Chords of orwell

Below is a complete list of all 11-odd-limit dyadic chords of 11-limit orwell temperament. Note that there are many common chords, for example 8:10:12:15, which are not listed; in this case due to 15/8 not being in the 11-odd-limit. Every chord listed has multiple inversions; only one is listed, that being the inversion where all notes are a nonnegative number of subminor third generators above the root.

Typing the chords requires consideration of the fact that orwell equates certain 11-odd-limit consonances – it conflates 14/11 with 9/7, 11/7 with 14/9, 12/11 with 11/10, and 11/6 with 20/11. If a transversal can be found which shows the chord to be essentially just, that transversal is listed along with a typing as otonal, utonal, or ambitonal. Sometimes multiple such transversals exist, in which case the chord is a plurichord, and the type is given for all possible interpretations. If the chord is essentially tempered, it is analyzed in terms of the transversal that requires the minimum amount of commas to be tempered out; if there is a tie between multiple transversals, it is analyzed in terms of the transversal which employs 9/7 and 11/10 over the root.

Those requiring tempering only by 225/224 are labeled marvel, by 99/98 mothwellsmic, by 121/120 biyatismic, by 176/175 valinorsmic, by 385/384 keenanismic, and by 540/539 swetismic. If it requires any two of 99/98, 176/175 and 225/224, it is labeled minerva, any two of 99/98, 121/120 or 540/539, big brother, any two of 121/120, 176/175 or 385/384, zeus, any two of 225/224, 385/384 or 540/539, unimarvel. If it requires both 99/98 and 385/384 it is labeled orwellian, and if it requires three independent commas among those discussed above, it is labeled orwell.

Orwell has mos of size 5, 9, 13, 22 and 31. The complaint is often made that orwell is lacking in low-complexity harmonies; however, even the orwell pentatonic scale has three triads and a tetrad. The 9-note mos is of course much better supplied, and with 13 notes we are rolling in chords and tossing them in the air, finding plenty of chords including even the inexorable major triad.

For diagrams showing how some of these chords might map to a keyboard such as the Axis-49, see Orwell on an isomorphic keyboard.

Triads

#	Generators	Transversal	Type	Comment
1	0–1–2	1–7/6–11/8	Mothwellsmic
2	0–1–3	1–7/6–8/5	Keenanismic
3	0–2–3	1–11/8–8/5	Keenanismic
4	0–2–5	1–11/10–11/8	Utonal
5	0–3–5	1–11/10–8/5	Otonal
6	0–1–6	1–7/6–14/11	Utonal
7	0–3–6	1–9/7–8/5	Marvel
8	0–5–6	1–12/11–14/11	Otonal
9	0–1–7	1–7/6–3/2	Otonal	6:7:9
10	0–2–7	1–11/8–3/2	Otonal
11	0–5–7	1–12/11–3/2	Utonal
12	0–6–7	1–9/7–3/2	Utonal	1/(9:7:6)
13	0–1–8	1–7/6–7/4	Utonal	1/(12:8:7)
14	0–2–8	1–11/8–7/4	Otonal
15	0–3–8	1–8/5–7/4	Valinorsmic
16	0–5–8	1–11/10–7/4	Valinorsmic
17	0–6–8	1–14/11–7/4	Utonal
18	0–7–8	1–3/2–7/4	Otonal	4:6:7
19	0–2–10	1–6/5–11/8	Keenanismic
20	0–3–10	1–6/5–8/5	Otonal	4:5:6
21	0–5–10	1–11/10–6/5	Otonal
22	0–7–10	1–6/5–3/2	Utonal	1/(6:5:4)
23	0–8–10	1–6/5–7/4	Keenanismic
24	0–1–11	1–7/6–7/5	Utonal	1/(7:6:5)
25	0–3–11	1–7/5–8/5	Otonal	4:5:7
26	0–5–11	1–11/10–7/5	Otonal
27	0–6–11	1–14/11–7/5	Utonal
28	0–8–11	1–7/5–7/4	Utonal	1/(10:8:7)
29	0–10–11	1–6/5–7/5	Otonal	5:6:7
30	0–1–12	1–7/6–18/11	Swetismic
31	0–2–12	1–11/8–18/11	Biyatismic
32	0–5–12	1–12/11–18/11	Otonal
33	0–6–12	1–9/7–18/11	Utonal
34	0–7–12	1–3/2–18/11	Utonal
35	0–10–12	1–6/5–18/11	Biyatismic
36	0–11–12	1–7/5–18/11	Swetismic
37	0–2–14	1–9/8–11/8	Otonal
38	0–3–14	1–9/8–8/5	Marvel
39	0–6–14	1–9/8–9/7	Utonal
40	0–7–14	1–9/8–3/2	Ambitonal	6:8:9, 8:9:12
41	0–8–14	1–9/8–7/4	Otonal
42	0–11–14	1–9/8–7/5	Marvel
43	0–12–14	1–9/8–18/11	Utonal
44	0–3–17	1–8/5–9/5	Otonal
45	0–5–17	1–11/10–9/5	Otonal
46	0–6–17	1–9/7–9/5	Utonal
47	0–7–17	1–3/2–9/5	Utonal	1/(9:6:5)
48	0–10–17	1–6/5–9/5	Otonal	6:9:10
49	0–11–17	1–7/5–9/5	Otonal
50	0–12–17	1–18/11–9/5	Utonal
51	0–14–17	1–9/8–9/5	Utonal

