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	{{interwiki\|de=Porcupine\|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. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, {{nowrap\|(10/9)<sup>3</sup> {{=}} (4/3)⋅(250/243)}}, and {{nowrap\|(10/9)<sup>5</sup> {{=}} (8/5)⋅(250/243)<sup>2</sup>}}. [[22edo\|3\22]] is a very recommendable generator, and [[Mos scale\|mos scales]] of 7, 8 and 15 notes make for some nice scale possibilities.		{{interwiki\|de=Porcupine\|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. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, {{nowrap\|(10/9)<sup>3</sup> {{=}} (4/3)⋅(250/243)}}, and {{nowrap\|(10/9)<sup>5</sup> {{=}} (8/5)⋅(250/243)<sup>2</sup>}}. [[22edo\|3\22]] is a very recommendable generator, and [[Mos scale\|mos scales]] of 7, 8 and 15 notes make for some nice scale possibilities.

	It most naturally manifests as a [[2.3.5.11 subgroup]] temperament, where it tempers out [[100/99]] and [[55/54]] equating the generator to 11/10 as well as 10/9.		It most naturally manifests as a [[2.3.5.11 subgroup]] temperament, where it tempers out [[100/99]] and [[55/54]] equating the generator to 11/10 as well as 10/9.

VectorGraphics: Created page with "{{interwiki|de=Porcupine|en=Porcupine family|es=|ja=}}{{Technical data page}}The '''porcupine family''' of temperaments tempers out the porcupine comma, 250/243, also called the maximal diesis. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, {{nowrap|(10/9)³ {{=}} (4/3)⋅(250/243)}}, and {{nowrap|(10/9)⁵ {{=}}..."

2025-05-11T07:19:56Z

Created page with "{{interwiki|de=Porcupine|en=Porcupine family|es=|ja=}}{{Technical data page}}The '''porcupine family''' of temperaments tempers out the porcupine comma, 250/243, also called the maximal diesis. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, {{nowrap|(10/9)3 {{=}} (4/3)⋅(250/243)}}, and {{nowrap|(10/9)5 {{=}}..."

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{{interwiki|de=Porcupine|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. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, {{nowrap|(10/9)3 {{=}} (4/3)⋅(250/243)}}, and {{nowrap|(10/9)5 {{=}} (8/5)⋅(250/243)2}}. [[22edo|3\22]] is a very recommendable generator, and [[Mos scale|mos scales]] of 7, 8 and 15 notes make for some nice scale possibilities.

It most naturally manifests as a [[2.3.5.11 subgroup]] temperament, where it tempers out [[100/99]] and [[55/54]] equating the generator to 11/10 as well as 10/9.

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

=== 5-limit ===
[[Subgroup]]: 2.3.5

[[Comma list]]: 250/243

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

: mapping generators: ~2, ~10/9

[[Optimal tuning|Optimal tunings]]:

* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.166

: [[error map]]: {{val|0.000 +5.547 -7.143}}

* [[POTE]]: ~2 = 1200.000, ~10/9 = 163.950

: error map: {{val|0.000 +6.194 -6.065}}

[[Tuning ranges]]:

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

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

[[Badness]] (Smith): 0.030778

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

Comma list: 55/54, 100/99

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 163.887
* POTE: ~2 = 1200.000, ~11/10 = 164.078

{{Optimal ET sequence|7, 15, 22, 73ce, 95ce|legend=0}}

Badness (Smith): 0.0097

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

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~88/65 = 518.086
* POTE: ~2 = 1200.000, ~88/65 = 518.209

{{Optimal ET sequence|7, 23bc, 30, 37, 44|legend=0}}

Badness (Smith): 0.0305

== Strong extensions ==
{{Main|Porcupine}}

=== Septimal 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. This extends porcupine to the full 11-limit:

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 163.105
* POTE: ~2 = 1200.000, ~11/10 = 162.747

Minimax tuning:

* 11-odd-limit: ~11/10 = {{monzo|1/6 -1/6 0 1/12}}

: eigenmonzo (unchanged-interval) basis: 2.9/7

Tuning ranges:

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

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

(11-limit) {{Optimal ET sequence|7, 15, 22, 37, 59|legend=0}}

Badness (Smith): 0.021562

==== Tridecimal porcupine ====
Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 163.442
* POTE: ~2 = 1200.000, ~11/10 = 162.708

Minimax tuning:

* 13- and 15-odd-limit: ~10/9 = {{monzo|1 0 0 0 -1/4}}

: eigenmonzo (unchanged-interval) basis: 2.11

Tuning ranges:

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

{{Optimal ET sequence|7, 15, 22f, 37f|legend=0}}

Badness (Smith): 0.021276

==== Porcupinefish ====
{{See also|The Biosphere}}Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 162.636
* POTE: ~2 = 1200.000, ~11/10 = 162.277

Minimax tuning:

* 13- and 15-odd-limit: ~10/9 = {{monzo|2/13 0 0 0 1/13 -1/13}}

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

Tuning ranges:

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

{{Optimal ET sequence|15, 22, 37|legend=0}}

Badness (Smith): 0.025314

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 163.378
* POTE: ~2 = 1200.000, ~11/10 = 162.482

Minimax tuning:

* 13- and 15-odd-limit: ~11/10 = {{monzo|1/14 0 0 -1/14 0 1/14}}

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

{{Optimal ET sequence|15f, 22f, 37, 59f|legend=0}}

Badness (Smith): 0.035130

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 163.678
* POTE: ~2 = 1200.000, ~11/10 = 163.688

Minimax tuning:

* 13- and 15-odd-limit: ~11/10 = {{monzo|1/6 -1/6 0 1/12}}

: eigenmonzo (unchanged-interval) basis: 2.9/7

{{Optimal ET sequence|7, 15f, 22|legend=0}}

Badness (Smith): 0.026043

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

Subgroup: 2.3.5.7.11

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

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:

* CTE: ~2 = 1200.000, ~11/10 = 164.321
* POTE: ~2 = 1200.000, ~11/10 = 164.552

Minimax tuning:

* 11-odd-limit: ~11/10 = {{monzo|2/11 0 1/11 -1/11}}

: eigenmonzo (unchanged-interval) basis: 2.7/5

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

{{Optimal ET sequence|7d, 15d, 22, 51|legend=0}}

Badness (Smith): 0.027268

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 164.478
* POTE: ~2 = 1200.000, ~11/10 = 164.953

{{Optimal ET sequence|7d, 22, 29, 51f, 80cdeff|legend=0}}

Badness (Smith): 0.026543

; Music

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

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

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 161.365
* POTE: ~2 = 1200.000, ~11/10 = 159.807

Minimax tuning:

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

{{Optimal ET sequence|7d, 8d, 15|legend=0}}

Badness (Smith): 0.022325

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 161.631
* POTE: ~2 = 1200.000, ~11/10 = 158.805

Minimax tuning:

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

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

Badness (Smith): 0.019389

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

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 165.925
* POTE: ~2 = 1200.000, ~11/10 = 165.981

Minimax tuning:

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

: eigenmonzo (unchanged-interval) basis: 2.3

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

{{Optimal ET sequence|7, 22d, 29, 65ce|legend=0}}

Badness (Smith): 0.049669

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 166.046
* POTE: ~2 = 1200.000, ~11/10 = 165.974

Minimax tuning:

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

: eigenmonzo (unchanged-interval) basis: 2.3

{{Optimal ET sequence|7, 22d, 29, 65cef|legend=0}}

Badness (Smith): 0.030233

== Hystrix ==
Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix 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: 2.3.5.7.11

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~11/10 = 164.768
* POTE: ~2 = 1200.000, ~11/10 = 158.750

{{Optimal ET sequence|7, 8d, 15d|legend=0}}

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|Fokker blocks]].

[[Subgroup]]: 2.3.5.7

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

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

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

[[Optimal tuning|Optimal tunings]]:

* [[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|1c, …, 6bcd, 7d|legend=1}}

[[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.

Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* CTE: ~7/5 = 600.000, ~9/7 = 435.528
* POTE: ~7/5 = 600.000, ~9/7 = 435.386

{{Optimal ET sequence|8d, 14c, 22|legend=1}} (7-limit)

{{Optimal ET sequence|8d, 14c, 22, 58ce|legend=0}}

Badness (Smith): 0.023095

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

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

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

Optimal tunings:

* CTE: ~7/5 = 600.000, ~9/7 = 436.309
* POTE: ~7/5 = 600.000, ~9/7 = 435.861

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

Badness (Smith): 0.021516

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

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

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

Optimal tunings:

* CTE: ~7/5 = 600.000, ~9/7 = 435.186
* POTE: ~7/5 = 600.000, ~9/7 = 437.078

{{Optimal ET sequence|14c, 22f|legend=0}}

Badness (Smith): 0.025233

== Hedgepig ==
Subgroup: 2.3.5.7.11

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

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

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

Optimal tunings:

* CTE: ~7/5 = 600.000, ~9/7 = 435.329
* POTE: ~7/5 = 600.000, ~9/7 = 435.425

{{Optimal ET sequence|22|legend=0}}

Badness (Smith): 0.068406

; Music

* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[Hedgehog14|hedgehog[14]]], 22edo tuning.

== Nautilus ==
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}}

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~21/20 = 81.802
* POTE: ~2 = 1200.000, ~21/20 = 82.504

{{Optimal ET sequence|14c, 15, 29, 44d|legend=0}}

Badness (Smith): 0.026023

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~21/20 = 81.912
* POTE: ~2 = 1200.000, ~21/20 = 82.530

{{Optimal ET sequence|14cf, 15, 29, 44d|legend=0}}

Badness (Smith): 0.022285

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~21/20 = 82.034
* POTE: ~2 = 1200.000, ~21/20 = 81.759

{{Optimal ET sequence|14c, 15, 29f, 44dff|legend=0}}

Badness (Smith): 0.029816

; Music

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

== Ammonite ==
Subgroup: 2.3.5.7.11

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

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:

* CTE: ~2 = 1200.000, ~9/7 = 454.505
* POTE: ~2 = 1200.000, ~9/7 = 454.512

{{Optimal ET sequence|8d, 21cd, 29, 37, 66|legend=1}} (7-limit)

{{Optimal ET sequence|8d, 21cde, 29, 37, 66|legend=0}}

Badness (Smith): 0.045694

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~13/10 = 454.480
* POTE: ~2 = 1200.000, ~13/10 = 454.529

{{Optimal ET sequence|8d, 21cdef, 29, 37, 66|legend=0}}

Badness (Smith): 0.027168

== Ceratitid ==
Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~36/35 = 54.702
* POTE: ~2 = 1200.000, ~36/35 = 54.376

{{Optimal ET sequence|1c, 21c, 22|legend=1}} (7-limit)

{{Optimal ET sequence|1ce, 21ce, 22|legend=0}}

Badness (Smith): 0.051319

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

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

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

Optimal tunings:

* CTE: ~2 = 1200.000, ~36/35 = 54.575
* POTE: ~2 = 1200.000, ~36/35 = 54.665

{{Optimal ET sequence|1ce, 21cef, 22|legend=0}}

Badness (Smith): 0.044739

User:VectorGraphics/Porcupine family - Revision history

VectorGraphics at 16:02, 11 May 2025