171edo: Difference between revisions

Line 3:

== 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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] but not enough to make it a [[The Riemann zeta function and tuning #Zeta EDO lists|gap edo]].

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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] but not enough to make it a [[The Riemann zeta function and tuning #Zeta EDO lists|gap edo]].

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 [[Schismatic family|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).

Line 11:

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 ~~edo~~ 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 EDO 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 ===

Line 258:

* [[Nestoria7]]

* [[Nestoria12]]

== Music ==

* [https://www.youtube.com/watch?v=7IKpuEqHTyk J.S.Bach Das Wohltemperirte Clavier Erster Teil 1 Präludium (171edo)]

== See also ==

* [[Ennealimmal-enneadecal equivalence continuum]]

== ~~Music~~ ==

== External links ==

* [https://~~www~~.~~youtube~~.com/~~watch~~?v=~~7IKpuEqHTyk J.S.Bach Das Wohltemperirte Clavier Erster Teil 1 Präludium (171edo)~~]

* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]

[[Category:171edo| ]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] but not enough to make it a [[The Riemann zeta function and tuning #Zeta EDO lists|gap edo]].
+edo 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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] but not enough to make it a [[The Riemann zeta function and tuning #Zeta EDO lists|gap edo]].
 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 [[Schismatic family|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).
@@ Line 11: / Line 11: @@
 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]].
-edo is an excellent edo for the [[Carlos Gamma]] scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents.
+edo is an excellent EDO 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 ===
@@ Line 258: / Line 258: @@
 * [[Nestoria7]]
 * [[Nestoria12]]
+== Music ==
+* [https://www.youtube.com/watch?v=7IKpuEqHTyk J.S.Bach Das Wohltemperirte Clavier Erster Teil 1 Präludium (171edo)]
 == See also ==
 * [[Ennealimmal-enneadecal equivalence continuum]]
-== Music ==
+== External links ==
-* [https://www.youtube.com/watch?v=7IKpuEqHTyk J.S.Bach Das Wohltemperirte Clavier Erster Teil 1 Präludium (171edo)]
+* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
 [[Category:171edo| ]]  <!-- main article -->