171edo

← 170edo

171edo

172edo →

Prime factorization	32 × 19
Step size	7.01754 ¢
Fifth	100\171 (701.754 ¢)
Semitones (A1:m2)	16:13 (112.3 ¢ : 91.23 ¢)
Consistency limit	13
Distinct consistency limit	13

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 7-limit microtemperament, 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 zeta gap. It is also almost consistent in the 17-odd-limit, only missing 15/11 and 22/15.

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). Therefore, 171et supports a number of notable 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 helmholtz temperament, tempering out 32805/32768, and the no-fives temperament tempering out [-59 39 0 -1⟩.

171edo is notably consistent to distance 6 in the 7-prime 15-odd-limit, and to distance 8 in the 9-odd-limit. No other edo is so consistent until 3125edo. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 7-limit commas by size. For example, one step represents 225/224, two steps 126/125, three steps 81/80, and four steps 64/63.

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. In the 2.3.13 subgroup, it provides the optimal patent val for glacier, and is generally a good 2.3.5.7.13 and better 2.3.5.7.13.17 subgroup temperament.

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

Prime harmonics

Approximation of prime harmonics in 171edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	-0.20	-0.35	-0.40	+3.07	+1.58	+0.31	-2.78	+3.30	+2.00	-1.18
	Relative (%)	+0.0	-2.9	-5.0	-5.8	+43.7	+22.5	+4.4	-39.6	+47.1	+28.5	-16.8
Steps (reduced)		171 (0)	271 (100)	397 (55)	480 (138)	592 (79)	633 (120)	699 (15)	726 (42)	774 (90)	831 (147)	847 (163)

Approximation of prime harmonics in 171edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+1.29	-0.99	+0.76	+1.16	-3.33	+0.48	-1.10	-2.11	+2.76	-3.23	+0.38
	Relative (%)	+18.3	-14.1	+10.9	+16.5	-47.4	+6.8	-15.6	-30.1	+39.3	-46.0	+5.3
Steps (reduced)		891 (36)	916 (61)	928 (73)	950 (95)	979 (124)	1006 (151)	1014 (159)	1037 (11)	1052 (26)	1058 (32)	1078 (52)

Subsets and supersets

171 factors into primes as 3² × 19, and 171edo 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, 7/6, and any one of primes 3, 5, or 7.

342edo, which doubles 171, provides an excellent correction for the 11th harmonic and is one of the most accurate 11-limit temperaments, with unmatched relative error up until 1848edo.

684edo, which quadruples it, achieves 17-odd-limit consistency.

Intervals

Main article: 171edo/Intervals

Notation

Ups and downs notation

171edo can be notated using ups and downs with quarter-tone accidentals:

Step offset	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33
Sharp symbol
Flat symbol

Approximation to JI

15-odd-limit intervals

The following tables show how 15-odd-limit intervals are represented in 171edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 171edo (direct approximation, even if inconsistent)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
9/7, 14/9	0.004	0.1
9/5, 10/9	0.052	0.7
7/5, 10/7	0.056	0.8
15/14, 28/15	0.145	2.1
5/3, 6/5	0.148	2.1
3/2, 4/3	0.201	2.9
7/6, 12/7	0.204	2.9
5/4, 8/5	0.349	5.0
9/8, 16/9	0.401	5.7
7/4, 8/7	0.405	5.8
15/8, 16/15	0.549	7.8
13/11, 22/13	1.490	21.2
13/8, 16/13	1.578	22.5
13/12, 24/13	1.778	25.3
13/10, 20/13	1.926	27.5
13/9, 18/13	1.979	28.2
13/7, 14/13	1.982	28.2
15/13, 26/15	2.127	30.3
11/8, 16/11	3.068	43.7
11/6, 12/11	3.269	46.6
15/11, 22/15	3.400	48.5
11/10, 20/11	3.417	48.7
11/9, 18/11	3.469	49.4
11/7, 14/11	3.473	49.5

15-odd-limit intervals in 171edo (patent val mapping)

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
9/7, 14/9	0.004	0.1
9/5, 10/9	0.052	0.7
7/5, 10/7	0.056	0.8
15/14, 28/15	0.145	2.1
5/3, 6/5	0.148	2.1
3/2, 4/3	0.201	2.9
7/6, 12/7	0.204	2.9
5/4, 8/5	0.349	5.0
9/8, 16/9	0.401	5.7
7/4, 8/7	0.405	5.8
15/8, 16/15	0.549	7.8
13/11, 22/13	1.490	21.2
13/8, 16/13	1.578	22.5
13/12, 24/13	1.778	25.3
13/10, 20/13	1.926	27.5
13/9, 18/13	1.979	28.2
13/7, 14/13	1.982	28.2
15/13, 26/15	2.127	30.3
11/8, 16/11	3.068	43.7
11/6, 12/11	3.269	46.6
11/10, 20/11	3.417	48.7
11/9, 18/11	3.469	49.4
11/7, 14/11	3.473	49.5
15/11, 22/15	3.617	51.5

Consistent circles

171edo contains consistent circles of 7/6, 6/5, and 9/7, each with 9, 19, and 171 notes respectively.

