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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]].
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]]). 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 }}.
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 (temperament)|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 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 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]].
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[[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency.
[[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency.
== 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 ===
=== 7-prime-limited odd-limit analysis ===
171edo is ''distinctly'' [[consistent]] and monotone up to the 7-prime-limited 45-odd-limit, i.e.
171edo is ''distinctly'' [[consistent]] and monotone up to the 7-prime-limited 45-odd-limit:
* when tempered using the patent val, the relative sizes of any two intervals are never conflated ''or'' reversed
* the direct approximation is equal to the approximation given by stacking patent val prime approximations, thus every interval has absolute error < 3.509c.
{{Databox
{{Databox
|collapse=true
|collapse=true
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  7-PL 45-OL odds:
  7-PL 45-OL odds:
  1 3 5 7 9 15 21 25 27 35 45
  1 3 5 7 9 15 21 25 27 35 45
   Mapping  Ratio
   Mapping  Ratio   Error
*)
*)
(*  7\171*) 36/35  
(*  7\171*) 36/35 (* +.352c *)
(*  9\171*) 28/27  
(*  9\171*) 28/27 (* +.197c *)
(* 10\171*) 25/24  
(* 10\171*) 25/24 (* -.497c *)
(* 12\171*) 21/20  
(* 12\171*) 21/20 (* -.257c *)
(* 16\171*) 16/15  
(* 16\171*) 16/15 (* +.549c *)
(* 17\171*) 15/14  
(* 17\171*) 15/14 (* -.145c *)
(* 19\171*) 27/25  
(* 19\171*) 27/25 (* +.096c *)
(* 22\171*) 35/32  
(* 22\171*) 35/32 (* -.754c *)
(* 26\171*) 10/9   
(* 26\171*) 10/9  (* +.052c *)
(* 28\171*) 28/25  
(* 28\171*) 28/25 (* +.293c *)
(* 29\171*) 9/8   
(* 29\171*) 9/8  (* -.401c *)
(* 33\171*) 8/7   
(* 33\171*) 8/7  (* -.405c *)
(* 38\171*) 7/6   
(* 38\171*) 7/6  (* -.204c *)
(* 42\171*) 32/27  
(* 42\171*) 32/27 (* +.602c *)
(* 43\171*) 25/21  
(* 43\171*) 25/21 (* -.092c *)
(* 45\171*) 6/5   
(* 45\171*) 6/5  (* +.148c *)
(* 54\171*) 56/45  
(* 54\171*) 56/45 (* +.345c *)
(* 55\171*) 5/4   
(* 55\171*) 5/4  (* -.349c *)
(* 61\171*) 32/25  
(* 61\171*) 32/25 (* +.698c *)
(* 62\171*) 9/7   
(* 62\171*) 9/7  (* +.004c *)
(* 64\171*) 35/27  
(* 64\171*) 35/27 (* -.152c *)
(* 67\171*) 21/16  
(* 67\171*) 21/16 (* -.605c *)
(* 71\171*) 4/3   
(* 71\171*) 4/3  (* +.201c *)
(* 74\171*) 27/20  
(* 74\171*) 27/20 (* -.253c *)
(* 78\171*) 48/35  
(* 78\171*) 48/35 (* +.553c *)
(* 81\171*) 25/18  
(* 81\171*) 25/18 (* -.296c *)
(* 83\171*) 7/5   
(* 83\171*) 7/5  (* -.056c *)
(* 84\171*) 45/32  
(* 84\171*) 45/32 (* -.750c *)
(* 87\171*) 64/45
(* 87\171*) 64/45
(* 88\171*) 10/7
(* 88\171*) 10/7
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}}
}}


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]]).
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]]).
 
== 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
|}


== Regular temperament properties ==
== Regular temperament properties ==
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| 2.88
| 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 [[441edo|441]] do we find a better equal temperaments in terms of absolute error, and not until [[3125edo|3125]] do we find one in terms of relative error.
* 171et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[441edo|441]] do we find a better equal temperaments in terms of absolute error, and not until [[3125edo|3125]] do we find one in terms of relative error.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
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| 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
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| 498.25
| 498.25
| 4/3
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]]
| [[Pontiac]]
|-
|-
| 1
| 1
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| 582.46
| 582.46
| 7/5
| 7/5
| [[Neptune]]
| [[Neptune (temperament)|Neptune]]
|-
|-
| 3
| 3
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| 182.46
| 182.46
| 10/9
| 10/9
| [[Terrain]] / [[domain]]
| [[Domain (temperament)|Domain]]
|-
|-
| 3
| 3
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| 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
Retrieved from "https://en.xen.wiki/w/171edo"