nearly pure 6/5 of 19EDO, with every 7-limit interval expressible in terms of 2, 6/5 and 7/6. 171EDO is much less accurate in the 11-limit, but still quite useful as it is a good tuning (emphasizing accuracy in the 7-limit) for the important rank-3 temperament jove, which tempers out 243/242 (rastma) and 441/440, not to mention 540/539 and 2401/2400. Jove can be extended by adding 364/363 for the 13-limit and 595/594 for the 17-limit, which 171EDO also supports.
Relative equal-step systems are 100EDF (step size 7.01955¢) and 271EDT (step size 7.01828¢).
Prime harmonics
Script error: No such module "primes_in_edo".
Intervals
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-271 171⟩ | [⟨171 271]] | +0.063 | 0.0633 | 0.90 |
| 2.3.5 | 32805/32768, [1 -27 18⟩ | [⟨171 271 397]] | +0.092 | 0.0660 | 0.94 |
| 2.3.5.7 | 2401/2400, 4375/4374, 32805/32768 | [⟨171 271 397 480]] | +0.105 | 0.0614 | 0.87 |
| 2.3.5.7.11 | 243/242, 441/440, 4375/4356, 16384/16335 | [⟨171 271 397 480 592]] (171) | -0.093 | 0.401 | 5.71 |
| 2.3.5.7.11.13 | 243/242, 364/363, 441/440, 625/624, 3584/3575 | [⟨171 271 397 480 592 633]] (171) | -0.149 | 0.386 | 5.50 |
| 2.3.5.7.11 | 385/384, 1331/1323, 1375/1372, 4375/4374 | [⟨171 271 397 480 591]] (171e) | +0.312 | 0.418 | 5.96 |
| 2.3.5.7.11.13 | 352/351, 385/384, 1001/1000, 1188/1183, 1331/1323 | [⟨171 271 397 480 591 633]] (171e) | +0.189 | 0.471 | 6.71 |
| 2.3.5.7.11.13 | 169/168, 325/324, 385/384, 1331/1323, 1375/1372 | [⟨171 271 397 480 591 632]] (171ef) | +0.505 | 0.576 | 8.21 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 5\171 | 35.09 | 234375/229376 | Gammic |
| 1 | 11\171 | 77.19 | 256/245 | Tertiaseptal / tertia (171e) |
| 1 | 17\171 | 119.30 | 15/14 | Septidiasemi / sedia |
| 1 | 20\171 | 140.35 | 243/224 | Tsaharuk |
| 1 | 25\171 | 175.44 | 448/405 | Sesquiquartififths / sesquart |
| 1 | 26\171 | 182.46 | 10/9 | Minortone / mitonic / mineral (171) / ore (171e) / goldmine (171ef) |
| 1 | 34\171 | 238.60 | 147/128 | Tokko |
| 1 | 46\171 | 322.81 | 3087/2560 | Senior / seniority |
| 1 | 49\171 | 343.86 | 8000/6561 | Geb |
| 1 | 56\171 | 392.98 | 2744/2187 | Emmthird |
| 1 | 61\171 | 428.07 | 2800/2187 | Osiris |
| 1 | 62\171 | 435.09 | 9/7 | Supermajor |
| 1 | 64\171 | 449.12 | 35/27 | Semidimi |
| 1 | 65\171 | 456.14 | 125/96 | Qak |
| 1 | 70\171 | 491.23 | 3645/2744 | Fifthplus |
| 1 | 71\171 | 498.25 | 4/3 | Helmholtz / pontiac |
| 1 | 83\171 | 582.46 | 7/5 | Neptune |
| 3 | 20\171 | 140.35 | 243/224 | Septichrome |
| 3 | 23\171 | 161.40 | 192/175 | Pnict |
| 3 | 71\171 (2\171) |
385.96 (14.04) |
5/4 (126/125) |
Mutt |
| 3 | 55\171 (2\171) |
498.25 (98.25) |
4/3 (200/189) |
Term / terminal / terminator |
| 9 | 7\171 | 49.12 | 36/35 | Ennealimmal (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171e) |
| 19 | 71\171 (1\171) |
498.25 (7.02) |
4/3 (225/224) |
Enneadecal |