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 {{monzo| 1 -5 3 }}, 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)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. 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 =
}}
The 5-limit parent comma for the '''porcupine family''' is [[250/243]], 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 fourth, with two more giving the minor sixth. 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 of 7, 8 and 15 notes make for some nice scale possibilities.


valid range: [150.000, 171.429] (8 to 7)
= Porcupine =


nice range: [157.821, 166.015]
[[Comma list]]: 250/243


strict range: [157.821, 166.015]
[[Mapping]]: [&lt;1 2 3|, &lt;0 -3 -5|]


[[POTE_tuning|POTE generator]]: ~27/25 = 163.950
[[POTE generator]]: ~27/25 = 163.950


Map: [&lt;1 2 3|, &lt;0 -3 -5|]
[[Tuning ranges]]:
* valid range: [150.000, 171.429] (1\8 to 1\7)
* nice range: [157.821, 166.015]
* strict range: [157.821, 166.015]


EDOs: {{EDOs| 7, 15, 22, 95c, 117bc, 139bc, 161bc, 183bc }}
{{Val list|legend=1| 7, 15, 22, 95c, 117bc, 139bc, 161bc, 183bc }}


Badness: 0.0308
[[Badness]]: 0.0308


== Seven limit children ==
== Extensions ==
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. That means [[64/63]], the Archytas comma, for [[#Porcupine|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]].
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]].


= Porcupine =
= Septimal porcupine =
{{main| Porcupine }}
{{main| Porcupine }}


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.
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.
 
Commas: 64/63, 250/243


valid range: [160.000, 163.636] (15 to 22)
[[Comma list]]: 64/63, 250/243


nice range: [157.821, 166.015]
[[Mapping]]: [&lt;1 2 3 2|, &lt;0 -3 -5 6|]


strict range: [160.000, 163.636]
{{Multival|legend=1| 3 5 -6 1 -18 -28 }}


[[POTE generator]]: ~10/9 = 162.880
[[POTE generator]]: ~10/9 = 162.880


7- and 9-limit minimax eigenmonzo: 9/7
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 9/7


Map: [&lt;1 2 3 2|, &lt;0 -3 -5 6|]
[[Tuning ranges]]:
* valid range: [160.000, 163.636] (2\15 to 3\22)
* nice range: [157.821, 166.015]
* strict range: [160.000, 163.636]


EDOs: {{EDOs| 7, 15, 22, 59, 81bd, 140bbd }}
{{Val list|legend=1| 7, 15, 22, 59, 81bd, 140bbd }}


Badness: 0.0411
[[Badness]]: 0.0411


== 11-limit ==
== 11-limit ==
Commas: 55/54, 64/63, 100/99
Comma list: 55/54, 64/63, 100/99
 
valid range: [160.000, 163.636] (15 to 22)
 
nice range: [150.637, 182.404]


strict range: [160.000, 163.636]
Mapping: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]


POTE generator: ~10/9 = 162.747
POTE generator: ~11/10 = 162.747


11-limit minimax eigenmonzo: 9/7
Minimax tuning:
* 11-odd-limit eigenmonzo: 9/7


Map: [&lt;1 2 3 2 4|, &lt;0 -3 -5 6 -4|]
Tuning ranges:
* valid range: [160.000, 163.636] (2\15 to 3\22)
* nice range: [150.637, 182.404]
* strict range: [160.000, 163.636]


EDOs: {{EDOs| 7, 15, 22, 37, 59 }}
Vals: {{val list| 7, 15, 22, 37, 59 }}


Badness: 0.0217
Badness: 0.0217


=== 13-limit ===
=== 13-limit ===
Commas: 40/39, 55/54, 64/63, 66/65
Comma list: 40/39, 55/54, 64/63, 66/65
 
valid range: [160.000, 163.636] (15 to 22f)
 
nice range: [138.573, 182.404]


strict range: [160.000, 163.636]
Mapping: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]


POTE generator: ~10/9 = 162.708
POTE generator: ~10/9 = 162.708


13- and 15-limit minimax eigenmonzo: 11/8
Minimax tuning:
* 13- and 15-odd-limit eigenmonzo: 11/8


