Kleismic family

(Redirected from Bikleismic)

The 5-limit parent comma for the kleismic family is 15625/15552, the kleisma. Its monzo is [-6 -5 6, and flipping that yields ⟨⟨6 5 -6]] for the wedgie. This tells us the generator is a classical minor third (6/5), and that to get to the interval class of major thirds will require five of these, and so to get to fifths will require six. In fact, (6/5)5 = 5/2 × 15625/15552. This 5-limit temperament (virtually a microtemperament) is commonly called hanson, and 14\53 is about perfect as a hanson generator, though 9\34 also makes sense, and 5\19 and 4\15 are possible. Other tunings include 72edo, 87edo and 140edo.

The second comma of the normal comma list defines which 7-limit family member we are looking at. 875/864, the keemic comma, gives keemun. 4375/4374, the ragisma, gives catakleismic. 5120/5103, hemifamity, gives countercata. 6144/6125, the porwell comma, gives hemikleismic. 245/243, sensamagic, gives clyde. 1029/1024, the gamelisma, gives tritikleismic. 2401/2400 the breedsma, gives quadritikleismic. Keemun, catakleismic and countercata all have octave period and use the minor third as a generator; catakleismic and countercata define the 7/4 more complexly but more accurately than keemun. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.

Hanson

Subgroup: 2.3.5

Comma list: 15625/15552

Mapping[1 0 1], 0 6 5]]

mapping generators: ~2, ~6/5
• CTE: ~2 = 1\1, ~6/5 = 317.0308
• POTE: ~2 = 1\1, ~6/5 = 317.007

Optimal ET sequence15, 19, 34, 53, 458, 511c, …, 882c

Cata

Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as 26/15. Notice 15625/15552 = (325/324)(625/624) and 325/324 = (625/624)(676/675). The S-expression-based comma list of the temperament is {S10/S12 = S25*S26, (S25,) S13/S15 = S26}. For the high-limit version of cata with a 1\5 period, see thunderclysmic.

Subgroup: 2.3.5.13

Comma list: 325/324, 625/624

Sval mapping: [1 0 1 0], 0 6 5 14]]

Optimal tunings:

• CTE: ~2 = 1\1, ~6/5 = 317.1110
• POTE: ~2 = 1\1, ~6/5 = 317.0756

Keemun

Subgroup: 2.3.5.7

Comma list: 49/48, 126/125

Mapping[1 0 1 2], 0 6 5 3]]

Wedgie⟨⟨6 5 3 -6 -12 -7]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.473

• 7-odd-limit diamond monotone: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
• 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 7- and 9-odd-limit diamond tradeoff: ~6/5 = [308.744, 322.942]

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 100/99

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.576

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 65/64, 100/99

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.611

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]

Kema

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 100/99

Mapping: [1 0 1 2 4 0], 0 6 5 3 -2 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.423

Tuning ranges:

• 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
• 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]

Kumbaya

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 49/48, 56/55, 66/65

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.595

Qeema

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 126/125

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

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 314.730

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 126/125

Mapping: [1 0 1 2 -1 0], 0 6 5 3 17 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 315.044

Darjeeling

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 77/75

Mapping: [1 0 1 2 0], 0 6 5 3 13]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.656

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 66/65, 77/75

Mapping: [1 0 1 2 0 0], 0 6 5 3 13 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.298

Catalan

Subgroup: 2.3.5.7

Comma list: 64/63, 15625/15552

Mapping[1 0 1 6], 0 6 5 -12]]

Wedgie⟨⟨6 5 -12 -6 -36 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.267

11-limit

Subgroup: 2.3.5.7.11

Comma list: 64/63, 100/99, 1331/1323

Mapping: [1 0 1 6 4], 0 6 5 -12 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 318.282

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
• 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 64/63, 100/99, 144/143, 275/273

Mapping: [1 0 1 6 4 0], 0 6 5 -12 -2 14]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 317.9159

Catakleismic

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 4375/4374

Mapping[1 0 1 -3], 0 6 5 22]]

Wedgie⟨⟨6 5 22 -6 18 37]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.732

2.3.5.7.13 subgroup

The S-expression-based comma list of this temperament is {S13, S15 = S25*S26*S27, S10/S12 = S25*S26(, S25, S26 = S13/S15, S27)}.

