Archytas clan
The archytas clan (or archy family) tempers out the Archytas' comma, 64/63. This means that four stacked 3/2 fifths equal a 9/7 major third. (Note the similarity in function to 81/80 in meantone, where four stacked 3/2 fifths equal a 5/4 major third.) This leads to tunings with 3's and 7's quite sharp, such as those of 22edo.
Archy
Subgroup: 2.3.7
Comma list: 64/63
Sval mapping: [⟨1 0 6], ⟨0 1 -2]]
- sval mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]
- gencom: [2 3/2; 64/63]
Optimal ET sequence: 2, 3, 5, 12, 17, 22, 137bdd, 159bddd, 181bbddd
Badness: 0.00464
Scales: archy5, archy7, archy12
Overview to extensions
Adding 245/243 to the list of commas gives superpyth; 2430/2401 gives quasisuper; 36/35 gives dominant; 360/343 gives schism; 6860/6561 gives ultrapyth; 33614/32805 gives quasiultra; 16/15 gives mother. These all use the same generators as archy.
50/49 gives pajara with a semioctave period. 126/125 gives augene with a 1/3-octave period. 28/27 gives blacksmith with a 1/5-octave period. 686/675 gives beatles, splitting the fifth in two. 250/243 gives porcupine, splitting the fourth in three. 4375/4374 gives modus, splitting the fifth in four. 3125/3087 gives passion, splitting the fourth in five.
Discussed under their respective 5-limit families are:
- Mother → Father family
- Dominant → Meantone family
- Augene → Augmented family
- Porcupine → Porcupine family
- Pajara → Diaschismic family
- Blacksmith → Limmic temperaments
- Catalan → Kleismic family
- Modus → Tetracot family
- Passion → Passion family
- Immunized → Immunity family
- Suhajira → Neutral clan
- Brightstone → Magic family
The rest are considered below.
Supra
Subgroup: 2.3.7.11
Comma list: 64/63, 99/98
Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]
Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]
- gencom: [2 3/2; 64/63 99/98]
Optimal tunings:
- CTE: ~3/2 = 708.4564
- POTE: ~3/2 = 707.192
Optimal ET sequence: 5, 12, 17, 39d, 56d
Badness: 0.00933
Supraphon
Subgroup: 2.3.7.11.13
Comma list: 64/63, 78/77, 99/98
Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]
Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]
- gencom: [2 3/2; 64/63 78/77 99/98]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 707.3440
- POTE: ~2 = 1\1, ~3/2 = 706.137
Badness: 0.0111
Superpyth
In the 5-limit, superpyth tempers out 20480/19683. This temperament has a fifth generator of ~3/2 = ~710¢ and ~5/4 is found at +9 generator steps, as an augmented second (C-D#). It also has a weak extension, bipyth (10cd & 22), tempering out the same 5-limit comma as the superpyth, but with a half-octave period and the jubilisma (50/49) rather than the Archytas comma tempered out.
Subgroup: 2.3.5
Comma list: 20480/19683
Mapping: [⟨1 0 -12], ⟨0 1 9]]
Optimal ET sequence: 5, 17, 22, 49, 120b, 169bbc
Badness: 0.135141
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 245/243
Mapping: [⟨1 0 -12 6], ⟨0 1 9 -2]]
Wedgie: ⟨⟨1 9 -2 12 -6 -30]]
Optimal ET sequence: 5, 17, 22, 27, 49, 174bbcddd
Badness: 0.032318
11-limit
The canonical extension to the 13-limit finds the ~11/8 at +16 generator steps, as a double augmented second (C-Dx) and finds the ~13/8 at +13 generator steps, as a double augmented fourth (C-Fx).
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 245/243
Mapping: [⟨1 0 -12 6 -22], ⟨0 1 9 -2 16]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 709.5143
- POTE: ~2 = 1\1, ~3/2 = 710.175
Optimal ET sequence: 22, 27e, 49
Badness: 0.024976
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 91/90, 100/99
Mapping: [⟨1 0 -12 6 -22 -17], ⟨0 1 9 -2 16 13]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 709.8362
- POTE: ~2 = 1\1, ~3/2 = 710.479
Optimal ET sequence: 22, 27e, 49, 76bcde
Badness: 0.024673
Thomas
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 100/99, 169/168, 245/243
Mapping: [⟨1 1 -3 4 -6 4], ⟨0 2 18 -4 32 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~16/13 = 354.7594
- POTE: ~2 = 1\1, ~16/13 = 355.036
Optimal ET sequence: 27e, 44, 71d, 98bde
Badness: 0.049183
Suprapyth
Suprapyth finds the ~11/8 at the diminished fifth (C-Gb), and finds the ~13/8 at the diminished seventh (C-Bbb).
