Pontiac

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Pontiac temperament is a 7-limit (and higher) temperament of the schismatic family. It is an extension of helmholtz temperament beyond the 5-limit but with the same simple chain-of-fifths structure (so that standard notation may be used). As in helmholtz temperament, 5/4 is mapped to the diminished fourth (e.g. A-D♭), and the new mapping specific to pontiac is that 7/4 is mapped to the quintuple augmented third (e.g. A-Cxxx). This makes pontiac a ragismic temperament.

Immediate 11-limit extensions include helenoid (53&65), mapping 11/8 to -30 fifths, ponta (53&171), mapping 11/8 to -83 fifths, and pontic (118&171), mapping 11/8 to +88 fifths.

Interval chain

Fifth
generator
Cents* Approximate Ratios
7-limit 17-limit Extension
Helenoid Ponta Pontic
0 0.00 1/1
1 701.76 3/2
2 203.51 9/8
3 905.27 27/16 22/13
4 407.03 81/64
5 1108.78 243/128, 256/135
6 610.54 64/45
7 112.30 16/15
8 814.05 8/5
9 315.81 6/5
10 1017.57 9/5
11 519.32 27/20
12 21.08 81/80
13 722.84 243/160
14 224.59 256/225
15 926.35 128/75
16 428.11 32/25
17 1129.86 48/25
18 631.62 36/25
19 133.38 27/25
20 835.13 81/50
21 336.89 175/144 17/14 17/14 17/14
22 1038.65 175/96 20/11
23 540.40 175/128 15/11
24 42.16 128/125
25 743.92 192/125 20/13 20/13 20/13
26 245.67 144/125 15/13 15/13 15/13
27 947.43 140/81
28 449.19 35/27 22/17
29 1150.94 35/18
30 652.70 35/24 16/11
31 154.46 35/32 12/11
32 856.21 105/64 18/11
33 357.97 315/256 16/13 16/13 16/13
34 1059.73 448/243 24/13 24/13 24/13
35 561.48 112/81 18/13 18/13 18/13
36 63.24 28/27
37 765.00 14/9
38 266.75 7/6
39 968.51 7/4
40 470.27 21/16
41 1172.02 63/32
42 673.78 189/128
43 175.54 448/405
44 877.29 224/135
45 379.05 56/45
46 1080.81 28/15
47 582.56 7/5
48 84.32 21/20
49 786.08 63/40 11/7
50 287.83 189/160 13/11
51 989.59 567/320
52 491.35 896/675
53 1193.10 448/225
54 694.86 112/75
55 196.62 28/25
56 898.37 42/25
57 400.13 63/50
58 1101.89 189/100 17/9 17/9 17/9
59 603.64 567/400 17/12 17/12 17/12
60 105.40 1225/1152 17/16 17/16 17/16
61 807.16 1225/768
62 308.91 448/375
63 1010.67 224/125
64 512.43 168/125
65 14.18 126/125
66 715.94 189/125
67 217.70 245/216 17/15 17/15 17/15
68 919.45 245/144 17/10 17/10 17/10
69 421.21 245/192 14/11
70 1122.97 245/128
71 624.72 735/512, 896/625
72 126.48 672/625 14/13 14/13 14/13
73 828.24 392/243
74 329.99 98/81
75 1031.75 49/27 20/11
76 533.51 49/36 15/11
77 35.26 49/48
78 737.02 49/32
79 238.78 147/128
80 940.53 441/256
81 442.29 1323/1024, 1568/1215
82 1144.05 784/405
83 645.80 196/135 16/11
84 147.56 49/45 12/11
85 849.32 49/30 18/11
86 351.07 49/40 11/9
87 1052.83 147/80 11/6
88 554.59 441/320 11/8
89 56.34 1323/1280
90 758.10 3136/2025 17/11
91 259.86 784/675
92 961.61 392/225
93 463.37 98/75 17/13 17/13 17/13
94 1165.13 49/25
95 666.88 147/100 22/15
96 168.64 441/400 11/10
97 870.40 1323/800
98 372.15 3969/3200
99 1073.91 6272/3375
100 575.67 1568/1125
101 77.42 392/375
102 779.18 196/125
103 280.94 147/125
104 982.69 441/250
105 484.45 1323/1000
106 1186.21 1715/864
107 687.96 1715/1152
108 189.72 1715/1536
109 891.48 1715/1024, 3136/1875
110 393.23 784/625
111 1094.99 1176/625
112 596.75 343/243
113 98.50 343/324
114 800.26 343/216
115 302.02 343/288
116 1003.77 343/192
117 505.53 343/256
118 7.29 1029/1024
119 709.04 3087/2048, 4704/3125,
5488/3645
120 210.80 1372/1215
121 912.56 686/405 22/13
122 414.31 343/270 14/11
123 1116.07 343/180
124 617.83 343/240
125 119.58 343/320

