Below are listed the dyadic chords of 11-limit hemififths temperament. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 540/539 are swetismic, by 441/440 werckismic, by 896/891 pentacircle, by 243/242 rastmic, and by 1344/1331 hemimin. Those requiring tempering by any two of 540/539, 441/440 or 243/242 are labeled jove. Those requiring both 441/440 and 896/891 are labeled pele. Those requiring any two of 243/242, 896/891 or 1344/1331 are labeled parahemif. If the full hemififths is required because of the tempering out of three independent hemififths commas, the chord is labeled "hemififths".
A striking feature of these hemififths chords is that essentially just chords tend to be of higher complexity than essentially tempered chords. Hemififths has MOS of size 7, 10, 17 and 24, and even seven notes are well-supplied with chords, mostly but by no means entirely essentially tempered chords. Extending consideration to the 13-limit adds even more such chords.
Chords are named with ups and downs. The tilde ~ means mid, half-way between major and minor. The comma (the actual punctuation mark) is pronounced "add", thus C~,7 is "C mid add7".
Hemififth's genchain
Genspan
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
15
|
Ratio
|
1/1
|
11/9
|
3/2
|
11/6
|
9/8
|
11/8
|
27/16
|
33/32
|
14/11
|
14/9
|
40/21
|
7/6
|
10/7
|
7/4
|
15/14
|
21/16
|
Interval
|
P1
|
~3
|
P5
|
~7
|
M2
|
~4
|
M6
|
^1
|
M3
|
^5
|
M7
|
^2
|
A4
|
^6
|
A1
|
^3
|
Triads
#
|
Chord
|
Transversal
|
Type
|
Name
|
1
|
0-1-2
|
1-11/9-3/2
|
rastmic
|
~
|
2
|
0-1-3
|
1-11/9-11/6
|
utonal
|
~7no5
|
3
|
0-2-3
|
1-3/2-11/6
|
otonal
|
~7no3
|
4
|
0-1-4
|
1-11/9-9/8
|
rastmic
|
|
5
|
0-2-4
|
1-3/2-9/8
|
ambitonal
|
2 or 4
|
6
|
0-3-4
|
1-11/6-9/8
|
rastmic
|
|
7
|
0-1-5
|
1-11/9-11/8
|
utonal
|
~6sus2
|
8
|
0-2-5
|
1-3/2-11/8
|
otonal
|
~7sus4
|
9
|
0-3-5
|
1-11/6-11/8
|
utonal
|
~2
|
10
|
0-4-5
|
1-9/8-11/8
|
otonal
|
~,7no5
|
11
|
0-3-8
|
1-11/6-14/11
|
hemimin
|
,~7no5
|
12
|
0-4-8
|
1-9/8-14/11
|
pentacircle
|
,9no5
|
13
|
0-5-8
|
1-11/8-14/11
|
hemimin
|
|
14
|
0-1-9
|
1-11/9-14/9
|
otonal
|
|
15
|
0-4-9
|
1-9/8-14/9
