User:VectorGraphics/Porcupine family

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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. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, (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.

It most naturally manifests as a 2.3.5.11 subgroup temperament, where it tempers out 100/99 and 55/54 equating the generator to 11/10 as well as 10/9.

Porcupine

Main article: Porcupine

5-limit

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:

Optimal ET sequence: 7, 15, 22, 95c

Badness (Smith): 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 = 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

Strong extensions

Main article: Porcupine

Septimal 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. This extends porcupine to the full 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⟩
eigenmonzo (unchanged-interval) basis: 2.9/7

Tuning ranges:

  • 11-odd-limit diamond monotone: ~11/10 = [160.000, 163.636] (2\15 to 3\22)
  • 11-odd-limit diamond tradeoff: ~11/10 = [150.637, 182.404]

(7-limit) Optimal ET sequence: 7, 15, 22, 37, 59, 81bd

(11-limit) Optimal ET sequence: 7, 15, 22, 37, 59

Badness (Smith): 0.021562

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⟩
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 (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⟩
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 (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⟩
eigenmonzo (unchanged-interval) 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⟩
eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 7, 15f, 22

Badness (Smith): 0.026043

Porky

Porky can be described as 7d & 22, suggesting a less sharp perfect fifth. 7\51 is a good generator.

Subgroup: 2.3.5.7.11

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 = 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⟩
eigenmonzo (unchanged-interval) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 29, 51, 73c (7-limit)

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

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: 2.3.5.7.11

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 tunings:

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

Minimax tuning:

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

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

Badness (Smith): 0.019389

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: 2.3.5.7.11

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⟩
eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65c, 94cd (7-limit)

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⟩
eigenmonzo (unchanged-interval) 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: 2.3.5.7.11

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

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 = 1200.000, ~10/9 = 161.341
error map: ⟨0.000 +14.023 +6.982 -91.507]
  • POTE: ~2 = 1200.000, ~10/9 = 169.112
error map: ⟨0.000 -9.291 -31.873 -107.050]

Optimal ET sequence: 1c, …, 6bcd, 7d

Badness (Smith): 0.059866

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)). 22edo provides the obvious (i.e the only patent val) tuning, but if you are looking for an alternative you could try the ⟨146 232 338 411] (146bccdd) val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14-note mos gives scope for harmony while stopping well short of 22. A related temperament is echidna, which offers much more accuracy. They merge on 22edo.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨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 = 600.000, ~9/7 = 435.528
  • POTE: ~7/5 = 600.000, ~9/7 = 435.386

Optimal ET sequence: 8d, 14c, 22 (7-limit)

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]]

Wedgie: ⟨⟨ 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: 22

Badness (Smith): 0.068406

Music

Nautilus

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 = 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

Ammonite

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

Optimal ET sequence: 8d, 21cd, 29, 37, 66 (7-limit)

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

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: 1c, 21c, 22 (7-limit)

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

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