Porcupine family: Difference between revisions

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<span style="display: block; text-align: right;">[[de:Porcupine]]</span>
{{interwiki
-----
| de = Porcupine
The 5-limit parent comma for the porcupine family is 250/243, the maximal [[diesis]] or porcupine comma. Its [[monzo]] is |1 -5 3&gt;, and flipping that yields &lt;&lt;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 fourth, with two more giving the minor sixth. In fact, (10/9)^3 = 4/3 * 250/243, and (10/9)^5 = 8/5 * (250/243)^2. 3\22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.
| en = Porcupine family
| es =
| 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.  


valid range: [150.000, 171.429] (8 to 7)
== Porcupine ==
{{Main| Porcupine }}


nice range: [157.821, 166.015]
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.


strict range: [157.821, 166.015]
[[Subgroup]]: 2.3.5


[[POTE_tuning|POTE generator]]: ~27/25 = 163.950
[[Comma list]]: 250/243


Map: [&lt;1 2 3|, &lt;0 -3 -5|]
{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}


EDOs: [[7edo|7]], [[15edo|15]], [[22edo|22]], [[95edo|95c]], [[117edo|117bc]], [[139edo|139bc]], [[161edo|161bc]], [[183edo|183bc]]
: mapping generators: ~2, ~10/9


Badness: 0.0308
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5444{{c}}, ~10/9 = 163.8881{{c}}
: [[error map]]: {{val| -0.456 +5.469 -7.121 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
: error map: {{val| 0.000 +5.859 -6.624 }}


==Seven limit children==
[[Tuning ranges]]:
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit|7-limit]] family member we are looking at. That means [[64/63|64/63]], the [[64/63|Archytas comma]], for [[Porcupine_family#Porcupine|porcupine]], [[36/35|36/35]], the [[septimal_quarter_tone|septimal quarter tone]], for [[Porcupine_family#Hystrix|hystrix]], [[50/49|50/49]], the [[jubilisma|jubilisma]], for [[Porcupine_family#Hedgehog|hedgehog]], and [[49/48|49/48]], the [[slendro_diesis|slendro diesis]], for [[Porcupine_family#Nautilus|nautilus]].
* [[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]


=Porcupine=
{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
[[Porcupine]], with wedgie &lt;&lt;3 5 -6 1 -18 -28||, 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.


Commas: 64/63, 250/243
[[Badness]] (Sintel): 0.722


valid range: [160.000, 163.636] (15 to 22)
=== Overview to extensions ===
==== 7-limit extensions ====
The second comma defines which [[7-limit]] family member we are looking at.  
* [[#Hystrix|Hystrix]] adds [[36/35]], the mint comma, for an exotemperament tuning around 8d-edo;
* [[#Opossum|Opossum]] adds [[28/27]], the trienstonic comma, for a tuning between 8d-edo and 15edo;
* [[#Septimal porcupine|Septimal porcupine]] adds [[64/63]], the archytas comma, for a tuning between 15edo and 22edo;
* [[#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.  


nice range: [157.821, 166.015]
Those all share the same generator with porcupine.  


strict range: [160.000, 163.636]
[[#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.  


[[POTE_tuning|POTE generator]]: ~10/9 = 162.880
Temperaments discussed elsewhere include:
* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].


7- and 9-limit minimax eigenmonzo: 9/7
==== Subgroup extensions ====
Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|S11]] }}, the temperament thus extends naturally to the 2.3.5.11 [[subgroup]], sometimes known as ''porkypine'', given right below.


Map: [&lt;1 2 3 2|, &lt;0 -3 -5 6|]
=== 2.3.5.11 subgroup (porkypine) ===
Subgroup: 2.3.5.11


EDOs: 7, 15, 22, [[59edo|59]], [[81edo|81bd]], [[140edo|140bbd]]
Comma list: 55/54, 100/99


Badness: 0.0411
Subgroup-val mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}


==11-limit==
Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
Commas: 55/54, 64/63, 100/99


valid range: [160.000, 163.636] (15 to 22)
Optimal tunings:  
* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}


nice range: [150.637, 182.404]
{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}


strict range: [160.000, 163.636]
Badness (Sintel): 0.303


POTE generator: ~10/9 = 162.747
==== Undecimation ====
Subgroup: 2.3.5.11.13


11-limit minimax eigenmonzo: 9/7
Comma list: 55/54, 100/99, 512/507


Map: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]
Subgroup-val mapping: {{mapping| 1 -1 -2 0 5 | 0 6 10 8 -3 }}


EDOs: [[7edo|7]], 15, 22, [[37edo|37]], [[59edo|59]]
: mapping generators: ~2, ~88/65


Badness: 0.0217
Optimal tunings:  
* WE: ~2 = 1199.4791{{c}}, ~88/65 = 517.9845{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~88/65 = 518.1740{{c}}


==13-limit==
{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
Commas: 40/39, 55/54, 64/63, 66/65


valid range: [160.000, 163.636] (15 to 22f)
Badness (Sintel): 1.21


nice range: [138.573, 182.404]
== Septimal porcupine ==
{{Main| Porcupine }}


strict range: [160.000, 163.636]
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.


