Pentacircle clan

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

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

sval mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8345, ~7/4 = 969.8722

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

Badness: 0.0205 × 10^-3

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.

Julius aka 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:

• Melpomene → Didymus rank-3 family
• Apollo → Marvel family
• Sensamagic → Sensamagic family
• Pele → Hemifamity family
• Uni → Hemimage family
• Julius or varda → Diaschismic rank-3 family
• Parahemif → Rastmic rank-3 clan
• Canta → Canou family

Considered below are tolerant, kujuku, and terrapyth.

Parapyth

Main article: Parapyth

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.8563, ~7/4 = 969.9074

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

Badness: 0.101 × 10^-3

Etypyth

Subgroup: 2.3.7.11.13.17

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

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0315, ~7/4 = 970.6051

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

Badness: 0.325 × 10^-3

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]]

mapping generators: ~2, ~3, ~7

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.1814, ~7/4 = 970.6217

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

Badness: 5.35 × 10^-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.1691, ~7/4 = 970.8432

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

Badness: 2.48 × 10^-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.1628, ~7/4 = 970.6620

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

Badness: 1.52 × 10^-3

Pentafrost

Pentafrost tempers out the frostma in addition to 896/891 which also means that the schisma is tempered out, mapping prime 5 to 8 perfect fourths and -1 octaves.

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]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 701.883, ~7/4 = 964.864
• CWE: ~2 = 1\1, ~3/2 = 701.903, ~7/4 = 964.614
• CE: ~2 = 1\1, ~3/2 = 702.006, ~7/4 = 964.085

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

Badness: 1.58 × 10^-3

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]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

• CTE: ~2 = 1\1, ~3/2 = 702.106, ~7/4 = 962.655
• CWE: ~2 = 1\1, ~3/2 = 702.145, ~7/4 = 962.175
• CE: ~2 = 1\1, ~3/2 = 702.360, ~7/4 = 962.210

Optimal ET sequence: 12f, 24, 29, 41

Badness: 1.59 × 10^-3

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]]

mapping generators: ~2, ~3, ~7

Optimal tunings:

• CTE: 2 = 1\1, ~3/2 = 701.783, ~7/4 = 966.113
• CWE: 2 = 1\1, ~3/2 = 701.753, ~7/4 = 966.445
• CE: 2 = 1\1, ~3/2 = 701.770, ~7/4 = 965.771

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

Badness: 2.62 × 10^-3

Tolerant

7-limit

Subgroup: 2.3.5.7

Comma list: 179200/177147

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

mapping generators: ~2, ~3, ~5

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 703.9571, ~5/4 = 386.8863

Optimal ET sequence: 41, 80, 87, 121, 167, 208, 329b, 375b, 537b, 583b, 704bd

Badness: 0.165 × 10^-3

11-limit

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]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 704.0412, ~5/4 = 387.2927

Optimal ET sequence: 41, 80, 87, 121, 167, 208, 334be, 375be, 542bce

Badness: 1.039 × 10^-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 703.9605, ~5/4 = 386.9831

Optimal ET sequence: 41, 46, 80, 87, 121, 167, 208, 375be, 583bef

Badness: 1.021 × 10^-3

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 tuning (POTE): ~2 = 1\1, ~3/2 = 704.0831, ~5/4 = 387.3269

Optimal ET sequence: 41, 46, 75e, 80, 87, 121, 167, 288beg

Badness: 0.982 × 10^-3

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 tuning (CTE): ~2 = 1\1, ~121/70 = 951.4956, ~5/4 = 386.7868

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

Badness: 2.26 × 10^-3

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 tuning (CTE): ~2 = 1\1, ~26/15 = 951.8367, ~5/4 = 386.4048

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

Badness: 1.06 × 10^-3

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 tuning (CTE): ~2 = 1\1, ~26/15 = 951.8015, ~5/4 = 386.9912

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

Badness: 1.24 × 10^-3

Trienparapyth

Trienparapyth can be described as the no-17's 23-limit 80 & 87 & 109 temperament. It splits the ~4/3 generator of parapythic into three ~11/10's by tempering out 4000/3993 = S10/S11 in the 11-limit and it 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:

• CTE: ~2 = 1\1, ~11/10 = 165.4134, ~5/4 = 386.8872
• CWE: ~2 = 1\1, ~11/10 = 165.3593, ~5/4 = 387.8093

Optimal ET sequence: 7d, 14e, 15d, 22, 51, 58, 80, 87, 145, 167, 312ce, 479bce

Badness: 1.26 × 10^-3

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:

• CTE: ~2 = 1\1, ~11/10 = 165.3975, ~5/4 = 386.7908
• CWE: ~2 = 1\1, ~11/10 = 165.3802, ~5/4 = 387.8759

Optimal ET sequence: 7d, 22, 29, 51f, 51cde, 58, 80, 87, 145, 167, 225ce, 254, 312ce

Badness: 1.23 × 10^-3

no-17's 19-limit

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:

• CTE: ~2 = 1\1, ~11/10 = 165.2990, ~5/4 = 386.3154
• CWE: ~2 = 1\1, ~11/10 = 165.2976, ~5/4 = 387.7451

Optimal ET sequence: 7d, 22, 29, 51fh, 51cde, 58h, 80, 87, 138cdeh, 167h

Badness: 1.22 × 10^-3

no-17's 23-limit

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

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

• CTE: ~2 = 1\1, ~11/10 = 165.2579, ~5/4 = 386.1446
• CWE: ~2 = 1\1, ~11/10 = 165.2679, ~5/4 = 387.7240

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

Badness: 1.04 × 10^-3