Porcupine family: Difference between revisions

Latest revision as of 10:23, 29 June 2025

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:

CTE: ~2 = 1200.000, ~10/9 = 164.166

error map: ⟨0.000 +5.547 -7.143]

POTE: ~2 = 1200.000, ~10/9 = 163.950

error map: ⟨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

Badness (Smith): 0.030778

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 → Very low accuracy temperaments.
Jamesbond → 7th-octave temperaments.

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

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

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

Badness (Smith): 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 = 1200.000, ~88/65 = 518.086
POTE: ~2 = 1200.000, ~88/65 = 518.209

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

Badness (Smith): 0.0305

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:

CTE: ~2 = 1200.000, ~10/9 = 163.203

error map: ⟨0.000 +8.435 -2.330 +10.394]

POTE: ~2 = 1200.000, ~10/9 = 162.880

error map: ⟨0.000 +9.405 -0.714 +8.455]

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 (Smith): 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 = 1200.000, ~11/10 = 163.105
POTE: ~2 = 1200.000, ~11/10 = 162.747

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 (Smith): 0.021562

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:

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

Minimax tuning:

13- and 15-odd-limit: ~10/9 = [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, 37f

Badness (Smith): 0.021276

Porcupinefish

See also: The Biosphere

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨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 = [2/13 0 0 0 1/13 -1/13⟩

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)
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 (Smith): 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 = 1200.000, ~11/10 = 163.378
POTE: ~2 = 1200.000, ~11/10 = 162.482

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 (Smith): 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 = 1200.000, ~11/10 = 163.678
POTE: ~2 = 1200.000, ~11/10 = 163.688

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 (Smith): 0.026043

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:

CTE: ~2 = 1200.000, ~10/9 = 161.306

error map: ⟨0.000 +14.126 +7.155 -20.583]

POTE: ~2 = 1200.000, ~10/9 = 159.691

error map: ⟨0.000 +18.971 +15.229 -6.048]

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15

Badness (Smith): 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]]

Optimal tunings:

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

Minimax tuning:

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

Optimal ET sequence: 7d, 8d, 15

Badness (Smith): 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 tunings:

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

Minimax tuning:

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

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

Badness (Smith): 0.019389

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:

CTE: ~2 = 1200.000, ~10/9 = 164.391

error map: ⟨0.000 +4.871 -8.270 +0.913]

POTE: ~2 = 1200.000, ~10/9 = 164.412

error map: ⟨0.000 +4.809 -8.375 +0.580]

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, 29, 51, 73c

Badness (Smith): 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]]

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 = [2/11 0 1/11 -1/11⟩

unchanged-interval (eigenmonzo) basis: 2.7/5

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

Badness (Smith): 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 = 1200.000, ~11/10 = 164.478
POTE: ~2 = 1200.000, ~11/10 = 164.953

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

Badness (Smith): 0.026543

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:

CTE: ~2 = 1200.000, ~10/9 = 166.094

error map: ⟨0.000 -0.236 -16.783 -9.607]

POTE: ~2 = 1200.000, ~10/9 = 166.041

error map: ⟨0.000 -0.077 -16.516 -10.299]

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, 94cd

Badness (Smith): 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 = 1200.000, ~11/10 = 165.925
POTE: ~2 = 1200.000, ~11/10 = 165.981

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 (Smith): 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 = 1200.000, ~11/10 = 166.046
POTE: ~2 = 1200.000, ~11/10 = 165.974

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 (Smith): 0.030233

Hystrix

Hystrix provides a less complex avenue to the 7-limit, with the generator taking on the role of approximating 8/7. Unfortunately in temperaments as in life you get what you pay for, and hystrix is very high in error due to the large disparity between typical porcupine generators and a justly-tuned 8/7, and is usually considered an exotemperament. A generator of 2\15 or 9\68 can be used for hystrix.

Subgroup: 2.3.5.7

Comma list: 36/35, 160/147

Mapping: [⟨1 2 3 3], ⟨0 -3 -5 -1]]

Optimal tunings:

CTE: ~2 = 1200.000, ~10/9 = 165.185

error map: ⟨0.000 +2.491 -12.236 +65.990]

POTE: ~2 = 1200.000, ~10/9 = 158.868

error map: ⟨0.000 +21.442 +19.348 +72.306]

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 (Smith): 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 = 1200.000, ~11/10 = 164.768
POTE: ~2 = 1200.000, ~11/10 = 158.750

Optimal ET sequence: 7, 8d, 15d

Badness (Smith): 0.026790

Hedgehog

See also: Sensamagic clan and 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)). Its ploidacot is diploid omega-tricot.

