Temperament family page revision proposal - Revision history

Lériendil: Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "

2025-06-21T06:57:39Z

Text replacement - "eigenmonzo (unchanged-interval) " to "unchanged-interval (eigenmonzo) "

← Older revision		Revision as of 06:57, 21 June 2025
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	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit]]: ~10/9 = {{monzo\| 3/5 0 -1/5 }}		* [[7-odd-limit]]: ~10/9 = {{monzo\| 3/5 0 -1/5 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5
	* [[9-odd-limit]]: ~10/9 = {{monzo\| 1/6 -1/6 0 1/12 }}		* [[9-odd-limit]]: ~10/9 = {{monzo\| 1/6 -1/6 0 1/12 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.9/7

	[[Tuning ranges]]:		[[Tuning ranges]]:
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	Minimax tuning:		Minimax tuning:
	* 11-odd-limit: ~11/10 = {{monzo\| 1/6 -1/6 0 1/12 }}		* 11-odd-limit: ~11/10 = {{monzo\| 1/6 -1/6 0 1/12 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.9/7		: unchanged-interval (eigenmonzo) basis: 2.9/7

	Tuning ranges:		Tuning ranges:
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	Minimax tuning:		Minimax tuning:
	* 13- and 15-odd-limit: ~10/9 = {{monzo\| 1 0 0 0 -1/4 }}		* 13- and 15-odd-limit: ~10/9 = {{monzo\| 1 0 0 0 -1/4 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.11		: unchanged-interval (eigenmonzo) basis: 2.11

	Tuning ranges:		Tuning ranges:
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	Minimax tuning:		Minimax tuning:
	* 13- and 15-odd-limit: ~10/9 = {{monzo\| 2/13 0 0 0 1/13 -1/13 }}		* 13- and 15-odd-limit: ~10/9 = {{monzo\| 2/13 0 0 0 1/13 -1/13 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.13/11		: unchanged-interval (eigenmonzo) basis: 2.13/11

	Tuning ranges:		Tuning ranges:
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	Minimax tuning:		Minimax tuning:
	* 13- and 15-odd-limit: ~11/10 = {{monzo\| 1/14 0 0 -1/14 0 1/14 }}		* 13- and 15-odd-limit: ~11/10 = {{monzo\| 1/14 0 0 -1/14 0 1/14 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.13/7		: unchanged-interval (eigenmonzo) basis: 2.13/7

	Optimal ET sequence: {{Optimal ET sequence\| 15f, 22f, 37, 59f }}		Optimal ET sequence: {{Optimal ET sequence\| 15f, 22f, 37, 59f }}
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	Minimax tuning:		Minimax tuning:
	* 13- and 15-odd-limit: ~11/10 = {{monzo\| 1/6 -1/6 0 1/12 }}		* 13- and 15-odd-limit: ~11/10 = {{monzo\| 1/6 -1/6 0 1/12 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.9/7		: unchanged-interval (eigenmonzo) basis: 2.9/7

	Optimal ET sequence: {{Optimal ET sequence\| 7, 15f, 22 }}		Optimal ET sequence: {{Optimal ET sequence\| 7, 15f, 22 }}
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	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit\|7-]] and [[9-odd-limit]] [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7		* [[7-odd-limit\|7-]] and [[9-odd-limit]] [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7

	{{Optimal ET sequence\|legend=1\| 7d, 8d, 15 }}		{{Optimal ET sequence\|legend=1\| 7d, 8d, 15 }}
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	Minimax tuning:		Minimax tuning:
	* 11-odd-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.7		* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7

	Optimal ET sequence: {{Optimal ET sequence\| 7d, 8d, 15 }}		Optimal ET sequence: {{Optimal ET sequence\| 7d, 8d, 15 }}
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	Minimax tuning:		Minimax tuning:
	* 13- and 15-odd-limit ~~eigenmonzo (~~unchanged-interval) basis: 2.7		* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7