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 325/324

Sval mapping: [1 0 1 -3 0], 0 6 5 22 14]]

Optimal tuning (CTE): ~2 = 1\1, ~6/5 = 316.8865

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 4375/4374

Mapping: [1 0 1 -3 9], 0 6 5 22 -21]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.719

Tuning ranges:

• 11-odd-limit diamond monotone range: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
• 11-odd-limit diamond tradeoff range: ~6/5 = [315.641, 317.263]

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 385/384

Mapping: [1 0 1 -3 9 0], 0 6 5 22 -21 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.738

Tuning ranges:

• 13- and 15-odd-limit diamond monotone: ~6/5 = [315.789, 316.981] (5\19 to 14\53)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]

Cataclysmic

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 2200/2187

Mapping: [1 0 1 -3 -5], 0 6 5 22 32]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.042

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 275/273

Mapping: [1 0 1 -3 -5 0], 0 6 5 22 32 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.036

Catalytic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4374

Mapping: [1 0 1 -3 -10], 0 6 5 22 51]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.653

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 325/324, 1716/1715

Mapping: [1 0 1 -3 -10 0], 0 6 5 22 51 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 316.639

Cataleptic

Subgroup: 2.3.5.7.11

Comma list: 100/99, 225/224, 864/847

Mapping: [1 0 1 -3 4], 0 6 5 22 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.083

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 676/675

Mapping: [1 0 1 -3 4 0], 0 6 5 22 -2 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.118

Bikleismic

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 4375/4356

Mapping: [2 0 2 -6 -1], 0 6 5 22 15]]

mapping generators: ~99/70, ~6/5

Optimal tuning (POTE): ~99/70 = 1\2, ~6/5 = 316.721

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 325/324

Mapping: [2 0 2 -6 -1 0], 0 6 5 22 15 14]]

Optimal tuning (POTE): ~55/39 = 1\2, ~6/5 = 316.726

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [2 0 2 -6 -1 0 5], 0 6 5 22 15 14 6]]

Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 225/224, 243/242, 325/324

Mapping: [2 0 2 -6 -1 0 5 -1], 0 6 5 22 15 14 6 18]]

Optimal tuning (POTE): ~17/12 = 1\2, ~6/5 = 316.726

Countercata

Subgroup: 2.3.5.7

Comma list: 5120/5103, 15625/15552

Mapping[1 0 1 11], 0 6 5 -31]]

Wedgie⟨⟨6 5 -31 -6 -66 -86]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.121

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2200/2187, 3388/3375

Mapping: [1 0 1 11 -5], 0 6 5 -31 32]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162

Tuning ranges:

• 11-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
• 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 317.370]

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 385/384, 625/624

Mapping: [1 0 1 11 -5 0], 0 6 5 -31 32 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.162

Tuning ranges:

• 13-odd-limit diamond monotone: ~6/5 = [316.981, 317.647] (14\53 to 9\34)
• 15-odd-limit diamond monotone: ~6/5 = [316.981, 317.241] (14\53 to 23\87)
• 13- and 15-odd-limit diamond tradeoff: ~6/5 = [315.641, 318.309]

Metakleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 179200/177147

Mapping[1 0 1 -12], 0 6 5 56]]

Wedgie⟨⟨6 5 56 -6 72 116]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.314

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 2200/2187, 14700/14641

Mapping: [1 0 1 -12 -5], 0 6 5 56 32]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 625/624

Mapping: [1 0 1 -12 -5 0], 0 6 5 56 32 14]]

Optimal tuning (POTE): ~2 = 1\1, ~6/5 = 317.311

Hemikleismic

Subgroup: 2.3.5.7

Comma list: 4000/3969, 6144/6125

Mapping[1 0 1 4], 0 12 10 -9]]

Wedgie⟨⟨12 10 -9 -12 -48 -49]]

Optimal tuning (POTE): ~2 = 1\1, ~35/32 = 158.649

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 4000/3969

Mapping: [1 0 1 4 2], 0 12 10 -9 11]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.677

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 275/273, 325/324

Mapping: [1 0 1 4 2 0], 0 12 10 -9 11 28]]

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 158.655

Clyde

Subgroup: 2.3.5.7

Comma list: 245/243, 3136/3125

Mapping[1 6 6 12], 0 -12 -10 -25]]

mapping generators: ~2, ~9/7

Wedgie⟨⟨12 10 25 -12 6 30]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.335

[[1 0 0 0, [6/25 0 0 12/25, [6/5 0 0 2/5, [0 0 0 1]
eigenmonzo (unchanged-interval) basis: 2.7

Algebraic generator: real root of 5x3 - 6x - 3, the Poussami generator. Approximately 441.309 cents. Associated recurrence relationship quickly converges.

