Syntonic–31 equivalence continuum: Difference between revisions
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The '''syntonic-31 equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with a [[31st-octave temperaments|31-comma ({{monzo| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]]. | The '''syntonic-31 equivalence continuum''' is a continuum of 5-limit temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with a [[31st-octave temperaments|31-comma ({{monzo| -49 31 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments supported by [[31edo]]. | ||
All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ {{monzo|-49 31}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] 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 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones. | All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ {{monzo|-49 31}}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[31edo]] 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 7.46781…, and temperaments having ''n'' near this value tend to be the most accurate ones. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments in the continuum | |+ style="font-size: 105%;" | Temperaments in the continuum | ||
|- | |- | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
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{| class="wikitable" | {| class="wikitable" | ||
|+ Notable temperaments of fractional ''n'' | |+ style="font-size: 105%;" | Notable temperaments of fractional ''n'' | ||
|- | |- | ||
! Temperament !! ''n'' !! Comma | ! Temperament !! ''n'' !! Comma | ||