171edo: Difference between revisions

Line 7:

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]]). Therefore, 171et [[support]]s a number of notable 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 }}.

~~All intervals with an~~ odd limit ~~below 10~~,~~000,000~~ in the 7-limit ~~are mapped consistently by 171edo~~. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]].

171edo is notably [[Consistency#Consistency to distance d|consistent to distance '''6''']] in the [[7-limit|7-prime]] [[15-odd-limit]], and to distance '''8''' in the [[9-odd-limit]]. No other edo is so consistent until [[3125edo]]. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]].

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~~ [[2.3.5.7.13 subgroup|2.3.5.7.13]] and 2.3.5.7.13.17 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 good [[2.3.5.7.13 subgroup|2.3.5.7.13]] and better 2.3.5.7.13.17 subgroup temperament.

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.

Line 18:

=== 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 and is one of the most accurate [[11-limit]] temperaments, with unmatched relative error up until [[1848edo]].

[[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency.

== Intervals ==

@@ Line 7: / Line 7: @@
 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]]). Therefore, 171et [[support]]s a number of notable 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 }}.
-All intervals with an odd limit below 10,000,000 in the 7-limit are mapped consistently by 171edo. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]].
+edo is notably [[Consistency#Consistency to distance d|consistent to distance '''6''']] in the [[7-limit|7-prime]] [[15-odd-limit]], and to distance '''8''' in the [[9-odd-limit]]. No other edo is so consistent until [[3125edo]]. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]].
-edo 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 [[2.3.5.7.13 subgroup|2.3.5.7.13]] and 2.3.5.7.13.17 subgroup temperament.
+edo 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 good [[2.3.5.7.13 subgroup|2.3.5.7.13]] and better 2.3.5.7.13.17 subgroup temperament.
 edo 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.
@@ Line 18: / Line 18: @@
 === Subsets and supersets ===
-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.
+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 and is one of the most accurate [[11-limit]] temperaments, with unmatched relative error up until [[1848edo]].
+[[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency.
 == Intervals ==