	Optimal ET sequence: {{Optimal ET sequence\| 7d, 8d, 15, 38bceff }}		Optimal ET sequence: {{Optimal ET sequence\| 7d, 8d, 15, 38bceff }}
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	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit\|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo\| 2/11 0 1/11 -1/11 }}		* [[7-odd-limit\|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo\| 2/11 0 1/11 -1/11 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7/5		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.7/5

	{{Optimal ET sequence\|legend=1\| 7d, 15d, 22, 29, 51, 73c }}		{{Optimal ET sequence\|legend=1\| 7d, 15d, 22, 29, 51, 73c }}
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	Minimax tuning:		Minimax tuning:
	* 11-odd-limit: ~11/10 = {{monzo\| 2/11 0 1/11 -1/11 }}		* 11-odd-limit: ~11/10 = {{monzo\| 2/11 0 1/11 -1/11 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.7/5		: unchanged-interval (eigenmonzo) basis: 2.7/5

	Optimal ET sequence: {{Optimal ET sequence\| 7d, 15d, 22, 51 }}		Optimal ET sequence: {{Optimal ET sequence\| 7d, 15d, 22, 51 }}
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	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit\|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo\| 2/3 -1/3 }}		* [[7-odd-limit\|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo\| 2/3 -1/3 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.3		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.3

	{{Optimal ET sequence\|legend=1\| 7, 22d, 29, 65c, 94cd }}		{{Optimal ET sequence\|legend=1\| 7, 22d, 29, 65c, 94cd }}
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	Minimax tuning:		Minimax tuning:
	* 11-odd-limit: ~11/10 = {{monzo\| 2/3 -1/3 }}		* 11-odd-limit: ~11/10 = {{monzo\| 2/3 -1/3 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.3		: unchanged-interval (eigenmonzo) basis: 2.3

	Optimal ET sequence: {{Optimal ET sequence\| 7, 22d, 29, 65ce }}		Optimal ET sequence: {{Optimal ET sequence\| 7, 22d, 29, 65ce }}
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	Minimax tuning:		Minimax tuning:
	* 13- and 15-odd-limit: ~11/10 = {{monzo\| 2/3 -1/3 }}		* 13- and 15-odd-limit: ~11/10 = {{monzo\| 2/3 -1/3 }}
	: ~~eigenmonzo (~~unchanged-interval) basis: 2.3		: unchanged-interval (eigenmonzo) basis: 2.3

	Optimal ET sequence: {{Optimal ET sequence\| 7, 22d, 29, 65cef }}		Optimal ET sequence: {{Optimal ET sequence\| 7, 22d, 29, 65cef }}
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	[[Minimax tuning]]:		[[Minimax tuning]]:
	* [[7-odd-limit\|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo\| 3/5 0 -1/5 }}		* [[7-odd-limit\|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo\| 3/5 0 -1/5 }}
	: [[Eigenmonzo basis\|~~eigenmonzo (~~unchanged-interval) basis]]: 2.5		: [[Eigenmonzo basis\|unchanged-interval (eigenmonzo) basis]]: 2.5

	{{Optimal ET sequence\|legend=1\| 7, 8d, 15d }}		{{Optimal ET sequence\|legend=1\| 7, 8d, 15d }}

VectorGraphics: Operation Loosen Underpants

2025-06-12T20:51:57Z

Operation Loosen Underpants

← Older revision		Revision as of 20:51, 12 June 2025
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	The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans\|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.		The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans\|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.

	Its [[monzo]] is {{monzo\| 1 -5 3 }}~~, and flipping that yields {{multival\| 3 5 1 }} for the [[wedgie]]~~. ~~This tells us the~~ [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo\|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.		Its [[monzo]] is {{monzo\| 1 -5 3 }}. The [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo\|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.

	Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.		Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.
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	{{Mapping\|legend=1\| 1 2 3 2 \| 0 -3 -5 6 }}		{{Mapping\|legend=1\| 1 2 3 2 \| 0 -3 -5 6 }}

	~~{{Multival\|legend=1\| 3 5 -6 1 -18 -28 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 1 2 3 4 \| 0 -3 -5 -9 }}		{{Mapping\|legend=1\| 1 2 3 4 \| 0 -3 -5 -9 }}

	~~{{Multival\|legend=1\| 3 5 9 1 6 7 }}~~

	[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063		[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
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	{{Mapping\|legend=1\| 1 2 3 5 \| 0 -3 -5 -16 }}		{{Mapping\|legend=1\| 1 2 3 5 \| 0 -3 -5 -16 }}

	~~{{Multival\|legend=1\| 3 5 16 1 17 23 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 1 2 3 1 \| 0 -3 -5 13 }}		{{Mapping\|legend=1\| 1 2 3 1 \| 0 -3 -5 13 }}

	~~{{Multival\|legend=1\| 3 5 -13 1 -29 -44 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 1 2 3 3 \| 0 -3 -5 -1 }}		{{Mapping\|legend=1\| 1 2 3 3 \| 0 -3 -5 -1 }}

	~~{{Multival\|legend=1\| 3 5 1 1 -7 -12 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	{{Mapping\|legend=1\| 1 2 3 3 \| 0 -3 -5 -2 }}		{{Mapping\|legend=1\| 1 2 3 3 \| 0 -3 -5 -2 }}

	~~{{Multival\|legend=1\| 3 5 2 1 -5 -9 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	: mapping generators: ~7/5, ~9/7		: mapping generators: ~7/5, ~9/7

	~~{{Multival\|legend=1\| 6 10 10 2 -1 -5 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	: mapping generators: ~2, ~21/20		: mapping generators: ~2, ~21/20

	~~{{Multival\|legend=1\| 6 10 3 2 -12 -21 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	: mapping generators: ~2, ~9/7		: mapping generators: ~2, ~9/7

	~~{{Multival\|legend=1\| 9 15 19 3 5 2 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
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	: mapping generators: ~2, ~36/35		: mapping generators: ~2, ~36/35

	~~{{Multival\|legend=1\| 9 15 4 3 -19 -33 }}~~

	[[Optimal tuning]]s:		[[Optimal tuning]]s:

BudjarnLambeth: Categorised uncategorised page

2024-09-11T03:22:37Z

Categorised uncategorised page

← Older revision		Revision as of 03:22, 11 September 2024
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	Badness: 0.044739		Badness: 0.044739

			[[Category:Xenharmonic Wiki]]

Lériendil: Created page with "'''''This article is a fork of Porcupine family used as a testing ground for potential revisions and cleanups to temperament data pages.''''' The '''porcupine family''' i..."

2024-08-16T07:29:23Z

Created page with "'''''This article is a fork of Porcupine family used as a testing ground for potential revisions and cleanups to temperament data pages.''''' The '''porcupine family''' i..."

New page

'''''This article is a fork of [[Porcupine family]] used as a testing ground for potential revisions and cleanups to temperament data pages.'''''

The '''porcupine family''' is the [[rank]]-2 [[Temperament families and clans|family of temperaments]] whose [[5-limit]] parent [[comma]] is [[250/243]], also called the maximal diesis or porcupine comma.

Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 3 5 1 }} for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, (10/9)3 = 4/3 × 250/243, and (10/9)5 = 8/5 × (250/243)2. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.

Notice 250/243 = ([[55/54]])([[100/99]]), the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine''.

The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. That means
* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].

Temperaments discussed elsewhere include [[7th-octave temperaments #Jamesbond|jamesbond]].

