Schismic–Mercator equivalence continuum: Difference between revisions
m Cleanup |
53-comma by itself defines a temperament |
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| Line 68: | Line 68: | ||
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}}) | ||
== Mercator == | |||
{{see also| Mercator's comma }} | |||
Comma list: {{monzo| -84 53 }} | |||
POTE genrator: ~5/4 = 386.264 | |||
Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}] | |||
Wedgie: {{wedgie| 0 53 84 }} | |||
{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }} | |||
Badness: 0.2843 | |||
== Counterschismic == | == Counterschismic == | ||
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Comma list: {{monzo| -69 45 -1 }} | Comma list: {{monzo| -69 45 -1 }} | ||
Mapping: [{{val| 1 2 21 }}, {{val| 0 -1 -45 | POTE generator: ~3/2 = 701.9175 | ||
Mapping: [{{val| 1 2 21 }}, {{val| 0 -1 -45 }}] | |||
Wedgie: {{wedgie| 1 45 69 }} | Wedgie: {{wedgie| 1 45 69 }} | ||
{{Val list|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}} | {{Val list|legend=1| 53, 412, 465, 518, 571, 624, 677, 730, 2973, 3703, 4433, 5163, 11056}} | ||
Revision as of 06:49, 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)
Mercator
Comma list: [-84 53⟩
POTE genrator: ~5/4 = 386.264
Mapping: [⟨53 84 123], ⟨0 0 1]]
Wedgie: ⟨⟨0 53 84]]
Badness: 0.2843
Counterschismic
Comma list: [-69 45 -1⟩
POTE generator: ~3/2 = 701.9175
Mapping: [⟨1 2 21], ⟨0 -1 -45]]
Wedgie: ⟨⟨1 45 69]]
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⟩