22edo provides an obvious tuning, which happens to be 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

Optimal tunings:

CTE: ~7/5 = 600.000, ~9/7 = 435.258

error map: ⟨0.000 +3.819 -10.024 +7.464]

POTE: ~7/5 = 600.000, ~9/7 = 435.648

error map: ⟨0.000 +4.989 -8.074 +9.414]

Optimal ET sequence: 8d, 14c, 22

Badness (Smith): 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]]

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, 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: [⟨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

Badness (Smith): 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 = 600.000, ~9/7 = 435.186
POTE: ~7/5 = 600.000, ~9/7 = 437.078

Optimal ET sequence: 14c, 22f

Badness (Smith): 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]]

Optimal tunings:

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

Optimal ET sequence: 22

Badness (Smith): 0.068406

Music

Phobos Light by Chris Vaisvil – in hedgehog[14], 22edo tuning.

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:

CTE: ~2 = 1200.000, ~21/20 = 81.914

error map: ⟨0.000 +6.559 -5.457 -14.569]

POTE: ~2 = 1200.000, ~21/20 = 82.505

error map: ⟨0.000 +3.012 -11.368 -16.342]

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

Badness (Smith): 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]]

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

Badness (Smith): 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 = 1200.000, ~21/20 = 81.912
POTE: ~2 = 1200.000, ~21/20 = 82.530

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

Badness (Smith): 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 = 1200.000, ~21/20 = 82.034
POTE: ~2 = 1200.000, ~21/20 = 81.759

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

Badness (Smith): 0.029816

Music

Nautilus Reverie by Igliashon Calvin Jones-Coolidge

Ammonite

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 5 8 10], ⟨0 -9 -15 -19]]

mapping generators: ~2, ~9/7

Optimal tunings:

CTE: ~2 = 1200.000, ~9/7 = 454.550

error map: ⟨0.000 +7.095 -4.564 -5.276]

POTE: ~2 = 1200.000, ~9/7 = 454.448

error map: ⟨0.000 +8.009 -3.040 -3.346]

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

Badness (Smith): 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]]

Optimal tunings:

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

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

Badness (Smith): 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 = 1200.000, ~13/10 = 454.480
POTE: ~2 = 1200.000, ~13/10 = 454.529

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

Badness (Smith): 0.027168

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:

CTE: ~2 = 1200.000, ~36/35 = 54.804

error map: ⟨0.000 +4.809 -8.374 +11.958]

POTE: ~2 = 1200.000, ~36/35 = 54.384

error map: ⟨0.000 +8.585 -2.081 +13.636]

Optimal ET sequence: 1c, 21c, 22

Badness (Smith): 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 = 1200.000, ~36/35 = 54.702
POTE: ~2 = 1200.000, ~36/35 = 54.376

Optimal ET sequence: 1ce, 21ce, 22

Badness (Smith): 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 = 1200.000, ~36/35 = 54.575
POTE: ~2 = 1200.000, ~36/35 = 54.665

