User:VectorGraphics/Porcupine family/Draft 1

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.


The porcupine family of temperaments tempers out the porcupine comma, 250/243, also called the maximal diesis. This comma splits 4/3 into three equal parts, and 6/5 makes up two of those parts. Thus, the generator is mapped to 10/9. Mathematically, (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 scales of 7, 8 and 15 notes make for some nice scale possibilities.

It most naturally manifests as a 2.3.5.11 subgroup temperament, where it tempers out 100/99 and 55/54 equating the generator to 11/10 as well as 10/9.

Porcupine

Main article: Porcupine

5-limit

Subgroup: 2.3.5

Comma list: 250/243

Mapping: [⟨1 2 3], ⟨0 -3 -5]]

mapping generators: ~2, ~10/9

Optimal tunings:

  • CTE: ~2 = 1200.000, ~10/9 = 164.166
error map: ⟨0.000 +5.547 -7.143]
  • POTE: ~2 = 1200.000, ~10/9 = 163.950
error map: ⟨0.000 +6.194 -6.065]

Tuning ranges:

Optimal ET sequence: 7, 15, 22, 95c

Badness (Smith): 0.030778

2.3.5.11 subgroup (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]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 163.887
  • POTE: ~2 = 1200.000, ~11/10 = 164.078

Optimal ET sequence: 7, 15, 22, 73ce, 95ce

Badness (Smith): 0.0097

Strong extensions

Main article: Porcupine
Map to strong full 13-limit extensions
Extension Mapping of 7 Mapping of 13 Tuning range*
Porcupinefish +6 -17 ↑ 22
Porky -16 +5 ↑ 29

↓ 22

Coendou 13 ↓ 29

* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions

Porcupinefish

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Porcupinefish (or "septimal porcupine" in its 11-limit form) uses six of its minor tone generator steps to get to 7/4. Here, we share the same mapping of 7/4 in terms of fifths as archy. 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. This extends porcupine to the full 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 163.105
  • POTE: ~2 = 1200.000, ~11/10 = 162.747

Minimax tuning:

  • 11-odd-limit: ~11/10 = [1/6 -1/6 0 1/12⟩
eigenmonzo (unchanged-interval) basis: 2.9/7

Tuning ranges:

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

(7-limit) Optimal ET sequence: 7, 15, 22, 37, 59, 81bd

(11-limit) Optimal ET sequence: 7, 15, 22, 37, 59

Badness (Smith): 0.021562

13-limit

In the 13-limit, porcupinefish maps 13/8 to -17 generators.

See also: The Biosphere
Subgroup: 2.3.5.7.11.13Comma list: 55/54, 64/63, 91/90, 100/99

Mapping: [⟨1 2 3 2 4 6], ⟨0 -3 -5 6 -4 -17]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 162.636
  • POTE: ~2 = 1200.000, ~11/10 = 162.277

Minimax tuning:

  • 13- and 15-odd-limit: ~10/9 = [2/13 0 0 0 1/13 -1/13⟩
eigenmonzo (unchanged-interval) basis: 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]

Optimal ET sequence: 15, 22, 37

Badness (Smith): 0.025314

Porky

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Porky can be described as 7d & 22, suggesting a less sharp perfect fifth. 7\51 is a good generator.

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]]

Wedgie: ⟨⟨ 3 5 16 4 1 17 -4 23 -8 -44 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 164.321
  • POTE: ~2 = 1200.000, ~11/10 = 164.552

Minimax tuning:

  • 11-odd-limit: ~11/10 = [2/11 0 1/11 -1/11⟩
eigenmonzo (unchanged-interval) basis: 2.7/5

Optimal ET sequence: 7d, 15d, 22, 29, 51, 73c (7-limit)

Optimal ET sequence: 7d, 15d, 22, 51

Badness (Smith): 0.027268

13-limit

As the porcupinefish mapping is inaccurate with a sharply tuned generator, this alternate mapping becomes more accurate at this point. Thus, 13-limit porky as a whole can be seen as reversing the tradeoff between 7 and 13 found in porcupinefish.

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 164.478
  • POTE: ~2 = 1200.000, ~11/10 = 164.953

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

Badness (Smith): 0.026543

Music

Coendou

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Coendou can be described as 7 & 29, suggesting an even less sharp or near-just perfect fifth. 9\65 is a good generator.

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 165.925
  • POTE: ~2 = 1200.000, ~11/10 = 165.981

Minimax tuning:

  • 11-odd-limit: ~11/10 = [2/3 -1/3⟩
eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65c, 94cd (7-limit)

Optimal ET sequence: 7, 22d, 29, 65ce

Badness (Smith): 0.049669

13-limit

Coendou shares the mapping of 13 with porky.

