Schismic–countercommatic equivalence continuum: Difference between revisions
ArrowHead294 (talk | contribs) m ArrowHead294 moved page Schismic-countercommatic equivalence continuum to Schismic–countercommatic equivalence continuum: The dash in titles like these should be an en dash, not a hyphen-minus, since "schismic" does not modify "countercommatic" |
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The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]]. | The '''schismic–countercommatic equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[32805/32768|schismas (32805/32768)]] with the [[41-comma|Pythagorean countercomma ({{monzo| 65 -41 }})]]. This continuum is theoretically interesting in that these are all [[5-limit]] [[microtemperament]]s [[support]]ed by [[41edo]]. | ||
All temperaments in the continuum satisfy (32805/32768)<sup>''n''</sup> ~ {{monzo| 65 -41 }}. 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 41edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap|(32805/32768)<sup>''n''</sup> ~ {{monzo| 65 -41 }}}}. 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 41edo due to it being the unique equal temperament that [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 10.1575233481…, and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example: | The Pythagorean countercomma is the characteristic 3-limit comma tempered out in 41edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of steps to obtain the interval class of [[3/1|harmonic 3]] in the generator chain. For example: | ||
* [[Cotoneum]] (''n'' = 1) is generated by a fifth; | * [[Cotoneum]] ({{nowrap|''n'' {{=}} 1}}) is generated by a fifth; | ||
* [[Newt]] (''n'' = 2) splits its fifth in two; | * [[Newt]] ({{nowrap|''n'' {{=}} 2}}) splits its fifth in two; | ||
* Etc. | * Etc. | ||
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{| class="wikitable center-1" | {| class="wikitable center-1" | ||
|+ Temperaments of integer ''n'' | |+ style="font-size: 105%;" | Temperaments of integer ''n'' | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
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Examples of temperaments with fractional values of ''n'': | Examples of temperaments with fractional values of ''n'': | ||
* [[Marvel temperaments #Septimin|Septimin]] (''n'' = | * [[Marvel temperaments #Septimin|Septimin]] ({{nowrap|''n'' {{=}} −{{sfrac|11|2}}}}) | ||
* [[Shibboleth family|Shibboleth]] (''n'' = | * [[Shibboleth family|Shibboleth]] ({{nowrap|''n'' {{=}} −{{sfrac|9|2}}}}) | ||
* [[Mirkwai clan #Pluto|Pluto]] (''n'' = | * [[Mirkwai clan #Pluto|Pluto]] ({{nowrap|''n'' {{=}} −{{sfrac|7|2}}}}) | ||
* 3737 & 5585 (''n'' = 31 | * 3737 & 5585 ({{nowrap|''n'' {{=}} {{sfrac|31|3}} {{=}} 10.{{overline|3}}}}) | ||
* 1277 & 2513 (''n'' = 21 | * 1277 & 2513 ({{nowrap|''n'' {{=}} −{{sfrac|21|2}}}}) | ||
== Rodan (5-limit) == | == Rodan (5-limit) == | ||