Pentacircle clan

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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 pentacircle clan of rank-3 temperaments tempers out the pentacircle comma, 896/891. This has the effect of identifying 14/11 at the Pythagorean major third.

For the rank-4 pentacircle temperament, see Rank-4 temperament #Pentacircle (896/891).

Parapythic

Main article: Parapyth

Parapyth, by the original definition, is the 2.3.7.11.13-subgroup temperament tempering out 352/351 and 364/363. We begin by looking at the 2.3.7.11 restriction thereof.

Subgroup: 2.3.7.11

Comma list: 896/891

Subgroup-val mapping: [⟨1 0 0 7], ⟨0 1 0 -4], ⟨0 0 1 1]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

  • WE: ~2 = 1199.3774 ¢, ~3/2 = 703.4693 ¢, ~7/4 = 969.3690 ¢
error map: ⟨-0.623 +0.892 -0.702 +1.061]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 703.7426 ¢, ~7/4 = 969.0476 ¢
error map: ⟨0.000 +1.788 +0.222 +2.759]

Optimal ET sequence: 12, 17, 36, 41, 58, 63, 104, 225e, 266e, 370bee, 699bbdeee

Badness (Sintel): 0.299

Overview to extensions

Subgroup extensions

By tempering out 896/891, we have mapped 14/11 to the major third, suggesting a slightly sharp fifth. This makes the minor third very close to the flat-of-Pythagorean 13/11, and extending the temperament to include harmonic 13 this way implies we temper out 352/351. In fact, 896/891 = (352/351)⋅(364/363), so it is a very natural interpretation, giving rise to the 2.3.7.11.13 subgroup temperament shown below.

Full 11-limit extensions

The second comma in the comma list determines how we extend parapyth to include the harmonic 5.

Pele adds 441/440 and finds the harmonic 5 by equating the syntonic comma (81/80) with the septimal comma (64/63). Together with the slightly sharp fifth this extension makes for one of the most natural interpretations. Sensamagic adds 245/243 or 385/384, a traditional RTT favorite. Apollo adds 100/99 or 225/224, and is even simpler than sensamagic. Pentafrost adds 245/242. Uni adds 540/539. Melpomene adds 56/55 or 81/80. Terrapyth adds 585640/583443, a complex entry that finds the harmonic 5 at the triple augmented unison (AAA1). These all have the same lattice structure as parapyth.

Varda adds 176/175, splitting the octave into two. Parahemif adds 243/242, splitting the perfect fifth into two. Kujuku adds 14700/14641, splitting the perfect twelfth into two. Tolerant adds 2200/2187, splitting the ~33/32 into two. Finally, canta adds 472392/471625, splitting the ~14/9 into three.

Temperaments discussed elsewhere are:

Considered below are tolerant, kujuku, and terrapyth.

Parapyth

Main article: Parapyth

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

Subgroup-val mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3706 ¢, ~3/2 = 703.4872 ¢, ~7/4 = 969.3987 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8328 ¢, ~7/4 = 969.1612 ¢

Optimal ET sequence: 12f, 17, 41, 46, 58, 87, 104, 266ef, 329bef, 370beef, 474beef, 595bdeeeff, 699bbdeeeff

Badness (Sintel): 0.266

Etypyth

Subgroup: 2.3.7.11.13.17

Comma list: 352/351, 364/363, 442/441

Subgroup-val mapping: [⟨1 0 0 7 12 -13], ⟨0 1 0 -4 -7 9], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3607 ¢, ~3/2 = 703.6564 ¢, ~7/4 = 970.0880 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.0139 ¢, ~7/4 = 969.8715 ¢

Optimal ET sequence: 12f, 17g, 29g, 41g, 46, 58, 75e, 104, 121, 225e

Badness (Sintel): 0.536

Terrapyth

Terrapyth tempers out the leapday comma, and can be described as 29 & 46 & 121.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 585640/583443

