171edo: Difference between revisions
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== Theory == | == Theory == | ||
171edo is a remarkable | 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 [[zeta gap edo|zeta gap]]. | ||
Remarkable 5-limit commas 171et tempers out are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -29 -11 20 }} ([[gammic comma]]), and remarkable 7-limit commas 171et 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 171et [[support]]s a number of 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor]], [[enneadecal]], [[neptune]], [[mitonic]], and [[mutt]]. It notably provides the [[optimal patent val]] for the rank-3 [[horwell]] temperament, and is also an excellent tuning for the 5-limit [[ | Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -29 -11 20 }} ([[gammic comma]]), and remarkable 7-limit commas 171et 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 171et [[support]]s a number of 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor]], [[enneadecal]], [[neptune]], [[mitonic]], and [[mutt]]. It notably provides the [[optimal patent val]] for the rank-3 [[horwell]] temperament, and is also an excellent tuning for the 5-limit [[helmholtz|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]). | ||
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. Alternatively, the 171e val can be used, which tempers out [[385/384]]. | |||
171edo is an excellent tuning for the [[Carlos Gamma]] scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents. | |||
171edo is an excellent | |||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|171|columns=15}} | {{Harmonics in equal|171|columns=15}} | ||
=== Subsets and supersets === | |||
171 factors into primes as 3<sup>2</sup> × 19, and 171edo 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, 7/6, and any one of primes 3, 5, or 7. | |||
== Intervals == | == Intervals == | ||
| Line 22: | Line 23: | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
| Line 32: | Line 33: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -271 171 }} | | {{monzo| -271 171 }} | ||
| | | {{mapping| 171 271 }} | ||
| +0.063 | | +0.063 | ||
| 0.0633 | | 0.0633 | ||
| Line 39: | Line 40: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{monzo| 1 -27 18 }} | | 32805/32768, {{monzo| 1 -27 18 }} | ||
| | | {{mapping| 171 271 397 }} | ||
| +0.092 | | +0.092 | ||
| 0.0660 | | 0.0660 | ||
| Line 46: | Line 47: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, 32805/32768 | | 2401/2400, 4375/4374, 32805/32768 | ||
| | | {{mapping| 171 271 397 480 }} | ||
| +0.105 | | +0.105 | ||
| 0.0614 | | 0.0614 | ||
| Line 53: | Line 54: | ||
| style="border-top: double;" | 2.3.5.7.11 | | style="border-top: double;" | 2.3.5.7.11 | ||
| style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335 | | style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335 | ||
| style="border-top: double;" | | | style="border-top: double;" | {{mapping| 171 271 397 480 592 }} | ||
| style="border-top: double;" | −0.093 | | style="border-top: double;" | −0.093 | ||
| style="border-top: double;" | 0.401 | | style="border-top: double;" | 0.401 | ||
| Line 60: | Line 61: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 243/242, 364/363, 441/440, 625/624, 2200/2197 | | 243/242, 364/363, 441/440, 625/624, 2200/2197 | ||
| | | {{mapping| 171 271 397 480 592 633 }} | ||
| −0.149 | | −0.149 | ||
| 0.386 | | 0.386 | ||
| Line 67: | Line 68: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197 | | 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197 | ||
| | | {{mapping| 171 271 397 480 592 633 699 }} | ||
| −0.138 | | −0.138 | ||
| 0.358 | | 0.358 | ||
| Line 74: | Line 75: | ||
| style="border-top: double;" | 2.3.5.7.11 | | style="border-top: double;" | 2.3.5.7.11 | ||
| style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374 | | style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374 | ||
| style="border-top: double;" | | | style="border-top: double;" | {{mapping| 171 271 397 480 591 }} (171e) | ||
| style="border-top: double;" | +0.312 | | style="border-top: double;" | +0.312 | ||
| style="border-top: double;" | 0.418 | | style="border-top: double;" | 0.418 | ||
| Line 81: | Line 82: | ||
| style="border-top: double;" | 2.3.5.7.13 | | style="border-top: double;" | 2.3.5.7.13 | ||
| style="border-top: double;" | 625/624, 729/728, 2205/2197, 2401/2400 | | style="border-top: double;" | 625/624, 729/728, 2205/2197, 2401/2400 | ||
| style="border-top: double;" | | | style="border-top: double;" | {{mapping| 171 271 397 480 633 }} | ||
| style="border-top: double;" | −0.001 | | style="border-top: double;" | −0.001 | ||
