Porcupine family: Difference between revisions

Latest revision as of 17:44, 30 September 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:

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

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:

Oxygen (+21/20) → Very low accuracy temperaments
Hedgehog (+50/49 or 245/243) → Stearnsmic clan
Jamesbond (+25/24) → Whitewood family

The rest are considered in this page.

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

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:

7-odd-limit: ~10/9 = [3/5 0 -1/5⟩

unchanged-interval (eigenmonzo) basis: 2.5

9-odd-limit: ~10/9 = [1/6 -1/6 0 1/12⟩

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

Undecimation

This weak extension is used by most patent vals that support porcupine and tune 13 well.

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 64/63, 100/99, 169/168

Mapping: [⟨1 -1 -2 8 0 5], ⟨0 6 10 -12 8 -3]]

mapping generators: ~2, ~35/26

Optimal tunings:

WE: ~2 = 1198.4569 ¢, ~35/26 = 517.9375 ¢
CWE: ~2 = 1200.0000 ¢, ~35/26 = 518.5822 ¢

Optimal ET sequence: 7, 30, 37, 81bd

Badness (Sintel): 1.68

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:

7- and 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.7

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:

7- and 9-odd-limit: ~10/9 = [2/11 0 1/11 -1/11⟩

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

Improvisation in 29edo (2024) by Budjarn Lambeth – in Palace scale, 29edo tuning

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:

7- and 9-odd-limit: ~10/9 = [2/3 -1/3⟩

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:

7- and 9-odd-limit: ~10/9 = [3/5 0 -1/5⟩

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

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

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 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

Porcupine family: Difference between revisions

Latest revision as of 17:44, 30 September 2026

Contents

Porcupine

Overview to extensions

7-limit extensions

Subgroup extensions

2.3.5.11 subgroup (porkypine)

