171edo - Revision history

FloraC: /* Regular temperament properties */ wtf is with this

2026-05-19T09:52:15Z

Regular temperament properties: wtf is with this

← Older revision		Revision as of 09:52, 19 May 2026
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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 ===

FloraC: /* Regular temperament properties */ - typo

2026-05-19T09:51:40Z

Regular temperament properties: - typo

← Older revision		Revision as of 09:51, 19 May 2026
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	\| 182.46		\| 182.46
	\| 10/9		\| 10/9
	\| [[[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)		\| [[Mitonic]] / mineral (171) / ore (171e) / goldmine (171ef)
	\|-		\|-
	\| 1		\| 1

FloraC: /* Regular temperament properties */ update

2026-05-19T09:51:08Z

Regular temperament properties: update

← Older revision		Revision as of 09:51, 19 May 2026
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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
Line 346:		Line 346:
	\| 182.46		\| 182.46
	\| 10/9		\| 10/9
	\| [[~~Terrain]] / [[domain~~]]		\| [[Domain (temperament)\|Domain]]
	\|-		\|-
	\| 3		\| 3
Line 364:		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

Inthar: /* 7-prime-limited odd-limit analysis */

2026-04-19T22:23:24Z

7-prime-limited odd-limit analysis

← Older revision		Revision as of 22:23, 19 April 2026
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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]]).

	== Regular temperament properties ==		== Regular temperament properties ==

Overthink: move to JI approx. section

2026-04-19T18:11:02Z

move to JI approx. section

@@ Line 23: / Line 23: @@
 [[684edo]], which quadruples it, achieves [[17-odd-limit]] consistency.
 === 7-prime-limited odd-limit analysis ===
@@ Line 97: / Line 140: @@
 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 ==

Inthar: /* 7-prime-limited odd-limit analysis */

2026-04-19T00:41:54Z

7-prime-limited odd-limit analysis

← Older revision		Revision as of 00:41, 19 April 2026
Line 25:		Line 25:

	=== 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
Line 36:		Line 34:
	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

Inthar: /* 7-prime-limited odd-limit analysis */

2026-04-18T19:44:42Z

7-prime-limited odd-limit analysis

← Older revision		Revision as of 19:44, 18 April 2026
Line 98:		Line 98:
	}}		}}

	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.		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 ==		== Intervals ==

Inthar: /* 7-prime-limited odd-limit analysis */

2026-04-18T19:41:09Z

7-prime-limited odd-limit analysis

← Older revision		Revision as of 19:41, 18 April 2026
Line 36:		Line 36:
	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
			*)
	(* 7\171*) 36/35		(* 7\171*) 36/35
	(* 9\171*) 28/27		(* 9\171*) 28/27

Inthar: /* Theory */

2026-04-18T19:40:24Z

Theory

← Older revision		Revision as of 19:40, 18 April 2026
Line 23:		Line 23:

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

			=== 7-prime-limited odd-limit analysis ===
			171edo is ''distinctly'' [[consistent]] and monotone up to the 7-prime-limited 45-odd-limit, i.e.
			* 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
			\|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*)
			(* 7\171*) 36/35
			(* 9\171*) 28/27
			(* 10\171*) 25/24
			(* 12\171*) 21/20
			(* 16\171*) 16/15
			(* 17\171*) 15/14
			(* 19\171*) 27/25
			(* 22\171*) 35/32
			(* 26\171*) 10/9
			(* 28\171*) 28/25
			(* 29\171*) 9/8
			(* 33\171*) 8/7
			(* 38\171*) 7/6
			(* 42\171*) 32/27
			(* 43\171*) 25/21
			(* 45\171*) 6/5
			(* 54\171*) 56/45
			(* 55\171*) 5/4
			(* 61\171*) 32/25
			(* 62\171*) 9/7
			(* 64\171*) 35/27
			(* 67\171*) 21/16
			(* 71\171*) 4/3
			(* 74\171*) 27/20
			(* 78\171*) 48/35
			(* 81\171*) 25/18
			(* 83\171*) 7/5
			(* 84\171*) 45/32
			(* 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.

	== Intervals ==		== Intervals ==

Overthink: italic instead of bold

2026-04-10T17:36:03Z

italic instead of bold

← Older revision		Revision as of 17:36, 10 April 2026
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 }}.		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 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]].

	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 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.

← Older revision		Revision as of 17:36, 10 April 2026
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 }}.		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 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]].

	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 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.