Schismic–Mercator equivalence continuum: Difference between revisions
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All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[53edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). | All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo|-84 53}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[53edo]] (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them). | ||
For a similar but | For a similar but perhaps more intuitive and practical concept, see [[Syntonic-chromatic equivalence continuum]]. | ||
{| class="wikitable center-1 center-2" | |||
|+ Temperaments in the continuum | |||
|- | |||
! rowspan="2" | ''n'' | |||
! rowspan="2" | Temperament | |||
! colspan="2" | Comma | |||
|- | |||
! Ratio | |||
! Monzo | |||
|- | |||
| 0 | |||
| [[Mercator's comma|Mercator]] | |||
| | |||
| {{monzo|-84 53}} | |||
|- | |||
| 1 | |||
| Counterschismic | |||
| | |||
| {{monzo|-69 45 -1}} | |||
|- | |||
| 2 | |||
| [[Very high accuracy temperaments#Monzismic|Monzismic]] | |||
| | |||
| [[Monzisma|{{monzo|54 -37 2}}]] | |||
|- | |||
| 3 | |||
| [[Tricot]] | |||
| | |||
| {{monzo| 39 -29 3}} | |||
|- | |||
| 4 | |||
| [[Vulture]] | |||
| | |||
| {{monzo| 24 -21 4 }} | |||
|- | |||
| 5 | |||
| [[Amity]] | |||
| [[1600000/1594323]] | |||
| {{monzo| 9 -13 5 }} | |||
|- | |||
| 6 | |||
| [[Kleismic]] | |||
| [[15625/15552]] | |||
| {{monzo|-6 -5 6}} | |||
|- | |||
| 7 | |||
| [[Orson]] | |||
| [[Semicomma|2109375/2097152]] | |||
| {{monzo|-21 3 7 }} | |||
|- | |||
| … | |||
| … | |||
| … | |||
| … | |||
|- | |||
| ∞ | |||
| [[Schismic]] | |||
| [[32805/32768]] | |||
| {{monzo| -15 8 1}} | |||
|} | |||
== Counterschismic == | == Counterschismic == | ||
Revision as of 05:20, 1 March 2021
The syntonic-chromatic equivalence continuum is a continuum of temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩).
All temperaments in the continuum satisfy (32805/32768)n ~ [-84 53⟩. Varying n results in different temperaments listed in the table below. It converges to schismic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 53edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them).
For a similar but perhaps more intuitive and practical concept, see Syntonic-chromatic equivalence continuum.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 0 | Mercator | [-84 53⟩ | |
| 1 | Counterschismic | [-69 45 -1⟩ | |
| 2 | Monzismic | [54 -37 2⟩ | |
| 3 | Tricot | [39 -29 3⟩ | |
| 4 | Vulture | [24 -21 4⟩ | |
| 5 | Amity | 1600000/1594323 | [9 -13 5⟩ |
| 6 | Kleismic | 15625/15552 | [-6 -5 6⟩ |
| 7 | Orson | 2109375/2097152 | [-21 3 7⟩ |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Counterschismic
Comma: [-69 45 -1⟩
Map: [<1 2 21|, <0 -1 -45|]
Wedgie: <<1 45 69||
POTE generator: ~3/2 = 701.9175
EDOs: 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056
Badness: 0.09123