This is intended to be a map of all interesting rank-2 temperaments that are compatible with octave equivalence. The only rank-2 temperaments not appearing here should be ones like Bohlen-Pierce that completely lack octaves.
Please make sure each fraction of an octave is always the mediant of the ones directly above and below.
One period per octave
Two periods per octave
| Generator
|
Cents
|
Comments
|
| 0\2
|
|
|
|
|
|
|
|
|
|
|
|
0.000
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1\26
|
46.154
|
|
|
|
|
|
|
|
|
|
|
|
|
1\24
|
|
50.000
|
Shrutar
|
|
|
|
|
|
|
|
|
|
|
|
|
2\46
|
52.174
|
Shrutar
|
|
|
|
|
|
|
|
|
|
|
1\22
|
|
|
54.545
|
Shrutar
|
|
|
|
|
|
|
|
|
|
1\20
|
|
|
|
60.000
|
|
|
|
|
|
|
|
|
|
1\18
|
|
|
|
|
66.667
|
|
|
|
|
|
|
|
|
|
|
2\34
|
|
|
|
70.588
|
Vishnu
|
|
|
|
|
|
|
|
|
|
|
|
|
7\118
|
71.186
|
Vishnu
|
|
|
|
|
|
|
|
|
|
|
|
5\84
|
|
71.429
|
Vishnu
|
|
|
|
|
|
|
|
|
|
|
3\50
|
|
|
72.000
|
|
|
|
|
|
|
|
|
1\16
|
|
|
|
|
|
75.000
|
|
|
|
|
|
|
|
|
|
2\30
|
|
|
|
|
80.000
|
|
|
|
|
|
|
|
|
|
|
3\44
|
|
|
|
81.818
|
|
|
|
|
|
|
|
|
|
|
|
4\58
|
|
|
82.759
|
Harry
|
|
|
|
|
|
|
|
|
|
|
|
|
9\130
|
83.077
|
Harry
|
|
|
|
|
|
|
|
|
|
|
|
5\72
|
|
83.333
|
Harry
|
|
|
|
|
|
|
1\14
|
|
|
|
|
|
|
85.714
|
|
|
|
|
|
|
|
|
2\26
|
|
|
|
|
|
92.308
|
Injera
|
|
|
|
|
|
|
|
|
3\38
|
|
|
|
|
94.737
|
Injera
|
|
|
|
|
|
1\12
|
|
|
|
|
|
|
|
100.000
|
Injera/diaschismic
|
|
|
|
|
|
|
|
|
|
5\58
|
|
|
|
103.448
|
Diaschismic
|
|
|
|
|
|
|
|
|
4\46
|
|
|
|
|
104.348
|
Diaschismic/srutal
|
|
|
|
|
|
|
|
|
|
7\80
|
|
|
|
105.000
|
Srutal
|
|
|
|
|
|
|
|
3\34
|
|
|
|
|
|
105.882
|
Srutal/pajara
|
|
|
|
|
|
|
|
|
5\56
|
|
|
|
|
107.143
|
Pajara
|
|
|
|
|
|
|
2\22
|
|
|
|
|
|
|
109.091
|
Pajara
|
|
|
|
|
|
|
|
3\32
|
|
|
|
|
|
112.500
|
|
|
|
|
|
1\10
|
|
|
|
|
|
|
|
|
120.000
|
Octokaidecal/Nimona
|
|
|
|
|
|
|
3\28
|
|
|
|
|
|
|
128.571
|
Octokaidecal/Nimona
|
|
|
|
|
|
2\18
|
|
|
|
|
|
|
|
133.333
|
Octokaidecal/Nimona
|
|
|
|
|
|
|
3\26
|
|
|
|
|
|
|
138.462
|
|
|
|
|
|
|
|
|
|
6\60
|
|
|
|
|
140.000
|
|
|
|
|
|
|
|
|
|
|
11\94
|
|
|
|
140.126
|
Fifive
|
|
|
|
|
|
|
|
|
|
|
15\128
|
|
|
140.625
|
Fifive
|
|
|
|
|
|
|
|
4\34
|
|
|
|
|
|
141.176
|
Fifive
|
|
|
|
|
|
|
|
|
5\42
|
|
|
|
|
142.857
|
|
|
|
|
1\8
|
|
|
|
|
|
|
|
|
|
150.000
|
|
|
|
|
|
|
|
4\30
|
|
|
|
|
|
|
160.000
|
|
|
|
|
|
|
3\22
|
|
|
|
|
|
|
|
163.636
|
Hedgehog/echidna
|
|
|
|
|
|
|