|
pentacircle
|
7~4no5
|
16
|
0-5-9
|
1-11/8-14/9
|
pentacircle
|
|
17
|
0-8-9
|
1-14/11-14/9
|
utonal
|
(^5)
|
18
|
0-2-11
|
1-3/2-7/6
|
otonal
|
|
19
|
0-3-11
|
1-11/6-7/6
|
otonal
|
|
20
|
0-8-11
|
1-14/11-7/6
|
utonal
|
|
21
|
0-9-11
|
1-14/9-7/6
|
utonal
|
|
22
|
0-1-12
|
1-11/9-10/7
|
swetismic
|
|
23
|
0-3-12
|
1-11/6-10/7
|
swetismic
|
v4(b5)
|
24
|
0-4-12
|
1-9/8-10/7
|
werckismic
|
|
25
|
0-8-12
|
1-14/11-10/7
|
werckismic
|
7(b5)no3
|
26
|
0-9-12
|
1-14/9-10/7
|
swetismic
|
~2(b5)
|
27
|
0-11-12
|
1-7/6-10/7
|
swetismic
|
~6(b5)no3
|
28
|
0-1-13
|
1-11/9-7/4
|
werckismic
|
|
29
|
0-2-13
|
1-3/2-7/4
|
otonal
|
|
30
|
0-4-13
|
1-9/8-7/4
|
otonal
|
|
31
|
0-5-13
|
1-11/8-7/4
|
otonal
|
|
32
|
0-8-13
|
1-14/11-7/4
|
utonal
|
|
33
|
0-9-13
|
1-14/9-7/4
|
utonal
|
|
34
|
0-11-13
|
1-7/6-7/4
|
utonal
|
|
35
|
0-12-13
|
1-10/7-7/4
|
werckismic
|
|
36
|
0-8-20
|
1-14/11-20/11
|
otonal
|
|
37
|
0-9-20
|
1-14/9-20/11
|
swetismic
|
|
38
|
0-11-20
|
1-7/6-20/11
|
swetismic
|
|
39
|
0-12-20
|
1-10/7-20/11
|
utonal
|
(b5)
|
40
|
0-1-21
|
1-11/9-10/9
|
otonal
|
|
41
|
0-8-21
|
1-14/11-10/9
|
werckismic
|
|
42
|
0-9-21
|
1-14/9-10/9
|
otonal
|
|
43
|
0-12-21
|
1-10/7-10/9
|
utonal
|
|
44
|
0-13-21
|
1-7/4-10/9
|
werckismic
|
|
45
|
0-20-21
|
1-20/11-10/9
|
utonal
|
|
46
|
0-2-23
|
1-3/2-5/3
|
otonal
|
|
47
|
0-3-23
|
1-11/6-5/3
|
otonal
|
|
48
|
0-11-23
|
1-7/6-5/3
|
otonal
|
|
49
|
0-12-23
|
1-10/7-5/3
|
utonal
|
|
50
|
0-20-23
|
1-20/11-5/3
|
utonal
|
|
51
|
0-21-23
|
1-10/9-5/3
|
utonal
|
|
52
|
0-2-25
|
1-3/2-5/4
|
otonal
|
|
53
|
0-4-25
|
1-9/8-5/4
|
otonal
|
|
54
|
0-5-25
|
1-11/8-5/4
|
otonal
|
|
55
|
0-12-25
|
1-10/7-5/4
|
utonal
|
|
56
|
0-13-25
|
1-7/4-5/4
|
otonal
|
|
57
|
0-20-25
|
1-20/11-5/4
|
utonal
|
|
58
|
0-21-25
|
1-10/9-5/4
|
utonal
|
|
59
|
0-23-25
|
1-5/3-5/4
|
utonal
|
|
Tetrads
#
|
Chord
|
Transversal
|
Type
|
Name
|
1
|
0-1-2-3
|
1-11/9-3/2-11/6
|
rastmic
|
~7
|
2
|
0-1-2-4
|
1-11/9-3/2-9/8
|
rastmic
|
~,9
|
3
|
0-1-3-4
|
1-11/9-11/6-9/8
|
rastmic
|
~9no5
|
4
|
0-2-3-4
|
1-3/2-11/6-9/8
|
rastmic
|
~9no3
|
5
|
0-1-2-5
|
1-11/9-3/2-11/8
|
rastmic
|
~6,9