Map: [&lt;1 2 3 2 4 4|, &lt;0 -3 -5 6 -4 -2|]
Tuning ranges:
* valid range: [160.000, 163.636] (15 to 22f)
* nice range: [138.573, 182.404]
* strict range: [160.000, 163.636]


EDOs: {{EDOs| 7, 15, 22f, 37f }}
Vals: {{val list| 7, 15, 22f, 37f }}


Badness: 0.0213
Badness: 0.0213
Line 84: Line 97:
{{see also| The Biosphere }}
{{see also| The Biosphere }}


Commas: 55/54, 64/63, 91/90, 100/99
Comma list: 55/54, 64/63, 91/90, 100/99
 
valid range: [160.000, 162.162] (15 to 37)
 
nice range: [150.637, 182.404]


strict range: [160.000, 162.162]
Mapping: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]


POTE generator: ~10/9 = 162.277
POTE generator: ~10/9 = 162.277


13- and 15-limit minimax eigenmonzo: 13/11
Minimax tuning:
* 13- and 15-odd-limit eigenmonzo: 13/11


Map: [&lt;1 2 3 2 4 6|, &lt;0 -3 -5 6 -4 -17|]
Tuning ranges:
* valid range: [160.000, 162.162] (15 to 37)
* nice range: [150.637, 182.404]
* strict range: [160.000, 162.162]


EDOs: {{EDOs| 15, 22, 37, 59, 96b }}
Vals: {{val list| 15, 22, 37, 59, 96b }}


Badness: 0.0253
Badness: 0.0253


=== Pourcup ===
=== Pourcup ===
Commas: 55/54, 64/63, 100/99, 196/195
Comma list: 55/54, 64/63, 100/99, 196/195
 
Mapping: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]


POTE generator: ~10/9 = 162.482
POTE generator: ~10/9 = 162.482


13- and 15-limit minimax eigenmonzo: 13/7
Minimax tuning:
 
* 13- and 15-odd-limit eigenmonzo: 13/7
Map: [&lt;1 2 3 2 4 1|, &lt;0 -3 -5 6 -4 20|]


EDOs: {{EDOs| 15f, 22f, 37 }}
Vals: {{val list| 15f, 22f, 37 }}


Badness: 0.0351
Badness: 0.0351


=== Porkpie ===
=== Porkpie ===
Commas: 55/54, 64/63, 65/63, 100/99
Comma list: 55/54, 64/63, 65/63, 100/99
 
Mapping: [&lt;1 2 3 2 4 3|, &lt;0 -3 -5 6 -4 5|]


POTE generator: ~10/9 = 163.688
POTE generator: ~10/9 = 163.688


13- and 15-limit minimax eigenmonzo: 9/7
Minimax tuning:
 
* 13- and 15-odd-limit eigenmonzo: 9/7
Map: [&lt;1 2 3 2 4 3|, &lt;0 -3 -5 6 -4 5|]


EDOs: {{EDOs| 7, 15f, 22 }}
Vals: {{val list| 7, 15f, 22 }}


Badness: 0.0260
Badness: 0.0260
Line 131: Line 146:
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.
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
[[Comma list]]: 36/35, 160/147
 
[[Mapping]]: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]


[[POTE generator]]: ~8/7 = 158.868
[[POTE generator]]: ~8/7 = 158.868


7- and 9-limit minimax eigenmonzo: 5/4
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 5/4


Map: [&lt;1 2 3 3|, &lt;0 -3 -5 -1|]
{{Val list|legend=1| 7, 8d, 15d }}


EDOs: {{EDOs| 7, 8d, 15d }}
[[Badness]]: 0.0449


Badness: 0.0449
== 11-limit ==
Comma list: 22/21, 36/35, 80/77


== 11-limit ==
Mapping: [&lt;1 2 3 3 4|, &lt;0 -3 -5 -1 -4|]
Commas: 22/21, 36/35, 80/77


