171edo
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 cents 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 zeta integral edo but not enough to make it a 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 microtemperament, tempering out 32805/32768, and the no-fives temperament tempering out [-59 39 0 -1⟩ (nanisma).
171 factors into primes as 32 × 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.
Relative equal-step systems are 100edf (step size 7.01955¢) and 271edt (step size 7.01828¢).
Prime harmonics
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