171edo: Difference between revisions

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== Theory ==
== 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 [[zeta gap edo|zeta gap]].
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 edo|zeta gap]]. It is also almost consistent in the 17-odd-limit, only missing [[15/11]] and [[22/15]].


Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -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 [[support]]s 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 [[helmholtz|schismatic microtemperament]], tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }} ([[nanisma]]).
Remarkable 5-limit commas 171et [[tempering out|tempers out]] are 32805/32768 ([[schisma]]), {{monzo| 1 -27 18 }} ([[ennealimma]]), {{monzo| -14 -19 19 }} ([[enneadeca]]), and {{monzo| -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 [[support]]s a number of notable 7-limit rank-2 temperaments: [[pontiac]], [[sesquiquartififths]], [[term]], [[ennealimmal]], [[tertiaseptal]], [[supermajor (temperament)|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)|helmholtz]] temperament, tempering out 32805/32768, and the no-fives temperament tempering out {{monzo| -59 39 0 -1 }}.


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 notably [[Consistency#Consistency to distance d|consistent to distance ''6'']] in the [[7-limit|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 comma|glacier]], and is generally a good [[2.3.5.7.13 subgroup|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.  
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 ===
=== Prime harmonics ===
{{Harmonics in equal|171|columns=15}}
{{Harmonics in equal|171|columns=11}}
{{Harmonics in equal|171|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 171edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
171 factors into primes as 3<sup>2</sup> × 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.
171 factors into primes as {{nowrap| 3<sup>2</sup> × 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 ==
== Intervals ==
{{Main| 171edo/Intervals }}
{{Main| 171edo/Intervals }}
== Notation ==
=== Ups and downs notation ===
171edo can be notated using [[Kite's ups and downs notation|ups and downs]] with quarter-tone accidentals:
{{Ups and downs sharpness|171|true}}
== Approximation to JI ==
=== 15-odd-limit intervals ===
{{Q-odd-limit intervals|171|15}}
=== Consistent circles ===
171edo contains consistent circles of [[7/6]], [[6/5]], and [[9/7]], each with 9, 19, and 171 notes respectively.
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Consistent circles in 171edo
|-
! Note<br>count
! [[Interval]]
! [[Closing error|Closing<br>error]]
! [[Circle #Definitions|Consistency]]
! Associated<br>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:
{{Databox
|collapse=true
|title=The 7-prime-limited 45-odd-limit, by 171edo mapping (SW3 format)
|text=
<pre>
(*
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
</pre>
}}
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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning Error
! colspan="2" | Tuning error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 32: Line 154:
|-
|-
| 2.3
| 2.3
| {{monzo| -271 171 }}
| {{Monzo| -271 171 }}
| {{mapping| 171 271 }}
| {{Mapping| 171 271 }}
| +0.063
| +0.063
| 0.0633
| 0.0633
Line 40: Line 162:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 1 -27 18 }}
| 32805/32768, {{monzo| 1 -27 18 }}
| {{mapping| 171 271 397 }}
| {{Mapping| 171 271 397 }}
| +0.092
| +0.092
| 0.0660
| 0.0660
Line 47: Line 169:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, 32805/32768
| 2401/2400, 4375/4374, 32805/32768
| {{mapping| 171 271 397 480 }}
| {{Mapping| 171 271 397 480 }}
| +0.105
| +0.105
| 0.0614
| 0.0614
| 0.87
| 0.87
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11
| 2.3.5.7.11
| style="border-top: double;" | 243/242, 441/440, 4375/4356, 16384/16335
| 243/242, 441/440, 4375/4356, 16384/16335
| style="border-top: double;" | {{mapping| 171 271 397 480 592 }}
| {{Mapping| 171 271 397 480 592 }}
| style="border-top: double;" | &minus;0.093
| −0.093
| style="border-top: double;" | 0.401
| 0.401
| style="border-top: double;" | 5.71
| 5.71
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 243/242, 364/363, 441/440, 625/624, 2200/2197
| 243/242, 364/363, 441/440, 625/624, 2200/2197
| {{mapping| 171 271 397 480 592 633 }}
| {{Mapping| 171 271 397 480 592 633 }}
| &minus;0.149
| −0.149
| 0.386
| 0.386
| 5.50
| 5.50
Line 68: Line 190:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197
| 243/242, 364/363, 375/374, 441/440, 595/594, 2200/2197
| {{mapping| 171 271 397 480 592 633 699 }}
| {{Mapping| 171 271 397 480 592 633 699 }}
| &minus;0.138
| −0.138
| 0.358
| 0.358
| 5.11
| 5.11
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11
| 2.3.5.7.11
| style="border-top: double;" | 385/384, 1331/1323, 1375/1372, 4375/4374
| 385/384, 1331/1323, 1375/1372, 4375/4374
| style="border-top: double;" | {{mapping| 171 271 397 480 591 }} (171e)
| {{Mapping| 171 271 397 480 591 }} (171e)
| style="border-top: double;" | +0.312
| +0.312
| style="border-top: double;" | 0.418
| 0.418
| style="border-top: double;" | 5.96
| 5.96
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.13
| 2.3.5.7.13
| style="border-top: double;" | 625/624, 729/728, 2205/2197, 2401/2400
| 625/624, 729/728, 2205/2197, 2401/2400
| style="border-top: double;" | {{mapping| 171 271 397 480 633 }}
| {{Mapping| 171 271 397 480 633 }}
| style="border-top: double;" | &minus;0.001
| −0.001
| style="border-top: double;" | 0.220
| 0.220
| style="border-top: double;" | 3.13
| 3.13
|-
|-
| 2.3.5.7.13.17
| 2.3.5.7.13.17
| 625/624, 729/728, 833/832, 1225/1224, 2205/2197
| 625/624, 729/728, 833/832, 1225/1224, 2205/2197
| {{mapping| 171 271 397 480 633 699 }}
| {{Mapping| 171 271 397 480 633 699 }}
| &minus;0.013
| −0.013
| 0.202
| 0.202
| 2.88
| 2.88
Line 98: Line 220:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods <br>per 8ve
|-
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
Line 139: Line 262:
| 182.46
| 182.46
| 10/9
| 10/9
| [[Minortone]] / [[mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)
| [[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)
|-
|-
| 1
| 1
Line 175: Line 298:
| 435.09
| 435.09
| 9/7
| 9/7
| [[Supermajor]]
| [[Supermajor (temperament)|Supermajor]]
|-
|-
| 1
| 1
Line 199: Line 322:
| 498.25
| 498.25
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]]
| [[Pontiac]]
|-
|-
| 1
| 1
Line 223: Line 346:
| 182.46
| 182.46
| 10/9
| 10/9
| [[Terrain]] / [[domain]]
| [[Domain (temperament)|Domain]]
|-
|-
| 3
| 3
Line 241: Line 364:
| 315.79<br>(49.12)
| 315.79<br>(49.12)
| 6/5<br>(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171ef)
| [[Ennealimmal]] / enneabiotic (171ef) / ennealympic (171) / ennealimnic (171) / ennealiminal (171ef)
|-
|-
| 9
| 9
Line 255: Line 378:
| [[Enneadecal]]
| [[Enneadecal]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==
* [[Nestoria7]]
* [[Nestoria7]]
* [[Nestoria12]]
* [[Nestoria12]]
* [[Diamond9]]: 26 3 4 5 7 10 7 9 12 5 12 9 7 10 7 5 4 3 26


