171edo: Difference between revisions

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| Augmented 1sn = 16\171 (112¢)

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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 [[cent]]s each, a size close to [[225/224]], the marvel comma.

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 [[cent]]s 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 [[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 ~~171EDO~~ 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 ~~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 family|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).

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

171 factors into primes as 3<sup>2</sup> × 19, and it shares the nearly pure [[7/6]] of [[9edo~~|9EDO~~]] and the nearly pure [[6/5]] of [[19edo~~|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.

171 factors into primes as 3<sup>2</sup> × 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.

~~Relative equal~~-~~step systems are ''~~[[~~100edf|100EDF~~]]'' (~~step size 7.01955¢~~) and ''[[~~271edt|271EDT~~]]~~'' (step size 7~~.~~01828¢)~~.

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]].

=== Prime harmonics ===

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== See also ==

* [[Ennealimmal-enneadecal equivalence continuum]]

* [[100edf]] (step size 7.01955¢)

* [[271edt]] (step size 7.01828¢)

[[Category:Equal divisions of the octave]]

[[Category:171edo| ]]

[[Category:Horwell]]

[[Category:Ennealimmal]]

[[Category:Enneadecal]]

@@ Line 7: / Line 7: @@
 | Augmented 1sn = 16\171  (112¢)
 }}
-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 [[cent]]s each, a size close to [[225/224]], the marvel comma.
+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 [[cent]]s each, a size close to [[225/224]], the marvel comma.
 == 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 171EDO 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 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 family|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).
+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 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 family|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).
-factors into primes as 3<sup>2</sup> × 19, and it shares the nearly pure [[7/6]] of [[9edo|9EDO]] and the nearly pure [[6/5]] of [[19edo|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.
+factors into primes as 3<sup>2</sup> × 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.
-Relative equal-step systems are ''[[100edf|100EDF]]'' (step size 7.01955¢) and ''[[271edt|271EDT]]'' (step size 7.01828¢).
+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]].
 === Prime harmonics ===
@@ Line 246: / Line 246: @@
 == See also ==
 * [[Ennealimmal-enneadecal equivalence continuum]]
+* [[100edf]] (step size 7.01955¢)
+* [[271edt]] (step size 7.01828¢)
 [[Category:Equal divisions of the octave]]
 [[Category:171edo| ]]  <!-- main article -->
+[[Category:Horwell]]
 [[Category:Ennealimmal]]
 [[Category:Enneadecal]]