Pontiac: Difference between revisions

| Comma basis = [[32805/32768]], [[4375/4374]]

| Edo join 1 = 53 | Edo join 2 = 65

'''Pontiac''' is a 7-limit (and higher) [[temperament]] of the [[Schismatic family #Pontiac|schismatic family]]. It is an extension of [[Helmholtz (temperament)|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♭), 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]]es 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.  

! colspan="4" | Approximate Ratios

! colspan="3" | 17-limit Extension

Using pontiac can be a challenge because it defies the tradition of tertian harmony in [[circle-of-fifths notation]]. The just major triad on C is C-Fb-G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C-vE-G.  

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.