|
6
|
0-1-3-5
|
1-11/9-11/6-11/8
|
utonal
|
|
7
|
0-2-3-5
|
1-3/2-11/6-11/8
|
ambitonal
|
|
8
|
0-1-4-5
|
1-11/9-9/8-11/8
|
rastmic
|
|
9
|
0-2-4-5
|
1-3/2-9/8-11/8
|
otonal
|
|
10
|
0-3-4-5
|
1-11/6-9/8-11/8
|
rastmic
|
~11no35
|
11
|
0-3-4-8
|
1-11/6-9/8-14/11
|
parahemif
|
9(~7)no5
|
12
|
0-3-5-8
|
1-11/6-11/8-14/11
|
hemimin
|
,~7~11no5
|
13
|
0-4-5-8
|
1-9/8-11/8-14/11
|
parahemif
|
|
14
|
0-1-4-9
|
1-11/9-9/8-14/9
|
parahemif
|
|
15
|
0-1-5-9
|
1-11/9-11/8-14/9
|
pentacircle
|
,~6,9no5
|
16
|
0-4-5-9
|
1-9/8-11/8-14/9
|
pentacircle
|
|
17
|
0-4-8-9
|
1-9/8-14/11-14/9
|
pentacircle
|
|
18
|
0-5-8-9
|
1-11/8-14/11-14/9
|
parahemif
|
|
19
|
0-2-3-11
|
1-3/2-11/6-7/6
|
otonal
|
|
20
|
0-3-8-11
|
1-11/6-14/11-7/6
|
hemimin
|
|
21
|
0-8-9-11
|
1-14/11-14/9-7/6
|
utonal
|
|
22
|
0-1-3-12
|
1-11/9-11/6-10/7
|
swetismic
|
|
23
|
0-1-4-12
|
1-11/9-9/8-10/7
|
jove
|
|
24
|
0-3-4-12
|
1-11/6-9/8-10/7
|
jove
|
|
25
|
0-3-8-12
|
1-11/6-14/11-10/7
|
hemififths
|
|
26
|
0-4-8-12
|
1-9/8-14/11-10/7
|
pele
|
|
27
|
0-1-9-12
|
1-11/9-14/9-10/7
|
swetismic
|
|
28
|
0-4-9-12
|
1-9/8-14/9-10/7
|
hemififths
|
|
29
|
0-8-9-12
|
1-14/11-14/9-10/7
|
jove
|
|
30
|
0-3-11-12
|
1-11/6-7/6-10/7
|
swetismic
|
|
31
|
0-8-11-12
|
1-14/11-7/6-10/7
|
jove
|
|
32
|
0-9-11-12
|
1-14/9-7/6-10/7
|
swetismic
|
|
33
|
0-1-2-13
|
1-11/9-3/2-7/4
|
jove
|
|
34
|
0-1-4-13
|
1-11/9-9/8-7/4
|
jove
|
|
35
|
0-2-4-13
|
1-3/2-9/8-7/4
|
otonal
|
|
36
|
0-1-5-13
|
1-11/9-11/8-7/4
|
werckismic
|
|
37
|
0-2-5-13
|
1-3/2-11/8-7/4
|
otonal
|
|
38
|
0-4-5-13
|
1-9/8-11/8-7/4
|
otonal
|
|
39
|
0-4-8-13
|
1-9/8-14/11-7/4
|
pentacircle
|
|
40
|
0-5-8-13
|
1-11/8-14/11-7/4
|
hemimin
|
|
41
|
0-1-9-13
|
1-11/9-14/9-7/4
|
werckismic
|
|
42
|
0-4-9-13
|
1-9/8-14/9-7/4
|
pentacircle
|
|
43
|
0-5-9-13
|
1-11/8-14/9-7/4
|
pentacircle
|
|
44
|
0-8-9-13
|
1-14/11-14/9-7/4
|
utonal
|
|
45
|
0-2-11-13
|
1-3/2-7/6-7/4
|
ambitonal
|
|
46
|
0-8-11-13
|
1-14/11-7/6-7/4
|
utonal
|
|
47
|
0-9-11-13
|
1-14/9-7/6-7/4
|
utonal
|
|
48
|
0-1-12-13
|
1-11/9-10/7-7/4
|
jove
|
|
49
|
0-4-12-13
|
1-9/8-10/7-7/4
|
werckismic
|
|
50
|
0-8-12-13
|