POTE generator: ~10/9 = 162.708
[[Subgroup]]: 2.3.5.7


13- and 15-limit minimax eigenmonzo: 11/8
[[Comma list]]: 64/63, 250/243


Map: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]
{{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}


EDOs: 7, 15, 22f, 37f
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1197.8178{{c}}, ~10/9 = 162.5839{{c}}
: [[error map]]: {{val| -2.182 +5.929 -5.780 +2.313 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}


Badness: 0.0213
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


==Porcupinefish==
[[Tuning ranges]]:
See also: [[The_Biosphere|The Biosphere]]
* 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]


Commas: 55/54, 64/63, 91/90, 100/99
{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}


valid range: [160.000, 162.162] (15 to 37)
[[Badness]] (Sintel): 1.04


nice range: [150.637, 182.404]
=== 11-limit ===
Subgroup: 2.3.5.7.11


strict range: [160.000, 162.162]
Comma list: 55/54, 64/63, 100/99


POTE generator: ~10/9 = 162.277
Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}


13- and 15-limit minimax eigenmonzo: 13/11
Optimal tunings:  
* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
Minimax tuning:  
* 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


EDOs: 15, 22, 37, 59, 96b
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]


Badness: 0.0253
{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}


==Pourcup==
Badness (Sintel): 0.713
Commas: 55/54, 64/63, 100/99, 196/195


POTE generator: ~10/9 = 162.482
==== Porcupinefowl ====
This extension used to be ''tridecimal porcupine''.  


13- and 15-limit minimax eigenmonzo: 13/7
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]
Comma list: 40/39, 55/54, 64/63, 66/65


EDOs: 15f, 22f, 37
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}


Badness: 0.0351
Optimal tunings:  
* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}


==Porkpie==
Minimax tuning:
Commas: 55/54, 64/63, 65/63, 100/99
* 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
: unchanged-interval (eigenmonzo) basis: 2.11


POTE generator: ~10/9 = 163.688
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]


13- and 15-limit minimax eigenmonzo: 9/7
{{Optimal ET sequence|legend=0| 7, 15, 22f }}


Map: [&lt;1 2 3 2 4 3|, &lt;0 -3 -5 6 -4 5|]
Badness (Sintel): 0.879


EDOs: 7, 15f, 22
==== Porcupinefish ====
{{See also| The Biosphere }}


Badness: 0.0260
Subgroup: 2.3.5.7.11.13


=Hystrix=
Comma list: 55/54, 64/63, 91/90, 100/99
Hystrix, with wedgie &lt;&lt;3 5 1 1 -7 -12||, provides a less complex avenue to the 7-limit. 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.


Commas: 36/35, 160/147
Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}


[[POTE_tuning|POTE generator]]: ~8/7 = 158.868
Optimal tunings:  
* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}


7- and 9-limit minimax eigenmonzo: 5/4
Minimax tuning:
* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
Tuning ranges:  
* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]


EDOs: 7, 8d, 15d
{{Optimal ET sequence|legend=0| 15, 22, 37 }}


Badness: 0.0449
Badness (Sintel): 1.05


==11-limit==
==== Pourcup ====
Commas: 22/21, 36/35, 80/77
Subgroup: 2.3.5.7.11.13


POTE generator: ~8/7 = 158.750
Comma list: 55/54, 64/63, 100/99, 196/195


Map: [&lt;1 2 3 3 4|, &lt;0 -3 -5 -1 -4|]
Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}


EDOs: 7, 8d, 15d
Optimal tunings:  
* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}


Badness: 0.0268
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
: unchanged-interval (eigenmonzo) basis: 2.13/7


=Porky=
{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
Commas: 225/224, 250/243


POTE generator: ~10/9 = 164.412
Badness (Sintel): 1.45


7- and 9-limit minimax eigenmonzo: 7/5
==== Porkpie ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
Comma list: 55/54, 64/63, 65/63, 100/99


Wedgie: &lt;&lt;3 5 16 1 17 23||
Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}


EDOs: 7d, 15d, 22, 29, 51, 73c
Optimal tunings:  
* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}


Badness: 0.0544
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
: unchanged-interval (eigenmonzo) basis: 2.9/7


==11-limit==
{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
Commas: 55/54, 100/99, 225/224


POTE generator: ~10/9 = 164.552
Badness (Sintel): 1.08


11-limit minimax eigenmonzo: 7/5
== Opossum ==
{{Main| Opossum }}


Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]
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.