Septimal porcupine

11-limit

Porcupinefowl

Porcupinefish

Pourcup

Porkpie

Undecimation

Opossum

11-limit

13-limit

Porky

11-limit

13-limit

Coendou

11-limit

13-limit

Hystrix

11-limit

Nautilus

11-limit

13-limit

Belauensis

Ammonite

11-limit

13-limit

Ceratitid

11-limit

13-limit

Navigation menu

@@ Line 1: / Line 1: @@
-{{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.
@@ Line 11: / Line 5: @@
 {{Main| Porcupine }}
-The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, {{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]]s of 7, 8 and 15 notes make for some nice scale possibilities.
+The [[generator]] of porcupine is a minor whole tone, the [[10/9]] interval, and three of these add up to a perfect fourth ([[4/3]]), with two more giving the minor sixth ([[8/5]]). In fact, {{nowrap| (10/9)<sup>3</sup> {{=}} (4/3)⋅(250/243) }}, and {{nowrap| (10/9)<sup>5</sup> {{=}} (8/5)⋅(250/243)<sup>2</sup> }}. Its [[ploidacot]] is omega-tricot. [[22edo|3\22]] is a very recommendable generator, and [[mos scale]]s of 7, 8 and 15 notes make for some nice scale possibilities.
 [[Subgroup]]: 2.3.5
@@ Line 18: / Line 12: @@
 {{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}
 : mapping generators: ~2, ~10/9
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.166
+* [[WE]]: ~2 = 1199.5444{{c}}, ~10/9 = 163.8881{{c}}
-: [[error map]]: {{val| 0.000 +5.547 -7.143 }}
+: [[error map]]: {{val| -0.456 +5.469 -7.121 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 163.950
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.0621{{c}}
-: error map: {{val| 0.000 +6.194 -6.065 }}
+: error map: {{val| 0.000 +5.859 -6.624 }}
 [[Tuning ranges]]:
@@ Line 33: / Line 26: @@
 {{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
-[[Badness]] (Smith): 0.030778
+[[Badness]] (Sintel): 0.722
 === 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 a tuning sharp of 8d-edo;
+* [[#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 flat of 29edo.
+* [[#Coendou|Coendou]] adds [[525/512]], the avicennma, for a tuning sharp of 29edo.
 Those all share the same generator with porcupine.
-[[#Nautilus|nautilus]] tempers out [[49/48]] and splits the generator in two. [[#Hedgehog|hedgehog]] tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.
+[[#Nautilus|Nautilus]] tempers out [[49/48]] and splits the generator in two. Hedgehog tempers out [[50/49]] with a semi-octave period. Finally, [[#Ammonite|ammonite]] tempers out [[686/675]] and [[#Ceratitid|ceratitid]] tempers out [[1728/1715]]. Those split the generator in three.
 Temperaments discussed elsewhere include:
-* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
+* ''[[Oxygen]]'' (+21/20) → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]]
-* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
+* ''[[Hedgehog]]'' (+50/49 or 245/243) → [[Stearnsmic clan #Hedgehog|Stearnsmic clan]]
+* ''[[Jamesbond]]'' (+25/24) → [[Whitewood family #Jamesbond|Whitewood family]]
+The rest are considered in this page.
 ==== Subgroup extensions ====
@@ Line 60: / Line 56: @@
 Comma list: 55/54, 100/99
-Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
+{{Mapping|legend=2| 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]
+{{Mapping|legend=4| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 163.887
+* WE: ~2 = 1200.3290{{c}}, ~11/10 = 164.1227{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 164.078
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.9951{{c}}
 {{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}
-Badness (Smith): 0.0097
+Badness (Sintel): 0.303
-==== 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|legend=0| 7, 23bc, 30, 37, 44 }}
-Badness (Smith): 0.0305
 == Septimal porcupine ==
@@ Line 101: / Line 78: @@
 {{Mapping|legend=1| 1 2 3 2 | 0 -3 -5 6 }}
-{{Multival|legend=1| 3 5 -6 1 -18 -28 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/9 = 163.203
+* [[WE]]: ~2 = 1197.8178{{c}}, ~10/9 = 162.5839{{c}}
-: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
+: [[error map]]: {{val| -2.182 +5.929 -5.780 +2.313 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 162.880
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 162.9493{{c}}