Tetrads

#	Generators	Transversal	Type	Comment
1	0–1–2–3	1–7/6–11/8–8/5	Orwellian
2	0–2–3–5	1–11/10–11/8–8/5	Keenanismic
3	0–1–3–6	1–7/6–9/7–8/5	Unimarvel
4	0–3–5–6	1–11/10–9/7–8/5	Unimarvel
5	0–1–2–7	1–7/6–11/8–3/2	Big brother
6	0–2–5–7	1–11/10–11/8–3/2	Biyatismic
7	0–1–6–7	1–7/6–9/7–3/2	Swetismic
8	0–5–6–7	1–11/10–9/7–3/2	Big brother
9	0–1–2–8	1–7/6–11/8–7/4	Mothwellsmic
10	0–1–3–8	1–7/6–8/5–7/4	Zeus
11	0–2–3–8	1–11/8–8/5–7/4	Orwell
12	0–2–5–8	1–11/10–11/8–7/4	Minerva
13	0–3–5–8	1–11/10–8/5–7/4	Valinorsmic
14	0–1–6–8	1–7/6–14/11–7/4	Utonal
15	0–3–6–8	1–9/7–8/5–7/4	Minerva
16	0–5–6–8	1–11/10–9/7–7/4	Orwell
17	0–1–7–8	1–7/6–3/2–7/4	Ambitonal	12:14:18:21, 14:18:21:24 9-odd-limit ASS
18	0–2–7–8	1–11/8–3/2–7/4	Otonal
19	0–5–7–8	1–11/10–3/2–7/4	Zeus
20	0–6–7–8	1–9/7–3/2–7/4	Mothwellsmic
21	0–2–3–10	1–6/5–11/8–8/5	Keenanismic
22	0–2–5–10	1–11/10–6/5–11/8	Zeus
23	0–3–5–10	1–11/10–6/5–8/5	Otonal
24	0–2–7–10	1–6/5–11/8–3/2	Zeus
25	0–5–7–10	1–12/11–6/5–3/2	Utonal
26	0–2–8–10	1–6/5–11/8–7/4	Orwellian
27	0–3–8–10	1–6/5–8/5–7/4	Zeus
28	0–5–8–10	1–11/10–6/5–7/4	Zeus
29	0–7–8–10	1–6/5–3/2–7/4	Keenanismic
30	0–1–3–11	1–7/6–7/5–8/5	Keenanismic
31	0–3–5–11	1–11/10–7/5–8/5	Otonal
32	0–1–6–11	1–7/6–14/11–7/5	Utonal
33	0–3–6–11	1–9/7–7/5–8/5	Minerva
34	0–5–6–11	1–11/10–9/7–7/5	Big brother
35	0–1–8–11	1–7/6–7/5–7/4	Utonal	1/(12:10:8:7)
36	0–3–8–11	1–7/5–8/5–7/4	Valinorsmic
37	0–5–8–11	1–11/10–7/5–7/4	Minerva
38	0–6–8–11	1–14/11–7/5–7/4	Utonal
39	0–3–10–11	1–6/5–7/5–8/5	Otonal	4:5:6:7
40	0–5–10–11	1–11/10–6/5–7/5	Otonal
41	0–8–10–11	1–6/5–7/5–7/4	Keenanismic
42	0–1–2–12	1–7/6–11/8–18/11	Big brother
43	0–2–5–12	1–11/10–11/8–18/11	Biyatismic
44	0–1–6–12	1–7/6–9/7–18/11	Big brother
45	0–5–6–12	1–12/11–14/11–18/11	Otonal
46	0–1–7–12	1–7/6–3/2–18/11	Big brother
47	0–2–7–12	1–11/8–3/2–18/11	Biyatismic