Optimal ET sequence: 1ce, 21cef, 22

Badness (Smith): 0.044739

@@ Line 5: / Line 5: @@
 | ja =
 }}
-The 5-limit parent comma for the '''porcupine family''' is [[250/243]], the maximal diesis or porcupine comma. Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 3 5 1 }} for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a fourth, with two more giving the minor sixth. In fact, (10/9)<sup>3</sup> = 4/3 × 250/243, and (10/9)<sup>5</sup> = 8/5 × (250/243)<sup>2</sup>. [[22edo|3\22]] is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.
+{{Technical data page}}
+The '''porcupine family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[porcupine comma]], [[250/243]], also called the maximal diesis.
-The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. That means
-* [[64/63]], the archytas comma, for [[#Septimal porcupine|septimal porcupine]],
-* [[36/35]], the septimal quarter tone, for [[#Hystrix|hystrix]],
-* [[50/49]], the jubilisma, for [[#Hedgehog|hedgehog]], and
-* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].
-All these 7-limit extensions notably share the same 2.3.5.11 subgroup, known as ''porkypine''.
-Temperaments discussed elsewhere include [[Dicot family #Jamesbond|jamesbond]].
 == Porcupine ==
 {{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> }}. 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 24: / Line 17: @@
 [[Comma list]]: 250/243
-[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}]
+{{Mapping|legend=1| 1 2 3 | 0 -3 -5 }}
 : mapping generators: ~2, ~10/9
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 164.1659
+[[Optimal tuning]]s:
+* [[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 monotone]]: ~10/9 = [150.000, 171.429] (1\8 to 1\7)
 * 5-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
-* 5-odd-limit diamond monotone and tradeoff: ~10/9 = [157.821, 166.015]
-{{Val list|legend=1| 7, 15, 22, 95c }}
+{{Optimal ET sequence|legend=1| 7, 15, 22, 95c }}
+[[Badness]] (Smith): 0.030778
+=== 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 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 sharp of 29edo.
-[[Badness]]: 0.030778
+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.
+Temperaments discussed elsewhere include:
+* [[Oxygen]] → [[Very low accuracy temperaments #Oxygen|Very low accuracy temperaments]].
+* [[Jamesbond]] → [[7th-octave temperaments #Jamesbond|7th-octave temperaments]].
+==== Subgroup extensions ====
+Noting that {{nowrap| 250/243 {{=}} ([[55/54]])⋅([[100/99]]) {{=}} S10<sup>2</sup>⋅[[121/120|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) ===
@@ Line 44: / Line 60: @@
 Comma list: 55/54, 100/99
-Sval mapping: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -4 }}]
+Sval mapping: {{mapping| 1 2 3 4 | 0 -3 -5 -4 }}
-Gencom mapping: [{{val| 1 2 3 0 4 }}, {{val| 0 -3 -5 0 -4 }}]
+Gencom mapping: {{mapping| 1 2 3 0 4 | 0 -3 -5 0 -4 }}
-Gencom: [2 10/9; 55/54, 100/99]
+: gencom: [2 10/9; 55/54, 100/99]
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.8867
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 163.887
+* POTE: ~2 = 1200.000, ~11/10 = 164.078
-Optimal GPV sequence: {{val list| 7, 15, 22, 73ce, 95ce }}
+{{Optimal ET sequence|legend=0| 7, 15, 22, 73ce, 95ce }}
-Badness: 0.0097
+Badness (Smith): 0.0097
 ==== Undecimation ====
@@ Line 61: / Line 79: @@
 Comma list: 55/54, 100/99, 512/507
-Sval mapping: [{{val| 1 5 8 8 2 }}, {{val| 0 -6 -10 -8 3 }}]
+Sval mapping: {{mapping| 1 5 8 8 2 | 0 -6 -10 -8 3 }}
 : sval mapping generators: ~2, ~65/44
-Optimal tuning (CTE): ~2 = 1\1, ~88/65 = 518.2094
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~88/65 = 518.086
+* POTE: ~2 = 1200.000, ~88/65 = 518.209
-Optimal GPV sequence: {{val list| 7, 23bc, 30, 37, 44 }}
+{{Optimal ET sequence|legend=0| 7, 23bc, 30, 37, 44 }}
-Badness: 0.0305
+Badness (Smith): 0.0305
 == Septimal porcupine ==
 {{Main| 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.
+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
@@ Line 80: / Line 100: @@
 [[Comma list]]: 64/63, 250/243
-[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -3 -5 6 }}]
+{{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
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 163.2032
+: [[error map]]: {{val| 0.000 +8.435 -2.330 +10.394 }}
