Porcupine family: Difference between revisions

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The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.

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The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and 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.

The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, {{nowrap| (10/9)3 {{=}} (4/3)⋅(250/243) }}, and {{nowrap| (10/9)5 {{=}} (8/5)⋅(250/243)2 }}. Its [[ploidacot]] is omega-tricot. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.

[[Subgroup]]: 2.3.5

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=== Overview to extensions ===

==== 7-limit extensions ====

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~~

The second comma defines which [[7-limit]] family member we are looking at.

* [[64/63]], the ~~archytas~~ comma, for [[#~~Septimal porcupine~~|~~septimal porcupine~~]],

* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;

* [[36/35]], the ~~septimal quarter tone~~, for [[#~~Hystrix~~|~~hystrix~~]],

* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;

* [[50/49]], the ~~jubilisma~~, for [[#~~Hedgehog~~|~~hedgehog~~]]~~, and~~

* [[#Septimal porcupine|Septimal porcupine]] adds [[64/63]], the archytas comma, for a tuning between 15edo and 22edo;

* [[49/48]], the ~~slendro diesis~~, for [[#~~Nautilus~~|~~nautilus~~]].

* [[#Porky|Porky]] adds [[225/224]], the marvel comma, for a tuning between 22edo and 29edo;

* [[#Coendou|Coendou]] adds [[525/512]], the avicennma, for a tuning sharp of 29edo.

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

Those all share the same generator with porcupine.

[[#Nautilus|nautilus]] tempers out [[49/48]] and splits the generator in two. [[#Hedgehog|hedgehog]] tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.

Temperaments discussed elsewhere include:

* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].

* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].

==== Subgroup extensions ====

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

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

Optimal tunings:

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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.

Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. Here, we share the same mapping of 7/4 in terms of fifths as [[archy]]. 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

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

[[Optimal tuning]]s:

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[[Minimax tuning]]:

* [[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 }}

: [[eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/7

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Tuning ranges]]:

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Minimax tuning:

* 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:

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Badness (Smith): 0.021562

==== ~~13-limit~~ ====

==== Porcupinefowl ====

This extension used to be ''tridecimal porcupine''.

Subgroup: 2.3.5.7.11.13

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Minimax tuning:

* 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:

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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

: unchanged-interval (eigenmonzo) basis: 2.13/11

Tuning ranges:

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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

: unchanged-interval (eigenmonzo) basis: 2.13/7

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Minimax tuning:

* 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

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== 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.

Opossum can be described as {{nowrap| 8d & 15 }}. 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

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

[[Optimal tuning]]s:

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[[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

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

~~{{Multival|legend=1| 3 5 9 4 1 6 -4 7 -8 -20 }}~~

Optimal tunings:

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Minimax tuning:

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

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

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Minimax tuning:

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

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

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== Porky ==

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

Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:

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[[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

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5

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

~~Wedgie: {{multival| 3 5 16 4 1 17 -4 23 -8 -44 }}~~

Optimal tunings:

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Minimax tuning:

* 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

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== Coendou ==

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

Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.

[[Subgroup]]: 2.3.5.7

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

[[Optimal tuning]]s:

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[[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

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

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Minimax tuning:

* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}

: ~~eigenmonzo (~~unchanged-interval) basis: 2.3

: unchanged-interval (eigenmonzo) basis: 2.3

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Minimax tuning:

* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}

: ~~eigenmonzo (~~unchanged-interval) basis: 2.3

: unchanged-interval (eigenmonzo) basis: 2.3

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

[[Optimal tuning]]s:

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[[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

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

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Badness (Smith): 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 = 1200.000, ~10/9 = 161.341

~~: [[error map]]: {{val| 0.000 +14.023 +6.982 -91.507 }}~~

* [[POTE]]: ~2 = 1200.000, ~10/9 = 169.112

~~: error map: {{val| 0.000 -9.291 -31.873 -107.050 }}~~

~~{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}~~

~~[[Badness]] (Smith): 0.059866~~

== Hedgehog ==

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.

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)). Its ploidacot is diploid omega-tricot.

