Porcupine family

de:Porcupine

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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.

valid range: [150.000, 171.429] (8 to 7)

nice range: [157.821, 166.015]

strict range: [157.821, 166.015]

POTE generator: 163.950

Map: [<1 2 3|, <0 -3 -5|]

EDOs: 15, 22, 95c, 117bc, 139bc, 161bc, 183bc

Badness: 0.0308

Seven limit children

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 porcupine, 36/35, the septimal quarter tone, for hystrix, 50/49, the jubilisma, for hedgehog, and 49/48, the slendro diesis, for nautilus.

Porcupine

Porcupine, with wedgie <<3 5 -6 1 -18 -28||, 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.

Commas: 250/243, 64/63

valid range: [160.000, 163.636] (15 to 22)

nice range: [157.821, 166.015]

strict range: [160.000, 163.636]

POTE generator: ~10/9 = 162.880

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

Map: [<1 2 3 2|, <0 -3 -5 6|]

EDOs: 22, 59, 81bd, 140bd

Badness: 0.0411

11-limit

Commas: 55/54, 64/63, 100/99

valid range: [160.000, 163.636] (15 to 22)

nice range: [150.637, 182.404]

strict range: [160.000, 163.636]

POTE generator: ~10/9 = 162.747

11-limit minimax eigenmonzo: 9/7

Map: [<1 2 3 2 4|, <0 -3 -5 6 -4|]

EDOs: 7, 15, 22, 37, 59

Badness: 0.0217

13-limit

Commas: 40/39, 55/54, 64/63, 66/65

valid range: [160.000, 163.636] (15 to 22f)

nice range: [138.573, 182.404]

strict range: [160.000, 163.636]

POTE generator: ~10/9 = 162.708

13- and 15-limit minimax eigenmonzo: 11/8

Map: [<1 2 3 2 4 4|, <0 -3 -5 6 -4 -2|]

EDOs: 7, 15, 22f, 37f

Badness: 0.0213

Porcupinefish

See also: The Biosphere

Commas: 55/54, 64/63, 91/90, 100/99

valid range: [160.000, 162.162] (15 to 37)

nice range: [150.637, 182.404]

strict range: [160.000, 162.162]

POTE generator: ~10/9 = 162.277

13- and 15-limit minimax eigenmonzo: 13/11

Map: [<1 2 3 2 4 6|, <0 -3 -5 6 -4 -17|]

EDOs: 15, 22, 37, 59, 96b

Badness: 0.0253

Badness: 0.0253

Pourcup

Commas: 55/54, 64/63, 100/99, 196/195

POTE generator: ~10/9 = 162.482

13- and 15-limit minimax eigenmonzo: 13/7

Map: [<1 2 3 2 4 1|, <0 -3 -5 6 -4 20|]

EDOs: 15f, 22f, 37

Badness: 0.0351

Porkpie

Commas: 55/54, 64/63, 65/63, 100/99

POTE generator: ~10/9 = 163.688

13- and 15-limit minimax eigenmonzo: 9/7

Map: [<1 2 3 2 4 3|, <0 -3 -5 6 -4 5|]

EDOs: 7, 15f, 22

Badness: 0.0260

Hystrix

Hystrix, with wedgie <<3 5 1 1 -7 -12||, 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.

Commas: 36/35, 160/147

POTE generator: 158.868

7- and 9-limit minimax eigenmonzo: 5/4

Map: [<1 2 3 3|, <0 -3 -5 -1|]

EDOs: 10d, 12, 13d, 15

Badness: 0.0449

Porky

Commas: 225/224, 250/243

POTE generator: ~10/9 = 164.412

7- and 9-limit minimax eigenmonzo: 7/5

Map: [<1 2 3 5|, <0 -3 -5 -16|]

Wedgie: <<3 5 16 1 17 23||

EDOS: 7, 8, 15, 22, 29, 51, 73

Badness: 0.0544

11-limit

Commas: 55/54, 100/99, 225/224

POTE generator: ~10/9 = 164.552

11-limit minimax eigenmonzo: 7/5

Map: [<1 2 3 5 4|, <0 -3 -5 -16 -4|]

EDOs: 7, 8, 15, 22, 29, 51, 73

Badness: 0.0273

13-limit

Commas: 55/54, 65/64, 91/90, 100/99

POTE generator: ~10/9 = 164.953

Map: [<1 2 3 5 4 3|, <0 -3 -5 -16 -4 5|]

EDOs: 22, 29, 51f, 80cdef

Badness: 0.0265

Coendou

Commas: 250/243, 525/512

POTE generator: ~10/9 = 166.041

7- and 9-limit minimax eigenmonzo: 3/2

Map: [<1 2 3 1|, <0 -3 -5 13|]