1-14/11-10/7-7/4
|
werckismic
|
|
51
|
0-9-12-13
|
1-14/9-10/7-7/4
|
jove
|
|
52
|
0-11-12-13
|
1-7/6-10/7-7/4
|
jove
|
|
53
|
0-8-9-20
|
1-14/11-14/9-20/11
|
swetismic
|
|
54
|
0-8-11-20
|
1-14/11-7/6-20/11
|
swetismic
|
|
55
|
0-9-11-20
|
1-14/9-7/6-20/11
|
swetismic
|
|
56
|
0-8-12-20
|
1-14/11-10/7-20/11
|
werckismic
|
|
57
|
0-9-12-20
|
1-14/9-10/7-20/11
|
swetismic
|
|
58
|
0-11-12-20
|
1-7/6-10/7-20/11
|
swetismic
|
|
59
|
0-1-9-21
|
1-11/9-14/9-10/9
|
otonal
|
|
60
|
0-8-9-21
|
1-14/11-14/9-10/9
|
werckismic
|
|
61
|
0-1-12-21
|
1-11/9-10/7-10/9
|
swetismic
|
|
62
|
0-8-12-21
|
1-14/11-10/7-10/9
|
werckismic
|
|
63
|
0-9-12-21
|
1-14/9-10/7-10/9
|
swetismic
|
|
64
|
0-1-13-21
|
1-11/9-7/4-10/9
|
werckismic
|
|
65
|
0-8-13-21
|
1-14/11-7/4-10/9
|
werckismic
|
|
66
|
0-9-13-21
|
1-14/9-7/4-10/9
|
werckismic
|
|
67
|
0-12-13-21
|
1-10/7-7/4-10/9
|
werckismic
|
|
68
|
0-8-20-21
|
1-14/11-20/11-10/9
|
werckismic
|
|
69
|
0-9-20-21
|
1-14/9-20/11-10/9
|
swetismic
|
|
70
|
0-12-20-21
|
1-10/7-20/11-10/9
|
utonal
|
|
71
|
0-2-3-23
|
1-3/2-11/6-5/3
|
otonal
|
|
72
|
0-2-11-23
|
1-3/2-7/6-5/3
|
otonal
|
|
73
|
0-3-11-23
|
1-11/6-7/6-5/3
|
otonal
|
|
74
|
0-3-12-23
|
1-11/6-10/7-5/3
|
swetismic
|
|
75
|
0-11-12-23
|
1-7/6-10/7-5/3
|
swetismic
|
|
76
|
0-11-20-23
|
1-7/6-20/11-5/3
|
swetismic
|
|
77
|
0-12-20-23
|
1-10/7-20/11-5/3
|
utonal
|
|
78
|
0-12-21-23
|
1-10/7-10/9-5/3
|
utonal
|
|
79
|
0-20-21-23
|
1-20/11-10/9-5/3
|
utonal
|
|
80
|
0-2-4-25
|
1-3/2-9/8-5/4
|
otonal
|
|
81
|
0-2-5-25
|
1-3/2-11/8-5/4
|
otonal
|
|
82
|
0-4-5-25
|
1-9/8-11/8-5/4
|
otonal
|
|
83
|
0-4-12-25
|
1-9/8-10/7-5/4
|
werckismic
|
|
84
|
0-2-13-25
|
1-3/2-7/4-5/4
|
otonal
|
|
85
|
0-4-13-25
|
1-9/8-7/4-5/4
|
otonal
|
|
86
|
0-5-13-25
|
1-11/8-7/4-5/4
|
otonal
|
|
87
|
0-12-13-25
|
1-10/7-7/4-5/4
|
werckismic
|
|
88
|
0-12-20-25
|
1-10/7-20/11-5/4
|
utonal
|
|
89
|
0-12-21-25
|
1-10/7-10/9-5/4
|
utonal
|
|
90
|
0-13-21-25
|
1-7/4-10/9-5/4
|
werckismic
|
|
91
|
0-20-21-25
|
1-20/11-10/9-5/4
|
utonal
|
|
92
|
0-2-23-25
|
1-3/2-5/3-5/4
|
ambitonal
|
|
93
|
0-12-23-25
|
1-10/7-5/3-5/4
|
utonal
|
|
94
|
0-20-23-25
