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. | ||