Jubilismic clan

The jubilismic clan tempers out the jubilisma, 50/49, which means 7/5 and 10/7 are both equated to the 600-cent tritone and the octave is divided in two.

Jubilic

The head of this clan, jubilic, is generated by ~5/4. That and a semioctave give ~7/4.

Subgroup: 2.5.7

Comma list: 50/49

Sval mapping: [⟨2 0 1], ⟨0 1 1]]

sval mapping generators: ~7/5, ~5

Gencom mapping: [⟨2 0 0 1], ⟨0 0 1 1]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~5/4 = 379.210 (~8/7 = 220.890)

error map: ⟨0.000 -7.104 +10.384]

• POTE: ~7/5 = 600.000, ~5/4 = 380.840 (~8/7 = 219.160)

error map: ⟨0.000 -5.474 +12.014]

Optimal ET sequence: 2, 4, 6, 16, 22, 60d

Badness (Smith): 0.00190

Overview to extensions

Lemba finds the perfect fifth three steps away by tempering out 1029/1024. Astrology, five steps away by tempering out 3125/3072. Decimal, two steps away by tempering out 25/24 and 49/48. Walid merges ~5/4 and ~4/3 by tempering out 16/15.

Diminished adds 36/35 and splits the ~7/5 period in a further two. Pajara adds 64/63 and slices the ~7/4 in two, with antikythera being every other step thereof. Dubbla adds 78125/73728 and slices the ~5/4 in two. Injera adds 81/80 and slices the ~5/1 in four. Octokaidecal adds 28/27. Bipelog adds 135/128. Those splits the generator into three in various ways. Hexe adds 128/125 and slices the period in three. Hedgehog adds 250/243. Elvis adds 8505/8192. Those slice the generator in five. Comic adds 2240/2187. Crepuscular adds 4375/4374. Those slice the generator in seven. Byhearted adds 19683/19208. Bipyth adds 20480/19683. Those slice the generator in nine.

Temperaments discussed elsewhere are:

• Decimal (+25/24) → Dicot family
• Diminished (+36/35) → Dimipent family
• Pajara (+64/63) → Diaschismic family
• Dubbla (+78125/73728) → Wesley family
• Injera (+81/80) → Meantone family
• Octokaidecal (+28/27) → Trienstonic clan
• Bipelog (+135/128) → Mavila family
• Hexe (+128/125) → Augmented family
• Hedgehog (+250/243) → Porcupine family
• Crepuscular (+4375/4374) → Fifive family
• Byhearted (+19683/19208) → Tetracot family

Considered below are lemba, astrology, walid, antikythera, doublewide, elvis, comic, and bipyth.

Lemba

Main article: Lemba

For the 5-limit version, see Miscellaneous 5-limit temperaments #Lemba.

Lemba tempers out 1029/1024, the gamelisma, and a stack of three ~8/7 generators gives an approximate perfect fifth.

Subgroup: 2.3.5.7

Comma list: 50/49, 525/512

Mapping: [⟨2 2 5 6], ⟨0 3 -1 -1]]

mapping generators: ~7/5, ~8/7

Wedgie: ⟨⟨ 6 -2 -2 -17 -20 1 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~8/7 = 232.927

error map: ⟨0.000 -3.175 -19.241 -1.753]

• POTE: ~7/5 = 600.000, ~8/7 = 232.089

error map: ⟨0.000 -5.689 -18.402 -0.915]

Optimal ET sequence: 10, 16, 26, 62c

Badness (Smith): 0.062208

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 385/384

Mapping: [⟨2 2 5 6 5], ⟨0 3 -1 -1 5]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~8/7 = 231.997
• POTE: ~7/5 = 600.000, ~8/7 = 230.974

Optimal ET sequence: 10, 16, 26

Badness (Smith): 0.041563

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 65/64, 78/77

Mapping: [⟨2 2 5 6 5 7], ⟨0 3 -1 -1 5 1]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~8/7 = 232.100
• POTE: ~7/5 = 600.000, ~8/7 = 230.966

Optimal ET sequence: 10, 16, 26

Badness (Smith): 0.025477

Astrology

Astrology tempers out 3125/3072, the magic comma, and a stack of five ~5/4 generators gives an approximate harmonic 3.

