Schismic–Mercator equivalence continuum: Difference between revisions
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| [[Very high accuracy temperaments#Monzismic|Monzismic]] | | [[Very high accuracy temperaments #Monzismic|Monzismic]] | ||
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| [[Monzisma|{{monzo|54 -37 2}}]] | | [[Monzisma|{{monzo|54 -37 2}}]] | ||
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Examples of temperaments with fractional values of ''n'': | Examples of temperaments with fractional values of ''n'': | ||
*3684 & 11105 (''n'' = 11/6 = 1.8{{overline|3}}) | * 3684 & 11105 (''n'' = 11/6 = 1.8{{overline|3}}) | ||
== Counterschismic == | == Counterschismic == | ||
{{see also| Counterschisma }} | |||
Comma: {{monzo|-69 45 -1}} | Comma list: {{monzo| -69 45 -1 }} | ||
Mapping: [{{val| 1 2 21 }}, {{val| 0 -1 -45| }}] | |||
Wedgie: | Wedgie: {{wedgie| 1 45 69 }} | ||
POTE generator: ~3/2 = 701.9175 | POTE generator: ~3/2 = 701.9175 | ||
{{Val list|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}} | |||
Badness: 0.09123 | Badness: 0.09123 | ||
== 3684 & 11105 == | == 3684 & 11105 == | ||
Comma: {{monzo|-339 230 -11}} | Comma list: {{monzo| -339 230 -11 }} | ||
Map: {{val|1 2 11}}, {{val|0 -11 -230}} | Map: [{{val|1 2 11}}, {{val|0 -11 -230}}] | ||
Wedgie: {{wedgie|11 230 339}} | Wedgie: {{wedgie|11 230 339}} | ||
| Line 92: | Line 93: | ||
POTE generator: 45.2769 | POTE generator: 45.2769 | ||
{{Val list|legend=1| 53, 3684, 11105 }} | |||
== Unnamed temperament (''n'' = 13/7) == | == Unnamed temperament (''n'' = 13/7) == | ||
Comma: {{monzo|393 -267 13}} | Comma list: {{monzo| 393 -267 13 }} | ||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:Temperament]] | [[Category:Temperament]] | ||
[[Category:Equivalence continua]] | [[Category:Equivalence continua]] | ||
Revision as of 06:39, 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). The just value of n is approximately 1.8503390493..., and temperaments having n near this value tend to be the most accurate ones.
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⟩ |
Examples of temperaments with fractional values of n:
- 3684 & 11105 (n = 11/6 = 1.83)
Counterschismic
Comma list: [-69 45 -1⟩
Mapping: [⟨1 2 21], ⟨0 -1 -45]]
Wedgie: ⟨⟨1 45 69]]
POTE generator: ~3/2 = 701.9175
Badness: 0.09123
3684 & 11105
Comma list: [-339 230 -11⟩
Map: [⟨1 2 11], ⟨0 -11 -230]]
Wedgie: ⟨⟨11 230 339]]
POTE generator: 45.2769
Unnamed temperament (n = 13/7)
Comma list: [393 -267 13⟩