|
|
11\80
|
|
|
|
|
165.000
|
Hedgehog/echidna
|
|
|
|
|
|
|
|
8\58
|
|
|
|
|
|
165.517
|
Hedgehog/echidna
|
|
|
|
|
|
|
5\36
|
|
|
|
|
|
|
166.667
|
|
|
|
|
|
2\14
|
|
|
|
|
|
|
|
|
171.429
|
|
|
|
|
|
|
3\20
|
|
|
|
|
|
|
|
180.000
|
|
|
|
|
|
|
|
|
7\46
|
|
|
|
|
|
182.609
|
Unidec/hendec
|
|
|
|
|
|
|
|
|
11\72
|
|
|
|
|
183.333
|
Unidec/hendec
|
|
|
|
|
|
|
4\26
|
|
|
|
|
|
|
184.615
|
Unidec/hendec
|
|
|
|
|
|
|
|
5\32
|
|
|
|
|
|
187.500
|
|
|
|
1\6
|
|
|
|
|
|
|
|
|
|
|
200.000
|
|
|
|
|
|
|
|
|
6\34
|
|
|
|
|
|
211.765
|
|
|
|
|
|
|
|
5\28
|
|
|
|
|
|
|
214.286
|
Antikythera
|
|
|
|
|
|
|
|
9\50
|
|
|
|
|
|
216.000
|
Antikythera/Wizard
|
|
|
|
|
|
|
|
|
13\72
|
|
|
|
|
216.667
|
Antikythera/Wizard
|
|
|
|
|
|
4\22
|
|
|
|
|
|
|
|
218.182
|
Antikythera/Wizard/Astrology
|
|
|
|
|
|
|
|
|
15\82
|
|
|
|
|
219.512
|
Astrology
|
|
|
|
|
|
|
|
11\60
|
|
|
|
|
|
220.000
|
Astrology
|
|
|
|
|
|
|
7\38
|
|
|
|
|
|
|
221.053
|
|
|
|
|
|
3\16
|
|
|
|
|
|
|
|
|
225.000
|
Lemba
|
|
|
|
|
|
5\26
|
|
|
|
|
|
|
|
230.769
|
Lemba
|
|
|
|
|
|
|
|
12\62
|
|
|
|
|
|
232.258
|
Lemba
|
|
|
|
|
|
|
7\36
|
|
|
|
|
|
|
233.333
|
|
|
|
|
|
|
|
|
9\46
|
|
|
|
|
|
234.783
|
Echidnic
|
|
|
|
|
|
|
|
|
|
|
29\148
|
|
|
235.135
|
Echidnic
|
|
|
|
|
|
|
|
|
|
20\102
|
|
|
|
235.294
|
Echidnic
|
|
|
|
|
|
|
|
|
11\56
|
|
|
|
|
235.714
|
|
|
|
|
2\10
|
|
|
|
|
|
|
|
|
|
240.000
|
Decimal
|
|
|
|
|
|
5\24
|
|
|
|
|
|
|
|
250.000
|
Decimal
|
|
|
|
|
|
|
8\38
|
|
|
|
|
|
|
252.632
|
Decimal
|
|
|
|
|
3\14
|
|
|
|
|
|
|
|
|
257.143
|
Decimal
|
|
|
|
|
|
4\18
|
|
|
|
|
|
|
|
266.667
|
|
|
|
|
|
|
|
5\22
|
|
|
|
|
|
|
272.727
|
Doublewide
|
|
|
|
|
|
|
|
|
|
16\70
|
|
|
|
274.286
|
Doublewide
|
|
|
|
|
|
|
|
|
11\48
|
|
|
|
|
275.000
|
Doublewide
|
|
|
|
|
|
|
|
6\26
|
|
|
|
|
|
276.923
|
|
| 1\4
|
|
|
|
|
|
|
|
|
|
|
|
300.000
|
|
Three periods per octave
| Generator
|
|
Cents
|
Comments
|
| 0\3
|
|
|
|
|
|
|
|
|
|
0.000
|
|
|
|
|
|
|
|
|
|
|
1\30
|
|
40.000
|
|
|
|
|
|
|
|
|
|
1\27
|
|
|
44.444
|
Semiaug
|
|
|
|
|
|
|
|
|
|
2\51
|
|
47.059
|
Semiaug
|
|
|
|
|
|
|
|
1\24
|
|
|
|
50.000
|
Semiaug
|
|
|
|
|
|
|
1\21
|
|
|
|
|
57.143
|
|
|
|
|
|
|
1\18
|
|
|
|
|
|
66.667
|
|
|
|
|
|
1\15
|
|
|
|
|
|
|
80.000
|
|
|
|
|
|
|
|
|
|
|
6\87
|
|
82.759
|
|
|
|
|
|
|
|
|
|
|
|
11\159
|
83.019
|
Tritikleismic
|
|
|
|
|
|
|
|
|
5\72
|
|
|
83.333
|
Tritikleismic
|
|
|
|
|
|
|
|
4\57
|
|
|
|
84.211
|
Tritikleismic