| style="border-top: double;" | 0.220 | | style="border-top: double;" | 0.220 | ||
| Line 88: | Line 89: | ||
| 2.3.5.7.13.17 | | 2.3.5.7.13.17 | ||
| 625/624, 729/728, 833/832, 1225/1224, 2205/2197 | | 625/624, 729/728, 833/832, 1225/1224, 2205/2197 | ||
| | | {{mapping| 171 271 397 480 633 699 }} | ||
| −0.013 | | −0.013 | ||
| 0.202 | | 0.202 | ||
| Line 99: | Line 100: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods <br>per 8ve | ! Periods <br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 222: | Line 223: | ||
| 182.46 | | 182.46 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Terrain]] / [[domain]] | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 234: | Line 235: | ||
| 498.25<br>(98.25) | | 498.25<br>(98.25) | ||
| 4/3<br>(200/189) | | 4/3<br>(200/189) | ||
| [[ | | [[Term (temperament)|Term]] / terminal / terminator | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 261: | Line 262: | ||
== Music == | == Music == | ||
; [[birdshite stalactite]] | ; [[birdshite stalactite]] | ||
* "it's probably gout" from ''clagworks / it's probably gout'' | * "it's probably gout" from ''clagworks / it's probably gout'' (2024) – [https://open.spotify.com/track/26bfjsdZ8quDTwAkGY5kQF Spotify] | [https://birdshitestalactite.bandcamp.com/track/its-probably-gout Bandcamp] | [https://www.youtube.com/watch?v=uycaqLtws_w YouTube] | ||
; [[レケム]] | ; [[レケム]] | ||
| Line 272: | Line 273: | ||
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律] | * [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律] | ||
[[Category:Horwell]] | [[Category:Horwell]] | ||
[[Category:Ennealimmal]] | [[Category:Ennealimmal]] | ||
Revision as of 07:50, 26 April 2024
| ← 170edo | 171edo | 172edo → |
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 about 7.02 cents each, a size close to 225/224, the marvel comma.
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 zeta gap.
Remarkable 5-limit commas 171et tempers out are 32805/32768 (schisma), [1 -27 18⟩ (ennealimma), [-14 -19 19⟩ (enneadeca), and [-29 -11 20⟩ (gammic comma), and remarkable 7-limit commas 171et 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 171et supports a number of 7-limit rank-2 temperaments: pontiac, sesquiquartififths, term, ennealimmal, tertiaseptal, supermajor, enneadecal, neptune, mitonic, and mutt. It notably provides the optimal patent val for the rank-3 horwell temperament, and 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).
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. Alternatively, the 171e val can be used, which tempers out 385/384.
171edo is an excellent tuning for the Carlos Gamma scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.20 | -0.35 | -0.40 | +3.07 | +1.58 | +0.31 | -2.78 | +3.30 | +2.00 | -1.18 | +1.29 | -0.99 | +0.76 | +1.16 |
| Relative (%) | +0.0 | -2.9 | -5.0 | -5.8 | +43.7 | +22.5 | +4.4 | -39.6 | +47.1 | +28.5 | -16.8 | +18.3 | -14.1 | +10.9 | +16.5 | |
| Steps (reduced) |
171 (0) |
271 (100) |
397 (55) |
480 (138) |
592 (79) |
633 (120) |
699 (15) |
726 (42) |
774 (90) |
831 (147) |
847 (163) |
891 (36) |
916 (61) |
928 (73) |
950 (95) | |
Subsets and supersets
171 factors into primes as 32 × 19, and 171edo 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, 7/6, and any one of primes 3, 5, or 7.
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]] | −0.093 | 0.401 | 5.71 |
| 2.3.5.7.11.13 | 243/242, 364/363, 441/440, 625/624, 2200/2197 | [⟨171 271 397 480 592 633]] | −0.149 | 0.386 | 5.50 |
| 2.3.5.7.11.13.17 | 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197 | [⟨171 271 397 480 592 633 699]] | −0.138 | 0.358 | 5.11 |
| 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 |
| 2.3.5.7.13 | 625/624, 729/728, 2205/2197, 2401/2400 | [⟨171 271 397 480 633]] | −0.001 | 0.220 | 3.13 |
| 2.3.5.7.13.17 | 625/624, 729/728, 833/832, 1225/1224, 2205/2197 | [⟨171 271 397 480 633 699]] | −0.013 | 0.202 | 2.88 |
- 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until 441 do we find a better equal temperaments in terms of absolute error, and not until 3125 do we find one in terms of relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | 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 | 26\171 | 182.46 | 10/9 | Terrain / domain |
| 3 | 55\171 (2\171) |
385.96 (14.04) |
5/4 (126/125) |
Mutt |
| 3 | 71\171 (14\171) |
498.25 (98.25) |
4/3 (200/189) |
Term / terminal / terminator |
| 9 | 45\171 (7\171) |
315.79 (49.12) |
6/5 (36/35) |
Ennealimmal (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171ef) |
| 9 | 10\171 | 70.17 | (336/323) | Enneasoteric (171f) |
| 19 | 71\171 (1\171) |
498.25 (7.02) |
4/3 (225/224) |
Enneadecal |