EDOs: 7d, 15d, 22, 29, 51, 73ce
[[Subgroup]]: 2.3.5.7


Badness: 0.0273
[[Comma list]]: 28/27, 126/125


==13-limit==
{{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
Commas: 55/54, 65/64, 91/90, 100/99


POTE generator: ~10/9 = 164.953
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.7927{{c}}, ~10/9 = 159.1315{{c}}
: [[error map]]: {{val| -4.207 +12.236 +5.407 -17.838 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7


EDOs: 7d, 22, 29, 51f, 80cdeff
{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}


Badness: 0.0265
[[Badness]] (Sintel): 1.03


=Coendou=
=== 11-limit ===
Commas: 250/243, 525/512
Subgroup: 2.3.5.7.11


POTE generator: ~10/9 = 166.041
Comma list: 28/27, 55/54, 77/75


7- and 9-limit minimax eigenmonzo: 3/2
Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
Optimal tunings:  
* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}


Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
Minimax tuning:  
* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


EDOs: 7, 29, 65c, 94cd
{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}


Badness: 0.1183
Badness (Sintel): 0.738


==11-limit==
=== 13-limit ===
Commas: 55/54, 100/99, 525/512
Subgroup: 2.3.5.7.11.13


POTE generator: ~10/9 = 165.981
Comma list: 28/27, 40/39, 55/54, 66/65


11-limit minimax eigenmonzo: 3/2
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}


Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]
Optimal tunings:  
* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}


EDOs: 7, 29, 65ce, 94cde
Minimax tuning:  
* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7


Badness: 0.0497
{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}


==13-limit==
Badness (Sintel): 0.801
Commas: 55/54, 65/64, 100/99, 105/104


POTE generator: ~10/9 = 165.974
== Porky ==
Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.  


13- and 15-limit minimax eigenmonzo: 3/2
[[Subgroup]]: 2.3.5.7


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
[[Comma list]]: 225/224, 250/243


EDOs: 7, 29, 65cef, 94cdef
{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}


Badness: 0.0302
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0685{{c}}, ~10/9 = 164.4215{{c}}
: [[error map]]: {{val| +0.068 +4.917 -8.216 +0.772 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}


=Hedgehog=
[[Minimax tuning]]:
Hedgehog, with wedgie &lt;&lt;6 10 10 2 -1 -5||, 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. 22edo provides the obvious tuning, but if you are looking for an alternative, you could try the &lt;146 232 338 411| 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.
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5


Commas: 50/49, 245/243
{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51, 73c }}


[[POTE_tuning|POTE generator]]: ~9/7 = 435.648
[[Badness]] (Sintel): 1.38


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Wedgie: &lt;&lt;6 10 10 2 -1 -5||
Comma list: 55/54, 100/99, 225/224


EDOs: 8d, 14c, 22, [[146edo|146bccdd]]
Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}


Badness: 0.0440
Optimal tunings:  
* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}


==11-limit==
Minimax tuning:
Commas: 50/49, 55/54, 99/98
* 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
: unchanged-interval (eigenmonzo) basis: 2.7/5


POTE generator: ~9/7 = 435.386
{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Badness (Sintel): 0.901


EDOs: 14c, 22, 58ce, 80ce, 102cde
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0231
Comma list: 55/54, 65/64, 91/90, 100/99


==13-limit==
Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
Commas: 50/49, 55/54, 65/63, 99/98


POTE generator: ~9/7 = 435.861
Optimal tunings:  
* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}


EDOs: 14cf, 22
Badness (Sintel): 1.10


Badness: 0.0215
; Music
* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning


==Urchin==
== Coendou ==
Commas: 40/39, 50/49, 55/54, 66/65
Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.


POTE generator: ~9/7 = 437.078
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
[[Comma list]]: 250/243, 525/512


EDOs: 14c, 22f
{{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}


Badness: 0.0252
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1202.6772{{c}}, ~10/9 = 166.4110{{c}}
: [[error map]]: {{val| +2.678 +4.166 -10.337 -2.806 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}


==Hedgepig==
[[Minimax tuning]]:
Commas: 50/49, 245/243, 385/384
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3


POTE generator: ~9/7 = 435.425
{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c }}


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
[[Badness]] (Sintel): 2.99


EDOs: 22, 80c, 102cd, 124cd
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0684
Comma list: 55/54, 100/99, 525/512


===Music===
Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by Chris Vaisvil in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.