-: error map: {{val| 0.000 +9.405 -0.714 +8.455 }}
+: error map: {{val| 0.000 +9.197 -1.060 +8.870 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 * [[9-odd-limit]]: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 [[Tuning ranges]]:
@@ Line 123: / Line 98: @@
 {{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
-[[Badness]] (Smith): 0.041057
+[[Badness]] (Sintel): 1.04
 === 11-limit ===
@@ Line 130: / Line 105: @@
 Comma list: 55/54, 64/63, 100/99
-Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
+{{Mapping|legend=0| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 163.105
+* WE: ~2 = 1198.3250{{c}}, ~11/10 = 162.5202{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 162.747
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8156{{c}}
 Minimax tuning:
 * 11-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
-: eigenmonzo (unchanged-interval) basis: 2.9/7
+: unchanged-interval (eigenmonzo) basis: 2.9/7
 Tuning ranges:
@@ Line 146: / Line 121: @@
 {{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}
-Badness (Smith): 0.021562
+Badness (Sintel): 0.713
 ==== Porcupinefowl ====
@@ Line 155: / Line 130: @@
 Comma list: 40/39, 55/54, 64/63, 66/65
-Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
+{{Mapping|legend=0| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 163.442
+* WE: ~2 = 1197.0054{{c}}, ~11/10 = 162.3022{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 162.708
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.8314{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
-: eigenmonzo (unchanged-interval) basis: 2.11
+: unchanged-interval (eigenmonzo) basis: 2.11
 Tuning ranges:
@@ Line 170: / Line 145: @@
 * 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]
-{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}
+{{Optimal ET sequence|legend=0| 7, 15, 22f }}
-Badness (Smith): 0.021276
+Badness (Sintel): 0.879
 ==== Porcupinefish ====
@@ Line 181: / Line 156: @@
 Comma list: 55/54, 64/63, 91/90, 100/99
-Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
+{{Mapping|legend=0| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 162.636
+* WE: ~2 = 1198.3206{{c}}, ~11/10 = 162.0502{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 162.277
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.3458{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~10/9 = {{monzo| 2/13 0 0 0 1/13 -1/13 }}
-: eigenmonzo (unchanged-interval) basis: 2.13/11
+: unchanged-interval (eigenmonzo) basis: 2.13/11
 Tuning ranges:
-* 13-odd-limit diamond monotone: ~10/9 = [160.000, 162.162] (2\15 to 5\37)
+* 13-odd-limit diamond monotone: ~11/10 = [160.000, 162.162] (2\15 to 5\37)
-* 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
+* 15-odd-limit diamond monotone: ~11/10 = 162.162 (5\37)
-* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
+* 13- and 15-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]
 {{Optimal ET sequence|legend=0| 15, 22, 37 }}
-Badness (Smith): 0.025314
+Badness (Sintel): 1.05
 ==== Pourcup ====
@@ Line 205: / Line 180: @@
 Comma list: 55/54, 64/63, 100/99, 196/195
-Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
+{{Mapping|legend=0| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 163.378
+* WE: ~2 = 1198.0537{{c}}, ~11/10 = 162.2183{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 162.482
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 162.4665{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
-: eigenmonzo (unchanged-interval) basis: 2.13/7
+: unchanged-interval (eigenmonzo) basis: 2.13/7
 {{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
-Badness (Smith): 0.035130
+Badness (Sintel): 1.45
 ==== Porkpie ====
@@ Line 224: / Line 199: @@
 Comma list: 55/54, 64/63, 65/63, 100/99
-Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
+{{Mapping|legend=0| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 163.678
+* WE: ~2 = 1200.0223{{c}}, ~11/10 = 163.6908{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 163.688
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 163.6874{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
-: eigenmonzo (unchanged-interval) basis: 2.9/7
+: unchanged-interval (eigenmonzo) basis: 2.9/7
