Hemififths/Chords

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, using pergen #4 (P8, P5/2) in the notation guide for rank-2 pergens. One up is 7 generators. The tilde ~ means mid, half-way between major and minor. ~4 = ^4 = vA4 and ~5 = v5 = ^d5. The comma (the actual punctuation mark) is pronounced "add", thus C~,7 is "C mid add 7". To facilitate chord naming, lifts and drops are also used. Enharmonic equivalences: vvA1 and v\m2. Cents: ^1 = 50¢ + 3.5c and /1 = 50¢ - 8.5c, where c = the amount in cents the tempered fifth exceeds 700¢. /1 = ~81/80 = ~64/63 and ^1 = ~33/32. To convert to 41edo, ^1 = 2\41 and /1 = 1\41.

Hemififth's genchain

Genspan	0	1	2	3	4	5	6	7	8	9	10	11	12	13	...	20	21	...	23	...	25
Cents (41edo)	0	351	702	1054	205	556	907	59	410	761	1112	263	615	966		1024	176		878		380
Ratio	1/1	11/9 16/13	3/2	11/6 24/13	9/8	11/8 18/13	27/16 22/13	28/27 33/32	14/11	14/9	40/21 21/11	7/6	10/7	7/4		20/11	10/9		5/3		5/4
Interval	P1	~3	P5	~7	M2	~4 \b5	M6	^1 \m2	M3	^5 \m6	M7	^M2 \m3	A4 \~5	^M6 \m7		A6 \~7	^A1 \M2		^A5 \M6		^A2 \M3

TODO: Complete the tables

Triads

#	Chord	Transversal	Type	Name	Inversion
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	~,9no5
5	0-2-4	1-3/2-9/8	ambitonal	2 or 4	1-4/3-9/8
6	0-3-4	1-11/6-9/8	rastmic	~9no35
7	0-1-5	1-11/9-11/8	utonal	~(\b5)
8	0-2-5	1-3/2-11/8	otonal	~7sus4no5	1-4/3-11/6
9	0-3-5	1-11/6-11/8	utonal	~2	1-12/11-3/2
10	0-4-5	1-9/8-11/8	otonal	~,7no5	1-11/9-16/9
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	(\b5)
14	0-1-9	1-11/9-14/9	otonal	~(^5)
15	0-4-9	1-9/8-14/9	pentacircle	7~4no5	1-11/8-16/9
16	0-5-9	1-11/8-14/9	pentacircle	2(~5) or /,7no5	1-9/8-13/9 1-9/7-16/9
17	0-8-9	1-14/11-14/9	utonal	(^5)
18	0-2-11	1-3/2-7/6	otonal	\m
19	0-3-11	1-11/6-7/6	otonal	\m~7no5
20	0-8-11	1-14/11-7/6	utonal	~2/6	1-12/11-12/7
21	0-9-11	1-14/9-7/6	utonal	/	1-9/7-3/2
22	0-1-12	1-11/9-10/7	swetismic	~(\~5)
23	0-3-12	1-11/6-10/7	swetismic	/(b5)	1-9/7-7/5
24	0-4-12	1-9/8-10/7	werckismic	2(\~5)
25	0-8-12	1-14/11-10/7	werckismic	(\~5)
26	0-9-12	1-14/9-10/7	swetismic	~2(b5)	1-12/11-7/5
27	0-11-12	1-7/6-10/7	swetismic	\m(\~5)
28	0-1-13	1-11/9-7/4	werckismic	~,\7no5
29	0-2-13	1-3/2-7/4	otonal	\7no3
30	0-4-13	1-9/8-7/4	otonal	\9no35
31	0-5-13	1-11/8-7/4	otonal	(~5)	1-14/11-16/11
32	0-8-13	1-14/11-7/4	utonal	,\7
33	0-9-13	1-14/9-7/4	utonal	/,2no5	1-9/8-9/7
34	0-11-13	1-7/6-7/4	utonal	\m7no5
35	0-12-13	1-10/7-7/4	werckismic	~(b5)	1-11/9-7/5
36	0-8-20	1-14/11-20/11	otonal	,\~7no5
37	0-9-20	1-14/9-20/11	swetismic
38	0-11-20	1-7/6-20/11	swetismic	\m\~7no5
39	0-12-20	1-10/7-20/11	utonal	(b5)
40	0-1-21	1-11/9-10/9	otonal	~,\9no5
41	0-8-21	1-14/11-10/9	werckismic	,\9no5
42	0-9-21	1-14/9-10/9	otonal	/(\~5)	1-9/7-10/7
43	0-12-21	1-10/7-10/9	utonal	\2(\~5)
44	0-13-21	1-7/4-10/9	werckismic	\2\m7no5
45	0-20-21	1-20/11-10/9	utonal
46	0-2-23	1-3/2-5/3	otonal	\m7no5	1-6/5-9/5
47	0-3-23	1-11/6-5/3	otonal
48	0-11-23	1-7/6-5/3	otonal	\m6no5
49	0-12-23	1-10/7-5/3	utonal	/dim	1-7/6-7/5
50	0-20-23	1-20/11-5/3	utonal
51	0-21-23	1-10/9-5/3	utonal	/7no3	1-3/2-9/5
52	0-2-25	1-3/2-5/4	otonal	\
53	0-4-25	1-9/8-5/4	otonal	\,9no5
54	0-5-25	1-11/8-5/4	otonal	\(\b5)
55	0-12-25	1-10/7-5/4	utonal	\(\~5)
56	0-13-25	1-7/4-5/4	otonal	\7no5
57	0-20-25	1-20/11-5/4	utonal	\,\~7no5
58	0-21-25	1-10/9-5/4	utonal	/9no35	1-9/5-9/4
59	0-23-25	1-5/3-5/4	utonal	/m	1-6/5-3/2

