Pontiac: Difference between revisions

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Explain the intervals in terms of commatic inflections
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| Subgroups = 2.3.5.7
| Subgroups = 2.3.5.7
| Comma basis = [[4375/4374]], [[32805/32768]]
| Comma basis = [[4375/4374]], [[32805/32768]]
| Generator = 3/2
| Edo join 1 = 53 | Edo join 2 = 171
| Mapping = 1; 1 -8 39
| Mapping = 1; 1 -8 39
| Generators = 3/2
| Generators tuning = 701.758
| Optimization method = CWE
| Pergen = (P8, P5)
| Pergen = (P8, P5)
| Edo join 1 = 53 | Edo join 2 = 171
| Optimization method = CWE
| Generator tuning = 701.758
| MOS scales = [[12L 17s]], [[12L 29s]], [[12L 41s]], [[53L 12s]]
| MOS scales = [[12L 17s]], [[12L 29s]], [[12L 41s]], [[53L 12s]]
| Odd limit 1 = 9 | Mistuning 1 = 0.401 | Complexity 1 = 53
| Odd limit 1 = 9 | Mistuning 1 = 0.401 | Complexity 1 = 53
| Odd limit 2 = (7-limit) 63 | Mistuning 2 = 0.716 | Complexity 2 = 118
| Odd limit 2 = 7-limit 81 | Mistuning 2 = 0.884 | Complexity 2 = 118
}}
}}
'''Pontiac''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the [[5-limit]] but with the same simple [[chain of fifths|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. C–F♭; a comma-flat major third), and the new mapping specific to pontiac is that [[7/4]] is mapped to the quintuple-augmented third (e.g. C–Exx#; a three-comma-sharp major sixth). This makes pontiac a [[ragismic microtemperaments|ragismic temperament]]. An excellent tuning for pontiac is [[171edo]], with a perfect fifth generator 100\171, and [[mos scale]]s of size 12, 17, 29, 41, 53, 65, and 118 are available.
'''Pontiac''' is a [[7-limit]] (and higher) [[regular temperament|temperament]] of the [[schismatic family]]. It is an [[extension]] of [[helmholtz (temperament)|helmholtz]] temperament beyond the [[5-limit]] but with the same simple [[chain of fifths|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. C–F♭; a comma-flat major third), and the new mapping specific to pontiac is that [[7/4]] is mapped to the quintuple-augmented third (e.g. C–Exx#; a three-comma-sharp major sixth). This makes pontiac a [[ragismic microtemperaments|ragismic temperament]]. An excellent tuning for pontiac is [[171edo]], with a perfect fifth generator 100\171, and [[mos scale]]s of size 12, 17, 29, 41, 53, 65, and 118 are available.
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Pontiac was named by [[Gene Ward Smith]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10729.html Yahoo! Tuning Group | Beep, orwell, and schismic]</ref>. For technical data see [[Schismatic family#Pontiac]].
Pontiac was named by [[Gene Ward Smith]] in 2004<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10729.html Yahoo! Tuning Group | Beep, orwell, and schismic]</ref>. For technical data see [[Schismatic family#Pontiac]].
 
__TOC__
{{Clear}}
== Interval chain ==
== Interval chain ==
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"

Latest revision as of 09:58, 9 February 2026

Pontiac
Subgroups 2.3.5.7
Comma basis 4375/4374, 32805/32768
Reduced mapping ⟨1; 1 -8 39]
ET join 53 & 171
Generators (CWE) ~3/2 = 701.758 ¢
MOS scales 12L 17s, 12L 29s, 12L 41s, 53L 12s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 9-odd-limit: 0.401 ¢;
7-limit 81-odd-limit: 0.884 ¢
Target scale size 9-odd-limit: 53 notes;
7-limit 81-odd-limit: 118 notes

Pontiac 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. C–F♭; a comma-flat major third), and the new mapping specific to pontiac is that 7/4 is mapped to the quintuple-augmented third (e.g. C–Exx#; a three-comma-sharp major sixth). This makes pontiac a ragismic temperament. An excellent tuning for pontiac is 171edo, with a perfect fifth generator 100\171, and mos scales of size 12, 17, 29, 41, 53, 65, and 118 are available.

Immediate 11-limit extensions include helenoid (53 & 65), mapping 11/8 to -30 fifths, ponta (171 & 224), mapping 11/8 to -83 fifths, and pontic (118 & 171), mapping 11/8 to +88 fifths.

Pontiac was named by Gene Ward Smith in 2004[1]. For technical data see Schismatic family#Pontiac.

Interval chain

# 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

Notation

Like in schismic, it is recommended to adopt an additional module of accidentals such as arrows to represent the comma step.

However, that which is considered sufficient to notate garibaldi may not be sufficient for pontiac when it comes to septimal and undecimal harmony, as 7/4 is a triple-up major sixth (C–^3A), which is still a lot of stacks of bending. The interval is often notated as a down-minor seventh such as in FJS and HEJI. Combination of these reasons suggests that another set of accidentals to represent 64/63, the septimal comma, or 5120/5103, the amount by which the septimal comma exceeds the syntonic comma, may be desired. Ponta, one notable extension to the 11-limit, identifies the undecimal quartertone of 33/32 by a stack of two septimal commas, and can benefit considerably from this new set of accidentals.

Tuning spectra

Helenoid

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		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

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		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

Eigenmonzo
(Unchanged-interval) 		Generator
(¢) 		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

Notes

  1. ↑ Yahoo! Tuning Group | Beep, orwell, and schismic
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