 {{Optimal ET sequence|legend=0| 7, 15f, 22 }}
-Badness (Smith): 0.026043
+Badness (Sintel): 1.08
+==== Undecimation ====
+This weak extension is used by most patent vals that support porcupine and tune 13 well.
+Subgroup: 2.3.5.7.11.13
+Comma list: 55/54, 64/63, 100/99, 169/168
+{{Mapping|legend=0| 1 -1 -2 8 0 5 | 0 6 10 -12 8 -3 }}
+: mapping generators: ~2, ~35/26
+Optimal tunings:
+* WE: ~2 = 1198.4569{{c}}, ~35/26 = 517.9375{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~35/26 = 518.5822{{c}}
+{{Optimal ET sequence|legend=0| 7, 30, 37, 81bd }}
+Badness (Sintel): 1.68
 == 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.
+{{Main| Opossum }}
+Opossum can be described as {{nowrap| 8d & 15 }}. Tempering out [[28/27]], the perfect fifth of three generator steps is conflated with not [[32/21]] as in porcupine but [[14/9]]. Three such fifths or nine generator steps octave reduced give a flat 7/4. 2\15 is a good generator.
 [[Subgroup]]: 2.3.5.7
@@ Line 246: / Line 241: @@
 {{Mapping|legend=1| 1 2 3 4 | 0 -3 -5 -9 }}
-{{Multival|legend=1| 3 5 9 1 6 7 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/9 = 161.306
+* [[WE]]: ~2 = 1195.7927{{c}}, ~10/9 = 159.1315{{c}}
-: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
+: [[error map]]: {{val| -4.207 +12.236 +5.407 -17.838 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 160.4589{{c}}
-: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}
+: error map: {{val| 0.000 +16.668 +11.392 -12.956 }}
 [[Minimax tuning]]:
-* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
 {{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
-[[Badness]] (Smith): 0.040650
+[[Badness]] (Sintel): 1.03
 === 11-limit ===
@@ Line 267: / Line 260: @@
 Comma list: 28/27, 55/54, 77/75
-Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
+{{Mapping|legend=0| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
-{{Multival|legend=1| 3 5 9 4 1 6 -4 7 -8 -20 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 161.365
+* WE: ~2 = 1196.2331{{c}}, ~11/10 = 159.3050{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 159.807
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.4644{{c}}
 Minimax tuning:
-* 11-odd-limit eigenmonzo (unchanged-interval) basis: 2.7
+* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
 {{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
-Badness (Smith): 0.022325
+Badness (Sintel): 0.738
 === 13-limit ===
@@ Line 287: / Line 278: @@
 Comma list: 28/27, 40/39, 55/54, 66/65
-Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
+{{Mapping|legend=0| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 161.631
+* WE: ~2 = 1193.5447{{c}}, ~11/10 = 157.9505{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 158.805
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 159.7600{{c}}
 Minimax tuning:
-* 13- and 15-odd-limit eigenmonzo (unchanged-interval) basis: 2.7
+* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
 {{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}
-Badness (Smith): 0.019389
+Badness (Sintel): 0.801
 == Porky ==
-Porky can be described as 7d & 22, suggesting a less sharp perfect fifth. 7\51 is a good generator.
+Porky can be described as {{nowrap| 22 & 29 }}, suggesting a less sharp perfect fifth. 7\51 is a good generator.
 [[Subgroup]]: 2.3.5.7
@@ Line 308: / Line 299: @@
 {{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
-{{Multival|legend=1| 3 5 16 1 17 23 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
+* [[WE]]: ~2 = 1200.0685{{c}}, ~10/9 = 164.4215{{c}}
-: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
+: [[error map]]: {{val| +0.068 +4.917 -8.216 +0.772 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 164.4060{{c}}
-: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}
+: error map: {{val| 0.000 +4.827 -8.344 +0.678 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
-{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
+{{Optimal ET sequence|legend=1| 7d, 15d, 22, 51, 73c }}
-[[Badness]] (Smith): 0.054389
+[[Badness]] (Sintel): 1.38
 === 11-limit ===
@@ Line 330: / Line 319: @@
 Comma list: 55/54, 100/99, 225/224
-Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