48	0–5–7–12	1–12/11–3/2–18/11	Ambitonal	11-odd-limit ASS
49	0–6–7–12	1–9/7–3/2–18/11	Utonal
50	0–2–10–12	1–6/5–11/8–18/11	Zeus
51	0–5–10–12	1–11/10–6/5–18/11	Biyatismic
52	0–7–10–12	1–6/5–3/2–18/11	Biyatismic
53	0–1–11–12	1–7/6–7/5–18/11	Swetismic
54	0–5–11–12	1–11/10–7/5–18/11	Big brother
55	0–6–11–12	1–9/7–7/5–18/11	Big brother
56	0–10–11–12	1–6/5–7/5–18/11	Big brother
57	0–2–3–14	1–9/8–11/8–8/5	Unimarvel
58	0–3–6–14	1–9/8–9/7–8/5	Marvel
59	0–2–7–14	1–9/8–11/8–3/2	Otonal
60	0–6–7–14	1–9/8–9/7–3/2	Utonal	1/(9:7:6:4)
61	0–2–8–14	1–9/8–11/8–7/4	Otonal
62	0–3–8–14	1–9/8–8/5–7/4	Minerva
63	0–6–8–14	1–9/8–9/7–7/4	Mothwellsmic
64	0–7–8–14	1–9/8–3/2–7/4	Otonal	4:6:7:9
65	0–3–11–14	1–9/8–7/5–8/5	Marvel
66	0–6–11–14	1–9/8–9/7–7/5	Minerva
67	0–8–11–14	1–9/8–7/5–7/4	Marvel
68	0–2–12–14	1–9/8–11/8–18/11	Biyatismic
69	0–6–12–14	1–9/8–9/7–18/11	Utonal
70	0–7–12–14	1–9/8–3/2–18/11	Utonal
71	0–11–12–14	1–9/8–7/5–18/11	Unimarvel
72	0–3–5–17	1–11/10–8/5–9/5	Otonal
73	0–3–6–17	1–9/7–8/5–9/5	Marvel
74	0–5–6–17	1–11/10–9/7–9/5	Swetismic
75	0–5–7–17	1–11/10–3/2–9/5	Biyatismic
76	0–6–7–17	1–9/7–3/2–9/5	Utonal	1/(9:7:6:5)
77	0–3–10–17	1–6/5–8/5–9/5	Otonal	4:5:6:9
78	0–5–10–17	1–11/10–6/5–9/5	Otonal
79	0–7–10–17	1–6/5–3/2–9/5	Ambitonal	10:12:15:18, 12:15:18:20 9-odd-limit ASS
80	0–3–11–17	1–7/5–8/5–9/5	Otonal	4:5:7:9
81	0–5–11–17	1–11/10–7/5–9/5	Otonal
82	0–6–11–17	1–9/7–7/5–9/5	Mothwellsmic
83	0–10–11–17	1–6/5–7/5–9/5	Otonal	5:6:7:9
84	0–5–12–17	1–11/10–18/11–9/5	Biyatismic
85	0–6–12–17	1–9/7–18/11–9/5	Utonal
86	0–7–12–17	1–3/2–18/11–9/5	Utonal
87	0–10–12–17	1–6/5–18/11–9/5	Biyatismic
88	0–11–12–17	1–7/5–18/11–9/5	Swetismic
89	0–3–14–17	1–9/8–8/5–9/5	Marvel
90	0–6–14–17	1–9/8–9/7–9/5	Utonal	1/(9:7:5:4)
91	0–7–14–17	1–9/8–3/2–9/5	Utonal	1/(9:6:5:4)
92	0–11–14–17	1–9/8–7/5–9/5	Marvel
93	0–12–14–17	1–9/8–18/11–9/5	Utonal

