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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]].
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.  


Remarkable 5-limit commas 171 EDO tempers out are 32805/32768 (schisma), 7629394531250/7625597484987 (ennealimmal comma), 19073486328125/19042491875328 (enneadecal comma), and 95367431640625/95105071448064 (gammic comma), and remarkable 7-limit commas 171 EDO 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, 171 EDO 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>.
== 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]].


171 factors into primes as 3^2 * 19, and it shares the nearly pure 7/6 of [[9 EDO]] and the nearly pure 6/5 of [[19 EDO]], with every 7-limit interval expressible in terms of 2, 6/5 and 7/6. 171 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-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 171 EDO also supports.
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]]).


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


== Prime intervals ==
Relative equal-step systems are ''[[100edf]]'' (step size 7.01955¢) and ''[[271edt]]'' (step size 7.01828¢).
 
=== Prime harmonics ===
{{Primes in edo|171}}
{{Primes in edo|171}}
== Intervals ==
{{Main| 171edo/Intervals }}


== Scales ==
== Scales ==
* [[Nestoria7]]
* [[Nestoria7]]
* [[Nestoria12]]
* [[Nestoria12]]


== See also ==
== See also ==
* [[171edo/Intervals|Table of 171 EDO intervals]]
* [[Ennealimmal-enneadecal equivalence continuum]]
* [[Ennealimmal-enneadecal equivalence continuum]]


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