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. | {{Infobox ET | ||
| Prime factorization = 3<sup>2</sup> × 19 | |||
| Step size = 6.81818¢ | |||
| Fifth = 100\171 (701.75¢) | |||
| Major 2nd = 29\171 (204¢) | |||
| Minor 2nd = 13\171 (91¢) | |||
| 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. | |||
== Theory == | == 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]]), | 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]]). | ||
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|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. | ||