|
|
|
|
|
|
|
3\42
|
|
|
|
|
85.714
|
|
|
|
|
|
|
2\27
|
|
|
|
|
|
88.889
|
Augmented/augene
|
|
|
|
|
|
|
3\39
|
|
|
|
|
92.308
|
Augmented/augene
|
|
|
|
1\12
|
|
|
|
|
|
|
|
100.000
|
Augmented/augene/august
|
|
|
|
|
|
3\33
|
|
|
|
|
|
109.091
|
August
|
|
|
|
|
2\21
|
|
|
|
|
|
|
114.286
|
August
|
|
|
|
|
|
3\30
|
|
|
|
|
|
120.000
|
|
|
|
1\9
|
|
|
|
|
|
|
|
|
133.333
|
|
|
|
|
|
|
4\33
|
|
|
|
|
|
145.455
|
|
|
|
|
|
3\24
|
|
|
|
|
|
|
150.000
|
Triforce
|
|
|
|
|
|
5\39
|
|
|
|
|
|
153.846
|
Triforce
|
|
|
|
2\15
|
|
|
|
|
|
|
|
160.000
|
Triforce
|
|
|
|
|
3/21
|
|
|
|
|
|
|
171.429
|
|
| 1\6
|
|
|
|
|
|
|
|
|
|
200.000
|
|
Four periods per octave
| Generator |
Cents |
Comments
|
| 0\4 |
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
1\76 |
15.789 |
|
|
|
|
|
|
|
|
1\72 |
|
16.667 |
Quadritikleismic
|
|
|
|
|
|
|
|
|
2\140 |
17.143 |
Quadritikleismic
|
|
|
|
|
|
|
|
1\68 |
|
17.647 |
Quadritikleismic
|
|
|
|
|
|
|
1\64 |
|
|
18.75 |
|
|
|
|
|
|
1\60 |
|
|
|
20 |
|
|
|
|
|
|
1\56 |
|
|
|
21.429 |
|
|
|
|
|
|
1\52 |
|
|
|
23.068 |
|
|
|
|
|
|
1\48 |
|
|
|
25 |
|
|
|
|
|
|
1\44 |
|
|
|
27.273 |
|
|
|
|
|
|
1\40 |
|
|
|
30 |
|
|
|
|
|
|
1\36 |
|
|
|
33.333 |
|
|
|
|
|
|
1\32 |
|
|
|
37.5 |
|
|
|
|
|
|
1\28 |
|
|
|
42.857 |
|
|
|
|
|
1\24 |
|
|
|
|
50 |
Darian calendar
|
|
|
|
1\20 |
|
|
|
|
|
60 |
|
|
|
1\16 |
|
|
|
|
|
|
75 |
|
|
1\12 |
|
|
|
|
|
|
|
100 |
Diminished
|
|
|
|
|
|
7\80 |
|
|
|
105 |
Diminished/Bidia
|
|
|
|
|
|
6\68 |
|
|
|
105.882 |
Diminished/Bidia
|
|
|
|
|
|
5\56 |
|
|
|
107.143 |
Diminished/Bidia
|
|
|
|
|
4\44 |
|
|
|
|
109.091 |
Diminished
|
|
|
|
3\32 |
|
|
|
|
|
112.5 |
Diminished
|
|
|
2\20 |
|
|
|
|
|
|
120 |
Diminished
|
| 1\8 |
|
|
|
|
|
|
|
|
150 |
Diminished
|
Five periods per octave
| Generator |
Cents |
Comments
|
| 0\5 |
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
1\35
|
|
|
|
34.286
|
|
|
|
|
|
|
1\30
|
|
|
|
|
40
|
|
|
|
|
|
|
|
2\55
|
|
|
|
43.636
|
|
|
|
|
|
1\25
|
|
|
|
|
|
48
|
|
|
|
|
|
|
|
3\70
|
|
|
|
51.429
|
|
|
|
|
|
|
2\45
|
|
|
|
|
53.3
|
|
|
|
|
|
|
|
3\65
|
|
|
|
55.385
|
|
|
|
|
1\20
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
|
4\75
|
|
|
|
64
|
|
|
|
|
|
|
3\55
|
|
|
|
|
65.455
|
|
|
|
|
|
|
|
5\90
|
|
|
|
66.667
|
|
|
|
|
|
2\35
|
|
|
|
|
|
68.571
|
|
|
|
|
|
|
|
5\85
|
|
|
|
70.588
|
|
|
|
|
|
|
3\50
|
|
|
|
|
72
|
|
|
|
|
|
|
|
4\65
|
|
|
|
73.846
|
|
|
1\15 |
|
|
|
|
|
|
