Middle Path table of five-limit rank two temperaments

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This table is an an updated version of Table 1 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.

The "main sequence" of this table comprises all possible five-limit 2-D cases where complexity/7.65 + damage/10 < 1. The 'exotemperaments' have larger damage and lend themselves to special measures such as custom inharmonic timbres in order to aid harmoniousness. The 'bonus temperaments' are more complex but have exceedingly small damage and sound like JI.

See also Middle Path table of seven-limit rank two temperaments, Middle Path table of eleven-limit rank two temperaments.

Vanishing Interval's Ratio V.I. Vector V.I. cents Temperament
name
TOP
period
TOP
generator
Mapp.
of 2
Mapp.
of 3
Mapp.
of 5
Cmplx. TOP
Dmg.
ETs
TWO EXOTEMPERAMENTS:
16/15 [4, -1, -1> 111.7 Father 1185.9 447.4 1, 0 2, -1 2, 1 1.37 14.13
27/25 [0, 3, -2> 133.2 Bug 1200.0 260.3 1, 0 2, -2 3, -3 1.63 14.18
MAIN SEQUENCE:
25/24 [-3, -1, 2> 70.7 Dicot 1207.66 353.22 1, 0 1, 2 2, 1 1.60 7.66 24c, 41cc, (48cc), 65ccc...
81/80 [-4, 4, -1> 21.5 Meantone 1201.70 504.13 1, 0 2, -1 4, -4 2.19 1.70 12, 19, (24), 26, 29c, 31, 33c, (36)...
128/125 [7, 0, -3> 41.1 Augmented 399.02 93.15 3, 0 5, -1 7, 0 2.42 2.94 12, 15, (24), 27, (30), 33, (36), 39...
135/128 [-7, 3, 1> 92.2 Mavila 1206.55 685.03 1, 0 1, 1 4, -3 2.44 6.55 37bc, 67bbccc, (74bbccc), 81bbcccc...
250/243 [1, -5, 3> 49.2 Porcupine 1196.91 1034.59 1, 0 -1, 3 -2, 5 2.75 3.09 15, 22, 29, (30), 36c, 37, 43c, (44)...
256/243 [8, -5, 0> 90.2 Blackwood 238.87 158.78 5, 0 8, 0 11, 1 2.76 5.67 15, 25, (30), 35bc, 40b, (45bc)...
648/625 [3, 4, -4> 62.6 Diminished 299.16 197.49 4, 0 7, -1 10, -1 3.23 3.36 12, (24), 28, 32c, (36), 40, 44c, (48c)...
2048/2025 [11, -4, -2> 19.6 Srutal 599.56 494.86 2, 0 4, -1 3, 2 3.81 0.89 12, 22, (24), 34, (36), (44), 46, (48c), 54...
3125/3072 [-10, -1, 5> 29.6 Magic 1201.28 380.80 1, 0 0, 5 2, 1 4.02 1.28 19, 22, (38), 41, (44), 54b, (57), 60...
6561/6250 [-1, 8, -5> 84.1 Ripple 1203.32 101.99 1, 0 2, -5 3, -8 4.38 3.32 12, (24), (36), 47, (48c), 59b, (60c)...
15625/15552 [-6, -5, 6> 8.11 Hanson 1200.29 317.07 1, 0 0, 6 1, 5 4.83 0.29 19, 34, (38), 49, 53, (57), 64b, (68)...
16875/16384 [-14, 3, 4> 51.1 Negri 1201.82 1075.68 1, 0 -2, 4 5, -3 4.86 1.82 19, 29, (38), 48, (57), (58c), 66b, 67c...
20000/19683 [5, -9, 4> 27.7 Tetracot 1199.03 176.11 1, 0 1, 4 1, 9 4.95 0.97 27, 34, 41, 48, (54c), 55c, 61, (68), 75...
20480/19683 [12, -9, 1> 68.7 Superpyth 1197.60 708.17 1, 0 1, 1 -3, 9 4.95 2.40 22, 27, (44), 49, (54c), (66), 71, 76bc...
32805/32768 [-15, 8, 1> 1.95 Helmholtz 1200.07 701.79 1, 0 1, 1 7, -8 5.20 0.07 41, 53, 65, 70c, 77, (82), 89, 94, 99c...
78732/78125 [2, 9, -7> 13.4 Sensi 1199.59 756.60 1, 0 6, -7 8, -9 5.63 0.41 19, (38), 46, (57), 65, 73, (76), 84...
262144/253125 [18, -4, -5> 60.6 Passion 1198.31 98.40 1, 0 2, -5 2, 4 6.23 1.69 73, 134bc, (146bc), 158bcc...
393216/390625 [17, 1, -8> 11.4 Würschmidt 1199.69 812.05 1, 0 7, -8 3, -1 6.44 0.31 34, 65, (68), 96, 99, (102)...
531441/524288 [-19, 12, 0> 23.5 Compton 100.05 15.13 12, 0 19, 0 28, -1 6.59 0.62 60, 72, 84, 96, 108, (120c), 132...
1600000/1594323 [9, -13, 5> 6.15 Amity 1199.85 860.38 1, 0 -2, 5 -7, 13 7.14 0.15 53, 99, (106), 145, 152, (159)...
2109375/2097152 [-21, 3, 7> 10.1 Orson 1200.24 271.65 1, 0 0, 7 3, -3 7.28 0.24 53, (106), 137, (159), 190...
TWO BONUS TEMPERAMENTS:
6115295232/6103515625 [23, 6, -14> 3.34 Vishnu 599.97 71.15 2, 0 4, -7 5, -3 11.27 0.05
274877906944/274658203125 [38, -2, -15> 1.38 Luna 1199.98 193.196 1, 0 4, -15 2, 2 13.17 0.02
ET TOP damage TOP octave Rank-2 temperaments supported
12 3.56 1197.67 meantone, augmented, diminished, srutal, ripple
15 5.67 1194.33 augmented, porcupine, blackwood
19 2.28 1202.28 meantone, magic, hanson, negri, sensi
22 3.22 1198.72 porcupine, srutal, magic, superpyth
24 (contorted) 3.56 1197.67 meantone, augmented, diminished, srutal, ripple
24c 7.87 1207.87 dicot
25 6.16 1194.83 blackwood
26 3.69 1203.69 meantone
27 2.94 1197.06 augmented, tetracot, superpyth
28 5.15 1205.15 diminished
29 3.47 1202.53 porcupine, negri
29c 5.89 1194.11 meantone
30 (contorted) 5.67 1194.33 augmented, porcupine, blackwood
31 1.81 1201.47 meantone
32c 5.61 1194.39 diminished
33 6.43 1200.54 augmented
33c 4.90 1204.90 meantone
34 1.24 1198.76 srutal, hanson, tetracot, wuerschmidt
35bc 9.43 1209.43 blackwood
36 (contorted) 3.56 1197.67 meantone, augmented, diminished, srutal, ripple
36c 4.25 1204.25 porcupine
37 3.64 1196.36 porcupine
37bc 6.62 1206.62 mavila
38 (contorted) 2.28 1202.28 meantone, magic, hanson, negri, sensi
mp5r2ms_cd.png

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 the size of the comma (vanishing interval). Note that the "exotemperaments" and "bonus temperaments" are not shown (a much larger range is needed to make them visible).