Subgroup: 2.3.5.7

Comma list: 50/49, 3125/3072

Mapping: [⟨2 0 4 5], ⟨0 5 1 1]]

mapping geenerators: ~7/5, ~5/4

Wedgie: ⟨⟨ 10 2 2 -20 -25 -1 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~5/4 = 380.355 (~8/7 = 219.645)

error map: ⟨0.000 -0.180 -5.959 +11.529]

• POTE: ~7/5 = 600.000, ~5/4 = 380.578 (~8/7 = 219.422)

error map: ⟨0.000 +0.937 -5.735 +11.752]

Optimal ET sequence: 6, 16, 22, 60d

Badness (Smith): 0.082673

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 176/175

Mapping: [⟨2 0 4 5 5], ⟨0 5 1 1 3]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~5/4 = 380.588 (~8/7 = 219.412)
• POTE: ~7/5 = 600.000, ~5/4 = 380.530 (~8/7 = 219.470)

Optimal ET sequence: 6, 16, 22

Badness (Smith): 0.039151

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/64, 78/77, 121/120

Mapping: [⟨2 0 4 5 5 8], ⟨0 5 1 1 3 -1]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~5/4 = 380.449 (~8/7 = 219.551)
• POTE: ~7/5 = 600.000, ~5/4 = 379.787 (~8/7 = 220.213)

Optimal ET sequence: 6, 16, 22, 38f

Badness (Smith): 0.034376

Music

• Astrology Percussion Quintet No 1⁠ ⁠^[dead link] play⁠ ⁠^[dead link] by Joel Taylor

Horoscope

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 66/65, 105/104, 121/120

Mapping: [⟨2 0 4 5 5 3], ⟨0 5 1 1 3 7]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~5/4 = 379.762 (~8/7 = 220.238)
• POTE: ~7/5 = 600.000, ~5/4 = 379.837 (~8/7 = 220.163)

Optimal ET sequence: 16, 22f, 38

Badness (Smith): 0.035284

Walid

Subgroup: 2.3.5.7

Comma list: 16/15, 50/49

Mapping: [⟨2 0 8 9], ⟨0 1 -1 -1]]

mapping generators: ~7/5, ~3

Wedgie: ⟨⟨ 2 -2 -2 -8 -9 1 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 754.204 (~15/14 = 154.204)

error map: ⟨0.000 +52.249 +59.482 +76.970]

• POTE: ~7/5 = 600.000, ~3/2 = 749.415 (~15/14 = 149.415)

error map: ⟨0.000 +47.460 +64.271 +81.759]

Optimal ET sequence: 2, 6, 8d

Badness (Smith): 0.048978

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 50/49

Mapping: [⟨2 0 8 9 7], ⟨0 1 -1 -1 0]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 754.205 (~15/14 = 154.205)
• POTE: ~7/5 = 600.000, ~3/2 = 749.756 (~15/14 = 149.756)

Optimal ET sequence: 2, 6, 8d

Badness (Smith): 0.029193

Antikythera

Named by Gene Ward Smith in 2011, antikythera is every other step of pajara.

Subgroup: 2.9.5.7

Comma list: 50/49, 64/63

Sval mapping: [⟨2 0 11 12], ⟨0 1 -1 -1]]

mapping generators: ~7/5, ~9

Gencom mapping: [⟨2 3 5 6], ⟨0 1/2 -1 -1]]

gencom: [7/5 8/7; 50/49 64/63]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~9/8 = 216.711

error map: ⟨0.000 +12.801 +3.025 +14.463]

• POTE: ~7/5 = 600.000, ~9/8 = 214.095

error map: ⟨0.000 +10.815 -0.409 +17.079]

Optimal ET sequence: 2, 4, 6, 16, 22, 28

RMS error: 2.572 cents

Badness (Smith): 0.00501

Doublewide

For the 5-limit version, see Superpyth–22 equivalence continuum #Doublewide (5-limit).

