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
Comma list: 250/243
Mapping: [⟨1 2 3], ⟨0 -3 -5]]
POTE generator: ~27/25 = 163.950
- 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
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.
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
- 7- and 9-odd-limit eigenmonzo: 9/7
- 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
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
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 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
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 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
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 eigenmonzo: 13/7
Vals: Template:Val list
Badness: 0.0351
Porkpie
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 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.
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
- 7- and 9-odd-limit eigenmonzo: 5/4
Badness: 0.0449
11-limit
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
Vals: Template:Val list
Badness: 0.0268
Porky
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
- 7- and 9-odd-limit eigenmonzo: 7/5
Badness: 0.0544
11-limit
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 eigenmonzo: 7/5
Vals: Template:Val list
Badness: 0.0273
13-limit
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
Vals: Template:Val list
Badness: 0.0265
Coendou
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
- 7- and 9-odd-limit eigenmonzo: 3/2
Badness: 0.1183
11-limit
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 eigenmonzo: 3/2
Vals: Template:Val list
Badness: 0.0497
13-limit
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 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.
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
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
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
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
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
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
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
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
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 = ~14/13 = 81.759
Vals: Template:Val list
Badness: 0.0298
- Music
Nautilus Reverie by Igliashon Calvin Jones-Coolidge
Ammonite
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
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
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
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
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
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