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#REDIRECT [[Schismatic family #Pontiac]]
{{Infobox regtemp
| Title = Pontiac
| Subgroups = 2.3.5.7
| Comma basis = [[4375/4374]], [[32805/32768]]
| Edo join 1 = 53 | Edo join 2 = 171
| Mapping = 1; 1 -8 39
| Generators = 3/2
| Generators tuning = 701.758
| Optimization method = CWE
| Pergen = (P8, P5)
| MOS scales = [[12L 17s]], [[12L 29s]], [[12L 41s]], [[53L 12s]]
| Odd limit 1 = 9 | Mistuning 1 = 0.401 | Complexity 1 = 53
| 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.


[[Category:Temperament]]
Immediate 11-limit extensions include helenoid ({{nowrap| 53 & 65 }}), mapping 11/8 to -30 fifths, ''ponta'' ({{nowrap| 171 & 224 }}), mapping 11/8 to -83 fifths, and ''pontic'' ({{nowrap| 118 & 171 }}), mapping 11/8 to +88 fifths.
[[Category:Schismatic]]
 
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 ==
{| class="wikitable center-1 right-2"
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="4" | Approximate ratios
|-
! rowspan="2" | 7-limit
! colspan="3" | 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, <br>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
|
|
|
|}
<nowiki>*</nowiki> 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 #Notation|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–^<sup>3</sup>A), 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 ===
{| class="wikitable center-1 center-2"
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Generator<br>(¢)
! 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 ===
{| class="wikitable center-1 center-2"
|-
! Eigenmonzo<br>(Unchanged-interval)
! Generator<br>(¢)
! 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 ===
{| class="wikitable center-1 center-2"
|-
! Eigenmonzo<br>(Unchanged-interval)
! Generator<br>(¢)
! 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 ==
 
[[Category:Pontiac| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Schismatic family]]
[[Category:Ragismic microtemperaments]]
[[Category:Horwell temperaments]]
Retrieved from "https://en.xen.wiki/w/Pontiac"