Pentads

#	Generators	Transversal	Type	Comment
1	0–1–2–3–8	1–7/6–11/8–8/5–7/4	Orwell
2	0–2–3–5–8	1–11/10–11/8–8/5–7/4	Orwell
3	0–1–3–6–8	1–7/6–9/7–8/5–7/4	Orwell
4	0–3–5–6–8	1–11/10–9/7–8/5–7/4	Orwell
5	0–1–2–7–8	1–7/6–11/8–3/2–7/4	Big brother
6	0–2–5–7–8	1–11/10–11/8–3/2–7/4	Orwell
7	0–1–6–7–8	1–7/6–9/7–3/2–7/4	Big brother
8	0–5–6–7–8	1–11/10–9/7–3/2–7/4	Orwell
9	0–2–3–5–10	1–11/10–6/5–11/8–8/5	Zeus
10	0–2–5–7–10	1–11/10–6/5–11/8–3/2	Zeus
11	0–2–3–8–10	1–6/5–11/8–8/5–7/4	Orwell
12	0–2–5–8–10	1–11/10–6/5–11/8–7/4	Orwell
13	0–3–5–8–10	1–6/5–11/10–8/5–7/4	Zeus
14	0–2–7–8–10	1–6/5–11/8–3/2–7/4	Orwell
15	0–5–7–8–10	1–11/10–6/5–3/2–7/4	Zeus
16	0–1–3–6–11	1–7/6–9/7–7/5–8/5	Orwell
17	0–3–5–6–11	1–11/10–9/7–7/5	Orwell
18	0–1–3–8–11	1–7/6–7/5–8/5–7/4	Zeus
19	0–3–5–8–11	1–11/10–8/5–7/5–7/4	Minerva
20	0–1–6–8–11	1–7/6–14/11–7/5–7/4	Utonal	1/(24:20:16:14:11)
21	0–3–6–8–11	1–9/7–7/5–8/5–7/4	Minerva
22	0–5–6–8–11	1–11/10–9/7–7/5–7/4	Orwell
23	0–3–5–10–11	1–11/10–6/5–7/5–8/5	Otonal	4:5:6:7:11
24	0–3–8–10–11	1–6/5–7/5–8/5–7/4	Zeus
25	0–5–8–10–11	1–11/10–6/5–7/5–7/4	Orwell
26	0–1–2–7–12	1–7/6–11/8–3/2–18/11	Big brother
27	0–2–5–7–12	1–11/10–11/8–3/2–18/11	Biyatismic
28	0–1–6–7–12	1–7/6–9/7–3/2–18/11	Big brother
29	0–5–6–7–12	1–11/10–9/7–3/2–18/11	Big brother
30	0–2–5–10–12	1–11/10–6/5–11/8–18/11	Zeus
31	0–2–7–10–12	1–6/5–11/8–3/2–18/11	Zeus
32	0–5–7–10–12	1–11/10–6/5–3/2–18/11	Biyatismic
33	0–1–6–11–12	1–7/6–9/7–7/5–18/11	Big brother
34	0–5–6–11–12	1–11/10–9/7–7/5–18/11	Big brother
35	0–5–10–11–12	1–11/10–6/5–7/5–18/11	Big brother
36	0–2–3–8–14	1–9/8–11/8–8/5–7/4	Orwell
37	0–3–6–8–14	1–9/8–9/7–8/5–7/4	Minerva
38	0–2–7–8–14	1–11/8–3/2–7/4–9/8	Otonal	4:6:7:9:11
39	0–6–7–8–14	1–9/8–9/7–3/2–7/4	Mothwellsmic
40	0–3–6–11–14	1–9/8–9/7–7/5–8/5	Minerva
41	0–3–8–11–14	1–9/8–7/5–8/5–7/4	Minerva
42	0–6–8–11–14	1–9/8–9/7–7/5–7/4	Minerva
43	0–2–7–12–14	1–9/8–11/8–3/2–18/11	Biyatismic
44	0–6–7–12–14	1–9/8–9/7–3/2–18/11	Utonal	1/(24:16:14:11:9)
45	0–6–11–12–14	1–9/8–9/7–7/5–18/11	Orwell
46	0–3–5–6–17	1–11/10–9/7–8/5–9/5	Unimarvel
47	0–5–6–7–17	1–11/10–9/7–3/2–9/5	Big brother
48	0–3–5–10–17	1–11/10–6/5–8/5–9/5	Otonal	4:5:6:9:11
49	0–5–7–10–17	1–11/10–6/5–3/2–9/5	Biyatismic
50	0–3–5–11–17	1–11/10–7/5–8/5–9/5	Otonal	4:5:7:9:11
51	0–3–6–11–17	1–9/7–7/5–8/5–9/5	Minerva
52	0–5–6–11–17	1–11/10–9/7–7/5–9/5	Big brother
53	0–3–10–11–17	1–6/5–7/5–8/5–9/5	Otonal	4:5:6:7:9
54	0–5–10–11–17	1–11/10–6/5–7/5–9/5	Otonal	5:6:7:9:11
55	0–5–6–12–17	1–11/10–9/7–18/11–9/5	Big brother
56	0–5–7–12–17	1–11/10–3/2–18/11–9/5	Biyatismic
57	0–6–7–12–17	1–9/7–3/2–18/11–9/5	Utonal	1/(24:20:14:11:9)
58	0–5–10–12–17	1–11/10–6/5–18/11–9/5	Biyatismic
59	0–7–10–12–17	1–6/5–3/2–18/11–9/5	Biyatismic
60	0–5–11–12–17	1–11/10–7/5–18/11–9/5	Big brother
61	0–6–11–12–17	1–9/7–7/5–18/11–9/5	Big brother
62	0–10–11–12–17	1–6/5–7/5–18/11–9/5	Big brother
63	0–3–6–14–17	1–9/8–9/7–8/5–9/5	Marvel
64	0–6–7–14–17	1–9/8–9/7–3/2–9/5	Utonal	1/(24:20:16:14:9)
65	0–3–11–14–17	1–9/8–7/5–8/5–9/5	Marvel
66	0–6–11–14–17	1–9/8–9/7–7/5–9/5	Minerva
67	0–6–12–14–17	1–9/8–9/7–18/11–9/5	Utonal	1/(20:16:14:11:9)
68	0–7–12–14–17	1–9/8–3/2–18/11–9/5	Utonal	1/(24:20:16:11:9)
69	0–11–12–14–17	1–9/8–7/5–18/11–9/5	Unimarvel