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 99/98
Mapping: [⟨1 0 -12 6 13], ⟨0 1 9 -2 -6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 709.3561
- POTE: ~2 = 1\1, ~3/2 = 709.495
Optimal ET sequence: 5, 17, 22
Badness: 0.032768
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 65/63, 99/98
Mapping: [⟨1 0 -12 6 13 18], ⟨0 1 9 -2 -6 -9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 708.7020
- POTE: ~2 = 1\1, ~3/2 = 708.703
Optimal ET sequence: 5f, 17, 22
Badness: 0.036336
Quasisuper
Quasisuper can be described as 17c & 22, with the ~5/4 mapped to -13 generator steps, as a double diminished fifth (C-Gbb).
Subgroup: 2.3.5.7
Comma list: 64/63, 2430/2401
Mapping: [⟨1 0 23 6], ⟨0 1 -13 -2]]
Wedgie: ⟨⟨1 -13 -2 -23 -6 32]]
Optimal ET sequence: 17c, 22, 61d
Badness: 0.063794
Quasisupra
Quasisupra can be viewed as an extension of the excellent 2.3.7.11 temperament supra, with the quasisuper mapping of 5 thrown in, rather than the superpyth mapping of 5 (which results in suprapyth).
Subgroup: 2.3.5.7.11
Comma list: 64/63, 99/98, 121/120
Mapping: [⟨1 0 23 6 13], ⟨0 1 -13 -2 -6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 708.7131
- POTE: ~2 = 1\1, ~3/2 = 708.205
Optimal ET sequence: 17c, 22, 39d, 61d
Badness: 0.032203
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 78/77, 91/90, 121/120
Mapping: [⟨1 0 23 6 13 18], ⟨0 1 -13 -2 -6 -9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 708.4106
- POTE: ~2 = 1\1, ~3/2 = 708.004
Optimal ET sequence: 17c, 22, 39d, 61df, 100bcdf
Badness: 0.030219
Quasisoup
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 2430/2401
Mapping: [⟨1 0 23 6 -22], ⟨0 1 -13 -2 16]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 709.0435
- POTE: ~2 = 1\1, ~3/2 = 709.021
Optimal ET sequence: 22
Badness: 0.083490
Ultrapyth
Ultrapyth can be viewed as an extension of the excellent 2.3.7.13/5 oceanfront temperament, mapping the ~5/4 to +14 fifths as a double augmented unison (C-Cx).
Subgroup: 2.3.5.7
Comma list: 64/63, 6860/6561
Mapping: [⟨1 0 -20 6], ⟨0 1 14 -2]]
Wedgie: ⟨⟨1 14 -2 20 -6 -44]]
Optimal ET sequence: 5, 27c, 32, 37
Badness: 0.108466
11-limit
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 2401/2376
Mapping: [⟨1 0 -20 6 21], ⟨0 1 14 -2 -11]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 713.2816
- POTE: ~2 = 1\1, ~3/2 = 713.395
Optimal ET sequence: 5, 32, 37
Badness: 0.068238
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 91/90, 1573/1568
Mapping: [⟨1 0 -20 6 21 -25], ⟨0 1 14 -2 -11 18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 713.3086
- POTE: ~2 = 1\1, ~3/2 = 713.500
Optimal ET sequence: 5, 32, 37
Badness: 0.049172
Ultramarine
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 3773/3645
Mapping: [⟨1 0 -20 6 -38], ⟨0 1 14 -2 26]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 713.3952
- POTE: ~2 = 1\1, ~3/2 = 713.791
Optimal ET sequence: 5e, 32e, 37, 79bce
Badness: 0.078068
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 91/90, 100/99, 847/845
Mapping: [⟨1 0 -20 6 -38 -25], ⟨0 1 14 -2 26 18]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 713.7079
- POTE: ~2 = 1\1, ~3/2 = 713.811
Optimal ET sequence: 5e, 32e, 37, 79bcef
Badness: 0.045653
Quasiultra
Quasiultra is to ultrapyth what quasisuper is to superpyth. It is the 27 & 32 temperament, mapping the ~5/4 to -18 fifths as a double diminished sixth (C-Abbb).
