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 {{=..."

2025-05-12T07:02:35Z

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

== Strong extensions ==
{{Main|Porcupine}}
{| class="wikitable"
|+Map to strong full 13-limit extensions
!Extension
!Mapping of 7
!Mapping of 13
!Tuning range*
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Porcupinefish|Porcupinefish]]
| +6
| -17
|↑ 22
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Porky|Porky]]
| -16
| rowspan="2" | +5
|↑ 29
↓ 22
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Coendou|Coendou]]
|13
|↓ 29
|}
<nowiki>*</nowiki> Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions

=== Porcupinefish ===
[[User:VectorGraphics/Porcupine family/Draft 1#Strong extensions|Return to the map]]

Porcupinefish (or "septimal porcupine" in its 11-limit form) 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

==== 13-limit ====
In the 13-limit, porcupinefish maps 13/8 to -17 generators.{{See also|The Biosphere}}Subgroup: 2.3.5.7.11.13Comma 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

=== Porky ===
[[User:VectorGraphics/Porcupine family/Draft 1#Strong extensions|Return to the map]]

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 ====
As the porcupinefish mapping is inaccurate with a sharply tuned generator, this alternate mapping becomes more accurate at this point. Thus, 13-limit porky as a whole can be seen as reversing the tradeoff between 7 and 13 found in porcupinefish.

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

=== Coendou ===
[[User:VectorGraphics/Porcupine family/Draft 1#Strong extensions|Return to the map]]

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 ====
Coendou shares the mapping of 13 with porky.

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

= Weak extensions =
{| class="wikitable"
|+Map to weak extensions
! rowspan="2" |Extensions
! rowspan="2" |Periods per octave
! rowspan="2" |Generator
! colspan="2" |Position of original generator
|-
!Number of generators
!Number of periods
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Hedgehog|Hedgehog]]
|period = 1/2 octave
|~9/7
| -1 generators
| +1 periods
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Undecimation|Undecimation]]
|period = octave
|~88/65
| +2 generators
| -1 periods
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Nautilus|Nautilus]]
|period = octave
|~21/20
| +2 generators
| +0 periods
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Ammonite|Ammonite]]
|period = octave
|~9/7
| +3 generators
| -1 periods
|-
|[[User:VectorGraphics/Porcupine family/Draft 1#Ceratitid|Ceratitid]]
|period = octave
|~36/35
| +3 generators
| +0 periods
|}

=== Hedgehog ===
{{See also|Sensamagic clan|Stearnsmic clan}}{{val|}} [[User:VectorGraphics/Porcupine family/Draft 1#Weak extensions|Return to the map]]

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, and collapses 5/4 and 7/4 to the same number of gensteps (in different periods). As such, it is best tuned around 165 cents and is also 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 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

==== Porkhog ====
Since hedgehog has the same sharply tuned generator as porky (with a different mapping for 7), it becomes reasonable to extend hedgehog with porky's mapping, mapping 13 to -5 gensteps.

[add temp data]

==== 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

=== Hedgepig ===
Hedgepig is a variant of hedgehog that uses a sharper and more accurate mapping of 11 available around the 165c tuning, and thus extends 2.3.5 porcupine instead of 2.3.5.11 porcupine.

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.

=== Undecimation ===
[[User:VectorGraphics/Porcupine family/Draft 1#Weak extensions|Return to the map]]

Undecimation is an extension to the 2.3.5.11.13 subgroup that splits the generator to introduce neutral intervals (and thus an obvious choice for mapping 13/8). It does this by stacking a flatly tuned fifth representing 65/44 twice to get to the generator. For whatever reason, the optimal tunings shown here are in the sharper range, even though tuning undecimation to 519.25 cents allows it to find 7 at 12 gensteps.

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

=== Nautilus ===
[[User:VectorGraphics/Porcupine family/Draft 1#Weak extensions|Return to the map]]

Nautilus splits the 10/9 generator into two 21/20s, making a much simpler mapping of 7/4 available. It can be seen as porcupine's generator chain expanded to include neutral intervals (like undecimation, but with a different generator), and as such has a mapping of 13/8 available at -19 steps.

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

; Music

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

=== Ammonite ===
[[User:VectorGraphics/Porcupine family/Draft 1#Weak extensions|Return to the map]]

Ammonite splits the porcupine generator (as ~1363.5 cents) into three parts representing 9/7.

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 ===
[[User:VectorGraphics/Porcupine family/Draft 1#Weak extensions|Return to the map]]

Ceratitid also splits the generator into three, and this time the more familiar neutral second is split into three parts representing 36/35.

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/Draft 1 - Revision history