POTE generator: ~8/7 = 158.750
POTE generator: ~8/7 = 158.750


Map: [&lt;1 2 3 3 4|, &lt;0 -3 -5 -1 -4|]
Vals: {{val list| 7, 8d, 15d }}
 
EDOs: {{EDOs| 7, 8d, 15d }}


Badness: 0.0268
Badness: 0.0268


= Porky =
= Porky =
Commas: 225/224, 250/243
[[Comma list]]: 225/224, 250/243


POTE generator: ~10/9 = 164.412
[[Mapping]]: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]


7- and 9-limit minimax eigenmonzo: 7/5
Wedgie: &lt;&lt;3 5 16 1 17 23||


Map: [&lt;1 2 3 5|, &lt;0 -3 -5 -16|]
[[POTE generator]]: ~10/9 = 164.412


Wedgie: &lt;&lt;3 5 16 1 17 23||
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 7/5


EDOs: {{EDOs| 7d, 15d, 22, 29, 51, 73c }}
{{Val list|legend=1| 7d, 15d, 22, 29, 51, 73c }}


Badness: 0.0544
[[Badness]]: 0.0544


== 11-limit ==
== 11-limit ==
Commas: 55/54, 100/99, 225/224
Comma list: 55/54, 100/99, 225/224
 
Mapping: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]


POTE generator: ~10/9 = 164.552
POTE generator: ~10/9 = 164.552


11-limit minimax eigenmonzo: 7/5
Minimax tuning:
 
* 11-odd-limit eigenmonzo: 7/5
Map: [&lt;1 2 3 5 4|, &lt;0 -3 -5 -16 -4|]


EDOs: {{EDOs| 7d, 15d, 22, 29, 51, 73ce }}
Vals: {{val list| 7d, 15d, 22, 29, 51, 73ce }}


Badness: 0.0273
Badness: 0.0273


== 13-limit ==
== 13-limit ==
Commas: 55/54, 65/64, 91/90, 100/99
Comma list: 55/54, 65/64, 91/90, 100/99
 
Mapping: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]


POTE generator: ~10/9 = 164.953
POTE generator: ~10/9 = 164.953


Map: [&lt;1 2 3 5 4 3|, &lt;0 -3 -5 -16 -4 5|]
Vals: {{val list| 7d, 22, 29, 51f, 80cdeff }}
 
EDOs: {{EDOs| 7d, 22, 29, 51f, 80cdeff }}


Badness: 0.0265
Badness: 0.0265


= Coendou =
= Coendou =
Commas: 250/243, 525/512
[[Comma list]]: 250/243, 525/512


POTE generator: ~10/9 = 166.041
[[Mapping]]: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]


7- and 9-limit minimax eigenmonzo: 3/2
Wedgie: &lt;&lt;3 5 -13 1 -29 -44||


Map: [&lt;1 2 3 1|, &lt;0 -3 -5 13|]
[[POTE generator]]: ~10/9 = 166.041


Wedgie: &lt;&lt;3 5 -13 1 -29 -44||
[[Minimax tuning]]:  
* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 3/2


EDOs: {{EDOs| 7, 29, 65c, 94cd }}
{{Val list|legend=1| 7, 29, 65c, 94cd }}


Badness: 0.1183
[[Badness]]: 0.1183


== 11-limit ==
== 11-limit ==
Commas: 55/54, 100/99, 525/512
Comma list: 55/54, 100/99, 525/512
 
Mapping: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]


POTE generator: ~10/9 = 165.981
POTE generator: ~10/9 = 165.981


11-limit minimax eigenmonzo: 3/2
Minimax tuning:
 
* 11-odd-limit eigenmonzo: 3/2
Map: [&lt;1 2 3 1 4|, &lt;0 -3 -5 13 -4|]


EDOs: {{EDOs| 7, 29, 65ce, 94cde }}
Vals: {{val list| 7, 29, 65ce, 94cde }}


Badness: 0.0497
Badness: 0.0497


== 13-limit ==
== 13-limit ==
Commas: 55/54, 65/64, 100/99, 105/104
Comma list: 55/54, 65/64, 100/99, 105/104
 
Mapping: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]


POTE generator: ~10/9 = 165.974
POTE generator: ~10/9 = 165.974


13- and 15-limit minimax eigenmonzo: 3/2
Minimax tuning:
* 13- and 15-odd-limit eigenmonzo: 3/2


Map: [&lt;1 2 3 1 4 3|, &lt;0 -3 -5 13 -4 5|]
Vals: {{val list| 7, 29, 65cef, 94cdef }}
 
EDOs: {{EDOs| 7, 29, 65cef, 94cdef }}


Badness: 0.0302
Badness: 0.0302
Line 237: Line 258:
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.
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.