+* [[POTE]]: ~2 = 1200.000, ~10/9 = 162.880
+: error map: {{val| 0.000 +9.405 -0.714 +8.455 }}
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
-: [[Eigenmonzo 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]]: 2.9/7
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
 [[Tuning ranges]]:
@@ Line 96: / Line 118: @@
 * 7-odd-limit [[diamond tradeoff]]: ~10/9 = [157.821, 166.015]
 * 9-odd-limit diamond tradeoff: ~10/9 = [157.821, 182.404]
-* 7- and 9-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]
-{{Val list|legend=1| 7, 15, 22, 37, 59, 81bd }}
+{{Optimal ET sequence|legend=1| 7, 15, 22, 37, 59, 81bd }}
-[[Badness]]: 0.041057
+[[Badness]] (Smith): 0.041057
 === 11-limit ===
@@ Line 107: / Line 128: @@
 Comma list: 55/54, 64/63, 100/99
-Mapping: [{{val| 1 2 3 2 4 }}, {{val| 0 -3 -5 6 -4 }}]
+Mapping: {{mapping| 1 2 3 2 4 | 0 -3 -5 6 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.1055
+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 basis: 2.9/7
+: 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]
-* 11-odd-limit diamond monotone and tradeoff: ~11/10 = [160.000, 163.636]
-Optimal GPV sequence: {{val list| 7, 15, 22, 37, 59 }}
+{{Optimal ET sequence|legend=0| 7, 15, 22, 37, 59 }}
-Badness: 0.021562
+Badness (Smith): 0.021562
+==== Porcupinefowl ====
+This extension used to be ''tridecimal porcupine''.
-==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
 Comma list: 40/39, 55/54, 64/63, 66/65
-Mapping: [{{val| 1 2 3 2 4 4 }}, {{val| 0 -3 -5 6 -4 -2 }}]
+Mapping: {{mapping| 1 2 3 2 4 4 | 0 -3 -5 6 -4 -2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.4425
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 163.442
+* POTE: ~2 = 1200.000, ~11/10 = 162.708
 Minimax tuning:
 * 13- and 15-odd-limit: ~10/9 = {{monzo| 1 0 0 0 -1/4 }}
-: Eigenmonzo basis: 2.11
+: unchanged-interval (eigenmonzo) basis: 2.11
 Tuning ranges:
@@ Line 141: / Line 167: @@
 * 15-odd-limit diamond monotone: ~11/10 = 163.636 (3\22)
 * 13- and 15-odd-limit diamond tradeoff: ~11/10 = [138.573, 182.404]
-* 13-odd-limit diamond monotone and tradeoff: ~11/10 = [160.000, 163.636]
-* 15-odd-limit diamond monotone and tradeoff: ~11/10 = 163.636
-Optimal GPV sequence: {{val list| 7, 15, 22f, 37f }}
+{{Optimal ET sequence|legend=0| 7, 15, 22f, 37f }}
-Badness: 0.021276
+Badness (Smith): 0.021276
 ==== Porcupinefish ====
@@ Line 155: / Line 179: @@
 Comma list: 55/54, 64/63, 91/90, 100/99
-Mapping: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]
+Mapping: {{mapping| 1 2 3 2 4 6 | 0 -3 -5 6 -4 -17 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 162.6361
+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 basis: 2.13/11
+: unchanged-interval (eigenmonzo) basis: 2.13/11
 Tuning ranges:
@@ Line 167: / Line 193: @@
 * 15-odd-limit diamond monotone: ~10/9 = 162.162 (5\37)
 * 13- and 15-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
-* 13-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 162.162]
-* 15-odd-limit diamond monotone and tradeoff: ~10/9 = 162.162
-Optimal GPV sequence: {{val list| 15, 22, 37 }}
+{{Optimal ET sequence|legend=0| 15, 22, 37 }}
-Badness: 0.025314
+Badness (Smith): 0.025314
 ==== Pourcup ====
@@ Line 179: / Line 203: @@
 Comma list: 55/54, 64/63, 100/99, 196/195
-Mapping: [{{val| 1 2 3 2 4 1 }}, {{val| 0 -3 -5 6 -4 20 }}]
+Mapping: {{mapping| 1 2 3 2 4 1 | 0 -3 -5 6 -4 20 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.3781
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 163.378
+* POTE: ~2 = 1200.000, ~11/10 = 162.482
 Minimax tuning:
 * 13- and 15-odd-limit: ~11/10 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}
-: Eigenmonzo basis: 2.13/7
+: unchanged-interval (eigenmonzo) basis: 2.13/7
-Optimal GPV sequence: {{val list| 15f, 22f, 37, 59f }}
+{{Optimal ET sequence|legend=0| 15f, 22f, 37, 59f }}
-Badness: 0.035130
+Badness (Smith): 0.035130
 ==== Porkpie ====
@@ Line 196: / Line 222: @@
 Comma list: 55/54, 64/63, 65/63, 100/99
-Mapping: [{{val| 1 2 3 2 4 3 }}, {{val| 0 -3 -5 6 -4 5 }}]
+Mapping: {{mapping| 1 2 3 2 4 3 | 0 -3 -5 6 -4 5 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 163.6778
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 163.678
+* POTE: ~2 = 1200.000, ~11/10 = 163.688
 Minimax tuning:
 * 13- and 15-odd-limit: ~11/10 = {{monzo| 1/6 -1/6 0 1/12 }}
-: Eigenmonzo basis: 2.9/7
+: unchanged-interval (eigenmonzo) basis: 2.9/7
-Optimal GPV sequence: {{val list| 7, 15f, 22 }}
+{{Optimal ET sequence|legend=0| 7, 15f, 22 }}
-Badness: 0.026043
+Badness (Smith): 0.026043
 == 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
 [[Comma list]]: 28/27, 126/125
-[[Mapping]]: [{{val| 1 2 3 4 }}, {{val| 0 -3 -5 -9 }}]
+{{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
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3063
+: [[error map]]: {{val| 0.000 +14.126 +7.155 -20.583 }}
+* [[POTE]]: ~2 = 1200.000, ~10/9 = 159.691
+: error map: {{val| 0.000 +18.971 +15.229 -6.048 }}
 [[Minimax tuning]]:
-* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis]]: 2.7
+* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
-{{Val list|legend=1| 7d, 8d, 15 }}
+{{Optimal ET sequence|legend=1| 7d, 8d, 15 }}
-[[Badness]]: 0.040650
+[[Badness]] (Smith): 0.040650
 === 11-limit ===
@@ Line 231: / Line 263: @@
 Comma list: 28/27, 55/54, 77/75
-Mapping: [{{val| 1 2 3 4 4 }}, {{val| 0 -3 -5 -9 -4 }}]
+Mapping: {{mapping| 1 2 3 4 4 | 0 -3 -5 -9 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.3646
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 161.365
+* POTE: ~2 = 1200.000, ~11/10 = 159.807
 Minimax tuning:
-* 11-odd-limit eigenmonzo basis: 2.7
+* 11-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
-Optimal GPV sequence: {{Val list| 7d, 8d, 15 }}
+{{Optimal ET sequence|legend=0| 7d, 8d, 15 }}
-Badness: 0.022325
+Badness (Smith): 0.022325
-=== 13-limit  ===
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
 Comma list: 28/27, 40/39, 55/54, 66/65
-Mapping: [{{val| 1 2 3 4 4 4 }}, {{val| 0 -3 -5 -9 -4 -2 }}]
+Mapping: {{mapping| 1 2 3 4 4 4 | 0 -3 -5 -9 -4 -2 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 161.6312
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 161.631
+* POTE: ~2 = 1200.000, ~11/10 = 158.805
 Minimax tuning:
-* 13- and 15-odd-limit eigenmonzo basis: 2.7
+* 13- and 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7
-Optimal GPV sequence: {{Val list| 7d, 8d, 15, 38bceff }}
+{{Optimal ET sequence|legend=0| 7d, 8d, 15, 38bceff }}
-Badness: 0.019389
+Badness (Smith): 0.019389
 == Porky ==
+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
 [[Comma list]]: 225/224, 250/243
-[[Mapping]]: [{{val| 1 2 3 5 }}, {{val| 0 -3 -5 -16 }}]
+{{Mapping|legend=1| 1 2 3 5 | 0 -3 -5 -16 }}
-{{Multival|legend=1| 3 5 16 1 17 23 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 164.3913
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~10/9 = 164.391
+: [[error map]]: {{val| 0.000 +4.871 -8.270 +0.913 }}
+* [[POTE]]: ~2 = 1200.000, ~10/9 = 164.412
+: error map: {{val| 0.000 +4.809 -8.375 +0.580 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}
-: [[Eigenmonzo basis]]: 2.7/5
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
-{{Val list|legend=1| 7d, 15d, 22, 29, 51, 73c }}
+{{Optimal ET sequence|legend=1| 7d, 15d, 22, 29, 51, 73c }}
-[[Badness]]: 0.054389
+[[Badness]] (Smith): 0.054389
 === 11-limit ===
@@ Line 282: / Line 322: @@
 Comma list: 55/54, 100/99, 225/224
-Mapping: [{{val| 1 2 3 5 4 }}, {{val| 0 -3 -5 -16 -4 }}]
+Mapping: {{mapping| 1 2 3 5 4 | 0 -3 -5 -16 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.3207
+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 basis: 2.7/5
+: unchanged-interval (eigenmonzo) basis: 2.7/5
-Optimal GPV sequence: {{val list| 7d, 15d, 22, 51 }}
+{{Optimal ET sequence|legend=0| 7d, 15d, 22, 51 }}
-Badness: 0.027268
+Badness (Smith): 0.027268
 === 13-limit ===
@@ Line 299: / Line 341: @@
 Comma list: 55/54, 65/64, 91/90, 100/99
-Mapping: [{{val| 1 2 3 5 4 3 }}, {{val| 0 -3 -5 -16 -4 5 }}]
+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 tuning (CTE): ~2 = 1\1, ~11/10 = 164.4782
+{{Optimal ET sequence|legend=0| 7d, 22, 29, 51f, 80cdeff }}
-Optimal GPV sequence: {{val list| 7d, 22, 29, 51f, 80cdeff }}
+Badness (Smith): 0.026543
-Badness: 0.026543
+; Music
+* [https://www.youtube.com/watch?v=CN4cLOyaVGE ''Improvisation in 29edo''] (2024) by [[Budjarn Lambeth]] – in Palace scale, 29edo tuning
 == Coendou ==
+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
 [[Comma list]]: 250/243, 525/512
