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'''Pontiac temperament''' is a 7-limit (and higher) temperament of the [[Schismatic family #Pontiac|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 garibaldi is that [[7/4]] is mapped to the quintuple augmented third (e.g. A-Cxxx). This makes pontiac a [[Ragismic microtemperaments|ragismic temperament]].  
{{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.


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.  
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.  


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"
! rowspan="3" | Fifth <br>generator
! rowspan="3" | #
! rowspan="3" | Cents*
! rowspan="3" | Cents*
! colspan="4" | Approximate Ratios
! colspan="4" | Approximate ratios
|-
|-
! rowspan="2" | 7-limit
! rowspan="2" | 7-limit
! colspan="3" | 17-limit Extension
! colspan="3" | 17-limit extension
|-
|-
! Helenoid
! Helenoid
Line 900: Line 917:
<nowiki>*</nowiki> in 7-limit POTE tuning
<nowiki>*</nowiki> in 7-limit POTE tuning


== Spectrum of pontiac tunings by eigenmonzos ==
== Notation ==
=== Helenoid mapping ===
Like in [[schismic]], it is recommended to adopt an additional module of accidentals such as arrows to represent the comma step.
Gencom: [2 4/3; 352/351 385/384 625/624 729/728]


Gencom map: [{{val|1 2 -1 19 -9 -10}}, {{val|0 -1 8 -39 30 33}}]
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.


{| class="wikitable center-1 right-2"
== Tuning spectra ==
=== Helenoid ===
{| class="wikitable center-1 center-2"
|-
|-
! Eigenmonzo
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Fifth
! Generator<br>(¢)
! Comments
! Comments
|-
|-
| 11/10
| 11/10
| 701.591
| 701.5907
|  
|  
|-
|-
| 15/11
| 15/11
| 701.607
| 701.6066
|  
|  
|-
|-
| 11/8
| 11/8
| 701.623
| 701.6227
|  
|  
|-
|-
| 12/11
| 12/11
| 701.633
| 701.6335
|  
|  
|-
|-
| 11/9
| 11/9
| 701.644
| 701.6435
|  
|  
|-
|-
| 16/15
| 16/15
| 701.676
| 701.6759
|  
|  
|-
|-
| 14/11
| 14/11
| 701.703
| 701.7030
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
| 22/17
| 701.7071
|
|-
|-
| 5/4
| 5/4
| 701.711
| 701.7108
|
|-
| 17/14
| 701.7205
|  
|  
|-
|-
| 6/5
| 6/5
| 701.738
| 701.7379
| 5-odd-limit minimax
| 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
| 15/14
Line 971: Line 1,017:
| 701.7648
| 701.7648
|  
|  
|-
| 17/13
| 701.7680
| 17-odd-limit minimax
|-
|-
| 14/13
| 14/13
| 701.782
| 701.7819
| 13 and 15-odd-limit minimax
| 13 and 15-odd-limit minimax
|-
|-
| 16/13
| 16/13
| 701.802
| 701.8022
|  
|  
|-
|-
| 13/12
| 13/12
| 701.807
| 701.8067
|  
|  
|-
|-
| 18/13
| 18/13
| 701.811
| 701.8109
|  
|  
|-
|-
| 13/10
| 13/10
| 701.831
| 701.8314
|  
|  
|-
|-
| 15/13
| 15/13
| 701.836
| 701.8362
|  
|  
|-
|-
| 4/3
| 4/3
| 701.955
| 701.9550
|  
|  
|-
|-
| 13/11
| 13/11
| 703.597
| 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:Ragismic microtemperaments]]
[[Category:Schismatic family]]
[[Category:Horwell temperaments]]
[[Category:Index of temperaments]]
Retrieved from "https://en.xen.wiki/w/Pontiac"