|
1-20/11-5/3-5/4
|
utonal
|
|
95
|
0-21-23-25
|
1-10/9-5/3-5/4
|
utonal
|
|
Pentads
#
|
Chord
|
Transversal
|
Type
|
Name
|
1
|
0-1-2-3-4
|
1-11/9-3/2-11/6-9/8
|
rastmic
|
~9
|
2
|
0-1-2-3-5
|
1-11/9-3/2-11/6-11/8
|
rastmic
|
~7,~11
|
3
|
0-1-2-4-5
|
1-11/9-3/2-9/8-11/8
|
rastmic
|
~,9~11 or ~11no7
|
4
|
0-1-3-4-5
|
1-11/9-11/6-9/8-11/8
|
rastmic
|
~11no5
|
5
|
0-2-3-4-5
|
1-3/2-11/6-9/8-11/8
|
rastmic
|
~11no3
|
6
|
0-3-4-5-8
|
1-11/6-9/8-11/8-14/11
|
parahemif
|
|
7
|
0-1-4-5-9
|
1-11/9-9/8-11/8-14/9
|
parahemif
|
|
8
|
0-4-5-8-9
|
1-9/8-11/8-14/11-14/9
|
parahemif
|
|
9
|
0-1-3-4-12
|
1-11/9-11/6-9/8-10/7
|
jove
|
|
10
|
0-3-4-8-12
|
1-11/6-9/8-14/11-10/7
|
hemififths
|
|
11
|
0-1-4-9-12
|
1-11/9-9/8-14/9-10/7
|
hemififths
|
|
12
|
0-4-8-9-12
|
1-9/8-14/11-14/9-10/7
|
hemififths
|
|
13
|
0-3-8-11-12
|
1-11/6-14/11-7/6-10/7
|
hemififths
|
|
14
|
0-8-9-11-12
|
1-14/11-14/9-7/6-10/7
|
jove
|
|
15
|
0-1-2-4-13
|
1-11/9-3/2-9/8-7/4
|
jove
|
|
16
|
0-1-2-5-13
|
1-11/9-3/2-11/8-7/4
|
jove
|
|
17
|
0-1-4-5-13
|
1-11/9-9/8-11/8-7/4
|
jove
|
|
18
|
0-2-4-5-13
|
1-3/2-9/8-11/8-7/4
|
otonal
|
|
19
|
0-4-5-8-13
|
1-9/8-11/8-14/11-7/4
|
parahemif
|
|
20
|
0-1-4-9-13
|
1-11/9-9/8-14/9-7/4
|
hemififths
|
|
21
|
0-1-5-9-13
|
1-11/9-11/8-14/9-7/4
|
pele
|
|
22
|
0-4-5-9-13
|
1-9/8-11/8-14/9-7/4
|
pentacircle
|
|
23
|
0-4-8-9-13
|
1-9/8-14/11-14/9-7/4
|
pentacircle
|
|
24
|
0-5-8-9-13
|
1-11/8-14/11-14/9-7/4
|
parahemif
|
|
25
|
0-8-9-11-13
|
1-14/11-14/9-7/6-7/4
|
utonal
|
|
26
|
0-1-4-12-13
|
1-11/9-9/8-10/7-7/4
|
jove
|
|
27
|
0-4-8-12-13
|
1-9/8-14/11-10/7-7/4
|
pele
|
|
28
|
0-1-9-12-13
|
1-11/9-14/9-10/7-7/4
|
jove
|
|
29
|
0-4-9-12-13
|
1-9/8-14/9-10/7-7/4
|
hemififths
|
|
30
|
0-8-9-12-13
|
1-14/11-14/9-10/7-7/4
|
jove
|
|
31
|
0-8-11-12-13
|
1-14/11-7/6-10/7-7/4
|
jove
|
|
32
|
0-9-11-12-13
|
1-14/9-7/6-10/7-7/4
|
jove
|
|
33
|
0-8-9-11-20
|
1-14/11-14/9-7/6-20/11
|
swetismic
|
|
34
|
0-8-9-12-20
|
1-14/11-14/9-10/7-20/11
|
jove
|
|
35
|
0-8-11-12-20
|
1-14/11-7/6-10/7-20/11
|
jove
|
|
36
|
0-9-11-12-20
|
1-14/9-7/6-10/7-20/11
|
swetismic
|
|
37
|
0-1-9-12-21
|
1-11/9-14/9-10/7-10/9
