Sqrtphi/Chords

Below are listed the 15-limit dyadic chords of sqrtphi temperament, subject to the condition that no interval smaller than 12/11, at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. Codimension two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it.

Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ = √((1+√5)/2), which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the "official" 13/8 of sqrtphi, φ is only 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.

Triads

Number	Chord	Transversal	Type
1	0-1-9	1-11/7-11/6	utonal
2	0-8-9	1-7/6-11/6	otonal
3	0-1-10	1-11/7-13/9	gentle
4	0-5-10	1-6/5-13/9	marveltwin
5	0-9-10	1-11/6-13/9	gentle
6	0-5-13	1-6/5-7/5	otonal
7	0-8-13	1-7/6-7/5	utonal
8	0-1-14	1-11/7-11/10	utonal
9	0-5-14	1-6/5-11/10	otonal
10	0-9-14	1-11/6-11/10	utonal
11	0-13-14	1-7/5-11/10	otonal
12	0-1-15	1-11/7-26/15	gentle
13	0-5-15	1-6/5-26/15	otonal
14	0-10-15	1-13/9-26/15	utonal
15	0-14-15	1-11/10-26/15	gentle
16	0-1-16	1-11/7-15/11	nicolic
17	0-8-16	1-7/6-15/11	swetismic
18	0-15-16	1-26/15-15/11	nicolic
19	0-5-21	1-6/5-18/11	utonal
20	0-8-21	1-7/6-18/11	swetismic
21	0-13-21	1-7/5-18/11	swetismic
22	0-16-21	1-15/11-18/11	otonal
23	0-1-22	1-11/7-9/7	otonal
24	0-8-22	1-7/6-9/7	swetismic
25	0-9-22	1-11/6-9/7	swetismic
26	0-13-22	1-7/5-9/7	swetismic
27	0-14-22	1-11/10-9/7	swetismic
28	0-21-22	1-18/11-9/7	utonal
29	0-9-25	1-11/6-5/4	otonal
30	0-10-25	1-13/9-5/4	island
31	0-15-25	1-26/15-5/4	island
32	0-16-25	1-15/11-5/4	utonal
33	0-5-30	1-6/5-3/2	utonal
34	0-8-30	1-7/6-3/2	otonal
35	0-9-30	1-11/6-3/2	otonal
36	0-14-30	1-11/10-3/2	otonal
37	0-15-30	1-26/15-3/2	island
38	0-16-30	1-15/11-3/2	utonal
39	0-21-30	1-18/11-3/2	utonal
40	0-22-30	1-9/7-3/2	utonal
41	0-25-30	1-5/4-3/2	otonal
42	0-1-31	1-11/7-13/11	gentle
43	0-9-31	1-11/6-13/11	gentle
44	0-10-31	1-13/9-13/11	utonal
45	0-15-31	1-26/15-13/11	utonal
46	0-16-31	1-15/11-13/11	otonal
47	0-21-31	1-18/11-13/11	otonal
48	0-22-31	1-9/7-13/11	gentle
49	0-30-31	1-3/2-13/11	gentle
50	0-5-35	1-6/5-9/5	otonal
51	0-10-35	1-13/9-9/5	marveltwin
52	0-13-35	1-7/5-9/5	otonal
53	0-14-35	1-11/10-9/5	otonal
54	0-21-35	1-18/11-9/5	utonal
55	0-22-35	1-9/7-9/5	utonal
56	0-25-35	1-5/4-9/5	marveltwin
57	0-30-35	1-3/2-9/5	utonal
58	0-8-38	1-7/6-7/4	utonal
59	0-13-38	1-7/5-7/4	utonal
60	0-16-38	1-15/11-7/4	swetismic
61	0-22-38	1-9/7-7/4	swetismic
62	0-25-38	1-5/4-7/4	otonal
63	0-30-38	1-3/2-7/4	otonal
64	0-1-39	1-11/7-11/8	utonal
65	0-8-39	1-7/6-11/8	gentle
66	0-9-39	1-11/6-11/8	utonal
67	0-14-39	1-11/10-11/8	utonal
68	0-25-39	1-5/4-11/8	otonal
69	0-30-39	1-3/2-11/8	otonal
70	0-31-39	1-13/11-11/8	gentle
71	0-38-39	1-7/4-11/8	otonal
72	0-10-45	1-13/9-13/10	utonal
73	0-14-45	1-11/10-13/10	otonal
74	0-15-45	1-26/15-13/10	utonal
75	0-30-45	1-3/2-13/10	otonal
76	0-31-45	1-13/11-13/10	utonal
77	0-35-45	1-9/5-13/10	otonal
78	0-15-60	1-26/15-9/8	island
79	0-21-60	1-18/11-9/8	utonal
80	0-22-60	1-9/7-9/8	utonal
81	0-25-60	1-5/4-9/8	otonal
82	0-30-60	1-3/2-9/8	ambitonal
83	0-35-60	1-9/5-9/8	utonal
84	0-38-60	1-7/4-9/8	otonal
85	0-39-60	1-11/8-9/8	otonal
86	0-45-60	1-13/10-9/8	island
87	0-10-70	1-13/9-13/8	utonal
88	0-25-70	1-5/4-13/8	otonal
89	0-31-70	1-13/11-13/8	utonal
90	0-35-70	1-9/5-13/8	marveltwin
91	0-39-70	1-11/8-13/8	otonal
92	0-45-70	1-13/10-13/8	utonal
93	0-60-70	1-9/8-13/8	otonal

