Quintupole

Todo: improve synopsis Add infobox. Rewrite intro.

Quintupole is a temperament for the 7, 11, 13, 17, and 19 prime limits. It is a member of quintaleap family, octagar temperaments, and mistismic temperaments. It has a semitone inverval which represents 18/17, 35/33, 52/49, 55/52 and 135/128 as a generator, five of them give a fourth (~2.2 cents flat of 4/3), sixteen give 5/2, and thirty-four give seventh harmonic. 121EDO is an excellent tuning for quintupole, with a semitone generator of 10\121, and MOS of 13, 25, 37, 49, 61, 73, 85, 97, or 109 notes are available.

Interval chain

Quintupole is considered as a cluster temperament with twelve clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents 100/99～120/119～171/170～196/195～209/208～210/209～221/220～225/224～289/288～324/323～441/440～513/512 all tempered together.

Number of generator	Cents value*	Approximate Ratios
0	0.000	1/1
1	99.164	18/17, 35/33
2	198.328	28/25
3	297.491	19/16, 25/21
4	396.655	34/27
5	495.819	4/3
6	594.983	24/17
7	694.146
8	793.310	19/12, 30/19
9	892.474
10	991.638	16/9
11	1090.801	15/8, 32/17
12	1189.965
13	89.129	19/18, 20/19, 21/20
14	188.293	19/17
15	287.457	13/11
16	386.620	5/4
17	485.784
18	584.948	7/5
19	684.112
20	783.275	11/7
21	882.439	5/3
22	981.603	30/17
23	1080.767	28/15
24	1179.930
25	79.094	22/21
26	178.258	10/9, 21/19
27	277.422	20/17
28	376.586
29	475.749	21/16
30	574.913
31	674.077	28/19
32	773.241
33	872.404
34	971.568	7/4
35	1070.732	13/7
36	1169.896
37	69.060	25/24, 26/25
38	168.223	11/10
39	267.387	7/6
40	366.551	21/17, 26/21
41	465.715
42	564.878
43	664.042	22/15
44	763.206	14/9
45	862.370	28/17
46	961.533
47	1060.697
48	1159.861	39/20
49	59.025	33/32
50	158.189
51	257.352	22/19
52	356.516
53	455.680	13/10
54	554.844	11/8
55	654.007
56	753.171
57	852.335
58	951.499	26/15
59	1050.662	11/6
60	1149.826	35/18
61	48.990
62	148.154
63	247.318
64	346.481	11/9
65	445.645	22/17
66	544.809	26/19
67	643.973
68	743.136
69	842.300	13/8
70	941.464
71	1040.628
72	1139.791
73	38.955
74	138.119	13/12
75	237.283
76	336.447	(close to 17/14)
77	435.610	(close to 9/7)
78	534.774
79	633.938	13/9
80	733.102	26/17

* in 19-limit POTE tuning

Tuning spectrum

Gencom: [2 18/17; 190/189 256/255 352/351 361/360 364/363 375/374]

Gencom mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]

Unchanged interval (eigenmonzo)	Generator (¢)	Comments
21/20	98.8052
19/15	98.8445
16/15	98.9335
18/17	98.9546
21/19	98.9718
21/16	98.9924
7/5	99.0285
19/14	99.0746
11/10	99.0791
8/7	99.0831
22/19	99.0942
11/8	99.0985
14/11	99.1246
13/10	99.1361
26/19	99.1366
16/13	99.1381
20/19	99.1385
15/11	99.1406
12/11	99.1417
5/4	99.1446
21/17	99.1456
7/6	99.1505	7 and 21-odd-limit minimax
15/14	99.1547
13/12	99.1699
19/16	99.1710
22/17	99.1748
15/13	99.1769
11/9	99.1782	11, 13, 15, 17 and 19-odd-limit minimax
14/13	99.1915
17/13	99.1947
17/14	99.1971
18/13	99.1977
9/7	99.2026	9-odd-limit minimax
22/21	99.2215
17/15	99.2415
26/21	99.2437
6/5	99.2552	5-odd-limit minimax
13/11	99.2807
20/17	99.3096
10/9	99.3232
24/19	99.4448
19/17	99.4684
24/17	99.4999
19/18	99.5079
17/16	99.5495
4/3	99.6090