Mapping: [⟨1 0 -31 0 7], ⟨0 1 21 0 -4], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3126 ¢, ~3/2 = 703.7780 ¢, ~7/4 = 970.0657 ¢
error map: ⟨-0.687 +1.136 -0.101 -0.135 +0.199]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1544 ¢, ~7/4 = 969.8575 ¢
error map: ⟨0.000 +2.199 +0.928 +1.032 +1.922]

Optimal ET sequence: 17c, 29, 46, 92de, 121, 167, 288be, 455bcde

Badness (Sintel): 6.43

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 9295/9261

Mapping: [⟨1 0 -31 0 7 12], ⟨0 1 0 21 0 4 -7], ⟨0 0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3695 ¢, ~3/2 = 703.7992 ¢, ~7/4 = 970.3331 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1459 ¢, ~7/4 = 970.0967 ¢

Optimal ET sequence: 17c, 29, 46, 75e, 92def, 121, 167, 288be

Badness (Sintel): 2.32

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 364/363, 442/441, 715/714

Mapping: [⟨1 0 -31 0 7 12 -13], ⟨0 1 0 21 0 4 -7 9], ⟨0 0 0 1 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3783 ¢, ~3/2 = 703.7980 ¢, ~7/4 = 970.1592 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.1406 ¢, ~7/4 = 969.9458 ¢

Optimal ET sequence: 17cg, 29g, 46, 75e, 92defg, 121, 167, 288beg

Badness (Sintel): 1.45

Pentafrost

Pentafrost tempers out the frostma in addition to 896/891 which also means that the schisma is tempered out, mapping prime 5 to eight perfect fourths minus an octave.

It was named by Tristan Bay in 2024 as a portmanteau of pentacircle and frost.

Subgroup: 2.3.5.7.11

Comma list: 245/242, 896/891

Mapping: [⟨1 0 15 0 7], ⟨0 1 -8 0 -4], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.1251 ¢, ~3/2 = 701.9850 ¢, ~7/4 = 964.6139 ¢
error map: ⟨+0.125 +0.155 -1.318 -3.962 +5.982]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9034 ¢, ~7/4 = 964.6143 ¢
error map: ⟨0.000 -0.052 -1.541 -4.212 +5.683]

Optimal ET sequence: 12, 24, 29, 36, 41, 106d

Badness (Sintel): 1.90

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 245/242, 352/351

Mapping: [⟨1 0 15 0 7 12], ⟨0 1 -8 0 -4 -7], ⟨0 0 0 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1200.2502 ¢, ~3/2 = 702.3077 ¢, ~7/4 = 962.1832 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1455 ¢, ~7/4 = 962.1748 ¢

Optimal ET sequence: 12f, 24, 29, 41

Badness (Sintel): 1.49

Permafrost

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 245/242, 896/891

Mapping: [⟨1 0 15 0 7 -3], ⟨0 1 -8 0 -4 6], ⟨0 0 0 1 1 -1]]

Optimal tunings:

  • WE: 2 = 1199.6241 ¢, ~3/2 = 701.5280 ¢, ~7/4 = 966.2056 ¢
  • CWE: 2 = 1200.000 ¢, ~3/2 = 701.7534 ¢, ~7/4 = 966.4455 ¢

Optimal ET sequence: 12, 17, 24, 36, 41, 77e

Badness (Sintel): 2.45

Tolerant

For the 7-limit version, see Miscellaneous 7-limit temperaments #Tolerant.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187

Mapping: [⟨1 0 0 -10 -3], ⟨0 1 0 11 7], ⟨0 0 1 -2 -2]]

mapping generators: ~2, ~3, ~5

Optimal tunings:

  • WE: ~2 = 1199.4396 ¢, ~3/2 = 703.7124 ¢, ~5/4 = 387.1118 ¢
error map: ⟨-0.560 +1.197 -0.323 -0.532 +0.445]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.9092 ¢, ~5/4 = 386.9306 ¢
error map: ⟨0.000 +1.951 +0.617 +0.281 +2.164]

