Syntonic–chromatic equivalence continuum: Difference between revisions

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The '''~~syntonic-chromatic~~ equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[7edo]].

The '''syntonic–chromatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[7edo]].

All temperaments in the continuum satisfy {{nowrap|(81/80)''n'' ~ 2187/2048}}. 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 7edo 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 5.2861…, and temperaments near this tend to be the most accurate ones.

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If we let {{nowrap|''k'' {{=}} ''n'' − 2}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} 2}}, {{nowrap|''k'' {{=}} −1}} means {{nowrap|''n'' {{=}} 1}}, etc. then the continuum corresponds to {{nowrap|(81/80)''k'' {{=}} 25/24}}. Some prefer this way of conceptualising it because:

* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "~~syntonic-chromatic~~ equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)0 ~ 1/1 ~ 25/24}}.

* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic–chromatic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)0 ~ 1/1 ~ 25/24}}.

* {{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 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity {{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.

* 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum.

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-The '''syntonic-chromatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[7edo]].
+The '''syntonic–chromatic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[7edo]].
 All temperaments in the continuum satisfy {{nowrap|(81/80)<sup>''n''</sup> ~ 2187/2048}}. 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 7edo 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 5.2861…, and temperaments near this tend to be the most accurate ones.
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 If we let {{nowrap|''k'' {{=}} ''n'' &minus; 2}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} 2}}, {{nowrap|''k'' {{=}} &minus;1}} means {{nowrap|''n'' {{=}} 1}}, etc. then the continuum corresponds to {{nowrap|(81/80)<sup>''k''</sup> {{=}} 25/24}}. Some prefer this way of conceptualising it because:
-* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic-chromatic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 25/24}}.
+* 25/24 is the classic chromatic semitone, notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could also logically be termed the "syntonic–chromatic equivalence continuum". This means that at {{nowrap|''k'' {{=}} 0}}, 5/4 and 6/5 are mapped to the same interval while 81/80 becomes independent of 25/24 (meaning 81/80 may or may not be tempered) because the relation becomes {{nowrap|(81/80)<sup>0</sup> ~ 1/1 ~ 25/24}}.
 * {{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 25/24 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be gravity {{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.
 * 25/24 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at unsigned infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum.