Syntonic–chromatic equivalence continuum: Difference between revisions

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The '''syntonic-chromatic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]].

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)]].

All temperaments in the continuum satisfy (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 tempers both commas and thus tempers all combinations of them. The just value of ''n'' is 5.2861…, and temperaments near this tend to be the most accurate ones.

2187/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]]. In each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain a harmonic 3 in the generator chain.

2187/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]]. In each case, we notice that ''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.

However, if we let ''k'' = ''n'' - 2 (meaning ''n'' = ''k'' + 2) so that ''k'' = 0 means ''n'' = 2, ''k'' = -1 means ''n'' = 1, etc. then the continuum corresponds to (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 ''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 (81/80)0 ~ 1/1 ~ 25/24.

* ''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 (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)

* ''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 (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = unsigned infinity). Temperaments corresponding to ''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.

* 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 continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[2187/2048|apotome (2187/2048)]].
+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)]].
 All temperaments in the continuum satisfy (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 tempers both commas and thus tempers all combinations of them. The just value of ''n'' is 5.2861…, and temperaments near this tend to be the most accurate ones.
-/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]]. In each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of steps to obtain a harmonic 3 in the generator chain.
+/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]]. In each case, we notice that ''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.
 However, if we let ''k'' = ''n'' - 2 (meaning ''n'' = ''k'' + 2) so that ''k'' = 0 means ''n'' = 2, ''k'' = -1 means ''n'' = 1, etc. then the continuum corresponds to (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 ''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 (81/80)<sup>0</sup> ~ 1/1 ~ 25/24.
-* ''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 (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
+* ''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 (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = unsigned infinity). Temperaments corresponding to ''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.
+* 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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