|
80 |
Blackwood/Blacksmith
|
|
|
|
|
|
|
5\70
|
|
|
|
85.714
|
Blackwood/Blacksmith
|
|
|
|
|
|
4\55
|
|
|
|
|
87.273
|
Blackwood/Blacksmith
|
|
|
|
|
|
|
7\95
|
|
|
|
88.421
|
Blackwood/Blacksmith
|
|
|
|
|
3\40
|
|
|
|
|
|
90
|
Blackwood/Blacksmith
|
|
|
|
|
|
|
8\105
|
|
|
|
91.429
|
Blackwood/Blacksmith
|
|
|
|
|
|
5\65
|
|
|
|
|
92.308
|
Blackwood/Blacksmith
|
|
|
|
|
|
|
7\90
|
|
|
|
93.333
|
Blackwood/Blacksmith
|
|
|
|
2\25
|
|
|
|
|
|
|
96
|
Blackwood/Blacksmith
|
|
|
|
|
|
5\60
|
|
|
|
|
100
|
Blackwood
|
|
|
|
|
|
|
8\95
|
|
|
|
101.053
|
Blackwood
|
|
|
|
|
3\35
|
|
|
|
|
|
102.857
|
Blackwood
|
|
|
|
|
|
|
7\80
|
|
|
|
105
|
Blackwood
|
|
|
|
|
|
4\45
|
|
|
|
|
106.667
|
Blackwood
|
|
|
|
|
|
|
5\55
|
|
|
|
109.091
|
Blackwood
|
|
|
|
|
|
|
|
6\65
|
|
|
110.769
|
Blackwood
|
|
|
|
|
|
|
|
|
7\75
|
|
112
|
Blackwood
|
|
|
|
|
|
|
|
|
|
8\85
|
112.941
|
Blackwood/Elderthing
|
|
|
|
|
|
|
|
|
|
9\95
|
113.684
|
Blackwood
|
|
|
|
|
|
|
|
|
|
10\105
|
114.286
|
Blackwood
|
| 1\10
|
|
|
|
|
|
|
|
|
120
|
Blackwood
|
Six periods per octave
| Generator |
Cents |
Comments
|
| 0\6 |
|
|
|
|
0 |
|
|
|
|
|
|
1\36
|
33.333
|
|
|
|
|
|
1\30
|
|
40
|
|
|
|
|
|
|
2\54
|
44.444
|
|
|
|
|
1\24
|
|
|
50
|
|
|
|
|
|
|
3\66
|
54.545
|
|
|
|
|
|
2\42
|
|
57.143
|
|
|
|
|
|
|
3\60
|
60
|
|
|
|
1\18
|
|
|
|
66.667
|
|
|
|
|
|
|
4\66
|
72.727
|
|
|
|
|
|
3\48
|
|
75
|
|
|
|
|
|
|
5\78
|
76.923
|
|
|
|
|
2\30
|
|
|
80
|
Hexe
|
|
|
|
|
|
5\72
|
83.333
|
Hexe
|
|
|
|
|
3\42
|
|
85.714
|
Hexe
|
|
|
|
|
|
4\54
|
88.889
|
Hexe
|
| 1\12
|
|
|
|
|
100
|
Hexe
|
Seven periods per octave
| Generator |
Cents |
Comments
|
| 0\7 |
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
1\91
|
|
|
|
13.187
|
|
|
|
|
|
|
|
1\84
|
|
|
|
14.286
|
Absurdity
|
|
|
|
|
|
|
|
|
|
4\329
|
14.59
|
Absurdity
|
|
|
|
|
|
|
|
|
3\245
|
|
14.69
|
|
|
|
|
|
|
|
|
2\161
|
|
|
14.907
|
|
|
|
|
|
|
|
1\77
|
|
|
|
15.584
|
|
|
|
|
|
|
|
1\70
|
|
|
|
17.143
|
|
|
|
|
|
|
|
1\63
|
|
|
|
19.048
|
Sevond
|
|
|
|
|
|
|
|
2\119
|
|
|
20.168
|
Sevond
|
|
|
|
|
|
|
1\56
|
|
|
|
21.429
|
Sevond
|
|
|
|
|
|
|
1\48
|
|
|
|
25
|
|
|
|
|
|
|
1\42
|
|
|
|
|
28.571
|
|
|
|
|
|
1\35
|
|
|
|
|
|
34.286
|
Whitewood
|
|
|
|
|
|
2\63
|
|
|
|
|
38.095
|
Whitewood
|
|
|
|
1\28
|
|
|
|
|
|
|
42.857
|
Whitewood
|
|
|
|
|
|
3\77
|
|
|
|
|
46.753
|
Whitewood
|
|
|
|
|
2\49
|
|
|
|
|
|
48.980
|
Whitewood
|
|
|
|
|
|
3\70
|
|
|
|
|
51.429