Subgroup: 2.3.5.7
Comma list: 64/63, 33614/32805
Mapping: [⟨1 0 31 6], ⟨0 1 -18 -2]]
Wedgie: ⟨⟨1 -18 -2 -31 -6 46]]
Optimal ET sequence: 27, 86bd, 113bcd, 140bbcd
Badness: 0.132
Schism
- See also: Schismatic family #Schism
Schism tempers out the schisma, mapping the ~5/4 to -8 fifths as a diminished fourth (C-Fb) as does any schismic temperament. 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
Subgroup: 2.3.5.7
Comma list: 64/63, 360/343
Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]
Wedgie: ⟨⟨1 -8 -2 -15 -6 18]]
Optimal ET sequence: 5c, 7c, 12
Badness: 0.056648
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
Optimal tunings:
- CTE: ~2 = 1\1, ~3/2 = 703.3833
- POTE: ~2 = 1\1, ~3/2 = 702.136
Optimal ET sequence: 5c, 7ce, 12, 29de
Badness: 0.037482
Beatles
- For the 5-limit version of this temperament, see High badness temperaments #Beatles.
Subgroup: 2.3.5.7
Comma list: 64/63, 686/675
Mapping: [⟨1 1 5 4], ⟨0 2 -9 -4]]
Wedgie: ⟨⟨2 -9 -4 -19 -12 16]]
Optimal ET sequence: 10, 17c, 27, 64b, 91bcd, 118bccd
Badness: 0.045872
- Music
11-limit
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 686/675
Mapping: [⟨1 1 5 4 10], ⟨0 2 -9 -4 -22]]
Optimal tunings:
- CTE: ~2 = 1\1, ~49/40 = 356.7189
- POTE: ~2 = 1\1, ~49/40 = 356.140
Optimal ET sequence: 10e, 17cee, 27e, 64be, 91bcdee
Badness: 0.045639
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 91/90, 100/99, 169/168
Mapping: [⟨1 1 5 4 10 4], ⟨0 2 -9 -4 -22 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~16/13 = 356.7223
- POTE: ~2 = 1\1, ~16/13 = 356.229
Optimal ET sequence: 10e, 27e, 37, 64be
Badness: 0.030161
Ringo
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63, 540/539
Mapping: [⟨1 1 5 4 2], ⟨0 2 -9 -4 5]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 355.9918
- POTE: ~2 = 1\1, ~11/9 = 355.419
Optimal ET sequence: 10, 17c, 27e
Badness: 0.032863
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 64/63, 78/77, 91/90
Mapping: [⟨1 1 5 4 2 4], ⟨0 2 -9 -4 5 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~11/9 = 356.0040
- POTE: ~2 = 1\1, ~11/9 = 355.456
Optimal ET sequence: 10, 17c, 27e
Badness: 0.022619
Beetle
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 686/675
Mapping: [⟨1 1 5 4 -1], ⟨0 2 -9 -4 15]]
Optimal tunings:
- CTE: ~2 = 1\1, ~49/40 = 356.6943
- POTE: ~2 = 1\1, ~49/40 = 356.710
Optimal ET sequence: 10, 27, 37
Badness: 0.058084
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 55/54, 64/63, 91/90, 169/168
Mapping: [⟨1 1 5 4 -1 4], ⟨0 2 -9 -4 15 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~16/13 = 356.6997
- POTE: ~2 = 1\1, ~16/13 = 356.701
Optimal ET sequence: 10, 27, 37
Badness: 0.033971
Progress
- For the 5-limit version of this temperament, see High badness temperaments #Progress.