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


[[POTE_tuning|POTE generator]]: ~9/7 = 435.648
[[Mapping]]: [&lt;2 1 1 2|, &lt;0 3 5 5|]


Map: [&lt;2 1 1 2|, &lt;0 3 5 5|]
Wedgie: &lt;&lt;6 10 10 2 -1 -5||


Wedgie: &lt;&lt;6 10 10 2 -1 -5||
[[POTE generator]]: ~9/7 = 435.648


EDOs: {{EDOs| 8d, 14c, 22, 146bccdd }}
{{Val list|legend=1| 8d, 14c, 22, 146bccdd }}


Badness: 0.0440
[[Badness]]: 0.0440


== 11-limit ==
== 11-limit ==
Commas: 50/49, 55/54, 99/98
Comma list: 50/49, 55/54, 99/98
 
Mapping: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]


POTE generator: ~9/7 = 435.386
POTE generator: ~9/7 = 435.386


Map: [&lt;2 1 1 2 4|, &lt;0 3 5 5 4|]
Vals: {{val list| 14c, 22, 58ce, 80ce, 102cde }}
 
EDOs: {{EDOs| 14c, 22, 58ce, 80ce, 102cde }}


Badness: 0.0231
Badness: 0.0231


=== 13-limit ===
=== 13-limit ===
Commas: 50/49, 55/54, 65/63, 99/98
Comma list: 50/49, 55/54, 65/63, 99/98
 
Mapping: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]


POTE generator: ~9/7 = 435.861
POTE generator: ~9/7 = 435.861


Map: [&lt;2 1 1 2 4 3|, &lt;0 3 5 5 4 6|]
Vals: {{val list| 14cf, 22 }}
 
EDOs: {{EDOs| 14cf, 22 }}


Badness: 0.0215
Badness: 0.0215


=== Urchin ===
=== Urchin ===
Commas: 40/39, 50/49, 55/54, 66/65
Comma list: 40/39, 50/49, 55/54, 66/65
 
Mapping: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]


POTE generator: ~9/7 = 437.078
POTE generator: ~9/7 = 437.078


Map: [&lt;2 1 1 2 4 6|, &lt;0 3 5 5 4 2|]
Vals: {{val list| 14c, 22f }}
 
EDOs: {{EDOs| 14c, 22f }}


Badness: 0.0252
Badness: 0.0252


== Hedgepig ==
== Hedgepig ==
Commas: 50/49, 245/243, 385/384
Comma list: 50/49, 245/243, 385/384
 
Mapping: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]


POTE generator: ~9/7 = 435.425
POTE generator: ~9/7 = 435.425


Map: [&lt;2 1 1 2 12|, &lt;0 3 5 5 -7|]
Vals: {{val list| 22, 80c, 102cd, 124cd }}
 
EDOs: {{EDOs| 22, 80c, 102cd, 124cd }}


Badness: 0.0684
Badness: 0.0684


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


= Nautilus =
= Nautilus =
Commas: 49/48, 250/243
[[Comma list]]: 49/48, 250/243


POTE generator: ~21/20 = 82.505
[[Mapping]]: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]


Map: [&lt;1 2 3 3|, &lt;0 -6 -10 -3|]
Wedgie: &lt;&lt;6 10 3 2 -12 -21||


Wedgie: &lt;&lt;6 10 3 2 -12 -21||
[[POTE generator]]: ~21/20 = 82.505


EDOs: {{EDOs| 15, 29, 43cd, 44d, 59d, 73cd, 102cd }}
{{Val list|legend=1| 15, 29, 43cd, 44d, 59d, 73cd, 102cd }}


== 11-limit ==
== 11-limit ==
Commas: 49/48, 55/54, 245/242
Comma list: 49/48, 55/54, 245/242
 
Mapping: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]


