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¢).

Intervals

Main article: 171edo/Intervals

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-271 171⟩	[⟨171 271]]	+0.063	0.0633	0.90
2.3.5	32805/32768, [1 -27 18⟩	[⟨171 271 397]]	+0.092	0.0660	0.94
2.3.5.7	2401/2400, 4375/4374, 32805/32768	[⟨171 271 397 480]]	+0.105	0.0614	0.87
2.3.5.7.11	243/242, 441/440, 4375/4356, 16384/16335	[⟨171 271 397 480 592]] (171)	-0.093	0.401	5.71
2.3.5.7.11	385/384, 1331/1323, 1375/1372, 4375/4374	[⟨171 271 397 480 591]] (171e)	+0.312	0.418	5.96

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	5\171	35.09	234375/229376	Gammic
1	20\171	140.35	243/224	Tsaharuk
1	25\171	175.44	448/405	Sesquiquartififths / sesquart
1	26\171	182.46	10/9	Minortone / mitonic / mineral (171) / ore (171e)
1	34\171	238.60	147/128	Tokko
1	46\171	322.81	3087/2560	Senior / seniority
1	62\171	435.09	9/7	Supermajor
1	64\171	449.12	35/27	Semidimi
1	65\171	456.14	125/96	Qak
1	71\171	498.25	4/3	Helmholtz / pontiac
1	83\171	582.46	7/5	Neptune
3	20\171	140.35	243/224	Septichrome
3	23\171	161.40	192/175	Pnict
3	71\171 (2\171)	385.96 (14.04)	5/4 (126/125)	Mutt
3	55\171 (2\171)	498.25 (98.25)	4/3 (200/189)	Term / terminal
9	7\171	49.12	36/35	Ennealimmal (171e) / Ennealimmia (171) / Ennealimnic (171) / Ennealiminal (171e)
19	71\171 (1\171)	498.25 (7.02)	4/3 (225/224)	Enneadecal

Scales

• Nestoria7
• Nestoria12

See also

• Ennealimmal-enneadecal equivalence continuum