Middle Path table of seven-limit rank two temperaments

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This table is an updated version of Table 2 of Paul Erlich's A Middle Path. The complexity is now measured a different way (which, however, is proportional to the original complexity for this table), and some temperament names have been updated.

This table comprises all possible 7-limit rank-2 cases where complexity/7.65 + damage/10 < 1.

Vanishing interval's ratios Temperament
name
TOP
period
TOP
generator
Mapp.
of 2
Mapp.
of 3
Mapp.
of 5
Mapp.
of 7
Cmplx TOP
Dmg
ETs
28/27, 49/48, 64/63, 256/243, 343/324, … Blackwood[note 1] 239.18 155.35 5, 0 8, 0 11, 1 14, 0 2.01 7.24 15, 25, (30), 35bc, 40b, (45bc), 50bc, (50b), …
36/35, 50/49, 126/125, 360/343, 648/625, … Diminished 298.53 197.08 4, 0 7, -1 10, -1 12, -1 2.46 5.87 12, (24d), (36d), (48cdd), 56cddd, (60cddd), …
36/35, 64/63, 81/80, 256/245, 729/700, … Dominant 1195.23 495.88 1, 0 2, -1 4, -4 2, 2 2.47 4.77 12, (24d), 29cd, (36d), 41cd, 46ccd, (48cdd), …
36/35, 128/125, 225/224, 405/392, 729/686, … August 399.99 107.31 3, 0 5, -1 7, 0 9, -2 2.58 5.87 12, (24d), 33, (36d), 45cd, (48cdd), 57cd, …
50/49, 64/63, 225/224, 2048/2025, … Pajara 598.45 491.88 2, 0 4, -1 3, 2 4, 2 3.23 3.11 22, 32, 34d, (44), 46d, 54, 56d, 58dd, (64bc), …
49/48, 81/80, 245/243, 1323/1280, … Godzilla[note 2] 1203.67 252.48 1, 0 2, -2 4, -8 3, -1 3.48 3.67 19, 33cd, (38d), 43d, 52cd, (57dd), 62dd, …
81/80, 126/125, 225/224, 3136/3125, … Meantone 1201.70 504.13 1, 0 2, -1 4, -4 7, -10 3.65 1.70 19, 31, (38d), 43, 50, 55d, (57dd), (62), 67d, …
50/49, 81/80, 405/392, 4000/3969, … Injera 600.89 507.28 2, 0 4, -1 8, -4 9, -4 3.70 3.58 26, 38, 50d, (52c), 62d, 64c, 66bcc, 74dd, …
49/48, 225/224, 525/512, 686/675, … Negri 1203.19 1078.35 1, 0 -2, 4 5, -3 1, 2 3.76 3.19 19, 29, (38d), 47d, 48d, (57dd), (58cd), …
64/63, 126/125, 128/125, 4000/3969, … Augene 399.02 90.59* 3, 0 5, -1 7, 0 8, 2 3.76 2.94 27, 39d, 42, 51cd, (54c), 57bc, 63cdd, 66cd, …
49/48, 126/125, 875/864, 1029/1000, … Keemun 1203.19 317.84 1, 0 0, 6 1, 5 2, 3 3.85 3.19 19, 34, (38d), 53d, (57dd), 61bcdd, (68d), …
81/80, 128/125, 648/625, 2048/2025, … Catler 99.81 75.22 12, 0 19, 0 28, 0 33, 1 3.98 3.56 36, 48c, 60cd, 60c, 72cd, (72c), 84cdd, 84c, …
50/49, 245/243, 250/243, 2430/2401, … Hedgehog 598.45 436.13 2, 0 1, 3 1, 5 2, 5 4.09 3.11 22, 36c, (44), 52bdd, 58c, (66d), (72cc), …
64/63, 245/243, 1728/1715, 2240/2187, … Superpyth 1197.60 708.17 1, 0 1, 1 -3, 9 4, -2 4.48 2.40 22, 27, (44), 49, (54c), (66d), 71d, 76bcd, …
126/125, 245/243, 686/675, 4375/4374, … Sensi 1198.39 755.23 1, 0 6, -7 8, -9 11, -13 4.49 1.61 19, 27, (38d), 46, (54c), (57dd), 65, 73, 76dd, …
50/49, 525/512, 1029/1024, 1875/1792, … Lemba 601.70 230.87 2, 0 2, 3 5, -1 6, -1 4.54 3.74 26, (52c), 62c, (78bcc), 88cc, 104bcc, …
64/63, 250/243, 875/864, 6144/6125, … Porcupine 1196.91 1034.59 1, 0 -1, 3 -2, 5 8, -6 4.59 3.09 22, 37, (44), 59, (66d), (74b), 81bd, (88bd), …
81/80, 525/512, 875/864, 4375/4374, … Flattone 1202.54 507.14 1, 0 2, -1 4, -4 -1, 9 4.77 2.54 26, 45, (52c), 64cd, 71bc, (78bcc), 83bcdd, …
225/224, 245/243, 875/864, 3125/3072, … Magic 1201.28 380.80 1, 0 0, 5 2, 1 -1, 12 4.82 1.28 22, 41, (44), 60, 63, (66d), 79d, (82), 85, …
50/49, 875/864, 1728/1715, 3125/3024, … Doublewide 599.28 326.96 2, 0 1, 4 3, 3 4, 3 4.84 3.27 22, (44), 48, (66d), 70c, 74c, (88bd), 92cd, …
49/48, 250/243, 4000/3969, 6125/5832, … Nautilus 1202.99 1119.69 1, 0 -4, 6 -7, 10 0, 3 4.85 3.48 29, (58cd), (87ccdd), (116ccddd), …
64/63, 686/675, 2401/2400, 6272/6075, … Beatles 1197.10 842.38 1, 0 3, -2 -4, 9 0, 4 5.24 2.90 27, (54c), (81bcd), (108bccd), 118bccd, …
81/80, 686/675, 1029/1000, 10976/10935, … Liese 1202.62 569.05 1, 0 3, -3 8, -12 8, -11 5.43 2.62 [almost 19], 74d, 93dd, 112bdd, 129dd, …
81/80, 1029/1024, 1728/1715, 8748/8575, … Cynder 1201.7 969.18 1, 0 4, -3 12, -12 2, 1 5.72 1.70 31, 57, (62), 88, (93), 98, (114bc), 119b, …
225/224, 1728/1715, 2430/2401, 6144/6125, … Orwell 1199.53 271.49 1, 0 0, 7 3, -3 1, 8 6.20 0.95 31, 53, (62), 75, 84, (93), (106), 115, (124b), …
225/224, 3125/3087, 4000/3969, 5120/5103, … Garibaldi 1200.76 702.64 1, 0 1, 1 7, -8 11, -14 6.30 0.91 41, 53, (82), 94, (106), 118d, (123c), 135, 147, …
126/125, 1728/1715, 2401/2400, 31104/30625, … Myna 1198.83 888.94 1, 0 9, -10 9, -9 8, -7 6.31 1.17 58, 89, (116c), 120, 143cd, 147c, 151, 174cd, …
225/224, 1029/1024, 2401/2400, 16875/16807, … Miracle 1200.63 116.72 1, 0 1, 6 3, -7 3, -2 6.55 0.63 41, 72, (82), 103, 113, (123c), 134, (144), …
A BONUS TEMPERAMENT:
2401/2400, 4375/4374, 250047/250000, … Ennealimmal 133.337 84.313 9, 0 13, 2 19, 3 24, 2 12.36 0.04

