Middle Path table of eleven-limit rank two temperaments

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This table is supposed to be the best guess at what would have appeared in A Middle Path Part 2, had Paul Erlich finished it. It is similar to the Middle Path table of five-limit rank two temperaments and Middle Path table of seven-limit rank two temperaments from Part 1. It is intended to comprise all possible 11-limit 2D cases where complexity/7.65 + damage/10 < 1.

Commas Name TOP

period

TOP

generator

Mapping Cmplx TOP

Dmg

ETs
36/35, 50/49, 64/63, 81/80, 126/125... Duodecim 99.59 31.48 [<12 19 28 34 42|, <0 0 0 0 -1|] 2.59 6.14 24d, 36d, 48cdde, (48cdd)...
45/44, 56/55, 81/80, 100/99, 125/121... Meanenneadecal 1198.56 503.60 [<1 2 4 7 6|, <0 -1 -4 -10 -6|] 3.06 5.32 31e, 50ee, (62eee), 69dee, 81eee...
36/35, 56/55, 64/63, 81/80, 99/98... Dominant 1195.04 495.58 [<1 2 4 2 1|, <0 -1 -4 2 6|] 3.08 4.96 29cde, 41cdee, 53cddeee...
49/48, 56/55, 77/75, 121/120, 176/175... Triforce 399.80 155.57 [<3 4 7 8 10|, <0 2 0 1 1|] 3.22 5.29 39, 54cd, 69bcd, (78cd), 93bccdd...
45/44, 50/49, 81/80, 99/98, 100/99... Injera 601.24 92.99 [<2 3 4 5 6|, <0 1 4 4 6|] 3.38 4.06 26, 38e, 50dee, (52c), 64ce, 66bcc...
33/32, 49/48, 77/75, 99/98, 176/175... Negric 1205.30 129.26 [<1 2 2 3 3|, <0 -4 3 -2 4|] 3.52 5.30 28de, (56bddeee), (84bbdddeeeee)...
56/55, 64/63, 100/99, 126/125, 176/175... Augene 398.88 87.10 [<3 5 7 8 10|, <0 -1 0 2 2|] 3.66 3.37 27e, 39dee, 42e, (54cee), 57bce...
45/44, 49/48, 81/80, 100/99, 125/121... Godzilla 1204.09 254.92 [<1 2 4 3 6|, <0 -2 -8 -1 -12|] 3.71 4.09 33cd, 52cd, (66bccdd), 71bcdd...
50/49, 55/54, 99/98, 100/99, 121/120... Hedgehog 599.09 163.15 [<2 4 6 7 8|, <0 -3 -5 -5 -4|] 3.75 3.23 22, 36ce, (44), 52bdd, 58ce, (66d)...
49/48, 55/54, 77/75, 126/125, 176/175... Darjeeling 1202.79 318.16 [<1 0 1 2 0|, <0 6 5 3 13|] 3.81 4.41 34e, 53dee, (68dee), 83ddee...
56/55, 64/63, 77/75, 121/120, 392/375... Progress 1195.64 559.36 [<1 3 0 0 3|, <0 -3 5 6 1|] 3.81 4.51 47bc, 62bcce, (94bbccce)...
55/54, 64/63, 100/99, 121/120, 176/175... Porcupine 1198.23 163.15 [<1 2 3 2 4|, <0 -3 -5 6 -4|] 3.90 3.18 22, 37, (44), 52b, 59, (66d), 67b...
49/48, 56/55, 100/99, 126/125, 343/330... Keemun 1200.82 318.18 [<1 0 1 2 4|, <0 6 5 3 -2|] 4.05 4.50 83dde, 117bddee, 151bdddeee...
49/48, 81/80, 126/125, 225/224, 245/243... Undevigintone 63.36 30.42 [<19 30 44 53 66|, <0 0 0 0 -1|] 4.07 3.83 38d, 57dd, 57ddee, 76dd, (76dde)...
50/49, 64/63, 99/98, 100/99, 176/175... Pajara 598.45 106.57 [<2 3 5 6 8|, <0 1 -2 -2 -6|] 4.17 3.11 22, 34d, (44), 46de, 56d, 58ddee...
49/48, 55/54, 100/99, 121/120, 250/243... Nautilus 1202.66 82.97 [<1 2 3 3 4|, <0 -6 -10 -3 -8|] 4.21 3.48 29, (58cde), 72ccddee, 73cde...
45/44, 81/80, 100/99, 125/121, 385/384... Flattone 1203.00 508.77 [<1 2 4 -1 6|, <0 -1 -4 9 -6|] 4.30 4.06 26, 45, (52c), 64cde, 71bc, (78bcc)...
50/49, 55/54, 64/63, 225/224, 385/384... Pajarous 599.29 108.86 [<2 3 5 6 6|, <0 1 -2 -2 5|] 4.44 3.27 22, (44), (66d), (88bd), 98bcd...
56/55, 81/80, 128/125, 176/175, 245/242... Catcall 99.81 31.73 [<12 19 28 34 42|, <0 0 0 -1 -1|] 4.48 3.56 36, 48ce, (72ce), 84cee, (96cceee)...