Tetrads

Number	Chord	Transversal	Type
1	0-1-9-10	1-11/7-11/6-13/9	gentle
2	0-1-9-14	1-11/7-11/6-11/10	utonal
3	0-5-13-14	1-6/5-7/5-11/10	otonal
4	0-1-10-15	1-11/7-13/9-26/15	gentle
5	0-5-10-15	1-6/5-13/9-26/15	marveltwin
6	0-1-14-15	1-11/7-11/10-26/15	gentle
7	0-5-14-15	1-6/5-11/10-26/15	gentle
8	0-1-15-16	1-11/7-26/15-15/11	island-gentle
9	0-5-13-21	1-6/5-7/5-18/11	swetismic
10	0-8-13-21	1-7/6-7/5-18/11	swetismic
11	0-8-16-21	1-7/6-15/11-18/11	swetismic
12	0-1-9-22	1-11/7-11/6-9/7	swetismic
13	0-8-9-22	1-7/6-11/6-9/7	swetismic
14	0-8-13-22	1-7/6-7/5-9/7	swetismic
15	0-1-14-22	1-11/7-11/10-9/7	swetismic
16	0-9-14-22	1-11/6-11/10-9/7	swetismic
17	0-13-14-22	1-7/5-11/10-9/7	swetismic
18	0-8-21-22	1-7/6-18/11-9/7	swetismic
19	0-13-21-22	1-7/5-18/11-9/7	swetismic
20	0-9-10-25	1-11/6-13/9-5/4	island-gentle
21	0-10-15-25	1-13/9-26/15-5/4	island
22	0-15-16-25	1-26/15-15/11-5/4	island-gentle
23	0-8-9-30	1-7/6-11/6-3/2	otonal
24	0-5-14-30	1-6/5-11/10-3/2	otonal
25	0-9-14-30	1-11/6-11/10-3/2	utonal
26	0-5-15-30	1-6/5-26/15-3/2	island
27	0-14-15-30	1-11/10-26/15-3/2	island-gentle
28	0-8-16-30	1-7/6-15/11-3/2	swetismic
29	0-15-16-30	1-26/15-15/11-3/2	island-gentle
30	0-5-21-30	1-6/5-18/11-3/2	utonal
31	0-8-21-30	1-7/6-18/11-3/2	swetismic
32	0-16-21-30	1-15/11-18/11-3/2	otonal
33	0-8-22-30	1-7/6-9/7-3/2	swetismic
34	0-9-22-30	1-11/6-9/7-3/2	swetismic
35	0-14-22-30	1-11/10-9/7-3/2	swetismic
36	0-21-22-30	1-18/11-9/7-3/2	utonal
37	0-9-25-30	1-11/6-5/4-3/2	otonal
38	0-15-25-30	1-26/15-5/4-3/2	island
39	0-16-25-30	1-15/11-5/4-3/2	utonal
40	0-1-9-31	1-11/7-11/6-13/11	gentle
41	0-1-10-31	1-11/7-13/9-13/11	gentle
42	0-9-10-31	1-11/6-13/9-13/11	gentle
43	0-1-15-31	1-11/7-26/15-13/11	gentle
44	0-10-15-31	1-13/9-26/15-13/11	utonal
45	0-1-16-31	1-11/7-15/11-13/11	island-gentle
46	0-15-16-31	1-26/15-15/11-13/11	nicolic
47	0-16-21-31	1-15/11-18/11-13/11	otonal
48	0-1-22-31	1-11/7-9/7-13/11	gentle
49	0-9-22-31	1-11/6-9/7-13/11	swetismic-gentle
50	0-21-22-31	1-18/11-9/7-13/11	gentle
51	0-9-30-31	1-11/6-3/2-13/11	gentle
52	0-15-30-31	1-26/15-3/2-13/11	island-gentle
53	0-16-30-31	1-15/11-3/2-13/11	gentle
54	0-21-30-31	1-18/11-3/2-13/11	gentle
55	0-22-30-31	1-9/7-3/2-13/11	gentle
56	0-5-10-35	1-6/5-13/9-9/5	marveltwin
57	0-5-13-35	1-6/5-7/5-9/5	otonal
58	0-5-14-35	1-6/5-11/10-9/5	otonal
59	0-13-14-35	1-7/5-11/10-9/5	otonal
60	0-5-21-35	1-6/5-18/11-9/5	utonal
61	0-13-21-35	1-7/5-18/11-9/5	swetismic
62	0-13-22-35	1-7/5-9/7-9/5	swetismic
63	0-14-22-35	1-11/10-9/7-9/5	swetismic
64	0-21-22-35	1-18/11-9/7-9/5	utonal
65	0-10-25-35	1-13/9-5/4-9/5	kleismic-marveltwin
66	0-5-30-35	1-6/5-3/2-9/5	ambitonal
67	0-14-30-35	1-11/10-3/2-9/5	otonal
68	0-21-30-35	1-18/11-3/2-9/5	utonal
69	0-22-30-35	1-9/7-3/2-9/5	utonal
70	0-25-30-35	1-5/4-3/2-9/5	marveltwin
71	0-8-13-38	1-7/6-7/5-7/4	utonal
72	0-8-16-38	1-7/6-15/11-7/4	swetismic
73	0-8-22-38	1-7/6-9/7-7/4	swetismic
74	0-13-22-38	1-7/5-9/7-7/4	swetismic
75	0-16-25-38	1-15/11-5/4-7/4	swetismic
76	0-8-30-38	1-7/6-3/2-7/4	ambitonal
77	0-16-30-38	1-15/11-3/2-7/4	swetismic
78	0-22-30-38	1-9/7-3/2-7/4	swetismic
79	0-25-30-38	1-5/4-3/2-7/4	otonal
80	0-1-9-39	1-11/7-11/6-11/8	utonal
81	0-8-9-39	1-7/6-11/6-11/8	gentle
82	0-1-14-39	1-11/7-11/10-11/8	utonal
83	0-9-14-39	1-11/6-11/10-11/8	utonal
84	0-9-25-39	1-11/6-5/4-11/8	otonal
85	0-8-30-39	1-7/6-3/2-11/8	gentle
86	0-9-30-39	1-11/6-3/2-11/8	ambitonal
87	0-14-30-39	1-11/10-3/2-11/8	utonal
88	0-25-30-39	1-5/4-3/2-11/8	otonal
89	0-1-31-39	1-11/7-13/11-11/8	gentle
90	0-9-31-39	1-11/6-13/11-11/8	gentle