-[[Mapping]]: [{{val| 1 2 3 1 }}, {{val| 0 -3 -5 13 }}]
+{{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
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 166.0938
+: [[error map]]: {{val| 0.000 -0.236 -16.783 -9.607 }}
+* [[POTE]]: ~2 = 1200.000, ~10/9 = 166.041
+: error map: {{val| 0.000 -0.077 -16.516 -10.299 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 2/3 -1/3 }}
-: [[Eigenmonzo basis]]: 2.3
+: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3
-{{Val list|legend=1| 7, 22d, 29, 65c, 94cd }}
+{{Optimal ET sequence|legend=1| 7, 22d, 29, 65c, 94cd }}
-[[Badness]]: 0.118344
+[[Badness]] (Smith): 0.118344
 === 11-limit ===
@@ Line 331: / Line 382: @@
 Comma list: 55/54, 100/99, 525/512
-Mapping: [{{val| 1 2 3 1 4 }}, {{val| 0 -3 -5 13 -4 }}]
+Mapping: {{mapping| 1 2 3 1 4 | 0 -3 -5 13 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 165.9246
+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 basis: 2.3
+: unchanged-interval (eigenmonzo) basis: 2.3
-Optimal GPV sequence: {{val list| 7, 22d, 29, 65ce }}
+{{Optimal ET sequence|legend=0| 7, 22d, 29, 65ce }}
-Badness: 0.049669
+Badness (Smith): 0.049669
 === 13-limit ===
@@ Line 348: / Line 401: @@
 Comma list: 55/54, 65/64, 100/99, 105/104
-Mapping: [{{val| 1 2 3 1 4 3 }}, {{val| 0 -3 -5 13 -4 5 }}]
+Mapping: {{mapping| 1 2 3 1 4 3 | 0 -3 -5 13 -4 5 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 166.0459
+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 basis: 2.3
+: unchanged-interval (eigenmonzo) basis: 2.3
-Optimal GPV sequence: {{val list| 7, 22d, 29, 65cef }}
+{{Optimal ET sequence|legend=0| 7, 22d, 29, 65cef }}
-Badness: 0.030233
+Badness (Smith): 0.030233
 == Hystrix ==
-Hystrix provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits.
+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
@@ Line 367: / Line 422: @@
 [[Comma list]]: 36/35, 160/147
-[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -1 }}]
+{{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
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 165.1845
+: [[error map]]: {{val| 0.000 +2.491 -12.236 +65.990 }}
+* [[POTE]]: ~2 = 1200.000, ~10/9 = 158.868
+: error map: {{val| 0.000 +21.442 +19.348 +72.306 }}
 [[Minimax tuning]]:
 * [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/9 = {{monzo| 3/5 0 -1/5 }}
-: [[Eigenmonzo basis]]: 2.5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
-{{Val list|legend=1| 7, 8d, 15d }}
+{{Optimal ET sequence|legend=1| 7, 8d, 15d }}
-[[Badness]]: 0.044944
+[[Badness]] (Smith): 0.044944
 === 11-limit ===
@@ Line 386: / Line 443: @@
 Comma list: 22/21, 36/35, 80/77
-Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -3 -5 -1 -4 }}]
+Mapping: {{mapping| 1 2 3 3 4 | 0 -3 -5 -1 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~11/10 = 164.7684
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~11/10 = 164.768
+* POTE: ~2 = 1200.000, ~11/10 = 158.750
-Optimal GPV sequence: {{val list| 7, 8d, 15d }}
+{{Optimal ET sequence|legend=0| 7, 8d, 15d }}
-Badness: 0.026790
+Badness (Smith): 0.026790
-== Oxygen ==
+== Hedgehog ==
-[[Subgroup]]: 2.3.5.7
+{{See also| Sensamagic clan | Stearnsmic clan }}
-[[Comma list]]: 21/20, 175/162
-[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -2 }}]
+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)). Its ploidacot is diploid omega-tricot.
-{{Multival|legend=1| 3 5 2 1 -5 -9 }}
+edo provides an obvious tuning, which happens to be the only [[patent val|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.
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~10/9 = 161.3408
-{{Val list|legend=1| 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. 22edo provides the obvious 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 419: / Line 464: @@
 [[Comma list]]: 50/49, 245/243
-[[Mapping]]: [{{val| 2 1 1 2 }}, {{val| 0 3 5 5 }}]
+{{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 tuning]] ([[CTE]]): ~7/5 = 1\2, ~9/7 = 435.2580
+{{Optimal ET sequence|legend=1| 8d, 14c, 22 }}
-{{Val list|legend=1| 8d, 14c, 22 }}
+[[Badness]] (Smith): 0.043983
-[[Badness]]: 0.043983
 === 11-limit ===
@@ Line 436: / Line 483: @@
 Comma list: 50/49, 55/54, 99/98
-Mapping: [{{val| 2 1 1 2 4 }}, {{val| 0 3 5 5 4 }}]
+Mapping: {{mapping| 2 1 1 2 4 | 0 3 5 5 4 }}
-Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.5281
+Optimal tunings:
+* CTE: ~7/5 = 600.000, ~9/7 = 435.528
+* POTE: ~7/5 = 600.000, ~9/7 = 435.386
-Optimal GPV sequence: {{val list| 8d, 14c, 22, 58ce }}
+{{Optimal ET sequence|legend=0| 8d, 14c, 22, 58ce }}
-Badness: 0.023095
+Badness (Smith): 0.023095
 ==== 13-limit ====
@@ Line 449: / Line 498: @@
 Comma list: 50/49, 55/54, 65/63, 99/98
-Mapping: [{{val| 2 1 1 2 4 3 }}, {{val| 0 3 5 5 4 6 }}]
+Mapping: {{mapping| 2 1 1 2 4 3 | 0 3 5 5 4 6 }}
-Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 436.3087
+Optimal tunings:
+* CTE: ~7/5 = 600.000, ~9/7 = 436.309
+* POTE: ~7/5 = 600.000, ~9/7 = 435.861
-Optimal GPV sequence: {{val list| 8d, 14cf, 22 }}
+{{Optimal ET sequence|legend=0| 8d, 14cf, 22 }}
-Badness: 0.021516
+Badness (Smith): 0.021516
 ==== Urchin ====
@@ Line 462: / Line 513: @@
 Comma list: 40/39, 50/49, 55/54, 66/65
-Mapping: [{{val| 2 1 1 2 4 6 }}, {{val| 0 3 5 5 4 2 }}]
+Mapping: {{mapping| 2 1 1 2 4 6 | 0 3 5 5 4 2 }}
-Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.1856
+Optimal tunings:
+* CTE: ~7/5 = 600.000, ~9/7 = 435.186
+* POTE: ~7/5 = 600.000, ~9/7 = 437.078
-Optimal GPV sequence: {{val list| 14c, 22f }}
+{{Optimal ET sequence|legend=0| 14c, 22f }}
-Badness: 0.025233
+Badness (Smith): 0.025233
 === Hedgepig ===
@@ Line 475: / Line 528: @@
 Comma list: 50/49, 245/243, 385/384
-Mapping: [{{val| 2 1 1 2 12 }}, {{val| 0 3 5 5 -7 }}]
+Mapping: {{mapping| 2 1 1 2 12 | 0 3 5 5 -7 }}
-Optimal tuning (CTE): ~7/5 = 1\2, ~9/7 = 435.3289
+Optimal tunings:
+* CTE: ~7/5 = 600.000, ~9/7 = 435.329
+* POTE: ~7/5 = 600.000, ~9/7 = 435.425
-Optimal GPV sequence: {{val list| 22 }}
+{{Optimal ET sequence|legend=0| 22 }}
-Badness: 0.068406
+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]]] to 22edo.
+* [http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 ''Phobos Light''] by [[Chris Vaisvil]] – in [[hedgehog14|hedgehog[14]]], 22edo tuning.
 == Nautilus ==
+Nautilus tempers out 49/48 and may be described as the {{nowrap| 14c & 15 }} temperament. Its ploidacot is omega-hexacot.
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 49/48, 250/243
-[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -6 -10 -3 }}]
+{{Mapping|legend=1| 1 2 3 3 | 0 -6 -10 -3 }}
 : 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
+: [[error map]]: {{val| 0.000 +6.559 -5.457 -14.569 }}
+* [[POTE]]: ~2 = 1200.000, ~21/20 = 82.505
+: error map: {{val| 0.000 +3.012 -11.368 -16.342 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~21/20 = 81.9143
+{{Optimal ET sequence|legend=1| 14c, 15, 29, 44d }}
-{{Val list|legend=1| 14c, 15, 29 }}
+[[Badness]] (Smith): 0.057420
-[[Badness]]: 0.057420
 === 11-limit ===
@@ Line 508: / Line 567: @@
 Comma list: 49/48, 55/54, 245/242
-Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -6 -10 -3 -8 }}]
+Mapping: {{mapping| 1 2 3 3 4 | 0 -6 -10 -3 -8 }}
-Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.8017
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~21/20 = 81.802
+* POTE: ~2 = 1200.000, ~21/20 = 82.504
-Optimal GPV sequence: {{val list| 14c, 15, 29 }}
+{{Optimal ET sequence|legend=0| 14c, 15, 29, 44d }}
-Badness: 0.026023
+Badness (Smith): 0.026023
 ==== 13-limit ====
@@ Line 521: / Line 582: @@
 Comma list: 49/48, 55/54, 91/90, 100/99
-Mapping: [{{val| 1 2 3 3 4 5 }}, {{val| 0 -6 -10 -3 -8 -19 }}]
+Mapping: {{mapping| 1 2 3 3 4 5 | 0 -6 -10 -3 -8 -19 }}
-Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 81.9123
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~21/20 = 81.912
+* POTE: ~2 = 1200.000, ~21/20 = 82.530
-Optimal GPV sequence: {{val list| 14cf, 15, 29 }}
+{{Optimal ET sequence|legend=0| 14cf, 15, 29, 44d }}
-Badness: 0.022285
+Badness (Smith): 0.022285
 ==== Belauensis ====
@@ Line 534: / Line 597: @@
 Comma list: 40/39, 49/48, 55/54, 66/65
-Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -6 -10 -3 -8 -4 }}]
+Mapping: {{mapping| 1 2 3 3 4 4 | 0 -6 -10 -3 -8 -4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~21/20 = 82.0342
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~21/20 = 82.034
+* POTE: ~2 = 1200.000, ~21/20 = 81.759
-Optimal GPV sequence: {{val list| 14c, 15 }}
+{{Optimal ET sequence|legend=0| 14c, 15, 29f, 44dff }}
-Badness: 0.029816