55/54, 99/98, 176/175, 225/224, 245/243... Telepathy 1199.00 381.39 [<1 0 2 -1 -1|, <0 5 1 12 14|] 4.52 3.15 22, 41e, (44), 63e, (66d), (82eee)...
55/54, 100/99, 121/120, 225/224, 250/243... Porky 1198.78 163.50 [<1 2 3 5 4|, <0 -3 -5 -16 -4|] 4.61 3.22 22, 29, (44), 51, (58cde), (66d)...
56/55, 100/99, 126/125, 245/243, 540/539... Sensis 1196.58 442.49 [<1 -1 -1 -2 2|, <0 7 9 13 4|] 4.68 3.42 27e, (54cee), 73ee, (81bcdeee)...
55/54, 64/63, 99/98, 245/243, 352/343... Suprapyth 1198.59 490.29 [<1 2 6 2 1|, <0 -1 -9 2 6|] 4.68 3.18 22, (44), (66d), (88bd), 93bde...
50/49, 64/63, 225/224, 245/243, 250/243... Vigintiduo 54.48 10.51 [<22 35 51 62 76|, <0 0 0 0 1|] 4.74 3.28 66de, 88bde, 110bdd, 110bddee...
81/80, 99/98, 126/125, 176/175, 225/224... Meantone 1201.61 504.02 [<1 2 4 7 11|, <0 -1 -4 -10 -18|] 4.96 1.74 31, 43, 50e, 55de, (62), 74, 81ee...
50/49, 55/54, 176/175, 352/343, 540/539... Fleetwood 599.28 272.31 [<2 5 6 7 11|, <0 -4 -3 -3 -9|] 4.96 3.27 22, (44), (66d), (88bd), (110bdde)...
50/49, 121/120, 176/175, 352/343, 385/384... Astrology 599.32 217.89 [<2 5 5 6 8|, <0 -5 -1 -1 -3|] 5.07 3.28 22, (44), (66d), (88bd), (110bdde)...
49/48, 55/54, 225/224, 441/440, 525/512... Negroni 1203.19 124.84 [<1 2 2 3 5|, <0 -4 3 -2 -15|] 5.11 3.19 77cddee, 106ccdddeee...
64/63, 99/98, 121/120, 352/343, 540/539... Quasisupra 1197.34 490.80 [<1 2 -3 2 1|, <0 -1 13 2 6|] 5.29 2.66 39d, 61d, (78cdd), 83d, 100bcdd...
99/98, 121/120, 176/175, 225/224, 385/384... Orwell 1201.25 271.43 [<1 0 3 1 3|, <0 7 -3 8 2|] 5.30 1.36 31, 53, (62), 75, 84e, (93e), 97de...
81/80, 99/98, 121/120, 441/440, 540/539... Squares 1201.70 426.46 [<1 3 8 6 7|, <0 -4 -16 -9 -10|] 5.37 1.70 31, 45e, (62), 76e, 79c, 90bcdee...
64/63, 100/99, 176/175, 245/243, 540/539... Superpyth 1197.60 489.43 [<1 2 6 2 10|, <0 -1 -9 2 -16|] 5.42 2.40 49, 71d, 76bcdee, 93bd, (98bde)...
121/120, 126/125, 176/175, 385/384, 441/440... Valentine 1200.87 77.63 [<1 1 2 3 3|, <0 9 5 -3 7|] 5.82 1.54 31, 46, (62), 77, (92), (93e), 107c...
81/80, 121/120, 176/175, 243/242, 385/384... Mohajira 1201.70 348.78 [<1 1 0 6 2|, <0 2 8 -11 5|] 6.01 1.70 31, (62), (93e), (124be), 148be...
100/99, 225/224, 245/243, 385/384, 540/539... Magic 1200.75 380.92 [<1 0 2 -1 6|, <0 5 1 12 -8|] 6.07 1.68 41, 63, (82), 104, (123c), (126c)...
81/80, 126/125, 225/224, 385/384, 540/539... Meanpop 1201.70 504.13 [<1 2 4 7 -2|, <0 -1 -4 -10 13|] 6.08 1.70 31, 50, (62), 81, (93e), (100d), 112b...
81/80, 121/120, 126/125, 225/224, 243/242... Migration 1201.70 348.78 [<1 1 0 -3 2|, <0 2 8 20 5|] 6.10 1.70 31, (62), (93e), 100de, (124be)...
99/98, 121/120, 126/125, 540/539, 891/875... Nusecond 1200.46 154.74 [<1 3 4 5 5|, <0 -11 -13 -17 -12|] 6.13 1.70 31, (62), (93e), 101, (124be), 132ce...
100/99, 225/224, 245/242, 441/440, 625/616... Cassandra 1201.36 497.94 [<1 2 -1 -3 -4|, <0 -1 8 14 18|] 6.19 1.79 152cd, 193ccd, 234ccd, 263cccdd...
225/224, 243/242, 385/384, 441/440, 540/539... Miracle 1200.63 116.72 [<1 1 3 3 2|, <0 6 -7 -2 15|] 7.14 0.63 72, (144), (216c), (288cd), (360bcd)...
A BONUS TEMPERAMENT:
2401/2400, 3025/3024, 4375/4374, 9801/9800... Hemiennealimmal 66.67 17.64 [<18 28 41 50 62|, <0 2 3 2 1|] 23.81 0.05

