Pontiac: Difference between revisions
53 & 171 is a better edo join. - gencom (irrelevant). Misc. cleanup and style improvements |
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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]] | ||
| | | 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) | ||
| 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 = | | 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♭), and the new mapping specific to pontiac is that [[7/4]] is mapped to the quintuple-augmented third (e.g. C–Exx#). This makes pontiac a [[ragismic microtemperaments|ragismic temperament]]. An excellent tuning for pontiac is [[171edo]], 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. | ||
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. | 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]]. | 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]]. | ||
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== Interval chain == | == Interval chain == | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
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== Notation == | == 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. | 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 == | == Tuning spectra == | ||