=Nautilus=
Optimal tunings:
Commas: 49/48, 250/243
* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}


POTE generator: ~21/20 = 82.505
Minimax tuning:
* 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}


Wedgie: &lt;&lt;6 10 3 2 -12 -21||
Badness (Sintel): 1.64


EDOs: 15, [[29edo|29]], 43cd, 44d, 59d, 73cd, [[102edo|102cd]]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 55/54, 65/64, 100/99, 105/104
Commas: 49/48, 55/54, 245/242


POTE generator: ~21/20 = 82.504
Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
Optimal tunings:  
* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}


EDOs: 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde
Minimax tuning:  
* 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
: unchanged-interval (eigenmonzo) basis: 2.3


==13-limit==
{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
Badness (Sintel): 1.25


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
== 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 is very high in [[error]] due to the large disparity between typical porcupine generators and a justly-tuned 8/7, and is usually considered an [[exotemperament]]. A generator of 2\15 or 9\68 can be used for hystrix.


EDOs: 15f, 29, 43cde, 44d, 59df, 73cde, 102cde
[[Subgroup]]: 2.3.5.7


Badness: 0.0223
[[Comma list]]: 36/35, 160/147


==Belauensis==
{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
Commas: 40/39, 49/48, 55/54, 66/65


POTE generator: ~21/20 = ~14/13 = 81.759
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1187.8599{{c}}, ~10/9 = 157.2605{{c}}
: [[error map]]: {{val| -12.140 +1.983 -9.037 +37.493 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}


Map: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5


EDOs: 14c, 15, 29f, 44df
{{Optimal ET sequence|legend=1| 7, 8d, 15d }}


Badness: 0.0298
[[Badness]] (Sintel): 1.14


[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[IgliashonJones|Igliashon Calvin Jones-Coolidge]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Ammonite=
Comma list: 22/21, 36/35, 80/77
Commas: 250/243, 686/675


POTE generator: ~9/7 = 454.448
Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
Optimal tunings:  
* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}


Wedgie: &lt;&lt;9 15 19 3 5 2||
{{Optimal ET sequence|legend=0| 7, 8d, 15d }}


EDOs: 29, 37, 66
Badness (Sintel): 0.886


Badness: 0.1077
== Hedgehog ==
{{See also| Sensamagic clan | Stearnsmic clan }}


==11-limit==
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 alpha-tricot.
Commas: 55/54, 100/99, 686/675


POTE generator: ~9/7 = 454.512
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.  


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
[[Subgroup]]: 2.3.5.7


EDOs: 29, 37, 66
[[Comma list]]: 50/49, 245/243


Badness: 0.0457
{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}


==13-limit==
: mapping generators: ~7/5, ~9/7
Commas: 55/54, 91/90, 100/99, 169/168


POTE generator: ~13/10 = 454.429
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6061{{c}}, ~9/7 = 435.3620{{c}}
: [[error map]]: {{val| -0.788 +3.737 -9.897 +7.197 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4483{{c}}
: error map: {{val| 0.000 +4.390 -9.072 +8.416 }}


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}


EDOs: 29, 37, 66
[[Badness]] (Sintel): 1.11


Badness: 0.0272
=== 11-limit ===
Subgroup: 2.3.5.7.11


=Ceratitid=
Comma list: 50/49, 55/54, 99/98
Commas: 250/243, 1728/1715


POTE generator: ~36/35 = 54.384
Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
Optimal tunings:  
* WE: ~7/5 = 600.1133{{c}}, ~9/7 = 435.4680{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4431{{c}}


Wedgie: &lt;&lt;9 15 4 3 -19 -33||
{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}


EDOs: 22
Badness (Sintel): 0.764


Badness: 0.115
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


==11-limit==
Comma list: 50/49, 55/54, 65/63, 99/98
Commas: 55/54, 100/99, 5324/5145


POTE generator: ~36/35 = 54.376
Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
Optimal tunings:  
* WE: ~7/5 = 600.3651{{c}}, ~9/7 = 436.1258{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.0483{{c}}


EDOs: 22
{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}


Badness: 0.0513
Badness (Sintel): 0.889


==13-limit==
==== Urchin ====
Commas: 55/54, 65/63, 100/99, 352/343
Subgroup: 2.3.5.7.11.13