Comparisons of closely related temperaments

36/35, 50/49, 64/63, 81/80, 126/125... Duodecim 99.59 31.48 [<12 19 28 34 42|, <0 0 0 0 -1|] 2.59 6.14 24d, 36d, 48cdde, (48cdd)...
56/55, 81/80, 128/125, 176/175, 245/242... Catcall 99.81 31.73 [<12 19 28 34 42|, <0 0 0 -1 -1|] 4.48 3.56 36, 48ce, (72ce), 84cee, (96cceee)...

Since the TOP tunings of duodecim and catcall are so similar, there is little practical use for duodecim temperament. If notes so altered from 12edo are available, there is no reason not to use them for ratios of 7 as well as ratios of 11.

45/44, 56/55, 81/80, 100/99, 125/121... Meanenneadecal 1198.56 503.60 [<1 2 4 7 6|, <0 -1 -4 -10 -6|] 3.06 5.32 31e, 50ee, (62eee), 69dee, 81eee...
36/35, 56/55, 64/63, 81/80, 99/98... Dominant 1195.04 495.58 [<1 2 4 2 1|, <0 -1 -4 2 6|] 3.08 4.96 29cde, 41cdee, 53cddeee...
45/44, 81/80, 100/99, 125/121, 385/384... Flattone 1203.00 508.77 [<1 2 4 -1 6|, <0 -1 -4 9 -6|] 4.30 4.06 26, 45, (52c), 64cde, 71bc, (78bcc)...
81/80, 99/98, 126/125, 176/175, 225/224... Meantone 1201.61 504.02 [<1 2 4 7 11|, <0 -1 -4 -10 -18|] 4.96 1.74 31, 43, 50e, 55de, (62), 74, 81ee...
81/80, 126/125, 225/224, 385/384, 540/539... Meanpop 1201.70 504.13 [<1 2 4 7 -2|, <0 -1 -4 -10 13|] 6.08 1.70 31, 50, (62), 81, (93e), (100d), 112b...