Wedgie: <<3 5 -13 1 -29 -44||

EDOs: 7, 29, 65c, 94cd

Badness: 0.1183

11-limit

Commas: 55/54, 100/99, 525/512

POTE generator: ~10/9 = 165.981

11-limit minimax eigenmonzo: 3/2

Map: [<1 2 3 1 4|, <0 -3 -5 13 -4|]

EDOs: 7, 29, 65ce, 94cde

Badness: 0.0497

13-limit

Commas: 55/54, 65/64, 100/99, 105/104

POTE generator: ~10/9 = 165.974

13- and 15-limit minimax eigenmonzo: 3/2

Map: [<1 2 3 1 4 3|, <0 -3 -5 13 -4 5|]

EDOs: 7, 29, 65cef, 94cdef

Badness: 0.0302

Hedgehog

Hedgehog, with wedgie <<6 10 10 2 -1 -5||, 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. 22et 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.

Commas: 50/49, 245/243

POTE generator: ~9/7 = 435.648

Map: [<2 1 1 2|, <0 3 5 5|]

Wedgie: <<6 10 10 2 -1 -5||

EDOs: 22, 146

Badness: 0.0440

11-limit

Commas: 50/49, 55/54, 99/98

POTE generator: ~9/7 = 435.386

Map: [<2 1 1 2 4|, <0 3 5 5 4|]

EDOs: 14c, 22, 58ce, 80ce, 102cde

Badness: 0.0231

13-limit

Commas: 50/49, 55/54, 65/63, 99/98

POTE generator: ~9/7 = 435.861

Map: [<2 1 1 2 4 3|, <0 3 5 5 4 6|]

EDOs: 14cf, 22

Badness: 0.0215

Urchin

Commas: 40/39, 50/49, 55/54, 66/65

POTE generator: ~9/7 = 437.078

Map: [<2 1 1 2 4 6|, <0 3 5 5 4 2|]

EDOs: 14c, 22f

Badness: 0.0252

Hedgepig

Commas: 50/49, 245/243, 385/384

POTE generator: ~9/7 = 435.425

Map: [<2 1 1 2 12|, <0 3 5 5 -7|]

EDOs: 22, 80c, 102cd, 124cd

Badness: 0.0684

Music

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

Nautilus

Commas: 49/48, 250/243

POTE generator: ~21/20 = 82.505

Map: [<1 2 3 3|, <0 -6 -10 -3|]

Wedgie: <<6 10 3 2 -12 -21||

EDOs: 15, 29, 43cd, 44d, 59d, 73cd, 102cd

11-limit

Commas: 49/48, 55/54, 245/242

POTE generator: ~21/20 = 82.504

Map: [<1 2 3 3 4|, <0 -6 -10 -3 -8|]

EDOs: 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde

13-limit

Commas: 49/48, 55/54, 91/90, 100/99

POTE generator: ~21/20 = 62.530

Map: [<1 2 3 3 4 5|, <0 -6 -10 -3 -8 -19|]

EDOs: 15f, 29, 43cde, 44d, 59df, 73cde, 102cde

Badness: 0.0223

Belauensis

Commas: 40/39, 49/48, 55/54, 66/65

POTE generator: ~21/20 = ~14/13 = 81.759

Map: [<1 2 3 3 4 4|, <0 -6 -10 -3 -8 -4|]

EDOs: 14c, 15, 29f, 44df

Badness: 0.0298

Nautilus Reverie by Igliashon Calvin Jones-Coolidge

Ammonite

Commas: 250/243, 686/675

POTE generator: ~9/7 = 454.448

Map: [<1 5 8 10|, <0 -9 -15 -19|]

Wedgie: <<9 15 19 3 5 2||

EDOs: 29, 37, 66

Badness: 0.1077

11-limit

Commas: 55/54, 100/99, 686/675

POTE generator: ~9/7 = 454.512

Map: [<1 5 8 10 8|, <0 -9 -15 -19 -12|]

EDOs: 29, 37, 66

Badness: 0.0457

13-limit

Commas: 55/54, 91/90, 100/99, 169/168

POTE generator: ~13/10 = 454.429

Map: [<1 5 8 10 8 9|, <0 -9 -15 -19 -12 -14|]

EDOs: 29, 37, 66

Badness: 0.0272

Ceratitid

Commas: 250/243, 1728/1715

POTE generator: ~36/35 = 54.384

Map: [<1 2 3 3|, <0 -9 -15 -4|]

Wedgie: <<9 15 4 3 -19 -33||

EDOs: 22

Badness: 0.115

11-limit

Commas: 55/54, 100/99, 5324/5145

POTE generator: ~36/35 = 54.376

Map: [<1 2 3 3 4|, <0 -9 -15 -4 -12|]

EDOs: 22

Badness: 0.0513

13-limit

Commas: 55/54, 65/63, 100/99, 352/343

POTE generator: ~36/35 = 54.665

Map: [<1 2 3 3 4 4|, <0 -9 -15 -4 -12 -7|]

EDOs: 22

Badness: 0.0447