+{{Mapping|legend=0| 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
+* WE: ~2 = 1200.8706{{c}}, ~11/10 = 164.6715{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 164.552
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.4810{{c}}
 Minimax tuning:
 * 11-odd-limit: ~11/10 = {{monzo| 2/11 0 1/11 -1/11 }}
-: eigenmonzo (unchanged-interval) basis: 2.7/5
+: unchanged-interval (eigenmonzo) basis: 2.7/5
 {{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}
-Badness (Smith): 0.027268
+Badness (Sintel): 0.901
 === 13-limit ===
@@ Line 351: / Line 338: @@
 Comma list: 55/54, 65/64, 91/90, 100/99
-Mapping: {{mapping| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
+{{Mapping|legend=0| 1 2 3 5 4 3 | 0 -3 -5 -16 -4 5 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 164.478
+* WE: ~2 = 1202.1557{{c}}, ~11/10 = 165.2494{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 164.953
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 164.8579{{c}}
 {{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
-Badness (Smith): 0.026543
+Badness (Sintel): 1.10
 ; Music
@@ Line 365: / Line 352: @@
 == Coendou ==
-Coendou can be described as 7 & 29, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.
+Coendou can be described as {{nowrap| 29 & 36c }}, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.
 [[Subgroup]]: 2.3.5.7
@@ Line 372: / Line 359: @@
 {{Mapping|legend=1| 1 2 3 1 | 0 -3 -5 13 }}
-{{Multival|legend=1| 3 5 -13 1 -29 -44 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/9 = 166.094
+* [[WE]]: ~2 = 1202.6772{{c}}, ~10/9 = 166.4110{{c}}
-: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
+: [[error map]]: {{val| +2.678 +4.166 -10.337 -2.806 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 166.0511{{c}}
-: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}
+: error map: {{val| 0.000 -0.108 -16.569 -10.161 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
-: [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
+{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c }}
-[[Badness]] (Smith): 0.118344
+[[Badness]] (Sintel): 2.99
 === 11-limit ===
@@ Line 394: / Line 379: @@
 Comma list: 55/54, 100/99, 525/512
-Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
+{{Mapping|legend=0| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 165.925
+* WE: ~2 = 1203.0245{{c}}, ~11/10 = 166.3991{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 165.981
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9714{{c}}
 Minimax tuning:
 * 11-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
-: eigenmonzo (unchanged-interval) basis: 2.3
+: unchanged-interval (eigenmonzo) basis: 2.3
 {{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}
-Badness (Smith): 0.049669
+Badness (Sintel): 1.64
 === 13-limit ===
@@ Line 413: / Line 398: @@
 Comma list: 55/54, 65/64, 100/99, 105/104
-Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
+{{Mapping|legend=0| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 166.046
+* WE: ~2 = 1202.9957{{c}}, ~11/10 = 166.3885{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 165.974
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9843{{c}}
 Minimax tuning:
 * 13- and 15-odd-limit: ~11/10 = {{monzo| 2/3 -1/3 }}
-: eigenmonzo (unchanged-interval) basis: 2.3
+: unchanged-interval (eigenmonzo) basis: 2.3
 {{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
-Badness (Smith): 0.030233
+Badness (Sintel): 1.25
 == Hystrix ==
@@ Line 435: / Line 420: @@
 {{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -1 }}
-{{Multival|legend=1| 3 5 1 1 -7 -12 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~10/9 = 165.185
+* [[WE]]: ~2 = 1187.8599{{c}}, ~10/9 = 157.2605{{c}}
-: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
+: [[error map]]: {{val| -12.140 +1.983 -9.037 +37.493 }}
-* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 161.2833{{c}}
-: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}
+: error map: {{val| 0.000 +14.195 +7.270 +69.891 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 {{Optimal ET sequence|legend=1| 7, 8d, 15d }}
-[[Badness]] (Smith): 0.044944
+[[Badness]] (Sintel): 1.14
 === 11-limit ===
@@ Line 457: / Line 440: @@
 Comma list: 22/21, 36/35, 80/77
-Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~11/10 = 164.768
+* WE: ~2 = 1189.2810{{c}}, ~11/10 = 157.3322{{c}}
-* POTE: ~2 = 1200.000, ~11/10 = 158.750
+* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 160.9603{{c}}
 {{Optimal ET sequence|legend=0| 7, 8d, 15d }}
-Badness (Smith): 0.026790
+Badness (Sintel): 0.886
-== Hedgehog ==
+== Nautilus ==
-{{See also| Sensamagic clan | Stearnsmic clan }}
+Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.
-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
-[[Comma list]]: 50/49, 245/243
-{{Mapping|legend=1| 2 1 1 2 | 0 3 5 5 }}
-: mapping generators: ~7/5, ~9/7
-{{Multival|legend=1| 6 10 10 2 -1 -5 }}
-[[Optimal tuning]]s:
-* [[CTE]]: ~7/5 = 600.000, ~9/7 = 435.258
-: [[error map]]: {{val| 0.000 +3.819 -10.024 +7.464 }}
-* [[POTE]]: ~7/5 = 600.000, ~9/7 = 435.648
-: error map: {{val| 0.000 +4.989 -8.074 +9.414 }}
-{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
-[[Badness]] (Smith): 0.043983
-=== 11-limit ===
-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|legend=1| 6 10 10 8 2 -1 -8 -5 -16 -12 }}
-Optimal tunings:
-* CTE: ~7/5 = 600.000, ~9/7 = 435.528
-* POTE: ~7/5 = 600.000, ~9/7 = 435.386
-{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
-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|legend=0| 8d, 14cf, 22 }}
-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|legend=0| 14c, 22f }}
-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|legend=1| 6 10 10 -14 2 -1 -43 -5 -67 -74 }}
-Optimal tunings:
-* CTE: ~7/5 = 600.000, ~9/7 = 435.329
-* POTE: ~7/5 = 600.000, ~9/7 = 435.425
-{{Optimal ET sequence|legend=0| 22 }}
-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
@@ Line 567: / Line 460: @@
 : mapping generators: ~2, ~21/20
-{{Multival|legend=1| 6 10 3 2 -12 -21 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~21/20 = 81.914
+* [[WE]]: ~2 = 1202.1642{{c}}, ~21/20 = 82.6542{{c}}
-: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
+: [[error map]]: {{val| +2.164 +6.448 -6.364 -10.296 }}
-* [[POTE]]: ~2 = 1200.000, ~21/20 = 82.505
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 82.2758{{c}}
-: error map: {{val| 0.000 +3.012 -11.368 -16.342 }}
+: error map: {{val| 0.000 +4.390 -9.072 -15.653 }}
-{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}
+{{Optimal ET sequence|legend=1| 14c, 15, 29 }}
-[[Badness]] (Smith): 0.057420
+[[Badness]] (Sintel): 1.45
 === 11-limit ===
@@ Line 585: / Line 476: @@
 Comma list: 49/48, 55/54, 245/242
-Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
-{{Multival|legend=1| 6 10 3 8 2 -12 -8 -21 -16 12 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/20 = 81.802
+* WE: ~2 = 1202.3781{{c}}, ~21/20 = 82.6673{{c}}
-* POTE: ~2 = 1200.000, ~21/20 = 82.504
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.2434{{c}}
-{{Optimal ET sequence|legend=0| 14c, 15, 29, 44d }}
+{{Optimal ET sequence|legend=0| 14c, 15, 29 }}
-Badness (Smith): 0.026023
+Badness (Sintel): 0.860
 ==== 13-limit ====
@@ Line 602: / Line 491: @@
 Comma list: 49/48, 55/54, 91/90, 100/99
-Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
+{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/20 = 81.912
+* WE: ~2 = 1202.4145{{c}}, ~21/20 = 82.6963{{c}}
-* POTE: ~2 = 1200.000, ~21/20 = 82.530
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 82.3130{{c}}
-{{Optimal ET sequence|legend=0| 14cf, 15, 29, 44d }}
+{{Optimal ET sequence|legend=0| 14cf, 15, 29 }}
-Badness (Smith): 0.022285
+Badness (Sintel): 0.921
 ==== Belauensis ====
@@ Line 617: / Line 506: @@
 Comma list: 40/39, 49/48, 55/54, 66/65
-Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
+{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/20 = 82.034
+* WE: ~2 = 1199.0072{{c}}, ~21/20 = 81.6911{{c}}
-* POTE: ~2 = 1200.000, ~21/20 = 81.759
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 81.8576{{c}}
-{{Optimal ET sequence|legend=0| 14c, 15, 29f, 44dff }}
+{{Optimal ET sequence|legend=0| 14c, 15 }}
-Badness (Smith): 0.029816
+Badness (Sintel): 1.23
 ; Music
-* [http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 ''Nautilus Reverie''] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]
+* [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 5 8 10 | 0 -9 -15 -19 }}