|
swetismic
|
|
38
|
0-8-9-12-21
|
1-14/11-14/9-10/7-10/9
|
jove
|
|
39
|
0-1-9-13-21
|
1-11/9-14/9-7/4-10/9
|
werckismic
|
|
40
|
0-8-9-13-21
|
1-14/11-14/9-7/4-10/9
|
werckismic
|
|
41
|
0-1-12-13-21
|
1-11/9-10/7-7/4-10/9
|
jove
|
|
42
|
0-8-12-13-21
|
1-14/11-10/7-7/4-10/9
|
werckismic
|
|
43
|
0-9-12-13-21
|
1-14/9-10/7-7/4-10/9
|
jove
|
|
44
|
0-8-9-20-21
|
1-14/11-14/9-20/11-10/9
|
jove
|
|
45
|
0-8-12-20-21
|
1-14/11-10/7-20/11-10/9
|
werckismic
|
|
46
|
0-9-12-20-21
|
1-14/9-10/7-20/11-10/9
|
swetismic
|
|
47
|
0-2-3-11-23
|
1-3/2-11/6-7/6-5/3
|
otonal
|
|
48
|
0-3-11-12-23
|
1-11/6-7/6-10/7-5/3
|
swetismic
|
|
49
|
0-11-12-20-23
|
1-7/6-10/7-20/11-5/3
|
swetismic
|
|
50
|
0-12-20-21-23
|
1-10/7-20/11-10/9-5/3
|
utonal
|
|
51
|
0-2-4-5-25
|
1-3/2-9/8-11/8-5/4
|
otonal
|
|
52
|
0-2-4-13-25
|
1-3/2-9/8-7/4-5/4
|
otonal
|
|
53
|
0-2-5-13-25
|
1-3/2-11/8-7/4-5/4
|
otonal
|
|
54
|
0-4-5-13-25
|
1-9/8-11/8-7/4-5/4
|
otonal
|
|
55
|
0-4-12-13-25
|
1-9/8-10/7-7/4-5/4
|
werckismic
|
|
56
|
0-12-13-21-25
|
1-10/7-7/4-10/9-5/4
|
werckismic
|
|
57
|
0-12-20-21-25
|
1-10/7-20/11-10/9-5/4
|
utonal
|
|
58
|
0-12-20-23-25
|
1-10/7-20/11-5/3-5/4
|
utonal
|
|
59
|
0-12-21-23-25
|
1-10/7-10/9-5/3-5/4
|
utonal
|
|
60
|
0-20-21-23-25
|
1-20/11-10/9-5/3-5/4
|
utonal
|
|
Hexads
#
|
Chord
|
Transversal
|
Type
|
Name
|
1
|
0-1-2-3-4-5
|
1-11/9-3/2-11/6-9/8-11/8
|
rastmic
|
~11
|
2
|
0-1-2-4-5-13
|
1-11/9-3/2-9/8-11/8-7/4
|
jove
|
|
3
|
0-1-4-5-9-13
|
1-11/9-9/8-11/8-14/9-7/4
|
hemififths
|
|
4
|
0-4-5-8-9-13
|
1-9/8-11/8-14/11-14/9-7/4
|
nofives
|
|
5
|
0-1-4-9-12-13
|
1-11/9-9/8-14/9-10/7-7/4
|
hemififths
|
|
6
|
0-4-8-9-12-13
|
1-9/8-14/11-14/9-10/7-7/4
|
hemififths
|
|
7
|
0-8-9-11-12-13
|
1-14/11-14/9-7/6-10/7-7/4
|
jove
|
|
8
|
0-8-9-11-12-20
|
1-14/11-14/9-7/6-10/7-20/11
|
jove
|
|
9
|
0-1-9-12-13-21
|
1-11/9-14/9-10/7-7/4-10/9
|
jove
|
|
10
|
0-8-9-12-13-21
|
1-14/11-14/9-10/7-7/4-10/9
|
jove
|
|
11
|
0-8-9-12-20-21
|
1-14/11-14/9-10/7-20/11-10/9
|
jove
|
|
12
|
0-2-4-5-13-25
|
1-3/2-9/8-11/8-7/4-5/4
|
otonal
|
|
13
|
0-12-20-21-23-25
|
1-10/7-20/11-10/9-5/3-5/4
|
utonal
|
|