Optimal ET sequence: 34d, 39d, 41, 80, 87, 121, 167, 208, 288be, 375be

Badness (Sintel): 1.25

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363

Mapping: [⟨1 0 0 -10 -3 2], ⟨0 1 0 11 7 4], ⟨0 0 1 -2 -2 -2]]

Optimal tunings:

  • WE: ~2 = 1199.5161 ¢, ~3/2 = 703.6767 ¢, ~5/4 = 386.8270 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.8968 ¢, ~5/4 = 386.8916 ¢

Optimal ET sequence: 34d, 41, 46, 75e, 80, 87, 121, 167, 208, 375be

Badness (Sintel): 0.955

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363

Mapping: [⟨1 0 0 -10 -3 2 8], ⟨0 1 0 11 7 4 -1], ⟨0 0 1 -2 -2 -2 -1]]

Optimal tunings:

  • WE: ~2 = 1199.3929 ¢, ~3/2 = 703.7268 ¢, ~5/4 = 387.1310 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.0472 ¢, ~5/4 = 387.3450 ¢

Optimal ET sequence: 34d, 41, 46, 75e, 80, 87, 121, 167, 288beg, 496bdeefggg

Badness (Sintel): 0.934

Kujuku

Kujuku splits the perfect twelfth into two. Scott Dakota has aliased this temperament SQPP (for semiquartal parapyth).

Subgroup: 2.3.5.7.11

Comma list: 896/891, 14700/14641

Mapping: [⟨1 0 0 -13 -6], ⟨0 2 0 17 9], ⟨0 0 1 1 1]]

mapping generators: ~2, ~121/70, ~5

Optimal tunings:

  • WE: ~2 = 1199.3881 ¢, ~121/70 = 951.4033 ¢, ~5/4 = 387.4865 ¢
error map: ⟨-0.612 +0.852 -0.051 -0.752 +1.246]
  • CWE: ~2 = 1200.0000 ¢, ~121/70 = 951.8708 ¢, ~5/4 = 387.2432 ¢
error map: ⟨0.000 +1.787 +0.930 +0.220 +2.762]

Optimal ET sequence: 24, 29, 34d, 53d, 58, 87, 121, 145, 179e, 208, 266e

Badness (Sintel): 2.72

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 676/675

Mapping: [⟨1 0 0 -13 -6 -1], ⟨0 2 0 17 9 3], ⟨0 0 1 1 1 1]]

Optimal tunings:

  • WE: ~2 = 1199.3660 ¢, ~26/15 = 951.3934 ¢, ~5/4 = 387.4050 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.8815 ¢, ~5/4 = 387.1043 ¢

Optimal ET sequence: 24, 29, 34d, 53d, 58, 87, 121, 179ef, 208, 266ef, 474beef

Badness (Sintel): 0.991

Complexity spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 676/675

Mapping: [⟨1 0 0 -13 -6 -1 8], ⟨0 2 0 17 9 3 -2], ⟨0 0 1 1 1 1 -1]]

Optimal tunings:

  • WE: ~2 = 1199.2826 ¢, ~26/15 = 951.3284 ¢, ~5/4 = 387.6639 ¢
  • CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.8791 ¢, ~5/4 = 387.7230 ¢

