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
- 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
- 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
- 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
- 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
- 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
- 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