Subgroup: 2.3.5.7

Comma list: 50/49, 875/864

Mapping: [⟨2 1 3 4], ⟨0 4 3 3]]

mapping generators: ~7/5, ~6/5

Wedgie: ⟨⟨ 8 6 6 -9 -13 -3 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~6/5 = 325.769 (~7/6 = 274.231)

error map: ⟨0.000 +1.120 -9.007 +8.481]

• POTE: ~7/5 = 600.000, ~6/5 = 325.719 (~7/6 = 274.281)

error map: ⟨0.000 +0.921 -9.156 +8.331]

Optimal ET sequence: 4, 14bd, 18, 22, 48

Badness (Smith): 0.043462

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 875/864

Mapping: [⟨2 1 3 4 8], ⟨0 4 3 3 -2]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~6/5 = 325.719 (~7/6 = 274.281)
• POTE: ~7/5 = 600.000, ~6/5 = 325.545 (~7/6 = 274.455)

Optimal ET sequence: 4, 18, 22, 48

Badness (Smith): 0.032058

Fleetwood

Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 176/175

Mapping: [⟨2 1 3 4 2], ⟨0 4 3 3 9]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~6/5 = 326.684 (~7/6 = 273.316)
• POTE: ~7/5 = 600.000, ~6/5 = 327.038 (~7/6 = 272.962)

Optimal ET sequence: 4e, …, 18e, 22

Badness (Smith): 0.035202

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 55/54, 65/63, 176/175

Mapping: [⟨2 1 3 4 2 3], ⟨0 4 3 3 9 8]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~6/5 = 327.450 (~7/6 = 272.540)
• POTE: ~7/5 = 600.000, ~6/5 = 327.841 (~7/6 = 272.159)

Optimal ET sequence: 4ef, …, 18e, 22

Badness (Smith): 0.031835

Cavalier

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 875/864

Mapping: [⟨2 1 3 4 1], ⟨0 4 3 3 11]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~6/5 = 324.238 (~7/6 = 275.762)
• POTE: ~7/5 = 600.000, ~6/5 = 323.427 (~7/6 = 276.573)

Optimal ET sequence: 4e, 22e, 26

Badness (Smith): 0.052899

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 325/324

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

Optimal tunings:

• CTE: ~7/5 = 600.000, ~6/5 = 324.187 (~7/6 = 275.813)
• POTE: ~7/5 = 600.000, ~6/5 = 323.396 (~7/6 = 276.604)

Optimal ET sequence: 4ef, 22ef, 26

Badness (Smith): 0.035040

Elvis

For the 5-limit version, see Miscellaneous 5-limit temperaments #Elvis.

Subgroup: 2.3.5.7

Comma list: 50/49, 8505/8192

Mapping: [⟨2 1 10 11], ⟨0 2 -5 -5]]

mapping generators: ~7/5, ~64/45

Wedgie: ⟨⟨ 4 -10 -10 -25 -27 5 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~64/45 = 645.313 (~64/63 = 45.313)

error map: ⟨0.000 -11.329 -12.879 +4.609]

• POTE: ~7/5 = 600.000, ~64/45 = 646.279 (~64/63 = 46.279)

error map: ⟨0.000 -9.397 -17.710 -0.222]

Optimal ET sequence: 2, 24c, 26

Badness (Smith): 0.141473

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 1344/1331

Mapping: [⟨2 1 10 11 8], ⟨0 2 -5 -5 -1]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~16/11 = 645.343 (~56/55 = 45.343)
• POTE: ~7/5 = 600.000, ~16/11 = 646.118 (~56/55 = 46.118)

Optimal ET sequence: 2, 24c, 26

Badness (Smith): 0.063212

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 1053/1024

Mapping: [⟨2 1 10 11 8 16], ⟨0 2 -5 -5 -1 -8]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~16/11 = 645.208 (~56/55 = 45.208)
• POTE: ~7/5 = 600.000, ~16/11 = 646.108 (~56/55 = 46.108)

Optimal ET sequence: 2f, 24cf, 26

Badness (Smith): 0.043997

Comic

For the 5-limit version, see Superpyth–22 equivalence continuum #Comic (5-limit).

