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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 TEMPS:
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).