Syntonic–limmic equivalence continuum: Difference between revisions

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* 16/15 is the diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at ''k'' = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)^0 = 1/1 = 16/15.

* ''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 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (''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.)

* 16/15 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 if we don't count non-integer ''k''.

* 16/15 is the simplest ratio to be tempered in the continuum.

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 * 16/15 is the diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at ''k'' = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)^0 = 1/1 = 16/15.
 * ''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 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (''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.)
-* 16/15 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 if we don't count non-integer ''k''.
+* 16/15 is the simplest ratio to be tempered in the continuum.
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