Subgroup: 2.3.5.7

Comma list: 50/49, 2240/2187

Mapping: [⟨2 1 -3 -2], ⟨0 2 7 7]]

mapping generators: ~7/5, ~40/27

Wedgie: ⟨⟨ 4 14 14 13 11 -7 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~40/27 = 653.872 (~28/27 = 53.872)

error map: ⟨0.000 +5.789 -9.211 +8.277]

• POTE: ~7/5 = 600.000, ~40/27 = 654.699 (~28/27 = 54.699)

error map: ⟨0.000 +7.444 -3.417 +14.070]

Optimal ET sequence: 2cd, …, 20cd, 22

Badness (Smith): 0.084395

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 2662/2625

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

Optimal tunings:

• CTE: ~7/5 = 600.000, ~22/15 = 654.289 (~28/27 = 54.289)
• POTE: ~7/5 = 600.000, ~22/15 = 655.184 (~28/27 = 55.184)

Optimal ET sequence: 2cde, …, 20cde, 22

Badness (Smith): 0.045052

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 65/63, 99/98, 968/945

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

Optimal tunings:

• CTE: ~7/5 = 600.000, ~22/15 = 654.547 (~28/27 = 54.547)
• POTE: ~7/5 = 600.000, ~22/15 = 654.435 (~28/27 = 54.435)

Optimal ET sequence: 2cde, 20cde, 22

Badness (Smith): 0.041470

Bipyth

For the 5-limit version, see Superpyth–22 equivalence continuum.

Subgroup: 2.3.5.7

Comma list: 50/49, 20480/19683

Mapping: [⟨2 0 -24 -23], ⟨0 1 9 9]]

mapping generators: ~7/5, ~3

Wedgie: ⟨⟨ 2 18 18 24 23 -9 ]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 708.695 (~15/14 = 108.695)

error map: ⟨0.000 +6.740 -8.058 +9.430]

• POTE: ~7/5 = 600.000, ~3/2 = 709.437 (~15/14 = 109.437)

error map: ⟨0.000 +7.482 -1.379 +16.109]

Optimal ET sequence: 10cd, 12cd, 22

Badness (Smith): 0.165033

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 121/120, 896/891

Mapping: [⟨2 0 -24 -23 -9], ⟨0 1 9 9 5]]

Optimal tunings:

• CTE: ~7/5 = 600.000, ~3/2 = 708.813 (~15/14 = 108.813)
• POTE: ~7/5 = 600.000, ~3/2 = 709.310 (~15/14 = 109.310)

Optimal ET sequence: 10cd, 12cde, 22

Badness (Smith): 0.070910

Sedecic

Subgroup: 2.3.5.7

Comma list: 50/49, 546875/524288

Mapping: [⟨16 0 37 45], ⟨0 1 0 0]]

Wedgie: ⟨⟨ 16 0 0 -37 -45 0 ]]

Optimal tunings:

• CTE: ~128/125 = 75.000, ~3/2 = 701.955 (~525/512 = 26.955)

error map: ⟨0.000 0.000 -11.314 +6.174]

• POTE: ~128/125 = 75.000, ~3/2 = 700.554 (~525/512 = 25.554)

error map: ⟨0.000 -1.401 -11.314 +6.174]

Optimal ET sequence: 16, 32, 48

Badness (Smith): 0.265972

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 385/384, 1331/1323

Mapping: [⟨16 0 37 45 30], ⟨0 1 0 0 1]]

Optimal tunings:

• CTE: ~22/21 = 75.000, ~3/2 = 701.844 (~45/44 = 26.844)
• POTE: ~22/21 = 75.000, ~3/2 = 700.331 (~45/44 = 25.331)

Optimal ET sequence: 16, 32, 48

Badness (Smith): 0.092774