POTE generator: ~21/20 = 82.504
POTE generator: ~21/20 = 82.504


Map: [&lt;1 2 3 3 4|, &lt;0 -6 -10 -3 -8|]
Vals: {{val list| 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde }}


EDOs: {{EDOs| 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde }}
=== 13-limit ===
Comma list: 49/48, 55/54, 91/90, 100/99


=== 13-limit ===
Mapping: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
Commas: 49/48, 55/54, 91/90, 100/99


POTE generator: ~21/20 = 62.530
POTE generator: ~21/20 = 62.530


Map: [&lt;1 2 3 3 4 5|, &lt;0 -6 -10 -3 -8 -19|]
Vals: {{val list| 15f, 29, 43cde, 44d, 59df, 73cde, 102cde }}
 
EDOs: {{EDOs| 15f, 29, 43cde, 44d, 59df, 73cde, 102cde }}


Badness: 0.0223
Badness: 0.0223


=== Belauensis ===
=== Belauensis ===
Commas: 40/39, 49/48, 55/54, 66/65
Comma list: 40/39, 49/48, 55/54, 66/65
 
Mapping: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]


POTE generator: ~21/20 = ~14/13 = 81.759
POTE generator: ~21/20 = ~14/13 = 81.759


Map: [&lt;1 2 3 3 4 4|, &lt;0 -6 -10 -3 -8 -4|]
Vals: {{val list| 14c, 15, 29f, 44df }}
 
EDOs: {{EDOs| 14c, 15, 29f, 44df }}


Badness: 0.0298
Badness: 0.0298


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


= Ammonite =
= Ammonite =
Commas: 250/243, 686/675
Comma list: 250/243, 686/675


POTE generator: ~9/7 = 454.448
Mapping: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]


Map: [&lt;1 5 8 10|, &lt;0 -9 -15 -19|]
Wedgie: &lt;&lt;9 15 19 3 5 2||


Wedgie: &lt;&lt;9 15 19 3 5 2||
POTE generator: ~9/7 = 454.448


EDOs: {{EDOs| 29, 37, 66 }}
{{Val list|legend=1| 29, 37, 66 }}


Badness: 0.1077
Badness: 0.1077


== 11-limit ==
== 11-limit ==
Commas: 55/54, 100/99, 686/675
Comma list: 55/54, 100/99, 686/675
 
Mapping: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]


POTE generator: ~9/7 = 454.512
POTE generator: ~9/7 = 454.512


Map: [&lt;1 5 8 10 8|, &lt;0 -9 -15 -19 -12|]
Vals: {{val list| 29, 37, 66 }}
 
EDOs: {{EDOs| 29, 37, 66 }}


Badness: 0.0457
Badness: 0.0457


== 13-limit ==
== 13-limit ==
Commas: 55/54, 91/90, 100/99, 169/168
Comma list: 55/54, 91/90, 100/99, 169/168
 
Mapping: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]


POTE generator: ~13/10 = 454.429
POTE generator: ~13/10 = 454.429


Map: [&lt;1 5 8 10 8 9|, &lt;0 -9 -15 -19 -12 -14|]
Vals: {{val list| 29, 37, 66 }}
 
EDOs: {{EDOs| 29, 37, 66 }}


Badness: 0.0272
Badness: 0.0272


= Ceratitid =
= Ceratitid =
Commas: 250/243, 1728/1715
[[Comma list]]: 250/243, 1728/1715


POTE generator: ~36/35 = 54.384
[[Mapping]]: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]


Map: [&lt;1 2 3 3|, &lt;0 -9 -15 -4|]
Wedgie: &lt;&lt;9 15 4 3 -19 -33||


Wedgie: &lt;&lt;9 15 4 3 -19 -33||
[[POTE generator]]: ~36/35 = 54.384


EDOs: {{EDOs| 22 }}
{{Val list|legend=1| 22 }}


Badness: 0.115
[[Badness]]: 0.115


== 11-limit ==
== 11-limit ==
Commas: 55/54, 100/99, 5324/5145
Comma list: 55/54, 100/99, 5324/5145
 
Mapping: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]