Subgroup: 2.3.5.7
Comma list: 64/63, 392/375
Mapping: [⟨1 0 5 6], ⟨0 3 -5 -6]]
- mapping generators: ~2, ~10/7
Wedgie: ⟨⟨3 -5 -6 -15 -18 0]]
Optimal ET sequence: 2, 13, 15, 32c
Badness: 0.066400
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63, 77/75
Mapping: [⟨1 0 5 6 4], ⟨0 3 -5 -6 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/7 = 638.8462
- POTE: ~2 = 1\1, ~10/7 = 637.915
Optimal ET sequence: 2, 13, 15, 32c, 47bc
Badness: 0.031036
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 64/63, 66/65, 77/75
Mapping: [⟨1 0 5 6 4 0], ⟨0 3 -5 -6 -1 7]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/7 = 637.8705
- POTE: ~2 = 1\1, ~10/7 = 637.635
Optimal ET sequence: 15, 17c, 32cf
Badness: 0.026214
Progressive
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 56/55, 64/63, 77/75
Mapping: [⟨1 0 5 6 4 9], ⟨0 3 -5 -6 -1 -10]]
Optimal tunings:
- CTE: ~2 = 1\1, ~10/7 = 637.7968
- POTE: ~2 = 1\1, ~10/7 = 636.761
Optimal ET sequence: 2f, 15f, 17c
Badness: 0.032721
Fervor
- For the 5-limit version of this temperament, see High badness temperaments #Fervor.
Subgroup: 2.3.5.7
Comma list: 64/63, 9604/9375
Mapping: [⟨1 4 -2 -2], ⟨0 -5 9 10]]
- mapping generators: ~2, ~7/5
Wedgie: ⟨⟨5 -9 -10 -26 -30 2]]
Optimal ET sequence: 2, 25, 27
Badness: 0.108455
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 64/63, 1350/1331
Mapping: [⟨1 4 -2 -2 3], ⟨0 -5 9 10 1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~7/5 = 577.2958
- POTE: ~2 = 1\1, ~7/5 = 577.850
Optimal ET sequence: 2, 25e, 27e
Badness: 0.052054
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 64/63, 78/77, 507/500
Mapping: [⟨1 4 -2 -2 3 -4], ⟨0 -5 9 10 1 16]]
Optimal tunings:
- CTE: ~2 = 1\1, ~7/5 = 577.3744
- POTE: ~2 = 1\1, ~7/5 = 578.060
Badness: 0.039705
Sixix
- See also: Dual-fifth temperaments #Dual-3 Sixix
Sixix is related to the kleismic family in a way similar to the one between meantone and mavila. In both cases the generator is nominally a 6/5 and the complexity to generate major and minor chords is the same, but in sixix it is tuned extremely sharply, to the point where the 3rd and 5th harmonics are reached by going down instead of up, inverting the logic of chord construction.
Subgroup: 2.3.5
Comma list: 3125/2916
Mapping: [⟨1 3 4], ⟨0 -5 -6]]
- mapping generators: ~2, ~6/5
Optimal ET sequence: 7, 25, 32, 39c
Badness: 0.153088
7-limit
Subgroup: 2.3.5.7
Comma list: 64/63, 3125/2916
Mapping: [⟨1 3 4 0], ⟨0 -5 -6 10]]
Wedgie: ⟨⟨5 6 -10 -2 -30 -40]]
Optimal ET sequence: 7, 18d, 25, 32
Badness: 0.158903
11-limit
Subgroup: 2.3.5.7.11
Comma list: 55/54, 64/63, 125/121
Mapping: [⟨1 3 4 0 6], ⟨0 -5 -6 10 -9]]
Optimal tunings:
- CTE: ~2 = 1\1, ~6/5 = 337.7486
- POTE: ~2 = 1\1, ~6/5 = 337.564
Optimal ET sequence: 7, 25e, 32
Badness: 0.070799
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 55/54, 64/63, 125/121
Mapping: [⟨1 3 4 0 6 4], ⟨0 -5 -6 10 -9 -1]]
Optimal tunings:
- CTE: ~2 = 1\1, ~6/5 = 337.7925
- POTE: ~2 = 1\1, ~6/5 = 337.483
Optimal ET sequence: 7, 25e, 32f
Badness: 0.046206
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 40/39, 55/54, 64/63, 85/84, 125/121
Mapping: [⟨1 3 4 0 6 4 1], ⟨0 -5 -6 10 -9 -1 11]]
Optimal tunings:
- CTE: ~2 = 1\1, ~6/5 = 337.6293
- POTE: ~2 = 1\1, ~6/5 = 337.513
Optimal ET sequence: 7, 25e, 32f
Badness: 0.039224