22edo provides an obvious tuning, which happens to be the only [[patent val|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

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

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

[[Optimal tuning]]s:

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

~~{{Multival|legend=1| 6 10 10 8 2 -1 -8 -5 -16 -12 }}~~

Optimal tunings:

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

~~{{Multival|legend=1| 6 10 10 -14 2 -1 -43 -5 -67 -74 }}~~

Optimal tunings:

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== Nautilus ==

Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~21/20

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

[[Optimal tuning]]s:

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

~~{{Multival|legend=1| 6 10 3 8 2 -12 -8 -21 -16 12 }}~~

Optimal tunings:

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== Ammonite ==

Ammonite adds 686/675 to the comma list and may be described as the {{nowrap| 8d & 29 }} temperament. Its ploidacot is epsilon-enneacot. [[37edo]] provides an obvious tuning.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~9/7

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

[[Optimal tuning]]s:

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

~~Wedgie: {{multival| 9 15 19 12 3 5 -12 2 -24 -32 }}~~

Optimal tunings:

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== Ceratitid ==

Ceratitid adds 1728/1715 to the comma list and may be described as the {{nowrap| 21c & 22 }} temperament. Its ploidacot is omega-enneacot. [[22edo]] provides an obvious tuning.

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~36/35

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

[[Optimal tuning]]s:

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Porcupine family| ]]

[[Category:Porcupine| ]]

[[Category:Rank 2]]

