Pontiac: Difference between revisions

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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 pontiac 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&~~amp;~~65), mapping 11/8 to -30 fifths, ''ponta'' (53&~~amp;171~~), mapping 11/8 to -83 fifths, and ''pontic'' (118&~~amp;~~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__

== Interval chain ==

{| class="wikitable center-1 right-2"

! rowspan="3" | ~~Fifth generator~~

! rowspan="3" | #

! rowspan="3" | Cents*

! colspan="4" | Approximate ~~Ratios~~

! colspan="4" | Approximate ratios

|-

! rowspan="2" | 7-limit

! colspan="3" | 17-limit ~~Extension~~

! colspan="3" | 17-limit extension

|-

! Helenoid

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|}

<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–^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 ===

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

{| class="wikitable center-1 center-2"

~~Gencom mapping: [{{val|1 2 -1 19 -9 -10 29}}, {{val|0 -1 8 -39 30 33 -60}}]~~

{| class="wikitable center-~~all~~"

|-

! [[~~eigenmonzo~~|~~eigenmonzo~~ (~~unchanged~~-interval]])

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval)]]

! ~~fifth~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 11/10

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=== Ponta ===

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

{| class="wikitable center-1 center-2"

~~Gencom mapping: [{{val|1 2 -1 19 -31 -10 29}}, {{val|0 -1 8 -39 83 33 -60}}]~~

{| class="wikitable center-~~all~~"

|-

! ~~eigenmonzo~~ (~~unchanged~~-interval)

! Eigenmonzo (Unchanged-interval)

! ~~fifth~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 16/15

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=== Pontic ===

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

{| class="wikitable center-1 center-2"

~~Gencom mapping: [{{val|1 2 -1 19 40 -10 29}}, {{val|0 -1 8 -39 -88 33 -60}}]~~

{| class="wikitable center-~~all~~"

|-

! ~~eigenmonzo~~ (~~unchanged~~-interval)

! Eigenmonzo (Unchanged-interval)

! ~~fifth~~ (¢)

! Generator (¢)

! ~~comments~~

! Comments

|-

| 22/17

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|}

== Notes ==

[[Category:Pontiac| ]]

[[Category:Rank-2 temperaments]]

[[Category:Schismatic family]]

[[Category:Ragismic microtemperaments]]

[[Category:~~Schismatic family]]~~

[[Category:Horwell temperaments]]

~~[[Category:Temperaments~~]]

@@ Line 1: / Line 1: @@
-'''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 pontiac 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&nbsp;17s]], [[12L&nbsp;29s]], [[12L&nbsp;41s]], [[53L&nbsp;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&amp;65), mapping 11/8 to -30 fifths, ''ponta'' (53&amp;171), mapping 11/8 to -83 fifths, and ''pontic'' (118&amp;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 ==
 {| class="wikitable center-1 right-2"
-! rowspan="3" | Fifth <br>generator
+! rowspan="3" | #
 ! rowspan="3" | Cents*
-! colspan="4" | Approximate Ratios
+! colspan="4" | Approximate ratios
 |-
 ! rowspan="2" | 7-limit
-! colspan="3" | 17-limit Extension
+! colspan="3" | 17-limit extension
 |-
 ! Helenoid
@@ Line 899: / Line 916: @@
 |}
 <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 ===
-Gencom: [2 4/3; 352/351 385/384 561/560 625/624 729/728]
+{| class="wikitable center-1 center-2"
-Gencom mapping: [{{val|1 2 -1 19 -9 -10 29}}, {{val|0 -1 8 -39 30 33 -60}}]
-{| class="wikitable center-all"
 |-
-! [[eigenmonzo|eigenmonzo<br>(unchanged-interval]])
+! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
-! fifth<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 | 11/10
@@ Line 1,038: / Line 1,056: @@
 === Ponta ===
-Gencom: [2 4/3; 375/374 540/539 625/624 729/728 2200/2197]
+{| class="wikitable center-1 center-2"
-Gencom mapping: [{{val|1 2 -1 19 -31 -10 29}}, {{val|0 -1 8 -39 83 33 -60}}]
-{| class="wikitable center-all"
 |-
-! eigenmonzo<br>(unchanged-interval)
+! Eigenmonzo<br>(Unchanged-interval)
-! fifth<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 | 16/15
@@ Line 1,174: / Line 1,188: @@
 === Pontic ===
-Gencom: [2 4/3; 441/440 595/594 625/624 729/728 2880/2873]
+{| class="wikitable center-1 center-2"
-Gencom mapping: [{{val|1 2 -1 19 40 -10 29}}, {{val|0 -1 8 -39 -88 33 -60}}]
-{| class="wikitable center-all"
 |-
-! eigenmonzo<br>(unchanged-interval)
+! Eigenmonzo<br>(Unchanged-interval)
-! fifth<br>(¢)
+! Generator<br>(¢)
-! comments
+! Comments
 |-
 | 22/17
@@ Line 1,309: / Line 1,319: @@
 |}
+== Notes ==
+[[Category:Pontiac| ]] <!-- main article -->
+[[Category:Rank-2 temperaments]]
+[[Category:Schismatic family]]
 [[Category:Ragismic microtemperaments]]
-[[Category:Schismatic family]]
+[[Category:Horwell temperaments]]
-[[Category:Temperaments]]