171edo: Difference between revisions

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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 }}.

~~All intervals with an~~ odd limit ~~below 10~~,~~000,000~~ in the 7-limit ~~are mapped consistently by 171edo~~. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify 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]].

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 ~~great~~ [[2.3.5.7.13 subgroup|2.3.5.7.13]] and 2.3.5.7.13.17 subgroup temperament.

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.

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=== Subsets and supersets ===

171 factors into primes as {{nowrap| 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.

171 factors into primes as {{nowrap| 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 ==

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| 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 ==

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| 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 ===

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| 182.46

| 10/9

| [[~~Minortone]] / [[mitonic~~]] / mineral (171) / ore (171e) / goldmine (171ef)

| [[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)

|-

| 1

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| 498.25

| 4/3

| [[~~Helmholtz (temperament)|Helmholtz]] / [[pontiac~~]]

| [[Pontiac]]

|-

| 1

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| 182.46

| 10/9

| [[~~Terrain]] / [[domain~~]]

| [[Domain (temperament)|Domain]]

|-

| 3

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| 315.79 (49.12)

| 6/5 (36/35)

| [[Ennealimmal]] (~~171e~~) / ~~ennealimmia~~ (171) / ennealimnic (171) / ennealiminal (171ef)

| [[Ennealimmal]] / enneabiotic (171ef) / ennealympic (171) / ennealimnic (171) / ennealiminal (171ef)

|-

| 9

@@ Line 7: / Line 7: @@
 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 }}.
-All intervals with an odd limit below 10,000,000 in the 7-limit are mapped consistently by 171edo. Because of its accuracy in the 7-limit, the 171et mapping is an excellent and relatively simple way to classify commas by size. For example, one step represents [[225/224]], two steps [[126/125]], three steps [[81/80]], and four steps [[64/63]].
+edo 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]].
-edo 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 great [[2.3.5.7.13 subgroup|2.3.5.7.13]] and 2.3.5.7.13.17 subgroup temperament.
+edo 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.
 edo 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.
@@ Line 18: / Line 18: @@
 === Subsets and supersets ===
-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.
+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 ==
@@ Line 62: / Line 66: @@
 | 62\171
 |}
+=== 7-prime-limited odd-limit analysis ===
+edo 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>
+(*
+-PL 45-OL odds:
+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 ==
@@ Line 138: / Line 216: @@
 | 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 ===
@@ Line 184: / Line 262: @@
 | 182.46
 | 10/9
-| [[Minortone]] / [[mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)
+| [[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)
 |-
 | 1
@@ Line 244: / Line 322: @@
 | 498.25
 | 4/3
-| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]]
+| [[Pontiac]]
 |-
 | 1
@@ Line 268: / Line 346: @@
 | 182.46
 | 10/9
-| [[Terrain]] / [[domain]]
+| [[Domain (temperament)|Domain]]
 |-
 | 3
@@ Line 286: / Line 364: @@
 | 315.79<br>(49.12)
 | 6/5<br>(36/35)
-| [[Ennealimmal]] (171e) / ennealimmia (171) / ennealimnic (171) / ennealiminal (171ef)
+| [[Ennealimmal]] / enneabiotic (171ef) / ennealympic (171) / ennealimnic (171) / ennealiminal (171ef)
 |-
 | 9