Porcupine family

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