Porcupine family

The porcupine family is the rank-2 family of temperaments whose 5-limit parent comma is 250/243, also called the maximal diesis or porcupine comma.

The generator 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. 3\22 is a very recommendable generator, and mos scales of 7, 8 and 15 notes make for some nice scale possibilities.

Notice 250/243 = (55/54)(100/99), the temperament thus extends naturally to the 2.3.5.11 subgroup, sometimes known as porkypine.

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

• 64/63, the archytas comma, for septimal porcupine,
• 36/35, the septimal quarter tone, for hystrix,
• 50/49, the jubilisma, for hedgehog, and
• 49/48, the slendro diesis, for nautilus.

Temperaments discussed elsewhere include jamesbond.

Porcupine

Main article: Porcupine

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

mapping generators: ~2, ~10/9

Optimal tunings:

• CTE: ~2 = 1\1, ~10/9 = 164.1659
• POTE: ~2 = 1\1, ~10/9 = 163.950

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: 0.030778

2.3.5.11 subgroup (porkypine)

Subgroup: 2.3.5.11

Comma list: 55/54, 100/99

Sval mapping: [⟨1 2 3 4], ⟨0 -3 -5 -4]]

Gencom mapping: [⟨1 2 3 0 4], ⟨0 -3 -5 0 -4]]

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

Optimal tunings:

• CTE: ~2 = 1\1, ~11/10 = 163.8867
• POTE: ~2 = 1\1, ~11/10 = 164.0777

Optimal ET sequence: 7, 15, 22, 73ce, 95ce

Badness: 0.0097

Undecimation

Subgroup: 2.3.5.11.13

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

Sval mapping: [⟨1 5 8 8 2], ⟨0 -6 -10 -8 3]]

sval mapping generators: ~2, ~65/44

Optimal tunings:

• CTE: ~2 = 1\1, ~88/65 = 518.0865
• POTE: ~2 = 1\1, ~88/65 = 518.2094

Optimal ET sequence: 7, 23bc, 30, 37, 44

Badness: 0.0305

Septimal porcupine

Main article: Porcupine

Septimal porcupine uses six of its minor tone generator steps to get to 7/4. For this to work you need a small minor tone such as 22edo provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.

Subgroup: 2.3.5.7

Comma list: 64/63, 250/243

Mapping: [⟨1 2 3 2], ⟨0 -3 -5 6]]

Wedgie: ⟨⟨ 3 5 -6 1 -18 -28 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~10/9 = 163.2032
• POTE: ~2 = 1\1, ~10/9 = 162.880

Minimax tuning:

• 7-odd-limit: ~10/9 = [3/5 0 -1/5⟩

eigenmonzo (unchanged-interval) basis: 2.5

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

eigenmonzo (unchanged-interval) 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: 0.041057

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:

• CTE: ~2 = 1\1, ~11/10 = 163.1055
• POTE: ~2 = 1\1, ~11/10 = 162.747

Minimax tuning:

• 11-odd-limit: ~11/10 = [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]

Optimal ET sequence: 7, 15, 22, 37, 59

Badness: 0.021562

13-limit

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:

• CTE: ~2 = 1\1, ~11/10 = 163.4425
• POTE: ~2 = 1\1, ~11/10 = 162.708

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [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

Badness: 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: [⟨1 2 3 2 4 6], ⟨0 -3 -5 6 -4 -17]]

Optimal tunings:

• CTE: ~2 = 1\1, ~11/10 = 162.6361
• POTE: ~2 = 1\1, ~11/10 = 162.277

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [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

Badness: 0.025314

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:

• CTE: ~2 = 1\1, ~11/10 = 163.3781
• POTE: ~2 = 1\1, ~11/10 = 162.482

Minimax tuning:

• 13- and 15-odd-limit: ~11/10 = [1/14 0 0 -1/14 0 1/14⟩

eigenmonzo (unchanged-interval) basis: 2.13/7

Optimal ET sequence: 15f, 22f, 37, 59f

Badness: 0.035130

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:

• CTE: ~2 = 1\1, ~11/10 = 163.6778
• POTE: ~2 = 1\1, ~11/10 = 163.688

Minimax tuning:

• 13- and 15-odd-limit: ~11/10 = [1/6 -1/6 0 1/12⟩

eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 7, 15f, 22

Badness: 0.026043

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

Comma list: 28/27, 126/125

Mapping: [⟨1 2 3 4], ⟨0 -3 -5 -9]]

Wedgie: ⟨⟨ 3 5 9 1 6 7 ]]

Optimal tuning (CTE): ~2 = 1\1, ~10/9 = 161.3063

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15

Badness: 0.040650

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]]

Wedgie: ⟨⟨ 3 5 9 4 1 6 -4 7 -8 -20 ]]

Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15

Badness: 0.022325

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 tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15, 38bceff

Badness: 0.019389

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

Comma list: 225/224, 250/243

Mapping: [⟨1 2 3 5], ⟨0 -3 -5 -16]]

Wedgie: ⟨⟨ 3 5 16 1 17 23 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~10/9 = 164.3913
• POTE: ~2 = 1\1, ~10/9 = 164.412

Minimax tuning:

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

eigenmonzo (unchanged-interval) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 29, 51, 73c

Badness: 0.054389

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]]

Wedgie: ⟨⟨ 3 5 16 4 1 17 -4 23 -8 -44 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~11/10 = 164.3207
• POTE: ~2 = 1\1, ~11/10 = 164.552

Minimax tuning:

• 11-odd-limit: ~11/10 = [2/11 0 1/11 -1/11⟩

eigenmonzo (unchanged-interval) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 51

Badness: 0.027268

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:

• CTE: ~2 = 1\1, ~11/10 = 164.4782
• POTE: ~2 = 1\1, ~11/10 = 164.953

Optimal ET sequence: 7d, 22, 29, 51f, 80cdeff

Badness: 0.026543

Music

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

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

Comma list: 250/243, 525/512

Mapping: [⟨1 2 3 1], ⟨0 -3 -5 13]]

Wedgie: ⟨⟨ 3 5 -13 1 -29 -44 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~10/9 = 166.0938
• POTE: ~2 = 1\1, ~10/9 = 166.041

Minimax tuning:

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

eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65c, 94cd

Badness: 0.118344

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:

• CTE: ~2 = 1\1, ~11/10 = 165.9246
• POTE: ~2 = 1\1, ~11/10 = 165.981

Minimax tuning:

• 11-odd-limit: ~11/10 = [2/3 -1/3⟩

eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65ce

Badness: 0.049669

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:

• CTE: ~2 = 1\1, ~11/10 = 166.0459
• POTE: ~2 = 1\1, ~11/10 = 165.974

Minimax tuning:

• 13- and 15-odd-limit: ~11/10 = [2/3 -1/3⟩

eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65cef

Badness: 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

Comma list: 36/35, 160/147

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -1]]

Wedgie: ⟨⟨ 3 5 1 1 -7 -12 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~10/9 = 165.1845
• POTE: ~2 = 1\1, ~10/9 = 158.868

Minimax tuning:

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

eigenmonzo (unchanged-interval) basis: 2.5

Optimal ET sequence: 7, 8d, 15d

Badness: 0.044944

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:

• CTE: ~2 = 1\1, ~11/10 = 164.7684
• POTE: ~2 = 1\1, ~11/10 = 158.750

Optimal ET sequence: 7, 8d, 15d

Badness: 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 blocks.

Subgroup: 2.3.5.7

Comma list: 21/20, 175/162

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -2]]

Wedgie: ⟨⟨ 3 5 2 1 -5 -9 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~10/9 = 161.3408
• POTE: ~2 = 1\1, ~10/9 = 169.112

Optimal ET sequence: 7d

Badness: 0.059866

Hedgehog

See also: Sensamagic clan

See also: 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 ⟨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: [⟨2 1 1 2], ⟨0 3 5 5]]

mapping generators: ~7/5, ~9/7

Wedgie: ⟨⟨ 6 10 10 2 -1 -5 ]]

Optimal tunings:

• CTE: ~7/5 = 1\2, ~9/7 = 435.2580
• POTE: ~7/5 = 1\2, ~9/7 = 435.648

Optimal ET sequence: 8d, 14c, 22

Badness: 0.043983

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 1 2 4], ⟨0 3 5 5 4]]

Wedgie: ⟨⟨ 6 10 10 8 2 -1 -8 -5 -16 -12 ]]