== Porcupine ==
{{Main| Porcupine }}

[[Subgroup]]: 2.3.5

[[Comma list]]: 250/243

{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}

: mapping generators: ~2, ~10/9

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 164.1659
* [[POTE]]: ~2 = 1\1, ~10/9 = 163.950

[[Tuning ranges]]:
* 5-odd-limit [[diamond monotone]]: ~10/9 = [150.000, 171.429] (1\8 to 1\7)
* 5-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]

{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}

[[Badness]]: 0.030778

=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}

Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}

: gencom: [2 10/9; 55/54, 100/99]

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 163.8867
* POTE: ~2 = 1\1, ~11/10 = 164.0777

Optimal ET sequence: {{Optimal ET sequence| 7, 15, 22, 73ce, 95ce }}

Badness: 0.0097

==== Undecimation ====
Subgroup: 2.3.5.11.13

Comma list: 55/54, 100/99, 512/507

Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}

: sval mapping generators: ~2, ~65/44

Optimal tunings:
* CTE: ~2 = 1\1, ~88/65 = 518.0865
* POTE: ~2 = 1\1, ~88/65 = 518.2094

Optimal ET sequence: {{Optimal ET sequence| 7, 23bc, 30, 37, 44 }}

Badness: 0.0305

== Septimal porcupine ==
{{Main| Porcupine }}

Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 64/63, 250/243

{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}

{{Multival|legend=1| 3 5 -6 1 -18 -28 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 163.2032
* [[POTE]]: ~2 = 1\1, ~10/9 = 162.880

[[Minimax tuning]]:
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~10/9 = [160.000, 163.636] (2\15 to 3\22)
* 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
* 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]

{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}

[[Badness]]: 0.041057

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 100/99

Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 163.1055
* POTE: ~2 = 1\1, ~11/10 = 162.747

Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: eigenmonzo (unchanged-interval) basis: 2.9/7

Tuning ranges:
* 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]

Optimal ET sequence: {{Optimal ET sequence| 7, 15, 22, 37, 59 }}

Badness: 0.021562

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 55/54, 64/63, 66/65

Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 163.4425
* POTE: ~2 = 1\1, ~11/10 = 162.708

Minimax tuning:
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: eigenmonzo (unchanged-interval) basis: 2.11

Tuning ranges:
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
* 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]

Optimal ET sequence: {{Optimal ET sequence| 7, 15, 22f, 37f }}

Badness: 0.021276

==== Porcupinefish ====
{{See also| The Biosphere }}

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 91/90, 100/99

Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 162.6361
* POTE: ~2 = 1\1, ~11/10 = 162.277

Minimax tuning:
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
: eigenmonzo (unchanged-interval) basis: 2.13/11

Tuning ranges:
* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]

Optimal ET sequence: {{Optimal ET sequence| 15, 22, 37 }}

Badness: 0.025314

==== Pourcup ====
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 100/99, 196/195

Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 163.3781
* POTE: ~2 = 1\1, ~11/10 = 162.482

Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
: eigenmonzo (unchanged-interval) basis: 2.13/7

Optimal ET sequence: {{Optimal ET sequence| 15f, 22f, 37, 59f }}

Badness: 0.035130

==== Porkpie ====
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 65/63, 100/99

Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 163.6778
* POTE: ~2 = 1\1, ~11/10 = 163.688

Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: {{Optimal ET sequence| 7, 15f, 22 }}

Badness: 0.026043

== Opossum ==
Opossum can be described as 7d & 8d. Tempering out [[28/27]], the perfect fifth of three generator steps is conflated with not [[32/21]] as in porcupine but [[14/9]]. Three such fifths or nine generator steps octave reduced give a flat 7/4. 2\15 is a good generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 28/27, 126/125

{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}

{{Multival|legend=1| 3 5 9 1 6 7 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7

{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}

[[Badness]]: 0.040650

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 28/27, 55/54, 77/75

Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646

Minimax tuning:
* 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7

Optimal ET sequence: {{Optimal ET sequence| 7d, 8d, 15 }}

Badness: 0.022325

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 55/54, 66/65

Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312

Minimax tuning:
* 13- and 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7

Optimal ET sequence: {{Optimal ET sequence| 7d, 8d, 15, 38bceff }}

Badness: 0.019389

== Porky ==
Porky can be described as 7d & 22, suggesting a less sharp perfect fifth. 7\51 is a good generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 250/243