POTE generator: ~36/35 = 54.376
POTE generator: ~36/35 = 54.376


Map: [&lt;1 2 3 3 4|, &lt;0 -9 -15 -4 -12|]
Vals: {{val list| 22 }}
 
EDOs: {{EDOs| 22 }}


Badness: 0.0513
Badness: 0.0513


== 13-limit ==
== 13-limit ==
Commas: 55/54, 65/63, 100/99, 352/343
Comma list: 55/54, 65/63, 100/99, 352/343
 
Mapping: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]


POTE generator: ~36/35 = 54.665
POTE generator: ~36/35 = 54.665


Map: [&lt;1 2 3 3 4 4|, &lt;0 -9 -15 -4 -12 -7|]
Vals: {{val list| 22 }}
 
EDOs: {{EDOs| 22 }}


Badness: 0.0447
Badness: 0.0447


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine family| ]] <!-- main article -->
[[Category:Porcupine]]
[[Category:Porcupine]]
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 14:25, 27 April 2021

The 5-limit parent comma for the porcupine family is 250/243, the maximal diesis or porcupine comma. Its monzo is [1 -5 3⟩, and flipping that yields ⟨⟨ 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.

Porcupine

Comma list: 250/243

Mapping: [<1 2 3|, <0 -3 -5|]

POTE generator: ~27/25 = 163.950

Tuning ranges:

  • valid range: [150.000, 171.429] (1\8 to 1\7)
  • nice range: [157.821, 166.015]
  • strict range: [157.821, 166.015]

Template:Val list

Badness: 0.0308

Extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. That means

Septimal porcupine

Main article: Porcupine

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.

Comma list: 64/63, 250/243

Mapping: [<1 2 3 2|, <0 -3 -5 6|]

Wedgie: ⟨⟨ 3 5 -6 1 -18 -28 ]]

POTE generator: ~10/9 = 162.880

Minimax tuning:

Tuning ranges:

  • valid range: [160.000, 163.636] (2\15 to 3\22)
  • nice range: [157.821, 166.015]
  • strict range: [160.000, 163.636]

Template:Val list

Badness: 0.0411

11-limit

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

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

POTE generator: ~11/10 = 162.747

Minimax tuning:

  • 11-odd-limit eigenmonzo: 9/7

Tuning ranges:

  • valid range: [160.000, 163.636] (2\15 to 3\22)
  • nice range: [150.637, 182.404]
  • strict range: [160.000, 163.636]

Vals: Template:Val list

Badness: 0.0217

13-limit

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

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

POTE generator: ~10/9 = 162.708

Minimax tuning:

  • 13- and 15-odd-limit eigenmonzo: 11/8

Tuning ranges:

  • valid range: [160.000, 163.636] (15 to 22f)
  • nice range: [138.573, 182.404]
  • strict range: [160.000, 163.636]

Vals: Template:Val list

Badness: 0.0213

Porcupinefish

See also: The Biosphere

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

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

POTE generator: ~10/9 = 162.277

Minimax tuning:

  • 13- and 15-odd-limit eigenmonzo: 13/11

Tuning ranges:

  • valid range: [160.000, 162.162] (15 to 37)
  • nice range: [150.637, 182.404]
  • strict range: [160.000, 162.162]

Vals: Template:Val list

Badness: 0.0253

Pourcup

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

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

POTE generator: ~10/9 = 162.482

Minimax tuning:

  • 13- and 15-odd-limit eigenmonzo: 13/7

Vals: Template:Val list

Badness: 0.0351

Porkpie

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

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

POTE generator: ~10/9 = 163.688

Minimax tuning:

  • 13- and 15-odd-limit eigenmonzo: 9/7

Vals: Template:Val list

Badness: 0.0260

Hystrix

Hystrix, with wedgie <<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.

