171edo: Difference between revisions

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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 7.01754 [[cent]]s each.

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 7.01754 [[cent]]s each.

== 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]]), 7629394531250/7625597484987 ([[ennealimma]]), 19073486328125/19042491875328 ([[enneadeca]]), and 95367431640625/95105071448064 ([[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 171EDO tempers out are 32805/32768 ([[schisma]]), 7629394531250/7625597484987 ([[ennealimma]]), 19073486328125/19042491875328 ([[enneadeca]]), and 95367431640625/95105071448064 ([[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]]).

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. ~~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|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.

Relative equal-step systems are ''[[100edf]]'' (step size 7.01955¢) and ''[[271edt]]'' (step size 7.01828¢).

Relative equal-step systems are ''[[100edf|100EDF]]'' (step size 7.01955¢) and ''[[271edt|271EDT]]'' (step size 7.01828¢).

=== Prime harmonics ===

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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 7.01754 [[cent]]s each.
+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 7.01754 [[cent]]s each.
 == 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]]), 7629394531250/7625597484987 ([[ennealimma]]), 19073486328125/19042491875328 ([[enneadeca]]), and 95367431640625/95105071448064 ([[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 171EDO tempers out are 32805/32768 ([[schisma]]), 7629394531250/7625597484987 ([[ennealimma]]), 19073486328125/19042491875328 ([[enneadeca]]), and 95367431640625/95105071448064 ([[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]]).
-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. 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|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.
-Relative equal-step systems are ''[[100edf]]'' (step size 7.01955¢) and ''[[271edt]]'' (step size 7.01828¢).
+Relative equal-step systems are ''[[100edf|100EDF]]'' (step size 7.01955¢) and ''[[271edt|271EDT]]'' (step size 7.01828¢).
 === Prime harmonics ===