+Badness (Smith): 0.029816
 ; Music
@@ Line 546: / Line 611: @@
 == Ammonite ==
+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]]: [{{val| 1 5 8 10 }}, {{val| 0 -9 -15 -19 }}]
+{{Mapping|legend=1| 1 5 8 10 | 0 -9 -15 -19 }}
 : mapping generators: ~2, ~9/7
-{{Multival|legend=1| 9 15 19 3 5 2 }}
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~9/7 = 454.550
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~9/7 = 454.5500
+: [[error map]]: {{val| 0.000 +7.095 -4.564 -5.276 }}
+* [[POTE]]: ~2 = 1200.000, ~9/7 = 454.448
+: error map: {{val| 0.000 +8.009 -3.040 -3.346 }}
-{{Val list|legend=1| 8d, 21cd, 29, 37, 66 }}
+{{Optimal ET sequence|legend=1| 8d, 21cd, 29, 37, 66 }}
-[[Badness]]: 0.107686
+[[Badness]] (Smith): 0.107686
 === 11-limit ===
@@ Line 567: / Line 636: @@
 Comma list: 55/54, 100/99, 686/675
-Mapping: [{{val| 1 5 8 10 8 }}, {{val| 0 -9 -15 -19 -12 }}]
+Mapping: {{mapping| 1 5 8 10 8 | 0 -9 -15 -19 -12 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 454.5050
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 454.505
+* POTE: ~2 = 1200.000, ~9/7 = 454.512
-Optimal GPV sequence: {{val list| 8d, 21cde, 29, 37, 66 }}
+{{Optimal ET sequence|legend=0| 8d, 21cde, 29, 37, 66 }}
-Badness: 0.045694
+Badness (Smith): 0.045694
 === 13-limit ===
@@ Line 580: / Line 651: @@
 Comma list: 55/54, 91/90, 100/99, 169/168
-Mapping: [{{val| 1 5 8 10 8 9 }}, {{val| 0 -9 -15 -19 -12 -14 }}]
+Mapping: {{mapping| 1 5 8 10 8 9 | 0 -9 -15 -19 -12 -14 }}
-Optimal tuning (CTE): ~2 = 1\1, ~13/10 = 454.4798
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~13/10 = 454.480
+* POTE: ~2 = 1200.000, ~13/10 = 454.529
-Optimal GPV sequence: {{val list| 8d, 21cdef, 29, 37, 66 }}
+{{Optimal ET sequence|legend=0| 8d, 21cdef, 29, 37, 66 }}
-Badness: 0.027168
+Badness (Smith): 0.027168
 == 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
 [[Comma list]]: 250/243, 1728/1715
-[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -9 -15 -4 }}]
+{{Mapping|legend=1| 1 2 3 3 | 0 -9 -15 -4 }}
 : 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
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 54.8040
+: [[error map]]: {{val| 0.000 +4.809 -8.374 +11.958 }}
+* [[POTE]]: ~2 = 1200.000, ~36/35 = 54.384
+: error map: {{val| 0.000 +8.585 -2.081 +13.636 }}
-{{Val list|legend=1| 1c, 21c, 22 }}
+{{Optimal ET sequence|legend=1| 1c, 21c, 22 }}
-[[Badness]]: 0.115304
+[[Badness]] (Smith): 0.115304
 === 11-limit ===
@@ Line 610: / Line 687: @@
 Comma list: 55/54, 100/99, 352/343
-Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -9 -15 -4 -12 }}]
+Mapping: {{mapping| 1 2 3 3 4 | 0 -9 -15 -4 -12 }}
-Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.7019
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~36/35 = 54.702
+* POTE: ~2 = 1200.000, ~36/35 = 54.376
-Optimal GPV sequence: {{val list| 1ce, 21ce, 22 }}
+{{Optimal ET sequence|legend=0| 1ce, 21ce, 22 }}
-Badness: 0.051319
+Badness (Smith): 0.051319
 === 13-limit ===
@@ Line 623: / Line 702: @@
 Comma list: 55/54, 65/63, 100/99, 352/343
-Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -9 -15 -4 -12 -7 }}]
+Mapping: {{mapping| 1 2 3 3 4 4 | 0 -9 -15 -4 -12 -7 }}
-Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 54.5751
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~36/35 = 54.575
+* POTE: ~2 = 1200.000, ~36/35 = 54.665
-Optimal GPV sequence: {{val list| 1ce, 21cef, 22 }}
+{{Optimal ET sequence|legend=0| 1ce, 21cef, 22 }}
-Badness: 0.044739
+Badness (Smith): 0.044739
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Porcupine family| ]] <!-- main article -->
-[[Category:Porcupine]]
+[[Category:Porcupine| ]] <!-- key article -->
 [[Category:Rank 2]]

Porcupine family: Difference between revisions

Latest revision as of 10:23, 29 June 2025

Porcupine

Overview to extensions

7-limit extensions

Subgroup extensions

2.3.5.11 subgroup (porkypine)

Undecimation

Septimal porcupine

11-limit

Porcupinefowl

Porcupinefish

Pourcup

Porkpie

Opossum

11-limit

13-limit

Porky

11-limit

13-limit

Coendou

11-limit

13-limit

Hystrix

11-limit

Hedgehog

11-limit

13-limit

Urchin

Hedgepig

Nautilus

11-limit

13-limit

Belauensis

Ammonite

11-limit

13-limit

Ceratitid

11-limit

13-limit

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