* Correction from the Xenharmonikon version, which gives the corner case generator 92.46 erroneously.

ET TOP damage TOP octave Rank-2 temperaments supported
12 Diminished, dominant, august
15 Blackwood
19 Godzilla, meantone, negri, keemun, sensi
22 Pajara, hedgehog, superpyth, porcupine, magic, doublewide
24d* Diminished, dominant, august
25 Blackwood
26 Injera, lemba, flattone
27 Augene, superpyth, sensi, beatles
29 Negri, nautilus
29cd Dominant
30* Blackwood
31 Meantone, cynder, orwell
32 Pajara
33 August
33cd Godzilla
34 Keemun
34d Pajara
35bc Blackwood
36 Catler
36c Hedgehog
36d* Diminished, dominant, august
37 Porcupine
38 Injera
38d* Godzilla, meantone, negri, keemun, sensi
39d Augene
40b Blackwood
41 Magic, garibaldi, miracle
41cd Dominant
42 Augene
43 Meantone
43d Godzilla
44* Pajara, hedgehog, superpyth, porcupine, magic, doublewide
45 Flattone
45bc Blackwood
45cd August
46 Sensi
46ccd Dominant
46d Pajara
47d Negri
48 Doublewide
48c Catler
48cdd* Diminished, dominant, august
48d Negri
49 Superpyth
50 Meantone
50b* Blackwood
50bc Blackwood
50d Injera
51cd Augene
52bdd Hedgehog
52c* Injera, lemba, flattone
52cd Semaphore
53 Orwell, garibaldi
53cdd Dominant
53d Keemun
54 Pajara
54c* Augene, superpyth, sensi, beatles
55b Blackwood
55d Meantone
56cddd Diminished
56d Pajara
57 Cynder
57bc Augene
57cd August
57dd* Godzilla, meantone, negri, keemun, sensi
58 Myna
58c Hedgehog
58ccdd* Dominant
58cd* Negri, nautilus
58dd Pajara
59 Porcupine
60 Magic
60b Blackwood
60bc* Blackwood
60c Catler
60cd Catler
60cddd* Diminished, dominant, august

* contorted

The bubble chart above (source) arranges the temperaments by complexity (x-axis) vs. damage (y-axis). Colors are arbitrary. The size of the bubble is proportional to (complexity × damage). Note that the "bonus temperament" ennealimmal is not shown (a much larger range would be needed to make it visible).

The same chart, only "zoomed in".

See also

Notes

  1. Given as blacksmith in A Middle Path.
  2. Given as semaphore in A Middle Path.