Syntonic–chromatic equivalence continuum: Difference between revisions

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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:

* 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=0 being n=2 could be more intuitive as when (81/80)^0 = 25/24 is used, 81/80 no longer becomes dependent on 25/24, and 25/24 is tempered (thus ''equating'' 5/4 and 6/5) and so it represents the border at which 5/4 and 6/5 swap places in the circle of fifths.

* 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.

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

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 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:
 * 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=0 being n=2 could be more intuitive as when (81/80)^0 = 25/24 is used, 81/80 no longer becomes dependent on 25/24, and 25/24 is tempered (thus ''equating'' 5/4 and 6/5) and so it represents the border at which 5/4 and 6/5 swap places in the circle of fifths.
 * 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.
 * 25/24 is the simplest ratio to be tempered in the continuum.

Syntonic–chromatic equivalence continuum: Difference between revisions

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