Tetrads

#	Chord	Transversal	Type	Name	Inversion
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	~,~11
6	0-1-3-5	1-11/9-11/6-11/8	utonal	2~6 or 4~9	1-9/8-3/2-13/8 1-4/3-3/2-24/11
7	0-2-3-5	1-3/2-11/6-11/8	ambitonal	~4~7
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	4~7	1-4/3-3/2-11/6
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	~2~11	1-12/11-11/8-3/2
13	0-4-5-8	1-9/8-11/8-14/11	parahemif	~,~9	1-11/9-3/2-13/12
14	0-1-4-9	1-11/9-9/8-14/9	parahemif	~,9(^5)
15	0-1-5-9	1-11/9-11/8-14/9	pentacircle	,~6,9no5	1-14/11-18/11-9/4
16	0-4-5-9	1-9/8-11/8-14/9	pentacircle	~,7(\b5)	1-11/9-11/8-16/9
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	\m~7
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	/,~6	1-9/7-3/2-18/11
22	0-1-3-12	1-11/9-11/6-10/7	swetismic	\m~6	1-7/6-3/2-13/8
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	9no5	1-9/8-14/11-16/9
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	~,\7
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	\9no3
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	~4,\7
38	0-4-5-13	1-9/8-11/8-7/4	otonal	~4\7,9 or ~,7(^5)	1-11/9-14/9-16/9
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	\m7
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	/,9	1-9/7-3/2-9/4
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	~,/6	1-11/9-3/2-12/7
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	\m6
73	0-3-11-23	1-11/6-7/6-5/3	otonal	\m6~7no5
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	\m6(\~5)
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	/7	1-9/7-3/2-9/5
79	0-20-21-23	1-20/11-10/9-5/3	utonal	/7~13no3	1-3/2-18/11-9/5
80	0-2-4-25	1-3/2-9/8-5/4	otonal	\,9
81	0-2-5-25	1-3/2-11/8-5/4	otonal	\,~11
82	0-4-5-25	1-9/8-11/8-5/4	otonal	\,9(\b5)
83	0-4-12-25	1-9/8-10/7-5/4	werckismic	,9(\~5)
84	0-2-13-25	1-3/2-7/4-5/4	otonal	\7
85	0-4-13-25	1-9/8-7/4-5/4	otonal	\9no5
86	0-5-13-25	1-11/8-7/4-5/4	otonal	\7~11no5
87	0-12-13-25	1-10/7-7/4-5/4	werckismic	\7(\~5)
88	0-12-20-25	1-10/7-20/11-5/4	utonal	,\7(b5)	1-14/11-7/5-7/4
89	0-12-21-25	1-10/7-10/9-5/4	utonal	/9no5	1-9/7-9/5-9/4
90	0-13-21-25	1-7/4-10/9-5/4	werckismic	\7\9no5
91	0-20-21-25	1-20/11-10/9-5/4	utonal	2~6^7no5	1-18/11-9/5-9/4
92	0-2-23-25	1-3/2-5/3-5/4	ambitonal	\6 or /m7	1-6/5-3/2-9/5
93	0-12-23-25	1-10/7-5/3-5/4	utonal	/m6 \m7(b5)	1-6/5-3/2-12/7 1-7/6-7/5-7/4
94	0-20-23-25	1-20/11-5/3-5/4	utonal	/m,~9	1-6/5-3/2-12/11
95	0-21-23-25	1-10/9-5/3-5/4	utonal	/9no3	1-3/2-9/5-9/4

Pentads

#	Chord	Transversal	Type	Name	Inversion
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	~11no9
3	0-1-2-4-5	1-11/9-3/2-9/8-11/8	rastmic	~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	~9(\m7)
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	\9~11no3
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	/,~6,9	1-9/7-3/2-18/11-9/4
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	\m6~7
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	/7~6	1-9/7-3/2-18/11-9/5
51	0-2-4-5-25	1-3/2-9/8-11/8-5/4	otonal	\,9~11
52	0-2-4-13-25	1-3/2-9/8-7/4-5/4	otonal	\9
53	0-2-5-13-25	1-3/2-11/8-7/4-5/4	otonal	\7~11
54	0-4-5-13-25	1-9/8-11/8-7/4-5/4	otonal	\9~11no5
55	0-4-12-13-25	1-9/8-10/7-7/4-5/4	werckismic	\9(\~5)
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	/9~6no5	1-9/7-18/11-9/5-9/4
58	0-12-20-23-25	1-10/7-20/11-5/3-5/4	utonal	/m6~9	1-6/5-3/2-12/7-13/6
59	0-12-21-23-25	1-10/7-10/9-5/3-5/4	utonal	/9	1-9/7-3/2-9/5-9/4
60	0-20-21-23-25	1-20/11-10/9-5/3-5/4	utonal	/9~6no3	1-3/2-18/11-9/5-9/4

Hexads

#	Chord	Transversal	Type	Name	Inversion
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	~11(\m7)
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	\9~11
13	0-12-20-21-23-25	1-10/7-20/11-10/9-5/3-5/4	utonal	/9~6	1-9/7-3/2-18/11-9/5-9/4