Syntonic–chromatic equivalence continuum: Difference between revisions

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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]]. The just value of ''n'' is 5.2861…, and temperaments near this tend to be the most accurate ones.

~~Note~~ that if you 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, which might be a preferred way of conceptualising it because:

2187/2048 has the advantage of being the characteristic 3-limit comma tempered out in 7edo. For each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if you 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, which might be a preferred way of conceptualising it because:

* 25/24 is the chromatic semitone, highly notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could ''also'' be termed the "syntonic-chromatic equivalence continuum".

* k=1 and upwards (up to a point) represent temperaments of 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) or at most ~~Absurdity~~ (k=5), with the only exception being ~~Meantone~~ (n = k = (unsigned) infinity) which is in a sense a simplicity that is the reverse of ~~Dicot~~'s as in ~~Meantone~~, 81/80 is tempered and 25/24 is no longer dependent on 81/80.

* ''k'' = 1 and upwards (up to a point) represent temperaments of 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) or at most absurdity (''k'' = 5), with the only exception being meantone (''n'' = ''k'' = (unsigned) infinity) which is in a sense a simplicity that is the reverse of dicot's as in meantone, 81/80 is tempered and 25/24 is no longer dependent on 81/80{{clarify}}.

* 25/24 is the simplest ratio to be tempered in the continuum.

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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]]. The just value of ''n'' is 5.2861…, and temperaments near this tend to be the most accurate ones.
-Note that if you 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, which might be a preferred way of conceptualising it because:
+/2048 has the advantage of being the characteristic 3-limit comma tempered out in 7edo. For each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if you 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, which might be a preferred way of conceptualising it because:
 * 25/24 is the chromatic semitone, highly notable in the 5-limit as the difference between 5/4 and 6/5, so this shifted continuum could ''also'' be termed the "syntonic-chromatic equivalence continuum".
-* k=1 and upwards (up to a point) represent temperaments of 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) or at most Absurdity (k=5), with the only exception being Meantone (n = k = (unsigned) infinity) which is in a sense a simplicity that is the reverse of Dicot's as in Meantone, 81/80 is tempered and 25/24 is no longer dependent on 81/80.
+* ''k'' = 1 and upwards (up to a point) represent temperaments of 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) or at most absurdity (''k'' = 5), with the only exception being meantone (''n'' = ''k'' = (unsigned) infinity) which is in a sense a simplicity that is the reverse of dicot's as in meantone, 81/80 is tempered and 25/24 is no longer dependent on 81/80{{clarify}}.
 * 25/24 is the simplest ratio to be tempered in the continuum.