These temperaments all temper out 81/80, making them extensions of 5-limit meantone. The differences are in the mappings of 7 and 11.

36/35, 56/55, 64/63, 81/80, 99/98... Dominant 1195.04 495.58 [<1 2 4 2 1|, <0 -1 -4 2 6|] 3.08 4.96 29cde, 41cdee, 53cddeee...
55/54, 64/63, 99/98, 245/243, 352/343... Suprapyth 1198.59 490.29 [<1 2 6 2 1|, <0 -1 -9 2 6|] 4.68 3.18 22, (44), (66d), (88bd), 93bde...
64/63, 99/98, 121/120, 352/343, 540/539... Quasisupra 1197.34 490.80 [<1 2 -3 2 1|, <0 -1 13 2 6|] 5.29 2.66 39d, 61d, (78cdd), 83d, 100bcdd...
64/63, 100/99, 176/175, 245/243, 540/539... Superpyth 1197.60 489.43 [<1 2 6 2 10|, <0 -1 -9 2 -16|] 5.42 2.40 49, 71d, 76bcdee, 93bd, (98bde)...

In contrast, these temperaments all temper out 64/63, making them extensions of archy temperament. The differences are in the mappings of 5 and 11. Temperaments in both this and the above list comprise the dominant family, which has only one representative here. Note that cassandra is in neither of the two families, because it preserves both 64/63 and 81/80 as non-vanishing intervals (and makes them both equal to the Pythagorean comma).

33/32, 49/48, 77/75, 99/98, 176/175... Negric 1205.30 129.26 [<1 2 2 3 3|, <0 -4 3 -2 4|] 3.52 5.30 28de, (56bddeee), (84bbdddeeeee)...
49/48, 55/54, 225/224, 441/440, 525/512... Negroni 1203.19 124.84 [<1 2 2 3 5|, <0 -4 3 -2 -15|] 5.11 3.19 77cddee, 106ccdddeee...

The difference is only in the mapping of 11. The two temperaments intersect in 19edo (using the 19e val tempering out 33/32), which is a fine tuning for negric (despite that it doesn't show up in the list), but sub-optimal for negroni (which does not temper out 33/32).

50/49, 64/63, 99/98, 100/99, 176/175... Pajara 598.45 106.57 [<2 3 5 6 8|, <0 1 -2 -2 -6|] 4.17 3.11 22, 34d, (44), 46de, 56d, 58ddee...
50/49, 55/54, 64/63, 225/224, 385/384... Pajarous 599.29 108.86 [<2 3 5 6 6|, <0 1 -2 -2 5|] 4.44 3.27 22, (44), (66d), (88bd), 98bcd...

The difference is only in the mapping of 11. The two temperaments intersect in 22edo, a fine tuning for both.

49/48, 55/54, 77/75, 126/125, 176/175... Darjeeling 1202.79 318.16 [<1 0 1 2 0|, <0 6 5 3 13|] 3.81 4.41 34e, 53dee, (68dee), 83ddee...
49/48, 56/55, 100/99, 126/125, 343/330... Keemun 1200.82 318.18 [<1 0 1 2 4|, <0 6 5 3 -2|] 4.05 4.50 83dde, 117bddee, 151bdddeee...

The difference is only in the mapping of 11. The two intersect in 15edo, which is, however, not a great tuning for either. In 19edo, darjeeling uses the 19e val, which tempers out 33/32, whereas keemun uses the patent val.

55/54, 64/63, 100/99, 121/120, 176/175... Porcupine 1198.23 163.15 [<1 2 3 2 4|, <0 -3 -5 6 -4|] 3.90 3.18 22, 37, (44), 52b, 59, (66d), 67b...
55/54, 100/99, 121/120, 225/224, 250/243... Porky 1198.78 163.50 [<1 2 3 5 4|, <0 -3 -5 -16 -4|] 4.61 3.22 22, 29, (44), 51, (58cde), (66d)...

