171edo: Difference between revisions
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171edo is a remarkable edo which serves as a [[7-limit]] [[microtemperament]], 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]]. It is also almost consistent in the 17-odd-limit, missing [[15/11]] and [[22/15]]. | 171edo is a remarkable edo which serves as a [[7-limit]] [[microtemperament]], 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]]. It is also almost consistent in the 17-odd-limit, missing [[15/11]] and [[22/15]]. | ||
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]]). 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 (temperament)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | 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]]). So 171et [[support]]s a number of 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|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 (temperament)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}. | ||
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]]. In the 2.3.13 subgroup, it provides the optimal patent val for [[Glacier comma|glacier]], and is generally a great [[2.3.5.13 subgroup|2.3.5.13]] and 2.3.5.13.17 subgroup temperament. | 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]]. In the 2.3.13 subgroup, it provides the optimal patent val for [[Glacier comma|glacier]], and is generally a great [[2.3.5.13 subgroup|2.3.5.13]] and 2.3.5.13.17 subgroup temperament. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
171 factors into primes as {{nowrap|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. [[342edo]], which doubles 171, provides an excellent correction for the 11th harmonic. | 171 factors into primes as {{nowrap| 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. [[342edo]], which doubles 171, provides an excellent correction for the 11th harmonic. | ||
== Intervals == | == Intervals == | ||
{{Main| 171edo/Intervals }} | {{Main| 171edo/Intervals }} | ||
== Notation == | == Notation == | ||
=== Ups and downs notation === | === Ups and downs notation === | ||
| Line 45: | Line 46: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 52: | Line 53: | ||
|- | |- | ||
| 2.3 | | 2.3 | ||
| {{ | | {{Monzo| -271 171 }} | ||
| {{ | | {{Mapping| 171 271 }} | ||
| +0.063 | | +0.063 | ||
| 0.0633 | | 0.0633 | ||
| Line 60: | Line 61: | ||
| 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 67: | Line 68: | ||
| 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 74: | Line 75: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 243/242, 441/440, 4375/4356, 16384/16335 | | 243/242, 441/440, 4375/4356, 16384/16335 | ||
| {{ | | {{Mapping| 171 271 397 480 592 }} | ||
| −0.093 | | −0.093 | ||
| 0.401 | | 0.401 | ||
| Line 81: | Line 82: | ||
| 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 88: | Line 89: | ||
| 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 95: | Line 96: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 385/384, 1331/1323, 1375/1372, 4375/4374 | | 385/384, 1331/1323, 1375/1372, 4375/4374 | ||
| {{ | | {{Mapping| 171 271 397 480 591 }} (171e) | ||
| +0.312 | | +0.312 | ||
| 0.418 | | 0.418 | ||
| Line 102: | Line 103: | ||
| 2.3.5.7.13 | | 2.3.5.7.13 | ||
| 625/624, 729/728, 2205/2197, 2401/2400 | | 625/624, 729/728, 2205/2197, 2401/2400 | ||
| {{ | | {{Mapping| 171 271 397 480 633 }} | ||
| −0.001 | | −0.001 | ||
| 0.220 | | 0.220 | ||
| Line 109: | Line 110: | ||
| 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 196: | Line 197: | ||
| 435.09 | | 435.09 | ||
| 9/7 | | 9/7 | ||
| [[Supermajor]] | | [[Supermajor (temperament)|Supermajor]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 276: | Line 277: | ||
| [[Enneadecal]] | | [[Enneadecal]] | ||
|} | |} | ||
<nowiki/>* [[Normal | <nowiki/>* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||