Syntonic–diatonic equivalence continuum: Difference between revisions

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The '''~~syntonic-diatonic~~ equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].

The '''syntonic–diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].

All temperaments in the continuum satisfy {{nowrap|(81/80)''n'' ~ 256/243}}. 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 5edo 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 4.1952…, and temperaments near this tend to be the most accurate ones.

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If we let {{nowrap|''k'' {{=}} ''n'' + 1}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} −1}}, {{nowrap|''k'' {{=}} 1}} means {{nowrap|''n'' {{=}} 0}}, etc. then the continuum corresponds to {{nowrap|(81/80)''k'' {{=}} 16/15}}. Some prefer this way of conceptualising it because:

* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "~~syntonic-diatonic~~ equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)0 ~ 1/1 ~ 16/15}}.

* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic–diatonic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)0 ~ 1/1 ~ 16/15}}.

* {{nowrap|''k'' {{=}} 1}} and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan {{nowrap|(''k'' {{=}} 4)}}, with the only exception being meantone {{nowrap|(''n'' {{=}} ''k'' {{=}} ∞)}}. (Temperaments corresponding to {{nowrap|''k'' {{=}} 0, −1, −2...}} are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)

* 16/15 is the simplest ratio to be tempered in the continuum.

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-The '''syntonic-diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].
+The '''syntonic–diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].
 All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 256/243}}. 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 5edo 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 4.1952…, and temperaments near this tend to be the most accurate ones.
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 If we let {{nowrap|''k'' {{=}} ''n'' + 1}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} &minus;1}}, {{nowrap|''k'' {{=}} 1}} means {{nowrap|''n'' {{=}} 0}}, etc. then the continuum corresponds to {{nowrap|(81/80)<sup>''k''</sup> {{=}} 16/15}}. Some prefer this way of conceptualising it because:
-* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 16/15}}.
+* 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic–diatonic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 16/15}}.
 * {{nowrap|''k'' {{=}} 1}} and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan {{nowrap|(''k'' {{=}} 4)}}, with the only exception being meantone {{nowrap|(''n'' {{=}} ''k'' {{=}} &infin;)}}. (Temperaments corresponding to {{nowrap|''k'' {{=}} 0, &minus;1, &minus;2...}} are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
 * 16/15 is the simplest ratio to be tempered in the continuum.