POTE generator: ~36/35 = 54.665
Comma list: 40/39, 50/49, 55/54, 66/65


Map: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]
Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}


EDOs: 22
Optimal tunings:  
* WE: ~7/5 = 598.3303{{c}}, ~9/7 = 435.8617{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 436.3485{{c}}


Badness: 0.0447
{{Optimal ET sequence|legend=0| 14c, 22f }}
[[Category:family]]
 
[[Category:porcupine]]
Badness (Sintel): 1.04
[[Category:theory]]
 
=== 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:
* WE: ~7/5 = 599.7917{{c}}, ~9/7 = 435.2737{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~9/7 = 435.4047{{c}}
 
{{Optimal ET sequence|legend=0| 22 }}
 
Badness (Sintel): 2.26
 
; Music
* [https://web.archive.org/web/20240624173512/http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|Hedgehog[14]]], 22edo tuning.
 
== 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
 
[[Comma list]]: 49/48, 250/243
 
{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}
 
: mapping generators: ~2, ~21/20
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.1642{{c}}, ~21/20 = 82.6542{{c}}
: [[error map]]: {{val| +2.164 +6.448 -6.364 -10.296 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}
 
{{Optimal ET sequence|legend=1| 14c, 15, 29 }}
 
[[Badness]] (Sintel): 1.45
 
=== 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:
* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 15, 29 }}
 
Badness (Sintel): 0.860
 
==== 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:
* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}
 
Badness (Sintel): 0.921
 
==== 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:
* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}
 
{{Optimal ET sequence|legend=0| 14c, 15 }}
 
Badness (Sintel): 1.23
 
; Music
* [https://web.archive.org/web/20201127013840/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones]]
 
== Ammonite ==
{{See also|Subgroup temperaments #Ammon}}
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
 
[[Comma list]]: 250/243, 686/675
 
{{Mapping|legend=1| 1 -4 -7 -9 | 0 9 15 19 }}
 
: mapping generators: ~2, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3342{{c}}, ~14/9 = 745.1379{{c}}
: [[error map]]: {{val| -0.666 +6.949 -4.584 -5.213 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}
 
{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
 
[[Badness]] (Sintel): 2.73
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 100/99, 686/675
 
Mapping: {{mapping| 1 -4 -7 -9 -4 | 0 9 15 19 12 }}
 
Optimal tunings:
* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
 
Badness (Sintel): 1.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 91/90, 100/99, 169/168
 
Mapping: {{mapping| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
 
Badness (Sintel): 1.12
 
== 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
 
[[Comma list]]: 250/243, 1728/1715
 
{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}
 
: mapping generators: ~2, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.6274{{c}}, ~36/35 = 54.2770{{c}}
: [[error map]]: {{val| -2.373 +4.807 -7.586 +6.948 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}
 
{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
 
[[Badness]] (Sintel): 2.92
 
=== 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:
* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
 
Badness (Sintel): 1.70
 
=== 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:
* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}
 
{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
 
Badness (Sintel): 1.85
 
[[Category:Temperament families]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine| ]] <!-- key article -->
[[Category:Rank 2]]

Latest revision as of 04:15, 5 February 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The porcupine family of temperaments tempers out the porcupine comma, 250/243, also called the maximal diesis.

Porcupine

Main article: 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)3 = (4/3)⋅(250/243), and (10/9)5 = (8/5)⋅(250/243)2. Its ploidacot is omega-tricot. 3\22 is a very recommendable generator, and mos scales of 7, 8 and 15 notes make for some nice scale possibilities.

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

mapping generators: ~2, ~10/9

Optimal tunings:

  • WE: ~2 = 1199.5444 ¢, ~10/9 = 163.8881 ¢
error map: ⟨-0.456 +5.469 -7.121]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 164.0621 ¢
error map: ⟨0.000 +5.859 -6.624]

Tuning ranges:

Optimal ET sequence: 7, 15, 22, 95c

Badness (Sintel): 0.722

Overview to extensions

7-limit extensions

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

  • Hystrix adds 36/35, the mint comma, for an exotemperament tuning around 8d-edo;
  • Opossum adds 28/27, the trienstonic comma, for a tuning between 8d-edo and 15edo;
  • Septimal porcupine adds 64/63, the archytas comma, for a tuning between 15edo and 22edo;
  • Porky adds 225/224, the marvel comma, for a tuning between 22edo and 29edo;
  • Coendou adds 525/512, the avicennma, for a tuning sharp of 29edo.

Those all share the same generator with porcupine.

