171edo: Difference between revisions

''171edo'' 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 7.016 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]].

171 supports a number of 7-limit rank-two temperaments: pontiac, with commas 4375/4374 and 32805/32768; sesquiquartififths, with 2401/2400 and 32805/32768; term, with 32805/32768 and 250047/250000; ennealimmal, with 2401/2400 and 4375/4374; tertiaseptal with 2401/2400 and 65635/65536; supermajor, with 4375/4374 and 52734275/52706752; enneadecal with 4375/4374 and 703125/702464; neptune, with 2401/2400 and 48828125/488771072; mitonic, with 4375/4374 and 2100875/2097152; and mutt, with 65635/65536 and 250047/250000. It is also an excellent tuning for the 5-limit schismatic microtemperament, tempering out 32805/32768, and the no-fives temperament tempering out <59 -39 0 1|.

171 factors into primes as 3^2 * 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. 171 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 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 also supports.