Syntonic–chromatic equivalence continuum: Difference between revisions

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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 [[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.

2187/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]], 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:

* [[Mavila]] (''n'' = 1) is generated by a fifth;

* [[Dicot]] (''n'' = 2) splits its fifth in two;

* [[Porcupine]] (''n'' = 3) splits its fourth in three;

* etc.

At ''n'' = 7, the corresponding temperament splits the ''octave'' into seven instead, as after a stack of seven syntonic commas, both the orders of 3 and 5 are multiples of 7 again.

~~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:

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.

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We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333…

We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333… The [[135/128|mavila comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

{| class="wikitable center-1"

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 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 [[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.
+/2048 is the characteristic [[3-limit]] comma tempered out in [[7edo]], 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:
+* [[Mavila]] (''n'' = 1) is generated by a fifth;
+* [[Dicot]] (''n'' = 2) splits its fifth in two;
+* [[Porcupine]] (''n'' = 3) splits its fourth in three;
+* etc.
+At ''n'' = 7, the corresponding temperament splits the ''octave'' into seven instead, as after a stack of seven syntonic commas, both the orders of 3 and 5 are multiples of 7 again.
-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:
+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.
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 |}
-We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333…
+We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''mavila/pelogic-chromatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.2333… The [[135/128|mavila comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.
 {| class="wikitable center-1"