== Music ==
== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=7IKpuEqHTyk "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]]
; [[birdshite stalactite]]
* "it's probably gout" from ''clagworks / it's probably gout'' (2024) – [https://open.spotify.com/track/26bfjsdZ8quDTwAkGY5kQF Spotify] | [https://birdshitestalactite.bandcamp.com/track/its-probably-gout Bandcamp] | [https://www.youtube.com/watch?v=uycaqLtws_w YouTube]
* "it's probably gout" from ''clagworks / it's probably gout'' (2024) – [https://open.spotify.com/track/26bfjsdZ8quDTwAkGY5kQF Spotify] | [https://birdshitestalactite.bandcamp.com/track/its-probably-gout Bandcamp] | [https://www.youtube.com/watch?v=uycaqLtws_w YouTube]
; [[レケム]]
* [https://www.youtube.com/watch?v=7IKpuEqHTyk J.S.Bach Das Wohltemperirte Clavier Erster Teil 1 Präludium (171edo)]


== See also ==
== See also ==
Line 273: Line 400:
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]


[[Category:Enneadecal]]
[[Category:Ennealimmal]]
[[Category:Gammic]]
[[Category:Horwell]]
[[Category:Horwell]]
[[Category:Ennealimmal]]
[[Category:Listen]]
[[Category:Enneadecal]]
[[Category:Mitonic]]
[[Category:Mutt]]
[[Category:Neptune]]
[[Category:Pontiac]]
[[Category:Pontiac]]
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]
[[Category:Term (temperament)]]
[[Category:Term (temperament)]]
[[Category:Tertiaseptal]]
[[Category:Tertiaseptal]]
[[Category:Gammic]]
[[Category:Neptune]]
[[Category:Mitonic]]
[[Category:Mutt]]
[[Category:Listen]]

Latest revision as of 09:52, 19 May 2026

← 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   
 
  
   
  
     
    
   
  
   
    
     
    
     
    
   
  
   
    
     
    
     
    
   
  
   
    
     
    
     
    
   
  
   
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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. 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

Music

Modern renderings

Johann Sebastian Bach

21st century

birdshite stalactite

See also

External links

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