Optimal tunings:

• CTE: ~7/5 = 1\2, ~9/7 = 435.5281
• POTE: ~7/5 = 1\2, ~9/7 = 435.386

Optimal ET sequence: 8d, 14c, 22, 58ce

Badness: 0.023095

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~7/5 = 1\2, ~9/7 = 436.3087
• POTE: ~7/5 = 1\2, ~9/7 = 435.861

Optimal ET sequence: 8d, 14cf, 22

Badness: 0.021516

Urchin

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~7/5 = 1\2, ~9/7 = 435.1856
• POTE: ~7/5 = 1\2, ~9/7 = 437.078

Optimal ET sequence: 14c, 22f

Badness: 0.025233

Hedgepig

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 1 1 2 12], ⟨0 3 5 5 -7]]

Wedgie: ⟨⟨ 6 10 10 -14 2 -1 -43 -5 -67 -74 ]]

Optimal tunings:

• CTE: ~7/5 = 1\2, ~9/7 = 435.3289
• POTE: ~7/5 = 1\2, ~9/7 = 435.425

Optimal ET sequence: 22

Badness: 0.068406

Music

• Phobos Light by Chris Vaisvil in hedgehog[14] to 22edo.

Nautilus

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

Wedgie: ⟨⟨ 6 10 3 2 -12 -21 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~21/20 = 81.9143
• POTE: ~2 = 1\1, ~21/20 = 82.505

Optimal ET sequence: 14c, 15, 29, 44d

Badness: 0.057420

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]]

Wedgie: ⟨⟨ 6 10 3 8 2 -12 -8 -21 -16 12 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~21/20 = 81.8017
• POTE: ~2 = 1\1, ~21/20 = 82.504

Optimal ET sequence: 14c, 15, 29, 44d

Badness: 0.026023

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:

• CTE: ~2 = 1\1, ~21/20 = 81.9123
• POTE: ~2 = 1\1, ~21/20 = 82.530

Optimal ET sequence: 14cf, 15, 29, 44d

Badness: 0.022285

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:

• CTE: ~2 = 1\1, ~21/20 = 82.0342
• POTE: ~2 = 1\1, ~21/20 = 81.759

Optimal ET sequence: 14c, 15, 29f, 44dff

Badness: 0.029816

Music

• Nautilus Reverie by Igliashon Calvin Jones-Coolidge

Ammonite

Subgroup: 2.3.5.7

Comma list: 250/243, 686/675

Mapping: [⟨1 5 8 10], ⟨0 -9 -15 -19]]

mapping generators: ~2, ~9/7

Wedgie: ⟨⟨ 9 15 19 3 5 2 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~9/7 = 454.5500
• POTE: ~2 = 1\1, ~9/7 = 454.448

Optimal ET sequence: 8d, 21cd, 29, 37, 66

Badness: 0.107686

11-limit

Subgroup: 2.3.5.7.11

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

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

Wedgie: ⟨⟨ 9 15 19 12 3 5 -12 2 -24 -32 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~9/7 = 454.5050
• POTE: ~2 = 1\1, ~9/7 = 454.512

Optimal ET sequence: 8d, 21cde, 29, 37, 66

Badness: 0.045694

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

• CTE: ~2 = 1\1, ~13/10 = 454.4798
• POTE: ~2 = 1\1, ~13/10 = 454.529

Optimal ET sequence: 8d, 21cdef, 29, 37, 66

Badness: 0.027168

Ceratitid

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

Wedgie: ⟨⟨ 9 15 4 3 -19 -33 ]]

Optimal tunings:

• CTE: ~2 = 1\1, ~36/35 = 54.8040
• POTE: ~2 = 1\1, ~36/35 = 54.384

Optimal ET sequence: 1c, 21c, 22

Badness: 0.115304

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:

• CTE: ~2 = 1\1, ~36/35 = 54.7019
• POTE: ~2 = 1\1, ~36/35 = 54.376

Optimal ET sequence: 1ce, 21ce, 22

Badness: 0.051319

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:

• CTE: ~2 = 1\1, ~36/35 = 54.5751
• POTE: ~2 = 1\1, ~36/35 = 54.665

Optimal ET sequence: 1ce, 21cef, 22

Badness: 0.044739