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

Temperaments discussed elsewhere include:

Subgroup extensions

Noting that 250/243 = (55/54)⋅(100/99) = S102⋅S11, the temperament thus extends naturally to the 2.3.5.11 subgroup, sometimes known as porkypine, given right below.

2.3.5.11 subgroup (porkypine)

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Subgroup-val mapping: [⟨1 2 3 4], ⟨0 -3 -5 -4]]

Gencom mapping: [⟨1 2 3 0 4], ⟨0 -3 -5 0 -4]]

Optimal tunings:

  • WE: ~2 = 1200.3290 ¢, ~11/10 = 164.1227 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 163.9951 ¢

Optimal ET sequence: 7, 15, 22, 73ce, 95ce

Badness (Sintel): 0.303

Undecimation

Subgroup: 2.3.5.11.13

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

Subgroup-val mapping: [⟨1 -1 -2 0 5], ⟨0 6 10 8 -3]]

mapping generators: ~2, ~88/65

Optimal tunings:

  • WE: ~2 = 1199.4791 ¢, ~88/65 = 517.9845 ¢
  • CWE: ~2 = 1200.0000 ¢, ~88/65 = 518.1740 ¢

Optimal ET sequence: 7, 23bc, 30, 37, 44

Badness (Sintel): 1.21

Septimal porcupine

Main article: Porcupine

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

Comma list: 64/63, 250/243

Mapping: [⟨1 2 3 2], ⟨0 -3 -5 6]]

Optimal tunings:

  • WE: ~2 = 1197.8178 ¢, ~10/9 = 162.5839 ¢
error map: ⟨-2.182 +5.929 -5.780 +2.313]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 162.9493 ¢
error map: ⟨0.000 +9.197 -1.060 +8.870]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5
unchanged-interval (eigenmonzo) 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: 7, 15, 22, 37, 59, 81bd

Badness (Sintel): 1.04

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 2 4], ⟨0 -3 -5 6 -4]]

Optimal tunings:

  • WE: ~2 = 1198.3250 ¢, ~11/10 = 162.5202 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.8156 ¢

Minimax tuning:

  • 11-odd-limit: ~11/10 = [1/6 -1/6 0 1/12⟩
unchanged-interval (eigenmonzo) 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: 7, 15, 22, 37, 59

Badness (Sintel): 0.713

Porcupinefowl

This extension used to be tridecimal porcupine.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 2 4 4], ⟨0 -3 -5 6 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1197.0054 ¢, ~11/10 = 162.3022 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.8314 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~10/9 = [1 0 0 0 -1/4⟩
unchanged-interval (eigenmonzo) 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: 7, 15, 22f

Badness (Sintel): 0.879

Porcupinefish

See also: The Biosphere

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 2 4 6], ⟨0 -3 -5 6 -4 -17]]

Optimal tunings:

  • WE: ~2 = 1198.3206 ¢, ~11/10 = 162.0502 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.3458 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~10/9 = [2/13 0 0 0 1/13 -1/13⟩
unchanged-interval (eigenmonzo) basis: 2.13/11

Tuning ranges:

  • 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
  • 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
  • 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]

Optimal ET sequence: 15, 22, 37

Badness (Sintel): 1.05

Pourcup

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 2 4 1], ⟨0 -3 -5 6 -4 20]]

Optimal tunings:

  • WE: ~2 = 1198.0537 ¢, ~11/10 = 162.2183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 162.4665 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [1/14 0 0 -1/14 0 1/14⟩
unchanged-interval (eigenmonzo) basis: 2.13/7

Optimal ET sequence: 15f, 22f, 37, 59f

Badness (Sintel): 1.45

Porkpie

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 2 4 3], ⟨0 -3 -5 6 -4 5]]

Optimal tunings:

  • WE: ~2 = 1200.0223 ¢, ~11/10 = 163.6908 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 163.6874 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [1/6 -1/6 0 1/12⟩
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 7, 15f, 22

Badness (Sintel): 1.08

Opossum

Main article: Opossum

Opossum can be described as 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

Comma list: 28/27, 126/125

Mapping: [⟨1 2 3 4], ⟨0 -3 -5 -9]]

Optimal tunings:

  • WE: ~2 = 1195.7927 ¢, ~10/9 = 159.1315 ¢
error map: ⟨-4.207 +12.236 +5.407 -17.838]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 160.4589 ¢
error map: ⟨0.000 +16.668 +11.392 -12.956]

Minimax tuning:

Optimal ET sequence: 7d, 8d, 15

Badness (Sintel): 1.03

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 4 4], ⟨0 -3 -5 -9 -4]]