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 3136/3125

Mapping: [1 6 6 12 -5], 0 -12 -10 -25 23]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.355

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 625/624

Mapping: [1 6 6 12 -5 14], 0 -12 -10 -25 23 -28]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 441.363

Tritikleismic

Subgroup: 2.3.5.7

Comma list: 1029/1024, 15625/15552

Mapping[3 0 3 10], 0 6 5 -2]]

mapping generators: ~63/50, ~6/5

Wedgie⟨⟨18 15 -6 -18 -60 -56]]

Optimal tuning (POTE): ~63/50 = 1\3, ~6/5 = 316.872

[[1 0 0 0, [2 0 6/7 -6/7, [8/3 0 5/7 -5/7, [8/3 0 -2/7 2/7]
eigenmonzo (unchanged-interval) basis: 2.7/5
[[1 0 0 0, [10/7 6/7 0 -3/7, [46/21 5/7 0 -5/14, [20/7 -2/7 0 1/7]
eigenmonzo (unchanged-interval) basis: 2.9/7

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 4000/3993

Mapping: [3 0 3 10 8], 0 6 5 -2 3]]

Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.881

Minimax tuning:

• 11-odd-limit: ~6/5 = [5/21 1/7 0 -1/14
[[1 0 0 0 0, [10/7 6/7 0 -3/7 0, [46/21 5/7 0 -5/14 0, [20/7 -2/7 0 1/7 0, [71/21 3/7 0 -3/14 0]
eigenmonzo (unchanged-interval) basis: 2.9/7

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 385/384, 625/624

Mapping: [3 0 3 10 8 0], 0 6 5 -2 3 14]]

Optimal tuning (POTE): ~44/35 = 1\3, ~6/5 = 316.9585

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 364/363, 375/374, 385/384

Mapping: [3 0 3 10 8 0 -2], 0 6 5 -2 3 14 18]]

Optimal tuning (POTE): ~34/27 = 1\3, ~6/5 = 316.9082

Subgroup: 2.3.5.7

Comma list: 2401/2400, 15625/15552

Mapping[4 0 4 7], 0 6 5 4]]

mapping generators: ~25/21, ~6/5

Wedgie⟨⟨24 20 16 -24 -42 -19]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9999

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 6250/6237

Mapping: [4 0 4 7 17], 0 6 5 4 -3]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9247

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 1375/1372

Mapping: [4 0 4 7 17 0], 0 6 5 4 -3 14]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9887

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 385/384, 442/441, 625/624

Mapping: [4 0 4 7 17 0 10], 0 6 5 4 -3 14 6]]

Optimal tuning (POTE): ~25/21 = 1\4, ~6/5 = 316.9846

Kleiboh

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3125/3087

Mapping[1 6 6 6], 0 -18 -15 -13]]

mapping generators: ~2, ~25/21

Wedgie⟨⟨18 15 13 -18 -30 -12]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.303

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 3125/3087

Mapping: [1 6 6 6 14], 0 -18 -15 -13 -43]]

Optimal tuning (POTE): ~2 = 1\1, ~25/21 = 294.181

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 275/273, 325/324, 540/539

Mapping: [1 6 6 6 14 14], 0 -18 -15 -13 -43 -42]]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 294.187

Marfifths

The marfifths temperament (19&140) tempers out the hemimage comma, 10976/10935. It splits the interval of major tenth (~10/3) into three marvelous fifth (112/75) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 15625/15552

Mapping[1 -6 -4 -17], 0 18 15 47]]

mapping generators: ~2, ~75/56

Wedgie⟨⟨18 15 47 -18 24 67]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.705

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 10976/10935

Mapping: [1 -6 -4 -17 22], 0 18 15 47 -44]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.684

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 10976/10935

Mapping: [1 -6 -4 -17 22 -14], 0 18 15 47 -44 42]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.686

Diatessic

The diatessic temperament (121 & 140) is closely related to the diatess tuning (generator: 505.727281 cents).