91	0-30-31-39	1-3/2-13/11-11/8	gentle
92	0-8-38-39	1-7/6-7/4-11/8	gentle
93	0-25-38-39	1-5/4-7/4-11/8	otonal
94	0-30-38-39	1-3/2-7/4-11/8	otonal
95	0-10-15-45	1-13/9-26/15-13/10	utonal
96	0-14-15-45	1-11/10-26/15-13/10	gentle
97	0-14-30-45	1-11/10-3/2-13/10	otonal
98	0-15-30-45	1-26/15-3/2-13/10	island
99	0-10-31-45	1-13/9-13/11-13/10	utonal
100	0-15-31-45	1-26/15-13/11-13/10	utonal
101	0-30-31-45	1-3/2-13/11-13/10	gentle
102	0-10-35-45	1-13/9-9/5-13/10	marveltwin
103	0-14-35-45	1-11/10-9/5-13/10	otonal
104	0-30-35-45	1-3/2-9/5-13/10	otonal
105	0-21-22-60	1-18/11-9/7-9/8	utonal
106	0-15-25-60	1-26/15-5/4-9/8	island
107	0-15-30-60	1-26/15-3/2-9/8	island
108	0-21-30-60	1-18/11-3/2-9/8	utonal
109	0-22-30-60	1-9/7-3/2-9/8	utonal
110	0-25-30-60	1-5/4-3/2-9/8	otonal
111	0-21-35-60	1-18/11-9/5-9/8	utonal
112	0-22-35-60	1-9/7-9/5-9/8	utonal
113	0-25-35-60	1-5/4-9/5-9/8	marveltwin
114	0-30-35-60	1-3/2-9/5-9/8	utonal
115	0-22-38-60	1-9/7-7/4-9/8	swetismic
116	0-25-38-60	1-5/4-7/4-9/8	otonal
117	0-30-38-60	1-3/2-7/4-9/8	otonal
118	0-25-39-60	1-5/4-11/8-9/8	otonal
119	0-30-39-60	1-3/2-11/8-9/8	otonal
120	0-38-39-60	1-7/4-11/8-9/8	otonal
121	0-15-45-60	1-26/15-13/10-9/8	island
122	0-30-45-60	1-3/2-13/10-9/8	island
123	0-35-45-60	1-9/5-13/10-9/8	island
124	0-10-25-70	1-13/9-5/4-13/8	island
125	0-10-31-70	1-13/9-13/11-13/8	utonal
126	0-10-35-70	1-13/9-9/5-13/8	marveltwin
127	0-25-35-70	1-5/4-9/5-13/8	marveltwin
128	0-25-39-70	1-5/4-11/8-13/8	otonal
129	0-31-39-70	1-13/11-11/8-13/8	gentle
130	0-10-45-70	1-13/9-13/10-13/8	utonal
131	0-31-45-70	1-13/11-13/10-13/8	utonal
132	0-35-45-70	1-9/5-13/10-13/8	marveltwin
133	0-25-60-70	1-5/4-9/8-13/8	otonal
134	0-35-60-70	1-9/5-9/8-13/8	marveltwin
135	0-39-60-70	1-11/8-9/8-13/8	otonal
136	0-45-60-70	1-13/10-9/8-13/8	island

