171edo: Difference between revisions
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''' | '''171 EDO''' is a remarkable division of the octave which serves as a microtemperament for the 7-limit, approximating the 9-limit tonality diamond within about 2/5 of a cent. It divides the octave into 171 parts of 7.01754 cents each. 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 (ennealimmal comma), 19073486328125/19042491875328 (enneadecal comma), 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-two temperaments: [[Schismatic family|pontiac]], [[Schismatic family|sesquiquartififths]], [[Schismatic family|term]], [[Ragismic microtemperaments|ennealimmal]], [[Breedsmic temperaments|tertiaseptal]], [[Ragismic microtemperaments|supermajor]] (tempering out 4375/4374 and 52734275/52706752), [[Ragismic microtemperaments|enneadecal]], [[Gammic family|neptune]] (tempering out 2401/2400 and 48828125/488771072), [[Ragismic microtemperaments|mitonic]] (tempering out 4375/4374 and 2100875/2097152), and [[Mutt family|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 |-59 39 0 -1>. | Remarkable 5-limit commas 171EDO tempers out are 32805/32768 (schisma), 7629394531250/7625597484987 (ennealimmal comma), 19073486328125/19042491875328 (enneadecal comma), 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-two temperaments: [[Schismatic family|pontiac]], [[Schismatic family|sesquiquartififths]], [[Schismatic family|term]], [[Ragismic microtemperaments|ennealimmal]], [[Breedsmic temperaments|tertiaseptal]], [[Ragismic microtemperaments|supermajor]] (tempering out 4375/4374 and 52734275/52706752), [[Ragismic microtemperaments|enneadecal]], [[Gammic family|neptune]] (tempering out 2401/2400 and 48828125/488771072), [[Ragismic microtemperaments|mitonic]] (tempering out 4375/4374 and 2100875/2097152), and [[Mutt family|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 |-59 39 0 -1>. | ||
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171 factors into primes as 3^2 * 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-three 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 3^2 * 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-three 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 ''100 [[EDF]]'' (step size 7.01955¢) and ''271 [[EDT]]'' (step size 7.01828¢). | |||
== Prime intervals == | == Prime intervals == | ||
{{Primes in edo|171|prec=2}} | {{Primes in edo|171|prec=2}} | ||
==Scales== | == Scales == | ||
* [[Nestoria7]] | * [[Nestoria7]] | ||
* [[Nestoria12]] | * [[Nestoria12]] | ||
==See also== | == See also == | ||
* [[Table of 171edo intervals]] | |||
* [[Ennealimmal-enneadecal equivalence continuum]] | * [[Ennealimmal-enneadecal equivalence continuum]] | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:171edo| ]] <!-- main article --> | [[Category:171edo| ]] <!-- main article --> | ||
[[Category:Scales]] | [[Category:Scales]] | ||