Optimal tunings:

  • WE: ~2 = 1196.2331 ¢, ~11/10 = 159.3050 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 160.4644 ¢

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15

Badness (Sintel): 0.738

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 4 4 4], ⟨0 -3 -5 -9 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1193.5447 ¢, ~11/10 = 157.9505 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 159.7600 ¢

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15, 38bceff

Badness (Sintel): 0.801

Porky

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

Subgroup: 2.3.5.7

Comma list: 225/224, 250/243

Mapping: [⟨1 2 3 5], ⟨0 -3 -5 -16]]

Optimal tunings:

  • WE: ~2 = 1200.0685 ¢, ~10/9 = 164.4215 ¢
error map: ⟨+0.068 +4.917 -8.216 +0.772]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 164.4060 ¢
error map: ⟨0.000 +4.827 -8.344 +0.678]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 51, 73c

Badness (Sintel): 1.38

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 5 4], ⟨0 -3 -5 -16 -4]]

Optimal tunings:

  • WE: ~2 = 1200.8706 ¢, ~11/10 = 164.6715 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 164.4810 ¢

Minimax tuning:

  • 11-odd-limit: ~11/10 = [2/11 0 1/11 -1/11⟩
unchanged-interval (eigenmonzo) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 51

Badness (Sintel): 0.901

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 5 4 3], ⟨0 -3 -5 -16 -4 5]]

Optimal tunings:

  • WE: ~2 = 1202.1557 ¢, ~11/10 = 165.2494 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 164.8579 ¢

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

Badness (Sintel): 1.10

Music

Coendou

Coendou can be described as 29 & 36c, 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: [⟨1 2 3 1], ⟨0 -3 -5 13]]

Optimal tunings:

  • WE: ~2 = 1202.6772 ¢, ~10/9 = 166.4110 ¢
error map: ⟨+2.678 +4.166 -10.337 -2.806]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 166.0511 ¢
error map: ⟨0.000 -0.108 -16.569 -10.161]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65c

Badness (Sintel): 2.99

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 1 4], ⟨0 -3 -5 13 -4]]

Optimal tunings:

  • WE: ~2 = 1203.0245 ¢, ~11/10 = 166.3991 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9714 ¢

Minimax tuning:

  • 11-odd-limit: ~11/10 = [2/3 -1/3⟩
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65ce

Badness (Sintel): 1.64

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 1 4 3], ⟨0 -3 -5 13 -4 5]]

Optimal tunings:

  • WE: ~2 = 1202.9957 ¢, ~11/10 = 166.3885 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9843 ¢

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [2/3 -1/3⟩
unchanged-interval (eigenmonzo) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65cef

Badness (Sintel): 1.25

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 is very high in error due to the large disparity between typical porcupine generators and a justly-tuned 8/7, and is usually considered an exotemperament. A generator of 2\15 or 9\68 can be used for hystrix.

Subgroup: 2.3.5.7

Comma list: 36/35, 160/147

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -1]]

Optimal tunings:

  • WE: ~2 = 1187.8599 ¢, ~10/9 = 157.2605 ¢
error map: ⟨-12.140 +1.983 -9.037 +37.493]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 161.2833 ¢
error map: ⟨0.000 +14.195 +7.270 +69.891]

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.5

Optimal ET sequence: 7, 8d, 15d

Badness (Sintel): 1.14

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -3 -5 -1 -4]]

Optimal tunings:

  • WE: ~2 = 1189.2810 ¢, ~11/10 = 157.3322 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 160.9603 ¢

Optimal ET sequence: 7, 8d, 15d

Badness (Sintel): 0.886

Hedgehog

See also: Sensamagic clan and 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)). Its ploidacot is diploid alpha-tricot.

22edo provides an obvious tuning, which happens to be the only patent-val tuning, but if you are looking for an alternative you could try the ⟨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: [⟨2 1 1 2], ⟨0 3 5 5]]

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

Optimal tunings:

  • WE: ~7/5 = 599.6061 ¢, ~9/7 = 435.3620 ¢
error map: ⟨-0.788 +3.737 -9.897 +7.197]
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4483 ¢
error map: ⟨0.000 +4.390 -9.072 +8.416]

Optimal ET sequence: 8d, 14c, 22

Badness (Sintel): 1.11

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 1 2 4], ⟨0 3 5 5 4]]

Optimal tunings:

  • WE: ~7/5 = 600.1133 ¢, ~9/7 = 435.4680 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4431 ¢