Hexads

#	Generators	Transversal	Type	Comment
1	0–2–3–5–8–10	1–11/10–6/5–11/8–8/5–7/4	Orwell
2	0–2–5–7–8–10	1–11/10–6/5–11/8–3/2–7/4	Orwell
3	0–1–3–6–8–11	1–7/6–9/7–7/5–8/5–7/4	Orwell
4	0–3–5–6–8–11	1–11/10–9/7–7/5–8/5–7/4	Orwell
5	0–3–5–8–10–11	1–11/10–6/5–7/5–8/5–7/4	Orwell
6	0–2–5–7–10–12	1–11/10–6/5–11/8–3/2–18/11	Zeus
7	0–3–6–8–11–14	1–9/8–9/7–7/5–8/5–7/4	Minerva
8	0–3–5–6–11–17	1–11/10–9/7–7/5–8/5–9/5	Orwell
9	0–3–5–10–11–17	1–11/10–6/5–7/5–8/5–9/5	Otonal	4:5:6:7:9:11
10	0–5–6–7–12–17	1–11/10–9/7–3/2–18/11–9/5	Big brother
11	0–5–7–10–12–17	1–11/10–6/5–3/2–18/11–9/5	Biyatismic
12	0–5–6–11–12–17	1–11/10–9/7–7/5–18/11–9/5	Big brother
13	0–5–10–11–12–17	1–11/10–6/5–7/5–18/11–9/5	Big brother
14	0–3–6–11–14–17	1–9/8–9/7–7/5–8/5–9/5	Minerva
15	0–6–7–12–14–17	1–9/8–9/7–3/2–18/11–9/5	Utonal	1/(24:20:16:14:11:9)
16	0–6–11–12–14–17	1–9/8–9/7–7/5–18/11–9/5	Orwell