Consistent circles in 171edo

Note count	Interval	Closing error	Consistency	Associated edostep
9	7/6	-26.2%	Normal	2\9 = 38\171
19	6/5	+40.1%	Normal	5\19 = 45\171
171	9/7	+8.8%	Strong	62\171

7-prime-limited odd-limit analysis

171edo is distinctly consistent and monotone up to the 7-prime-limited 45-odd-limit:

The 7-prime-limited 45-odd-limit, by 171edo mapping (SW3 format)

(*
 7-PL 45-OL odds:
 1 3 5 7 9 15 21 25 27 35 45
  Mapping   Ratio    Error
*)
(*  7\171*) 36/35 (* +.352c *)
(*  9\171*) 28/27 (* +.197c *)
(* 10\171*) 25/24 (* -.497c *)
(* 12\171*) 21/20 (* -.257c *)
(* 16\171*) 16/15 (* +.549c *)
(* 17\171*) 15/14 (* -.145c *)
(* 19\171*) 27/25 (* +.096c *)
(* 22\171*) 35/32 (* -.754c *)
(* 26\171*) 10/9  (* +.052c *)
(* 28\171*) 28/25 (* +.293c *)
(* 29\171*) 9/8   (* -.401c *)
(* 33\171*) 8/7   (* -.405c *)
(* 38\171*) 7/6   (* -.204c *)
(* 42\171*) 32/27 (* +.602c *)
(* 43\171*) 25/21 (* -.092c *)
(* 45\171*) 6/5   (* +.148c *)
(* 54\171*) 56/45 (* +.345c *)
(* 55\171*) 5/4   (* -.349c *)
(* 61\171*) 32/25 (* +.698c *)
(* 62\171*) 9/7   (* +.004c *)
(* 64\171*) 35/27 (* -.152c *)
(* 67\171*) 21/16 (* -.605c *)
(* 71\171*) 4/3   (* +.201c *)
(* 74\171*) 27/20 (* -.253c *)
(* 78\171*) 48/35 (* +.553c *)
(* 81\171*) 25/18 (* -.296c *)
(* 83\171*) 7/5   (* -.056c *)
(* 84\171*) 45/32 (* -.750c *)
(* 87\171*) 64/45
(* 88\171*) 10/7
(* 90\171*) 36/25
(* 93\171*) 35/24
(* 97\171*) 40/27
(*100\171*) 3/2
(*104\171*) 32/21
(*107\171*) 54/35
(*109\171*) 14/9
(*110\171*) 25/16
(*116\171*) 8/5
(*117\171*) 45/28
(*126\171*) 5/3
(*128\171*) 42/25
(*129\171*) 27/16
(*133\171*) 12/7
(*138\171*) 7/4
(*142\171*) 16/9
(*143\171*) 25/14
(*145\171*) 9/5
(*149\171*) 64/35
(*152\171*) 50/27
(*154\171*) 28/15
(*155\171*) 15/8
(*159\171*) 40/21
(*161\171*) 48/25
(*162\171*) 27/14
(*164\171*) 35/18
(*171\171*) 2/1

The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, ~49/48 = ~50/49 (this is characteristic of all ennealimmal tunings). However, 171edo remains consistent up to much higher 7-prime-limited odd-limits (much higher than even 99edo).

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]]	−0.093	0.401	5.71
2.3.5.7.11.13	243/242, 364/363, 441/440, 625/624, 2200/2197	[⟨171 271 397 480 592 633]]	−0.149	0.386	5.50
2.3.5.7.11.13.17	243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197	[⟨171 271 397 480 592 633 699]]	−0.138	0.358	5.11
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
2.3.5.7.13	625/624, 729/728, 2205/2197, 2401/2400	[⟨171 271 397 480 633]]	−0.001	0.220	3.13
2.3.5.7.13.17	625/624, 729/728, 833/832, 1225/1224, 2205/2197	[⟨171 271 397 480 633 699]]	−0.013	0.202	2.88

• 171et is lower in relative error than any previous equal temperaments in the 7-limit and in the 9-odd-limit. Not until 441 do we find a better equal temperaments 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 8ve	Generator*	Cents*	Associated ratio*	Temperament
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	[[[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	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	26\171	182.46	10/9	Domain
3	55\171 (2\171)	385.96 (14.04)	5/4 (126/125)	Mutt
3	71\171 (14\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 / enneabiotic (171ef) / ennealympic (171) / ennealimnic (171) / ennealiminal (171ef)
9	10\171	70.17	(336/323)	Enneasoteric (171f)
19	71\171 (1\171)	498.25 (7.02)	4/3 (225/224)	Enneadecal

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

• Nestoria7
• Nestoria12
• Diamond9: 26 3 4 5 7 10 7 9 12 5 12 9 7 10 7 5 4 3 26

Music

Modern renderings

Johann Sebastian Bach

• "Prelude" from Prelude and Fugue in C major, No. 1, BWV 846, from The Well-Tempered Clavier, Book I (1722) – rendered by レケム (2022)

21st century

birdshite stalactite

• "it's probably gout" from clagworks / it's probably gout (2024) – Spotify | Bandcamp | YouTube

See also

• Ennealimmal-enneadecal equivalence continuum

External links

• 171EDO / 171平均律