Optimal ET sequence: 24, 34d, 58, 87, 121, 179ef, 208g, 266efg

Badness (Sintel): 1.18

Trienparapyth

Named by Godtone in 2024, trienparapyth can be described as the 58 & 80 & 87 temperament, with an extension to the no-17's 23-limit. It splits the ~4/3 generator of parapythic into three ~11/10's by tempering out 4000/3993 (S10/S11) in the 11-limit. It further interprets (11/10)2 accurately as 23/19 in its full subgroup, tempering out 2300/2299 (S20/S22), or optionally less accurately as ~17/14, though because this mapping only really makes much sense in 80edo it is not included here; however, its connection to parapyth structure comes from later in the generator chain; specifically, from (11/10)7 onwards. We may simplify (11/10)7 as (4/3)2(11/10) = 88/45, the octave-complement of 45/44. Notice that parapythic wants a slightly flat ~4/3 corresponding to an 11/10 being tuned anywhere from around just (in an extremely sharp-for-parapyth tuning) to a little less than 1-cent sharp, a very narrow tuning range; therefore 88/45 is flattened so that 2/(11/10)7~45/44 is sharpened so that we can equate it with 40/39, tempering out (40/39)/(45/44) = 352/351, and because we know we want prime 19 later on, we equate this with 39/38 by tempering out the pinkanberry, 1521/1520 (S39). Next, for eight generator steps, observe that (11/10)9/(11/10)/2 = (4/3)3/(11/10)/2 = (32/27)/(11/10) = 320/297 is sharp of 15/14 by 896/891, which is reasonable to equate it with because in an optimal tuning 11/10 will be very slightly sharp so that the interval of eight generator steps is eight times as sharp. Thus, tempering out 896/891 and 4000/3993 defines trienparapyth in the 11-limit, which also tempers out 3388/3375, the 13-limit adds 352/351, the no-17's 19-limit equates 40/39 with 39/38 and the no-17's 23-limit equates 23/19 with (11/10)2 as already mentioned.

Structurally, trienparapyth is three copies of parapyth with the independent generator of 7 connected to an equivalent independent generator for 5 through the ~15/7 reached at (11/10)8 so that ~20/7 is reached at (11/10)11, and this is how the last generator can be either 5 or 7.

Subgroup: 2.3.5.7.11

Comma list: 896/891, 3388/3375

Mapping: [⟨1 2 0 2 1], ⟨0 -3 0 -11 1], ⟨0 0 1 1 1]]

mapping generators: ~2, ~11/10, ~5

Optimal tunings:

  • WE: ~2 = 1199.3706 ¢, ~11/10 = 165.2428 ¢, ~5/4 = 388.1147 ¢
error map: ⟨-0.629 +1.058 +0.542 -0.899 +0.151]
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.3593 ¢, ~5/4 = 387.8093 ¢
error map: ⟨0.000 +1.967 +1.496 +0.031 +1.851]

Optimal ET sequence: 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 1001/1000

Mapping: [⟨1 2 0 2 1 0], ⟨0 -3 0 -11 1 10], ⟨0 0 1 1 1 1]]

mapping generators: ~2, ~11/10, ~5

Optimal tunings:

  • WE: ~2 = 1199.4286 ¢, ~11/10 = 165.2932 ¢, ~5/4 = 388.2127 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.3802 ¢, ~5/4 = 387.8759 ¢

Optimal ET sequence: 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce

Badness (Sintel): 1.15

2.3.5.7.11.13.19 subgroup

Note 109edo is a good patent val tuning not listed in the optimal ET sequence here.

Subgroup: 2.3.5.7.11.13.19

Comma list: 286/285, 352/351, 364/363, 400/399

Mapping: [⟨1 2 0 2 1 0 0], ⟨0 -3 0 -11 1 10 14], ⟨0 0 1 1 1 1 1]]

mapping generators: ~2, ~11/10, ~5

Optimal tunings:

  • WE: ~2 = 1199.3123 ¢, ~11/10 = 165.2022 ¢, ~5/4 = 388.1654 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.2976 ¢, ~5/4 = 387.7451 ¢

Optimal ET sequence: 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h

Badness (Sintel): 1.20

2.3.5.7.11.13.19.23 subgroup

Subgroup: 2.3.5.7.11.13.19.23

Comma list: 208/207, 286/285, 352/351, 364/363, 400/399

Mapping: [⟨1 2 0 2 1 0 0 0], ⟨0 -3 0 -11 1 10 14 16], ⟨0 0 1 1 1 1 1 1]]

mapping generators: ~2, ~11/10, ~5

Optimal tunings:

  • WE: ~2 = 1199.2714 ¢, ~11/10 = 165.1718 ¢, ~5/4 = 388.1729 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.2679 ¢, ~5/4 = 387.7240 ¢

Optimal ET sequence: 22i, 29, 51fhi, 51cde, 58hi, 80, 87, 109, 138cdehi, 167hi

Badness (Sintel): 1.14

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