Quintupole

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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.

Temperament data

Quintupole temperament (109&121)

Subgroup: 2.3.5.7.11.13.17.19

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

Comma list:

  • 7-limit: 4000/3969, 458752/455625
  • 11-limit: 896/891, 1375/1372, 4375/4356
  • 13-limit: 352/351, 364/363, 625/624, 2704/2695
  • 17-limit: 256/255, 352/351, 364/363, 375/374, 442/441
  • 19-limit: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374

Constrained TE generator:

  • 7-limit: ~135/128 = 99.12306
  • 11-limit: ~35/33 = 99.10969
  • 13-limit: ~35/33 = 99.12210
  • 17-limit: ~18/17 = 99.12595
  • 19-limit: ~18/17 = 99.12598

POTE generator:

  • 7-limit: ~135/128 = 99.17546
  • 11-limit: ~35/33 = 99.15582
  • 13-limit: ~35/33 = 99.16472
  • 17-limit: ~18/17 = 99.17178
  • 19-limit: ~18/17 = 99.16377

TOP generators:

  • 7-limit: ~2 = 1199.32542, ~135/128 = 99.11971
  • 11-limit: ~2 = 1199.34360, ~35/33 = 99.10159
  • 13-limit: ~2 = 1199.32361, ~35/33 = 99.10883
  • 17-limit: ~2 = 1199.22471, ~18/17 = 99.10770
  • 19-limit: ~2 = 1199.36954, ~18/17 = 99.11167

Diamond monotone ranges:

  • 7 and 9-odd-limit: ~135/128 = [98.63014, 100.00000] (6\73 to 1\12)
  • 11-odd-limit: ~35/33 = [98.96907, 100.00000] (8\97 to 1\12)
  • 13 and 15-odd-limit: ~35/33 = [99.08257, 99.24812] (9\109 to 11\133)
  • 17, 19, and 21-odd-limit: ~18/17 = [99.08257, 99.24812] (9\109 to 11\133)

Diamond tradeoff ranges:

  • 7 and 9-odd-limit: ~135/128 = [99.02846, 99.60900]
  • 11 and 13-odd-limit: ~35/33 = [99.02846, 99.60900]
  • 15-odd-limit: ~35/33 = [98.93352, 99.60900]
  • 17-odd-limit: ~18/17 = [98.93352, 99.60900]
  • 19-odd-limit: ~18/17 = [98.84446, 99.60900]
  • 21-odd-limit: ~18/17 = [98.80517, 99.60900]

Diamond monotone and tradeoff ranges:

  • 7 and 9-odd-limit: ~135/128 = [99.02846, 99.60900]
  • 11-odd-limit: ~35/33 = [99.02846, 99.60900]
  • 13 and 15-odd-limit: ~35/33 = [99.08257, 99.24812]
  • 17, 19, and 21-odd-limit: ~18/17 = [99.08257, 99.24812]

Optimal ET sequences (Constrained TE):

Optimal ET sequences (POTE):

Badness:

  • 7-limit: 0.111620
  • 11-limit: 0.056501
  • 13-limit: 0.038431
  • 17-limit: 0.028721
  • 19-limit: 0.023818

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]]

Eigenmonzo
(unchanged-interval) 		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

Scales

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