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

Badness (Sintel): 0.764

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 1 1 2 4 3], ⟨0 3 5 5 4 6]]

Optimal tunings:

  • WE: ~7/5 = 600.3651 ¢, ~9/7 = 436.1258 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.0483 ¢

Optimal ET sequence: 8d, 14cf, 22

Badness (Sintel): 0.889

Urchin

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 1 1 2 4 6], ⟨0 3 5 5 4 2]]

Optimal tunings:

  • WE: ~7/5 = 598.3303 ¢, ~9/7 = 435.8617 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 436.3485 ¢

Optimal ET sequence: 14c, 22f

Badness (Sintel): 1.04

Hedgepig

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 1 2 12], ⟨0 3 5 5 -7]]

Optimal tunings:

  • WE: ~7/5 = 599.7917 ¢, ~9/7 = 435.2737 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~9/7 = 435.4047 ¢

Optimal ET sequence: 22

Badness (Sintel): 2.26

Music

Nautilus

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

Subgroup: 2.3.5.7

Comma list: 49/48, 250/243

Mapping: [⟨1 2 3 3], ⟨0 -6 -10 -3]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • WE: ~2 = 1202.1642 ¢, ~21/20 = 82.6542 ¢
error map: ⟨+2.164 +6.448 -6.364 -10.296]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 82.2758 ¢
error map: ⟨0.000 +4.390 -9.072 -15.653]

Optimal ET sequence: 14c, 15, 29

Badness (Sintel): 1.45

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -6 -10 -3 -8]]

Optimal tunings:

  • WE: ~2 = 1202.3781 ¢, ~21/20 = 82.6673 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 82.2434 ¢

Optimal ET sequence: 14c, 15, 29

Badness (Sintel): 0.860

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 5], ⟨0 -6 -10 -3 -8 -19]]

Optimal tunings:

  • WE: ~2 = 1202.4145 ¢, ~21/20 = 82.6963 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 82.3130 ¢

Optimal ET sequence: 14cf, 15, 29

Badness (Sintel): 0.921

Belauensis

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 4], ⟨0 -6 -10 -3 -8 -4]]

Optimal tunings:

  • WE: ~2 = 1199.0072 ¢, ~21/20 = 81.6911 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 81.8576 ¢

Optimal ET sequence: 14c, 15

Badness (Sintel): 1.23

Music

Ammonite

See also: Subgroup temperaments § Ammon

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

Subgroup: 2.3.5.7

Comma list: 250/243, 686/675

Mapping: [⟨1 -4 -7 -9], ⟨0 9 15 19]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.3342 ¢, ~14/9 = 745.1379 ¢
error map: ⟨-0.666 +6.949 -4.584 -5.213]
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 745.4994 ¢
error map: ⟨0.000 +7.540 -3.823 -4.337]

Optimal ET sequence: 8d, 21cd, 29, 37, 66

Badness (Sintel): 2.73

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -4 -7 -9 -4], ⟨0 9 15 19 12]]

Optimal tunings:

  • WE: ~2 = 1200.0141 ¢, ~14/9 = 745.4971 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 745.4894 ¢

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

Badness (Sintel): 1.51

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -4 -7 -9 -4 -5], ⟨0 9 15 19 12 14]]

Optimal tunings:

  • WE: ~2 = 1200.2478 ¢, ~14/9 = 745.6252 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 745.4904 ¢

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

Badness (Sintel): 1.12

Ceratitid

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

Subgroup: 2.3.5.7

Comma list: 250/243, 1728/1715

Mapping: [⟨1 2 3 3], ⟨0 -9 -15 -4]]

mapping generators: ~2, ~36/35

Optimal tunings:

  • WE: ~2 = 1197.6274 ¢, ~36/35 = 54.2770 ¢
error map: ⟨-2.373 +4.807 -7.586 +6.948]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 54.5489 ¢
error map: ⟨0.000 +7.105 -4.548 +12.978]

Optimal ET sequence: 1c, 21c, 22

Badness (Sintel): 2.92

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -9 -15 -4 -12]]

Optimal tunings:

  • WE: ~2 = 1198.2851 ¢, ~36/35 = 54.2986 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 54.4992 ¢

Optimal ET sequence: 1ce, 21ce, 22

Badness (Sintel): 1.70

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 4], ⟨0 -9 -15 -4 -12 -7]]

Optimal tunings:

  • WE: ~2 = 1200.3864 ¢, ~36/35 = 54.6830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 54.6396 ¢

Optimal ET sequence: 1ce, 21cef, 22

Badness (Sintel): 1.85

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