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 | ja =
 }}
+{{Technical data page}}
 The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.
@@ Line 10: / Line 11: @@
 {{Main| Porcupine }}
-The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and 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.
+The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, {{nowrap| (10/9)<sup>3</sup> {{=}} (4/3)⋅(250/243) }}, and {{nowrap| (10/9)<sup>5</sup> {{=}} (8/5)⋅(250/243)<sup>2</sup> }}. Its [[ploidacot]] is omega-tricot. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.
 [[Subgroup]]: 2.3.5
@@ Line 36: / Line 37: @@
 === Overview to extensions ===
 ==== 7-limit extensions ====
-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
+The second comma defines which [[7-limit]] family member we are looking at.
-* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
+* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;
-* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
+* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;
-* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
+* [[#Septimal porcupine|Septimal porcupine]] adds [[64/63]], the archytas comma, for a tuning between 15edo and 22edo;
-* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].
+* [[#Porky|Porky]] adds [[225/224]], the marvel comma, for a tuning between 22edo and 29edo;
+* [[#Coendou|Coendou]] adds [[525/512]], the avicennma, for a tuning sharp of 29edo.
-Temperaments discussed elsewhere include [[7th-octave temperaments #Jamesbond|jamesbond]].
+Those all share the same generator with porcupine.
+[[#Nautilus|nautilus]] tempers out [[49/48]] and splits the generator in two. [[#Hedgehog|hedgehog]] tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.
+Temperaments discussed elsewhere include:
+* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
+* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
 ==== Subgroup extensions ====
@@ Line 56: / Line 64: @@
 Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
 : gencom: [2 10/9; 55/54, 100/99]
 Optimal tunings:
@@ Line 86: / Line 94: @@
 {{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.
+Septimal porcupine uses six of its minor tone generator steps to get to [[7/4]]. Here, we share the same mapping of 7/4 in terms of fifths as [[archy]]. 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
@@ Line 93: / Line 101: @@
 {{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
-{{Multival|legend=1| 3 5 -6 1 -18 -28 }}
 [[Optimal tuning]]s:
@@ Line 104: / Line 110: @@
 [[Minimax tuning]]:
 * [[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 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 [[Tuning ranges]]:
@@ Line 130: / Line 136: @@
 Minimax tuning:
 * 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:
@@ Line 140: / Line 146: @@
 Badness (Smith): 0.021562
-==== 13-limit ====
+==== Porcupinefowl ====
+This extension used to be ''tridecimal porcupine''.
 Subgroup: 2.3.5.7.11.13
@@ Line 153: / Line 161: @@
 Minimax tuning:
 * 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:
@@ Line 179: / Line 187: @@
 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
+: unchanged-interval (eigenmonzo) basis: 2.13/11
 Tuning ranges:
@@ Line 203: / Line 211: @@
 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
+: unchanged-interval (eigenmonzo) basis: 2.13/7
 {{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
@@ Line 222: / Line 230: @@
 Minimax tuning:
 * 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|legend=0| 7, 15f, 22 }}
@@ Line 229: / Line 237: @@
 == 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.
+Opossum can be described as {{nowrap| 8d & 15 }}. 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
@@ Line 236: / Line 244: @@
 {{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
-{{Multival|legend=1| 3 5 9 1 6 7 }}
 [[Optimal tuning]]s:
@@ Line 246: / Line 252: @@
 [[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 }}
@@ Line 258: / Line 264: @@
 Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
-{{Multival|legend=1| 3 5 9 4 1 6 -4 7 -8 -20 }}
 Optimal tunings:
@@ Line 266: / Line 270: @@
 Minimax tuning:
-* 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7
+* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
 {{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
@@ Line 284: / Line 288: @@
 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|legend=0| 7d, 8d, 15, 38bceff }}
@@ Line 291: / Line 295: @@
 == Porky ==
-Porky can be described as 7d & 22, suggesting a less sharp perfect fifth. 7\51 is a good generator.
+Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.
 [[Subgroup]]: 2.3.5.7
@@ Line 298: / Line 302: @@
 {{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
-{{Multival|legend=1| 3 5 16 1 17 23 }}
 [[Optimal tuning]]s:
@@ Line 309: / Line 311: @@
 [[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
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
 {{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
@@ Line 321: / Line 323: @@
 Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
-Wedgie: {{multival| 3 5 16 4 1 17 -4 23 -8 -44 }}
 Optimal tunings:
@@ Line 330: / Line 330: @@
 Minimax tuning:
 * 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|legend=0| 7d, 15d, 22, 51 }}
@@ Line 355: / Line 355: @@
 == Coendou ==
-Coendou can be described as 7 & 29, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.
+Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.
 [[Subgroup]]: 2.3.5.7
@@ Line 362: / Line 362: @@
 {{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
-{{Multival|legend=1| 3 5 -13 1 -29 -44 }}
 [[Optimal tuning]]s:
@@ Line 373: / Line 371: @@
 [[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
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
 {{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
@@ Line 392: / Line 390: @@
 Minimax tuning:
 * 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|legend=0| 7, 22d, 29, 65ce }}
@@ Line 411: / Line 409: @@
 Minimax tuning:
 * 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|legend=0| 7, 22d, 29, 65cef }}
@@ Line 425: / Line 423: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
-{{Multival|legend=1| 3 5 1 1 -7 -12 }}
 [[Optimal tuning]]s:
@@ Line 436: / Line 432: @@
 [[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
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 {{Optimal ET sequence|legend=1| 7, 8d, 15d }}
@@ Line 456: / Line 452: @@
 Badness (Smith): 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 = 1200.000, ~10/9 = 161.341
-: [[error map]]: {{val| 0.000 +14.023 +6.982 -91.507 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 169.112
-: error map: {{val| 0.000 -9.291 -31.873 -107.050 }}
-{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}
-[[Badness]] (Smith): 0.059866
 == Hedgehog ==
 {{See also| Sensamagic clan | 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.
+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)). Its ploidacot is diploid omega-tricot.
+edo provides an obvious tuning, which happens to be the only [[patent val|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
@@ Line 490: / Line 467: @@
 : mapping generators: ~7/5, ~9/7
-{{Multival|legend=1| 6 10 10 2 -1 -5 }}
 [[Optimal tuning]]s:
@@ Line 509: / Line 484: @@
 Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
-{{Multival|legend=1| 6 10 10 8 2 -1 -8 -5 -16 -12 }}
 Optimal tunings:
@@ Line 556: / Line 529: @@
 Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
-{{Multival|legend=1| 6 10 10 -14 2 -1 -43 -5 -67 -74 }}
 Optimal tunings:
@@ Line 571: / Line 542: @@
 == Nautilus ==
+Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 578: / Line 551: @@
 : mapping generators: ~2, ~21/20
-{{Multival|legend=1| 6 10 3 2 -12 -21 }}
 [[Optimal tuning]]s:
@@ Line 597: / Line 568: @@
 Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
-{{Multival|legend=1| 6 10 3 8 2 -12 -8 -21 -16 12 }}
 Optimal tunings:
@@ Line 642: / Line 611: @@
 == Ammonite ==
+Ammonite adds 686/675 to the comma list and may be described as the {{nowrap| 8d & 29 }} temperament. Its ploidacot is epsilon-enneacot. [[37edo]] provides an obvious tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 649: / Line 620: @@
 : mapping generators: ~2, ~9/7
-{{Multival|legend=1| 9 15 19 3 5 2 }}
 [[Optimal tuning]]s:
@@ Line 668: / Line 637: @@
 Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
-Wedgie: {{multival| 9 15 19 12 3 5 -12 2 -24 -32 }}
 Optimal tunings:
@@ Line 695: / Line 662: @@
 == Ceratitid ==
+Ceratitid adds 1728/1715 to the comma list and may be described as the {{nowrap| 21c & 22 }} temperament. Its ploidacot is omega-enneacot. [[22edo]] provides an obvious tuning.
 [[Subgroup]]: 2.3.5.7
@@ Line 702: / Line 671: @@
 : mapping generators: ~2, ~36/35
-{{Multival|legend=1| 9 15 4 3 -19 -33 }}
 [[Optimal tuning]]s:
@@ Line 746: / Line 713: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Porcupine family| ]] <!-- main article -->
 [[Category:Porcupine| ]] <!-- key article -->
 [[Category:Rank 2]]