Porcupine family

The 5-limit parent comma for the porcupine family is 250/243, the maximal diesis or porcupine comma. Its monzo is [1 -5 3⟩, and flipping that yields ⟨⟨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)³ = 4/3 × 250/243, and (10/9)⁵ = 8/5 × (250/243)². 3\22 is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.

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.

All these 7-limit extensions notably share the same 2.3.5.11 subgroup, porkypine.

Temperaments discussed elsewhere include opossum, oxygen, and jamesbond.

Porcupine

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

POTE generator: ~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]
• 5-odd-limit diamond monotone and tradeoff: ~10/9 = [157.821, 166.015]

Optimal GPV sequence: 7, 15, 22, 95c

Badness: 0.030778

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]

POTE generator: ~11/10 = 164.0777

Optimal GPV sequence: 7, 15, 22, 37, 73ce, 95ce

Undecimation

Subgroup: 2.3.5.11.13

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

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

POTE generator: ~88/65 = 518.2094

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

Septimal porcupine

Main article: Porcupine

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]]

POTE generator: ~10/9 = 162.880

Minimax tuning:

• 7-odd-limit: ~10/9 = [3/5 0 -1/5⟩

Eigenmonzos (unchanged intervals): 2, 5/4

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

Eigenmonzos (unchanged intervals): 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]
• 7- and 9-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]

Optimal GPV sequence: 7, 15, 22, 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]]

POTE generator: ~10/9 = 162.747

Minimax tuning:

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

Eigenmonzos (unchanged intervals): 2, 9/7

Tuning ranges:

• 11-odd-limit diamond monotone: ~10/9 = [160.000, 163.636] (2\15 to 3\22)
• 11-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]
• 11-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]

Optimal GPV 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]]

POTE generator: ~10/9 = 162.708

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [1 0 0 0 -1/4⟩

Eigenmonzo (unchanged intervals): 2, 11/8

Tuning ranges:

• 13-odd-limit diamond monotone: ~10/9 = [160.000, 163.636] (2\15 to 3\22)
• 15-odd-limit diamond monotone: ~10/9 = 163.636 (3\22)
• 13- and 15-odd-limit diamond tradeoff: ~10/9 = [138.573, 182.404]
• 13-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]
• 15-odd-limit diamond monotone and tradeoff: ~10/9 = 163.636

Optimal GPV 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]]

POTE generator: ~10/9 = 162.277

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [2/13 0 0 0 1/13 -1/13⟩

Eigenmonzos (unchanged intervals): 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]
• 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: 15, 22, 37, 59, 96b

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]]

POTE generator: ~10/9 = 162.482

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [1/14 0 0 -1/14 0 1/14⟩

Eigenmonzos (unchanged intervals): 2, 14/13

Optimal GPV sequence: 15f, 22f, 37

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]]

POTE generator: ~10/9 = 163.688

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [1/6 -1/6 0 1/12⟩

Eigenmonzos (unchanged intervals): 2, 9/7

Optimal GPV sequence: 7, 15f, 22

Badness: 0.026043

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.

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]]

POTE generator: ~8/7 = 158.868

Minimax tuning:

• 7- and 9-odd-limit: ~8/7 = [3/5 0 -1/5⟩

Eigenmonzos (unchanged intervals): 2, 5/4

Optimal GPV 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]]

POTE generator: ~8/7 = 158.750

Optimal GPV sequence: 7, 8d, 15d

Badness: 0.026790

Porky

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]]

POTE generator: ~10/9 = 164.412

Minimax tuning:

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

Eigenmonzos (unchanged intervals): 2, 7/5

Optimal GPV 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]]

POTE generator: ~10/9 = 164.552

Minimax tuning:

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

Eigenmonzos (unchanged intervals): 2, 7/5

Optimal GPV sequence: 7d, 15d, 22, 29, 51, 73ce

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]]

POTE generator: ~10/9 = 164.953

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

Badness: 0.026543

Coendou

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]]

POTE generator: ~10/9 = 166.041

Minimax tuning:

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

Eigenmonzos (unchanged intervals): 2, 3

Optimal GPV sequence: 7, 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]]

POTE generator: ~10/9 = 165.981

Minimax tuning:

• 11-odd-limit: ~10/9 = [2/3 -1/3⟩

Eigenmonzos (unchanged intervals): 2, 3

Optimal GPV sequence: 7, 29, 65ce, 94cde

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]]

POTE generator: ~10/9 = 165.974

Minimax tuning:

• 13- and 15-odd-limit: ~10/9 = [2/3 -1/3⟩

Eigenmonzos (unchanged intervals): 2, 3

Optimal GPV sequence: 7, 29, 65cef, 94cdef

Badness: 0.030233

Hedgehog

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 ⟨146 232 338 411] 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

Comma list: 50/49, 245/243

Mapping: [⟨2 1 1 2], ⟨0 3 5 5]]

Wedgie: ⟨⟨6 10 10 2 -1 -5]]

POTE generator: ~9/7 = 435.648

Optimal GPV sequence: 8d, 14c, 22, 146bccdd

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]]

POTE generator: ~9/7 = 435.386

Optimal GPV sequence: 8d, 14c, 22, 58ce, 80ce, 102cde

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]]

POTE generator: ~9/7 = 435.861

Optimal GPV 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]]

POTE generator: ~9/7 = 437.078

Optimal GPV 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]]

POTE generator: ~9/7 = 435.425

Optimal GPV sequence: 22, 80c, 102cd, 124cd

Badness: 0.068406

Music

Phobos Light by Chris Vaisvil in Hedgehog[14] tuned to 22EDO.

Nautilus

Subgroup: 2.3.5.7

Comma list: 49/48, 250/243

Mapping: [⟨1 2 3 3], ⟨0 -6 -10 -3]]

Wedgie: ⟨⟨6 10 3 2 -12 -21]]

POTE generator: ~21/20 = 82.505

Optimal GPV sequence: 14c, 15, 29, 44d, 59d, 73cd, 102cd

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]]

POTE generator: ~21/20 = 82.504

Optimal GPV sequence: 14c, 15, 29, 44d, 59d, 73cde, 102cde

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]]

POTE generator: ~21/20 = 82.530

Optimal GPV sequence: 14cf, 15, 29, 44d, 59df, 73cde, 102cde

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]]

POTE generator: ~21/20 = 81.759

Optimal GPV sequence: 14c, 15, 29f, 44df

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]]

Wedgie: ⟨⟨9 15 19 3 5 2]]

POTE generator: ~9/7 = 454.448

Optimal GPV sequence: 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]]

POTE generator: ~9/7 = 454.512

Optimal GPV sequence: 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]]

POTE generator: ~13/10 = 454.529

Optimal GPV sequence: 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]]

Wedgie: ⟨⟨9 15 4 3 -19 -33]]

POTE generator: ~36/35 = 54.384

Optimal GPV 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]]

POTE generator: ~36/35 = 54.376

Optimal GPV 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]]

POTE generator: ~36/35 = 54.665

Optimal GPV sequence: 1ce, 21cef, 22

Badness: 0.044739