+{{Mapping|legend=1| 1 -4 -7 -9 | 0 9 15 19 }}
-: mapping generators: ~2, ~9/7
+: mapping generators: ~2, ~14/9
-{{Multival|legend=1| 9 15 19 3 5 2 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
+* [[WE]]: ~2 = 1199.3342{{c}}, ~14/9 = 745.1379{{c}}
-: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
+: [[error map]]: {{val| -0.666 +6.949 -4.584 -5.213 }}
-* [[POTE]]: ~2 = 1200.000, ~9/7 = 454.448
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 745.4994{{c}}
-: error map: {{val| 0.000 +8.009 -3.040 -3.346 }}
+: error map: {{val| 0.000 +7.540 -3.823 -4.337 }}
 {{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
-[[Badness]] (Smith): 0.107686
+[[Badness]] (Sintel): 2.73
 === 11-limit ===
@@ Line 656: / Line 546: @@
 Comma list: 55/54, 100/99, 686/675
-Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
+{{Mapping|legend=0| 1 -4 -7 -9 -4 | 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
+* WE: ~2 = 1200.0141{{c}}, ~14/9 = 745.4971{{c}}
-* POTE: ~2 = 1200.000, ~9/7 = 454.512
+* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4894{{c}}
 {{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
-Badness (Smith): 0.045694
+Badness (Sintel): 1.51
 === 13-limit ===
@@ Line 673: / Line 561: @@
 Comma list: 55/54, 91/90, 100/99, 169/168
-Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}
+{{Mapping|legend=0| 1 -4 -7 -9 -4 -5 | 0 9 15 19 12 14 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~13/10 = 454.480
+* WE: ~2 = 1200.2478{{c}}, ~14/9 = 745.6252{{c}}
-* POTE: ~2 = 1200.000, ~13/10 = 454.529
+* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 745.4904{{c}}
 {{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
-Badness (Smith): 0.027168
+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
@@ Line 691: / Line 581: @@
 : mapping generators: ~2, ~36/35
-{{Multival|legend=1| 9 15 4 3 -19 -33 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~36/35 = 54.804
+* [[WE]]: ~2 = 1197.6274{{c}}, ~36/35 = 54.2770{{c}}
-: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
+: [[error map]]: {{val| -2.373 +4.807 -7.586 +6.948 }}
-* [[POTE]]: ~2 = 1200.000, ~36/35 = 54.384
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 54.5489{{c}}
-: error map: {{val| 0.000 +8.585 -2.081 +13.636 }}
+: error map: {{val| 0.000 +7.105 -4.548 +12.978 }}
 {{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
-[[Badness]] (Smith): 0.115304
+[[Badness]] (Sintel): 2.92
 === 11-limit ===
@@ Line 709: / Line 597: @@
 Comma list: 55/54, 100/99, 352/343
-Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
+{{Mapping|legend=0| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~36/35 = 54.702
+* WE: ~2 = 1198.2851{{c}}, ~36/35 = 54.2986{{c}}
-* POTE: ~2 = 1200.000, ~36/35 = 54.376
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.4992{{c}}
 {{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
-Badness (Smith): 0.051319
+Badness (Sintel): 1.70
 === 13-limit ===
@@ Line 724: / Line 612: @@
 Comma list: 55/54, 65/63, 100/99, 352/343
-Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
+{{Mapping|legend=0| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~36/35 = 54.575
+* WE: ~2 = 1200.3864{{c}}, ~36/35 = 54.6830{{c}}
-* POTE: ~2 = 1200.000, ~36/35 = 54.665
+* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 54.6396{{c}}
 {{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
-Badness (Smith): 0.044739
+Badness (Sintel): 1.85
+[[Category:Porcupine family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Porcupine family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Porcupine| ]] <!-- key article -->
-[[Category:Rank 2]]

Porcupine family: Difference between revisions

Latest revision as of 17:44, 30 September 2026

Porcupine

Overview to extensions

7-limit extensions

Subgroup extensions

2.3.5.11 subgroup (porkypine)

Septimal porcupine

11-limit

Porcupinefowl

Porcupinefish

Pourcup

Porkpie

Undecimation

Opossum

11-limit

13-limit

Porky

11-limit

13-limit

Coendou

11-limit

13-limit

Hystrix

11-limit

Nautilus

11-limit

13-limit

Belauensis

Ammonite

11-limit

13-limit

Ceratitid

11-limit

13-limit

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