* in 7-limit POTE tuning

Tuning spectra

Helenoid

Gencom: [2 4/3; 352/351 385/384 561/560 625/624 729/728]

Gencom map: [1 2 -1 19 -9 -10 29], 0 -1 8 -39 30 33 -60]]

eigenmonzo
(unchanged interval
)
fifth
(¢)
comments
11/10 701.5907
15/11 701.6066
11/8 701.6227
12/11 701.6335
11/9 701.6435
16/15 701.6759
14/11 701.7030 11-odd-limit minimax
22/17 701.7071
5/4 701.7108
17/14 701.7205
6/5 701.7379 5-odd-limit minimax
17/15 701.7416
18/17 701.7422
20/17 701.7447
24/17 701.7458
17/16 701.7493
15/14 701.7512
9/7 701.7544
7/5 701.7556 7-odd-limit minimax
10/9 701.7596 9-odd-limit minimax
7/6 701.7598
8/7 701.7648
17/13 701.7680 17-odd-limit minimax
14/13 701.7819 13 and 15-odd-limit minimax
16/13 701.8022
13/12 701.8067
18/13 701.8109
13/10 701.8314
15/13 701.8362
4/3 701.9550
13/11 703.5968

Ponta

Gencom: [2 4/3; 375/374 540/539 625/624 729/728 2200/2197]

Gencom map: [1 2 -1 19 -31 -10 29], 0 -1 8 -39 83 33 -60]]

eigenmonzo
(unchanged interval)
fifth
(¢)
comments
16/15 701.6759
5/4 701.7108
17/14 701.7205
6/5 701.7379 5-odd-limit minimax
17/15 701.7416
18/17 701.7422
20/17 701.7447
24/17 701.7458
17/16 701.7493
15/14 701.7512
9/7 701.7544
7/5 701.7556 7-odd-limit minimax
10/9 701.7596 9-odd-limit minimax
7/6 701.7598
8/7 701.7648
17/13 701.7680
22/17 701.7737 17-odd-limit minimax
14/13 701.7819
14/11 701.7829 11, 13 and 15-odd-limit minimax
13/11 701.7842
11/8 701.7914
12/11 701.7933
11/9 701.7952
11/10 701.7999
15/11 701.8020
16/13 701.8022
13/12 701.8067
18/13 701.8109
13/10 701.8314
15/13 701.8362
4/3 701.9550

Pontic

Gencom: [2 4/3; 441/440 595/594 625/624 729/728 2880/2873]

Gencom map: [1 2 -1 19 40 -10 29], 0 -1 8 -39 -88 33 -60]]

eigenmonzo
(unchanged interval)
fifth
(¢)
comments
22/17 701.6558
16/15 701.6759
14/11 701.6835
5/4 701.7108
11/9 701.7140
15/11 701.7163
12/11 701.7168
11/10 701.7188
11/8 701.7195 11-odd-limit minimax
17/14 701.7205
6/5 701.7379 5-odd-limit minimax
17/15 701.7416
13/11 701.7421 13, 15 and 17-odd-limit minimax
18/17 701.7422
20/17 701.7447
24/17 701.7458
17/16 701.7493
15/14 701.7512
9/7 701.7544
7/5 701.7556 7-odd-limit minimax
10/9 701.7596 9-odd-limit minimax
7/6 701.7598
8/7 701.7648
17/13 701.7680
14/13 701.7819
16/13 701.8022
13/12 701.8067
18/13 701.8109
13/10 701.8314
15/13 701.8362
4/3 701.9550