171edo

Template:Infobox ET 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 about 7.02 cents each, a size close to 225/224, the marvel comma.

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 171et tempers out are 32805/32768 (schisma), | 1 -27 18 > (ennealimma), | -14 -19 19 > (enneadeca), and | -29 -11 20 > (gammic comma), and remarkable 7-limit commas 171et 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 171et supports a number of 7-limit rank-2 temperaments: pontiac, sesquiquartififths, term, ennealimmal, tertiaseptal, supermajor, enneadecal, neptune, mitonic, and mutt. It notably provides the optimal patent val for the rank-3 horwell temperament, and 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. Alternatively, the 171e val can be used, which tempers out 385/384.

171edo is an excellent EDO for Carlos Gamma scale, since the difference between 5 steps of 171edo and 1 step of Carlos Gamma is only -0.010823 cents.

Regular temperament properties

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

• 171et is lower in relative error than any previous ETs in the 7-limit. Not until 441 do we find a better ET in terms of absolute error, and not until 3125 do we find one in terms of relative error.

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	11\171	77.19	256/245	Tertiaseptal / tertia (171e)
1	17\171	119.30	15/14	Septidiasemi / sedia
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) / goldmine (171ef)
1	34\171	238.60	147/128	Tokko
1	46\171	322.81	3087/2560	Senior / seniority
1	49\171	343.86	8000/6561	Geb
1	56\171	392.98	2744/2187	Emmthird
1	61\171	428.07	2800/2187	Osiris
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	70\171	491.23	3645/2744	Fifthplus
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 / terminator
9	45\171 (7\171)	315.79 (49.12)	6/5 (36/35)	Ennealimmal (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171ef)
19	71\171 (1\171)	498.25 (7.02)	4/3 (225/224)	Enneadecal

Scales

• Nestoria7
• Nestoria12

See also

• Ennealimmal-enneadecal equivalence continuum
• 100edf (step size 7.01955¢)
• 271edt (step size 7.01828¢)