{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}

{{Multival|legend=1| 3 5 16 1 17 23 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 164.3913
* [[POTE]]: ~2 = 1\1, ~10/9 = 164.412

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5

{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}

[[Badness]]: 0.054389

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 100/99, 225/224

Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 164.3207
* POTE: ~2 = 1\1, ~11/10 = 164.552

Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: eigenmonzo (unchanged-interval) basis: 2.7/5

Optimal ET sequence: {{Optimal ET sequence| 7d, 15d, 22, 51 }}

Badness: 0.027268

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 91/90, 100/99

Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 164.4782
* POTE: ~2 = 1\1, ~11/10 = 164.953

Optimal ET sequence: {{Optimal ET sequence| 7d, 22, 29, 51f, 80cdeff }}

Badness: 0.026543

== Coendou ==
Coendou can be described as 7 & 29, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 250/243, 525/512

{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}

{{Multival|legend=1| 3 5 -13 1 -29 -44 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 166.0938
* [[POTE]]: ~2 = 1\1, ~10/9 = 166.041

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3

{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}

[[Badness]]: 0.118344

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 100/99, 525/512

Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 165.9246
* POTE: ~2 = 1\1, ~11/10 = 165.981

Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: {{Optimal ET sequence| 7, 22d, 29, 65ce }}

Badness: 0.049669

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/64, 100/99, 105/104

Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 166.0459
* POTE: ~2 = 1\1, ~11/10 = 165.974

Minimax tuning:
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: {{Optimal ET sequence| 7, 22d, 29, 65cef }}

Badness: 0.030233

== Hystrix ==
Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 160/147

{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}

{{Multival|legend=1| 3 5 1 1 -7 -12 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 165.1845
* [[POTE]]: ~2 = 1\1, ~10/9 = 158.868

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5

{{Optimal ET sequence|legend=1| 7, 8d, 15d }}

[[Badness]]: 0.044944

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 22/21, 36/35, 80/77

Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~11/10 = 164.7684
* POTE: ~2 = 1\1, ~11/10 = 158.750

Optimal ET sequence: {{Optimal ET sequence| 7, 8d, 15d }}

Badness: 0.026790

== Oxygen ==
Oxygen is perhaps not meant to be used as a serious temperament of harmony. Its comma basis suggests potential utility to construct [[Fokker block]]s.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 21/20, 175/162

{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}

{{Multival|legend=1| 3 5 2 1 -5 -9 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~10/9 = 161.3408
* [[POTE]]: ~2 = 1\1, ~10/9 = 169.112

{{Optimal ET sequence|legend=1| 7d }}

[[Badness]]: 0.059866

== Hedgehog ==
{{See also| Sensamagic clan }}
{{See also| Stearnsmic clan }}

Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. It is a strong extension of [[BPS]] (as BPS has no 2 or sqrt(2)). 22edo provides the obvious (i.e the only [[patent val]]) tuning, but if you are looking for an alternative you could try the {{val| 146 232 338 411 }} (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is [[echidna]], which offers much more accuracy. They merge on 22edo.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 245/243

{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}

: mapping generators: ~7/5, ~9/7

{{Multival|legend=1| 6 10 10 2 -1 -5 }}

[[Optimal tuning]]s:
* [[CTE]]: ~7/5 = 1\2, ~9/7 = 435.2580
* [[POTE]]: ~7/5 = 1\2, ~9/7 = 435.648

{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}

[[Badness]]: 0.043983

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 99/98

Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}

Optimal tunings:
* CTE: ~7/5 = 1\2, ~9/7 = 435.5281
* POTE: ~7/5 = 1\2, ~9/7 = 435.386