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 2200/2187, 5632/5625

Mapping: [1 -6 -4 -17 -37], 0 18 15 47 96]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 625/624, 1375/1372

Mapping: [1 -6 -4 -17 -37 -14], 0 18 15 47 96 42]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.740

Marf

The marf temperament (19 & 121) has a POTE generator which strongly approximates the marvelous fifth interval of 112/75.

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 15625/15552

Mapping: [1 -6 -4 -17 14], 0 18 15 47 -25]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.769

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 540/539, 625/624, 896/891

Mapping: [1 -6 -4 -17 14 -14], 0 18 15 47 -25 42]]

Optimal tuning (POTE): ~2 = 1\1, ~75/56 = 505.771

Marthirds

The marthirds temperament (19 & 193) tempers out the breeze comma (laquadru-atruyo comma), 2460375/2458624. It splits the interval of minor tenth (~12/5) into four marvelous major third (56/45) intervals, and uses it for a generator.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 2460375/2458624

Mapping[1 -6 -4 -19], 0 24 20 69]]

mapping generators: ~2, ~56/45

Wedgie⟨⟨24 20 69 -24 42 104]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.252

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 15625/15552, 19712/19683

Mapping: [1 -6 -4 -19 -43], 0 24 20 69 147]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.257

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 19712/19683

Mapping: [1 -6 -4 -19 -43 -14], 0 24 20 69 147 56]]

Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 379.256

Quartkeenlig

Quartkeenlig uses a generator in the 11-limit that is 33/32~36/35 tempered together, and is called so because it tempers out the quartisma by virtue of five 33/32's being with 7/6, keenanisma, 385/384, tempering 33/32 and 36/35 together, and liganellus comma (6250/6237). It can also be viewed as a regular temperament interpretation of stretched 23edo.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 117649/116640

Mapping[1 0 1 1], 0 36 30 41]]

mapping generator: ~2, ~36/35

Optimal tuning (CTE): ~2 = 1\1, ~36/35 = 52.8562

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 6250/6237, 67228/66825

Mapping: [1 0 1 1 5], 0 36 30 41 -35]]

Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8524

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 625/624, 16807/16731

Mapping: [1 0 1 1 5 0], 0 36 30 41 -35 84]]

Optimal tuning (CTE): ~2 = 1\1, ~33/32 = 52.8562

Novemkleismic

Subgroup: 2.3.5.7

Comma list: 15625/15552, 40353607/40310784

Mapping[9 0 9 11], 0 6 5 6]]

mapping generators: ~2592/2401, ~6/5

Wedgie⟨⟨54 45 54 -54 -66 -1]]

Optimal tuning (POTE): ~2592/2401 = 1\9, ~6/5 = 317.005

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4000/3993, 15625/15552

Mapping: [9 0 9 11 24], 0 6 5 6 3]]

Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.010

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 625/624, 1375/1372, 4000/3993

Mapping: [9 0 9 11 24 0], 0 6 5 6 3 14]]

Optimal tuning (POTE): ~250/231 = 1\9, ~6/5 = 317.086

Sqrtphi

The just value of sqrt (φ) is 416.545 cents.

Subgroup: 2.3.5.7

Comma list: 15625/15552, 16875/16807

Mapping[1 12 11 16], 0 -30 -25 -38]]

mapping generators: ~2, 125/98

Optimal tuning (POTE): ~2 = 1\1, ~125/98 = 416.603

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4375/4356

Mapping: [1 12 11 16 17], 0 -30 -25 -38 -39]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.604

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 625/624, 1375/1372

Mapping: [1 12 11 16 17 28], 0 -30 -25 -38 -39 -70]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 375/374, 540/539, 595/594

Mapping: [1 12 11 16 17 28 27], 0 -30 -25 -38 -39 -70 -66]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.585

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 364/363, 375/374, 400/399, 442/441, 595/594

Mapping: [1 12 11 16 17 28 27 -2], 0 -30 -25 -38 -39 -70 -66 18]]

Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 416.580