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.

Porcupine

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

POTE generator: ~27/25 = 163.950

Tuning ranges:

• valid range: [150.000, 171.429] (1\8 to 1\7)
• nice range: [157.821, 166.015]
• strict range: [157.821, 166.015]

Badness: 0.0308

Extensions

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.

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- and 9-odd-limit eigenmonzo: 9/7

Tuning ranges:

• valid range: [160.000, 163.636] (2\15 to 3\22)
• nice range: [157.821, 166.015]
• strict range: [160.000, 163.636]

Badness: 0.0411

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: ~11/10 = 162.747

Minimax tuning:

• 11-odd-limit eigenmonzo: 9/7

Tuning ranges:

• valid range: [160.000, 163.636] (2\15 to 3\22)
• nice range: [150.637, 182.404]
• strict range: [160.000, 163.636]

Vals: Template:Val list

Badness: 0.0217

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: ~11/10 = 162.708

Minimax tuning:

• 13- and 15-odd-limit eigenmonzo: 11/8

Tuning ranges:

• valid range: [160.000, 163.636] (15 to 22f)
• nice range: [138.573, 182.404]
• strict range: [160.000, 163.636]

Vals: Template:Val list

Badness: 0.0213

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: ~11/10 = 162.277

Minimax tuning:

• 13- and 15-odd-limit eigenmonzo: 13/11

Tuning ranges:

• valid range: [160.000, 162.162] (15 to 37)
• nice range: [150.637, 182.404]
• strict range: [160.000, 162.162]

Vals: Template:Val list

Badness: 0.0253

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: ~11/10 = 162.482

Minimax tuning:

• 13- and 15-odd-limit eigenmonzo: 13/7

Vals: Template:Val list

Badness: 0.0351

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: ~11/10 = 163.688

Minimax tuning:

• 13- and 15-odd-limit eigenmonzo: 9/7

Vals: Template:Val list

Badness: 0.0260

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: ~10/9 = 158.868

Minimax tuning:

• 7- and 9-odd-limit eigenmonzo: 5/4

Badness: 0.0449

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: ~11/10 = 158.750

Vals: Template:Val list

Badness: 0.0268

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 eigenmonzo: 7/5

Badness: 0.0544

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: ~11/10 = 164.552

Minimax tuning:

• 11-odd-limit eigenmonzo: 7/5

Vals: Template:Val list

Badness: 0.0273

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: ~11/10 = 164.953

Vals: Template:Val list

Badness: 0.0265

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 eigenmonzo: 3/2

Badness: 0.1183

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: ~11/10 = 165.981

Minimax tuning:

• 11-odd-limit eigenmonzo: 3/2

Vals: Template:Val list

Badness: 0.0497

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: ~11/10 = 165.974

Minimax tuning:

• 13- and 15-odd-limit eigenmonzo: 3/2

Vals: Template:Val list

Badness: 0.0302

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

Badness: 0.0440

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

Vals: Template:Val list

Badness: 0.0231

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

Vals: Template:Val list

Badness: 0.0215

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

Vals: Template:Val list

Badness: 0.0252

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

Vals: Template:Val list

Badness: 0.0684

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

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

Vals: Template:Val list

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 = 62.530

Vals: Template:Val list

Badness: 0.0223

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

Vals: Template:Val list

Badness: 0.0298

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

Badness: 0.1077

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

Vals: Template:Val list

Badness: 0.0457

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 = 454.429

Vals: Template:Val list

Badness: 0.0272

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

Badness: 0.115

11-limit

Subgroup: 2.3.5.7.11

Comma list: 55/54, 100/99, 5324/5145

Mapping: [⟨1 2 3 3 4], ⟨0 -9 -15 -4 -12]]

POTE generator: ~36/35 = 54.376

Vals: Template:Val list

Badness: 0.0513

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

Vals: Template:Val list

Badness: 0.0447