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/10 = 166.046
  • POTE: ~2 = 1200.000, ~11/10 = 165.974

Minimax tuning:

  • 13- and 15-odd-limit: ~11/10 = [2/3 -1/3⟩
eigenmonzo (unchanged-interval) basis: 2.3

Optimal ET sequence: 7, 22d, 29, 65cef

Badness (Smith): 0.030233

Weak extensions

Map to weak extensions
Extensions Periods per octave Generator Position of original generator
Number of generators Number of periods
Hedgehog period = 1/2 octave ~9/7 -1 generators +1 periods
Undecimation period = octave ~88/65 +2 generators -1 periods
Nautilus period = octave ~21/20 +2 generators +0 periods
Ammonite period = octave ~9/7 +3 generators -1 periods
Ceratitid period = octave ~36/35 +3 generators +0 periods

Hedgehog

See also: Sensamagic clan and Stearnsmic clan
⟨] Return to the map

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, and collapses 5/4 and 7/4 to the same number of gensteps (in different periods). As such, it is best tuned around 165 cents and is also a strong extension of BPS (as BPS has no 2 or sqrt(2)). 22edo provides the obvious (i.e the only patent val) tuning, but if you are looking for an alternative you could try the 146 232 338 411 (146bccdd) 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. A related temperament is echidna, which offers much more accuracy. They merge on 22edo.

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]]

Wedgie: ⟨⟨ 6 10 10 8 2 -1 -8 -5 -16 -12 ]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 435.528
  • POTE: ~7/5 = 600.000, ~9/7 = 435.386

Optimal ET sequence: 8d, 14c, 22 (7-limit)

Optimal ET sequence: 8d, 14c, 22, 58ce

Badness (Smith): 0.023095

Porkhog

Since hedgehog has the same sharply tuned generator as porky (with a different mapping for 7), it becomes reasonable to extend hedgehog with porky's mapping, mapping 13 to -5 gensteps.

[add temp data]

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]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 436.309
  • POTE: ~7/5 = 600.000, ~9/7 = 435.861

Optimal ET sequence: 8d, 14cf, 22

Badness (Smith): 0.021516

Hedgepig

Hedgepig is a variant of hedgehog that uses a sharper and more accurate mapping of 11 available around the 165c tuning, and thus extends 2.3.5 porcupine instead of 2.3.5.11 porcupine.

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]]

Wedgie: ⟨⟨ 6 10 10 -14 2 -1 -43 -5 -67 -74 ]]

Optimal tunings:

  • CTE: ~7/5 = 600.000, ~9/7 = 435.329
  • POTE: ~7/5 = 600.000, ~9/7 = 435.425

Optimal ET sequence: 22

Badness (Smith): 0.068406

Music

Undecimation

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Undecimation is an extension to the 2.3.5.11.13 subgroup that splits the generator to introduce neutral intervals (and thus an obvious choice for mapping 13/8). It does this by stacking a flatly tuned fifth representing 65/44 twice to get to the generator. For whatever reason, the optimal tunings shown here are in the sharper range, even though tuning undecimation to 519.25 cents allows it to find 7 at 12 gensteps.

Subgroup: 2.3.5.11.13

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

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

sval mapping generators: ~2, ~65/44

Optimal tunings:

  • CTE: ~2 = 1200.000, ~88/65 = 518.086
  • POTE: ~2 = 1200.000, ~88/65 = 518.209

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

Badness (Smith): 0.0305

Nautilus

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Nautilus splits the 10/9 generator into two 21/20s, making a much simpler mapping of 7/4 available. It can be seen as porcupine's generator chain expanded to include neutral intervals (like undecimation, but with a different generator), and as such has a mapping of 13/8 available at -19 steps.

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]]

Wedgie: ⟨⟨ 6 10 3 8 2 -12 -8 -21 -16 12 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 81.802
  • POTE: ~2 = 1200.000, ~21/20 = 82.504

Optimal ET sequence: 14c, 15, 29, 44d

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~21/20 = 81.912
  • POTE: ~2 = 1200.000, ~21/20 = 82.530

Optimal ET sequence: 14cf, 15, 29, 44d

Badness (Smith): 0.022285

Music

Ammonite

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Ammonite splits the porcupine generator (as ~1363.5 cents) into three parts representing 9/7.

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]]

Wedgie: ⟨⟨ 9 15 19 12 3 5 -12 2 -24 -32 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 454.505
  • POTE: ~2 = 1200.000, ~9/7 = 454.512

Optimal ET sequence: 8d, 21cd, 29, 37, 66 (7-limit)

Optimal ET sequence: 8d, 21cde, 29, 37, 66

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/10 = 454.480
  • POTE: ~2 = 1200.000, ~13/10 = 454.529

Optimal ET sequence: 8d, 21cdef, 29, 37, 66

Badness (Smith): 0.027168

Ceratitid

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Ceratitid also splits the generator into three, and this time the more familiar neutral second is split into three parts representing 36/35.

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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~36/35 = 54.702
  • POTE: ~2 = 1200.000, ~36/35 = 54.376

Optimal ET sequence: 1c, 21c, 22 (7-limit)

Optimal ET sequence: 1ce, 21ce, 22

Badness (Smith): 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]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~36/35 = 54.575
  • POTE: ~2 = 1200.000, ~36/35 = 54.665

Optimal ET sequence: 1ce, 21cef, 22

Badness (Smith): 0.044739

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