Schismic–Mercator equivalence continuum: Difference between revisions
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The '''schismic-Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments. | The '''schismic-Mercator equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with [[Mercator's comma|Mercator's comma ({{monzo|-84 53}})]]. This continuum is theoretically interesting in that these are all 5-limit microtemperaments. | ||
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). The just value of ''n'' is approximately 1. | 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). 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]]. | For a similar but perhaps more intuitive and practical concept, see [[Syntonic-chromatic equivalence continuum]]. | ||
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Examples of temperaments with fractional values of ''n'': | Examples of temperaments with fractional values of ''n'': | ||
* [[Quartonic]] (''n'' = 5.5) | |||
* [[Ditonic]] (''n'' = 6.5) | |||
* 53 & 3684 (''n'' = 11/6 = 1.8{{overline|3}}) | * 53 & 3684 (''n'' = 11/6 = 1.8{{overline|3}}) | ||
* 53 & 4190 (''n'' = 13/7 = 1.{{overline|857142}}) | * 53 & 4190 (''n'' = 13/7 = 1.{{overline|857142}}) | ||
== Mercator == | == Mercator == | ||
{{ | {{See also| Mercator's comma }} ''and [[Mercator family]]'' | ||
Comma list: {{monzo| -84 53 }} | Comma list: {{monzo| -84 53 }} | ||
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Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}] | Mapping: [{{val| 53 84 123 }}, {{val| 0 0 1 }}] | ||
{{Multival|legend=1| 0 53 84 }} | |||
{{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }} | {{Val list|legend=1| 53, 477, 530, 583, 636, 689, 742, 795, 848, 901, 1749, 2650 }} | ||
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== Counterschismic == | == Counterschismic == | ||
{{ | {{See also| Counterschisma }} | ||
Counterschismic is much like [[schismic]], but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in [[53edo]], of course. | Counterschismic is much like [[schismic]], but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in [[53edo]], of course. | ||
Revision as of 19:50, 22 February 2022
The schismic-Mercator equivalence continuum is a continuum of 5-limit temperaments which equate a number of schismas (32805/32768) with Mercator's comma ([-84 53⟩). This continuum is theoretically interesting in that these are all 5-limit microtemperaments.
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⟩ |
| 8 | Submajor | [-36 11 8⟩ | |
| 9 | Untriton | [-51 19 9⟩ | |
| … | … | … | … |
| ∞ | Schismic | 32805/32768 | [-15 8 1⟩ |
Examples of temperaments with fractional values of n:
Mercator
- and Mercator family
Comma list: [-84 53⟩
POTE generator: ~5/4 = 386.264
Mapping: [⟨53 84 123], ⟨0 0 1]]
Wedgie: ⟨⟨ 0 53 84 ]]
Badness: 0.2843
Counterschismic
Counterschismic is much like schismic, but the harmonic 5 is located at +45 fifths instead of schismic's -8. They unite in 53edo, of course.
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
53 & 3684
Comma list: [-339 230 -11⟩
POTE generator: ~10737418240/10460353203 = 45.2769
Mapping: [⟨1 2 11], ⟨0 -11 -230]]
Wedgie: ⟨⟨11 230 339]]
Badness: 0.276036
53 & 4190
Comma list: [393 -267 13⟩
POTE generator: ~[-60 41 -2⟩ = 407.5419
Mapping: [⟨1 6 93], ⟨0 -13 -267]]
Wedgie: ⟨⟨13 267 393]]
Badness: 0.173433