Optimal ET sequence: {{Optimal ET sequence| 8d, 14c, 22, 58ce }}

Badness: 0.023095

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 65/63, 99/98

Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}

Optimal tunings:
* CTE: ~7/5 = 1\2, ~9/7 = 436.3087
* POTE: ~7/5 = 1\2, ~9/7 = 435.861

Optimal ET sequence: {{Optimal ET sequence| 8d, 14cf, 22 }}

Badness: 0.021516

==== Urchin ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 66/65

Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}

Optimal tunings:
* CTE: ~7/5 = 1\2, ~9/7 = 435.1856
* POTE: ~7/5 = 1\2, ~9/7 = 437.078

Optimal ET sequence: {{Optimal ET sequence| 14c, 22f }}

Badness: 0.025233

=== Hedgepig ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 245/243, 385/384

Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}

Optimal tunings:
* CTE: ~7/5 = 1\2, ~9/7 = 435.3289
* POTE: ~7/5 = 1\2, ~9/7 = 435.425

Optimal ET sequence: {{Optimal ET sequence| 22 }}

Badness: 0.068406

; Music
* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] in [[hedgehog14|hedgehog[14]]] to 22edo.

== Nautilus ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 49/48, 250/243

{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}

: mapping generators: ~2, ~21/20

{{Multival|legend=1| 6 10 3 2 -12 -21 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~21/20 = 81.9143
* [[POTE]]: ~2 = 1\1, ~21/20 = 82.505

{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}

[[Badness]]: 0.057420

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 245/242

Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 81.8017
* POTE: ~2 = 1\1, ~21/20 = 82.504

Optimal ET sequence: {{Optimal ET sequence| 14c, 15, 29, 44d }}

Badness: 0.026023

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 91/90, 100/99

Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 81.9123
* POTE: ~2 = 1\1, ~21/20 = 82.530

Optimal ET sequence: {{Optimal ET sequence| 14cf, 15, 29, 44d }}

Badness: 0.022285

==== Belauensis ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 55/54, 66/65

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~21/20 = 82.0342
* POTE: ~2 = 1\1, ~21/20 = 81.759

Optimal ET sequence: {{Optimal ET sequence| 14c, 15, 29f, 44dff }}

Badness: 0.029816

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]

== Ammonite ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 250/243, 686/675

{{Mapping|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}

: mapping generators: ~2, ~9/7

{{Multival|legend=1| 9 15 19 3 5 2 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/7 = 454.5500
* [[POTE]]: ~2 = 1\1, ~9/7 = 454.448

{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}

[[Badness]]: 0.107686

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 100/99, 686/675

Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~9/7 = 454.5050
* POTE: ~2 = 1\1, ~9/7 = 454.512

Optimal ET sequence: {{Optimal ET sequence| 8d, 21cde, 29, 37, 66 }}

Badness: 0.045694

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 91/90, 100/99, 169/168

Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~13/10 = 454.4798
* POTE: ~2 = 1\1, ~13/10 = 454.529

Optimal ET sequence: {{Optimal ET sequence| 8d, 21cdef, 29, 37, 66 }}

Badness: 0.027168

== Ceratitid ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 250/243, 1728/1715

{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}

: mapping generators: ~2, ~36/35

{{Multival|legend=1| 9 15 4 3 -19 -33 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~36/35 = 54.8040
* [[POTE]]: ~2 = 1\1, ~36/35 = 54.384

{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}

[[Badness]]: 0.115304

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 55/54, 100/99, 352/343

Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~36/35 = 54.7019
* POTE: ~2 = 1\1, ~36/35 = 54.376

Optimal ET sequence: {{Optimal ET sequence| 1ce, 21ce, 22 }}

Badness: 0.051319

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 65/63, 100/99, 352/343

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~36/35 = 54.5751
* POTE: ~2 = 1\1, ~36/35 = 54.665

Optimal ET sequence: {{Optimal ET sequence| 1ce, 21cef, 22 }}

Badness: 0.044739