Pentads

Number	Chord	Transversal	Type
1	0-1-9-14-22	1-11/7-11/6-11/10-9/7	swetismic
2	0-8-13-21-22	1-7/6-7/5-18/11-9/7	swetismic
3	0-5-14-15-30	1-6/5-11/10-26/15-3/2	island-gentle
4	0-8-16-21-30	1-7/6-15/11-18/11-3/2	swetismic
5	0-8-9-22-30	1-7/6-11/6-9/7-3/2	swetismic
6	0-9-14-22-30	1-11/6-11/10-9/7-3/2	swetismic
7	0-8-21-22-30	1-7/6-18/11-9/7-3/2	swetismic
8	0-15-16-25-30	1-26/15-15/11-5/4-3/2	island-gentle
9	0-1-9-10-31	1-11/7-11/6-13/9-13/11	gentle
10	0-1-10-15-31	1-11/7-13/9-26/15-13/11	gentle
11	0-1-15-16-31	1-11/7-26/15-15/11-13/11	island-gentle
12	0-1-9-22-31	1-11/7-11/6-9/7-13/11	swetismic-gentle
13	0-15-16-30-31	1-26/15-15/11-3/2-13/11	island-gentle
14	0-16-21-30-31	1-15/11-18/11-3/2-13/11	gentle
15	0-9-22-30-31	1-11/6-9/7-3/2-13/11	swetismic-gentle
16	0-21-22-30-31	1-18/11-9/7-3/2-13/11	gentle
17	0-5-13-14-35	1-6/5-7/5-11/10-9/5	otonal
18	0-5-13-21-35	1-6/5-7/5-18/11-9/5	swetismic
19	0-13-14-22-35	1-7/5-11/10-9/7-9/5	swetismic
20	0-13-21-22-35	1-7/5-18/11-9/7-9/5	swetismic
21	0-5-14-30-35	1-6/5-11/10-3/2-9/5	otonal
22	0-5-21-30-35	1-6/5-18/11-3/2-9/5	utonal
23	0-14-22-30-35	1-11/10-9/7-3/2-9/5	swetismic
24	0-21-22-30-35	1-18/11-9/7-3/2-9/5	utonal
25	0-8-13-22-38	1-7/6-7/5-9/7-7/4	swetismic
26	0-8-16-30-38	1-7/6-15/11-3/2-7/4	swetismic
27	0-8-22-30-38	1-7/6-9/7-3/2-7/4	swetismic
28	0-16-25-30-38	1-15/11-5/4-3/2-7/4	swetismic
29	0-1-9-14-39	1-11/7-11/6-11/10-11/8	utonal
30	0-8-9-30-39	1-7/6-11/6-3/2-11/8	gentle
31	0-9-14-30-39	1-11/6-11/10-3/2-11/8	utonal
32	0-9-25-30-39	1-11/6-5/4-3/2-11/8	otonal
33	0-1-9-31-39	1-11/7-11/6-13/11-11/8	gentle
34	0-9-30-31-39	1-11/6-3/2-13/11-11/8	gentle
35	0-8-30-38-39	1-7/6-3/2-7/4-11/8	gentle
36	0-25-30-38-39	1-5/4-3/2-7/4-11/8	otonal
37	0-14-15-30-45	1-11/10-26/15-3/2-13/10	island-gentle
38	0-10-15-31-45	1-13/9-26/15-13/11-13/10	utonal
39	0-15-30-31-45	1-26/15-3/2-13/11-13/10	island-gentle
40	0-14-30-35-45	1-11/10-3/2-9/5-13/10	otonal
41	0-21-22-30-60	1-18/11-9/7-3/2-9/8	utonal
42	0-15-25-30-60	1-26/15-5/4-3/2-9/8	island
43	0-21-22-35-60	1-18/11-9/7-9/5-9/8	utonal
44	0-21-30-35-60	1-18/11-3/2-9/5-9/8	utonal
45	0-22-30-35-60	1-9/7-3/2-9/5-9/8	utonal
46	0-25-30-35-60	1-5/4-3/2-9/5-9/8	marveltwin
47	0-22-30-38-60	1-9/7-3/2-7/4-9/8	swetismic
48	0-25-30-38-60	1-5/4-3/2-7/4-9/8	otonal
49	0-25-30-39-60	1-5/4-3/2-11/8-9/8	otonal
50	0-25-38-39-60	1-5/4-7/4-11/8-9/8	otonal
51	0-30-38-39-60	1-3/2-7/4-11/8-9/8	otonal
52	0-15-30-45-60	1-26/15-3/2-13/10-9/8	island
53	0-30-35-45-60	1-3/2-9/5-13/10-9/8	island
54	0-10-25-35-70	1-13/9-5/4-9/5-13/8	kleismic-marveltwin
55	0-10-31-45-70	1-13/9-13/11-13/10-13/8	utonal
56	0-10-35-45-70	1-13/9-9/5-13/10-13/8	marveltwin
57	0-25-35-60-70	1-5/4-9/5-9/8-13/8	marveltwin
58	0-25-39-60-70	1-5/4-11/8-9/8-13/8	otonal
59	0-35-45-60-70	1-9/5-13/10-9/8-13/8	kleismic-marveltwin

Hexads

Number	Chord	Transversal	Type
1	0-21-22-30-35-60	1-18/11-9/7-3/2-9/5-9/8	utonal
2	0-25-30-38-39-60	1-5/4-3/2-7/4-11/8-9/8	otonal