|
Whitewood
|
|
|
1\21
|
|
|
|
|
|
|
|
57.143
|
Whitewood/Jamesbond/Septimal
|
|
|
|
|
|
4\77
|
|
|
|
|
62.338
|
Jamesbond/Septimal
|
|
|
|
|
3\56
|
|
|
|
|
|
64.286
|
Jamesbond/Septimal
|
|
|
|
|
|
5\91
|
|
|
|
|
65.934
|
Jamesbond/Septimal
|
|
|
|
2\35
|
|
|
|
|
|
|
68.571
|
Jamesbond/Septimal
|
|
|
|
|
|
5\84
|
|
|
|
|
71.429
|
Jamesbond/Septimal
|
|
|
|
|
3\49
|
|
|
|
|
|
73.469
|
Jamesbond/Septimal
|
|
|
|
|
|
4\63
|
|
|
|
|
76.190
|
Jamesbond/Septimal
|
| 1\14
|
|
|
|
|
|
|
|
|
85.714
|
Jamesbond/Septimal
|
Eight periods per octave
| Generator |
Cents |
Comments
|
| 0\8 |
|
|
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
|
1\80
|
|
|
15
|
|
|
|
|
|
|
|
|
|
|
|
2\152
|
|
15.789
|
Octoid
|
|
|
|
|
|
|
|
|
|
|
|
3\224
|
16.071
|
Octoid
|
|
|
|
|
|
|
|
|
1\72
|
|
|
|
16.667
|
Octoid
|
|
|
|
|
|
|
|
1\64
|
|
|
|
|
18.75
|
|
|
|
|
|
|
|
1\56
|
|
|
|
|
|
21.429
|
|
|
|
|
|
|
1\48
|
|
|
|
|
|
|
25
|
|
|
|
|
|
1\40
|
|
|
|
|
|
|
|
30
|
|
|
|
|
|
|
2\72
|
|
|
|
|
|
|
33.333
|
|
|
|
|
1\32
|
|
|
|
|
|
|
|
|
37.5
|
|
|
|
|
|
|
3\88
|
|
|
|
|
|
|
40.909
|
|
|
|
|
|
2\56
|
|
|
|
|
|
|
|
42.857
|
|
|
|
|
|
|
3\80
|
|
|
|
|
|
|
45
|
|
|
|
1\24
|
|
|
|
|
|
|
|
|
|
50
|
|
|
|
|
|
|
4\88
|
|
|
|
|
|
|
54.545
|
|
|
|
|
|
3\64
|
|
|
|
|
|
|
|
56.250
|
|
|
|
|
|
|
5\104
|
|
|
|
|
|
|
57.692
|
|
|
|
|
2\40
|
|
|
|
|
|
|
|
|
60
|
|
|
|
|
|
|
5\96
|
|
|
|
|
|
|
62.5
|
|
|
|
|
|
3\56
|
|
|
|
|
|
|
|
64.286
|
|
|
|
|
|
|
4\72
|
|
|
|
|
|
|
66.667
|
|
| 1\16
|
|
|
|
|
|
|
|
|
|
|
75
|
|
Nine periods per octave
- Ennealimmal - The generator is 49.02 cents, and don't forget the ".02" because it really is that accurate.
Twelve periods per octave
See also: Pythagorean family
Temperaments in this family are interesting because they can be thought of as 12edo with microtonal alterations.
- Compton - 3-limit as in 12edo; intervals of 5 are off by one generator. In the 7-limit (sometimes called waage), intervals of 7 are off by two generators. In the 11-limit, intervals of 11 are off by 3 generators. Thinking of 72edo might make this more concrete.
- Catler - 5-limit as in 12edo; intervals of 7 are off by one generator.
- Atomic - Does not temper out the schisma, so 3/2 is one schisma sharp of its 12edo value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the Pythagorean comma is twelve schismas, and hence 81/80, the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.
See also