Comma list: 36/35, 160/147

Mapping: [<1 2 3 3|, <0 -3 -5 -1|]

POTE generator: ~8/7 = 158.868

Minimax tuning:

Template:Val list

Badness: 0.0449

11-limit

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

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

POTE generator: ~8/7 = 158.750

Vals: Template:Val list

Badness: 0.0268

Porky

Comma list: 225/224, 250/243

Mapping: [<1 2 3 5|, <0 -3 -5 -16|]

Wedgie: <<3 5 16 1 17 23||

POTE generator: ~10/9 = 164.412

Minimax tuning:

Template:Val list

Badness: 0.0544

11-limit

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

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

POTE generator: ~10/9 = 164.552

Minimax tuning:

  • 11-odd-limit eigenmonzo: 7/5

Vals: Template:Val list

Badness: 0.0273

13-limit

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

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

POTE generator: ~10/9 = 164.953

Vals: Template:Val list

Badness: 0.0265

Coendou

Comma list: 250/243, 525/512

Mapping: [<1 2 3 1|, <0 -3 -5 13|]

Wedgie: <<3 5 -13 1 -29 -44||

POTE generator: ~10/9 = 166.041

Minimax tuning:

Template:Val list

Badness: 0.1183

11-limit

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

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

POTE generator: ~10/9 = 165.981

Minimax tuning:

  • 11-odd-limit eigenmonzo: 3/2

Vals: Template:Val list

Badness: 0.0497

13-limit

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

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

POTE generator: ~10/9 = 165.974

Minimax tuning:

  • 13- and 15-odd-limit eigenmonzo: 3/2

Vals: Template:Val list

Badness: 0.0302

Hedgehog

Hedgehog, with wedgie <<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 <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.

Comma list: 50/49, 245/243

Mapping: [<2 1 1 2|, <0 3 5 5|]

Wedgie: <<6 10 10 2 -1 -5||

POTE generator: ~9/7 = 435.648

Template:Val list

Badness: 0.0440

11-limit

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

Mapping: [<2 1 1 2 4|, <0 3 5 5 4|]

POTE generator: ~9/7 = 435.386

Vals: Template:Val list

Badness: 0.0231

13-limit

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

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

POTE generator: ~9/7 = 435.861

Vals: Template:Val list

Badness: 0.0215

Urchin

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

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

POTE generator: ~9/7 = 437.078

Vals: Template:Val list

Badness: 0.0252

Hedgepig

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

Mapping: [<2 1 1 2 12|, <0 3 5 5 -7|]

POTE generator: ~9/7 = 435.425

Vals: Template:Val list

Badness: 0.0684

Music

Phobos Light by Chris Vaisvil in Hedgehog[14] tuned to 22edo.

Nautilus

Comma list: 49/48, 250/243

Mapping: [<1 2 3 3|, <0 -6 -10 -3|]

Wedgie: <<6 10 3 2 -12 -21||

POTE generator: ~21/20 = 82.505

Template:Val list

11-limit

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

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

POTE generator: ~21/20 = 82.504

Vals: Template:Val list

13-limit

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

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

POTE generator: ~21/20 = 62.530

Vals: Template:Val list

Badness: 0.0223

Belauensis

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

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

POTE generator: ~21/20 = ~14/13 = 81.759

Vals: Template:Val list

Badness: 0.0298

Music

Nautilus Reverie by Igliashon Calvin Jones-Coolidge

Ammonite

Comma list: 250/243, 686/675

Mapping: [<1 5 8 10|, <0 -9 -15 -19|]

Wedgie: <<9 15 19 3 5 2||

POTE generator: ~9/7 = 454.448

Template:Val list

Badness: 0.1077

11-limit

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

Mapping: [<1 5 8 10 8|, <0 -9 -15 -19 -12|]

POTE generator: ~9/7 = 454.512

Vals: Template:Val list

Badness: 0.0457

13-limit

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

Mapping: [<1 5 8 10 8 9|, <0 -9 -15 -19 -12 -14|]

POTE generator: ~13/10 = 454.429

Vals: Template:Val list

Badness: 0.0272

Ceratitid

Comma list: 250/243, 1728/1715

Mapping: [<1 2 3 3|, <0 -9 -15 -4|]

Wedgie: <<9 15 4 3 -19 -33||

POTE generator: ~36/35 = 54.384

Template:Val list

Badness: 0.115

11-limit

Comma list: 55/54, 100/99, 5324/5145

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

POTE generator: ~36/35 = 54.376

Vals: Template:Val list

Badness: 0.0513

13-limit

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

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

POTE generator: ~36/35 = 54.665

Vals: Template:Val list

Badness: 0.0447

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