The difference is only in the mapping of 7. The two intersect in 22edo, which is probably the only reasonable incarnation of porky temperament.

55/54, 99/98, 176/175, 225/224, 245/243... Telepathy 1199.00 381.39 [<1 0 2 -1 -1|, <0 5 1 12 14|] 4.52 3.15 22, 41e, (44), 63e, (66d), (82eee)...
100/99, 225/224, 245/243, 385/384, 540/539... Magic 1200.75 380.92 [<1 0 2 -1 6|, <0 5 1 12 -8|] 6.07 1.68 41, 63, (82), 104, (123c), (126c)...

The difference is only in the mapping of 11. The two intersect in 22edo, which is a fine telepathy tuning but slightly sub-optimal for magic. Since telepathy is significantly higher in error, it can be regarded as an alternate version of magic that exists in 22edo.

81/80, 121/120, 176/175, 243/242, 385/384... Mohajira 1201.70 348.78 [<1 1 0 6 2|, <0 2 8 -11 5|] 6.01 1.70 31, (62), (93e), (124be), 148be...
81/80, 121/120, 126/125, 225/224, 243/242... Migration 1201.70 348.78 [<1 1 0 -3 2|, <0 2 8 20 5|] 6.10 1.70 31, (62), (93e), 100de, (124be)...

The TOP tunings of mohajira and migration are not merely close, but exactly equal, because the prime 7 does not affect the TOP tuning. The two temperaments intersect in 31edo, which is also near-optimal for both.

ET TOP damage TOP octave Rank-2 temperaments supported
22 hedgehog, porcupine, pajara, pajarous, telepathy, porky, suprapyth, fleetwood, astrology
24d duodecim
26 injera, flattone
27e augene, sensis
28de negric
29 nautilus, porky
29cde dominant
31 meantone, orwell, squares, valentine, mohajira, meanpop, migration, nusecond
31e meanenneadecal
33cd godzilla
34d pajara
34e darjeeling
36 catcall
36ce hedgehog
36d duodecim
37 porcupine
38d undevigintone
38e injera
39 triforce
39d quasisupra
39dee augene
41 magic
41cdee dominant
41e telepathy
42e augene
43 meantone
44* hedgehog, porcupine, pajara, pajarous, telepathy, porky, suprapyth, fleetwood, astrology
45 flattone
45e squares
46 valentine
46de pajara
47bc progress
48cdd* duodecim
48cdde duodecim
48ce catcall
49 superpyth
50 meanpop
50dee injera
50e meantone
50ee meanenneadecal
51 porky
52b porcupine
52bdd hedgehog
52c* injera, flattone
52cd godzilla
53 orwell
53cddeee dominant
53dee darjeeling
54cd triforce
54cee* augene, sensis
55de meantone
56bddeee* negric
56d pajara
57bce augene
57dd undevigintone
57ddee undevigintone
58cde* nautilus, porky
58ce hedgehog
58ccddee* dominant
58ddee pajara
59 porcupine
60cddd duodecim
60cdddee duodecim
61d quasisupra
62* meantone, orwell, squares, valentine, mohajira, meanpop, migration, nusecond
62bcce progress
62eee* meanenneadecal
63 magic
63e telepathy
64cd injera
64cde flattone
65ccdddeeee dominant
66bcc injera
66bccdd* godzilla
66d* hedgehog, porcupine, pajara, pajarous, telepathy, porky, suprapyth, fleetwood, astrology
66de vigintiduo
67b porcupine
68dde* pajara
68dee* darjeeling
69bcd triforce
69bcee augene
69dee meanenneadecal
70ccddeee dominant
70ddeee pajara
71bc flattone
71bcdd godzilla
71d superpyth
72 miracle
72bce augene
72ccddee nautilus
72ccee* hedgehog
72cddd* duodecim
72cddde* duodecim
72cdddeee duodecim
72ce* catcall