Syntonic–diatonic equivalence continuum: Difference between revisions

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256/243 has the advantage of being the characteristic [[3-limit]] comma tempered out in [[5edo]]. 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 we let ''k'' = ''n'' + 1 (meaning ''n'' = ''k'' - 1) so that ''k'' = 0 means ''n'' = -1, ''k'' = 1 means ''n'' = 0, etc. then the continuum corresponds to (81/80)<sup>''k''</sup> = 16/15, which might be a preferred way of conceptualising it because:

* 25/24 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.

* 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''.

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 /243 has the advantage of being the characteristic [[3-limit]] comma tempered out in [[5edo]]. 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 we let ''k'' = ''n'' + 1 (meaning ''n'' = ''k'' - 1) so that ''k'' = 0 means ''n'' = -1, ''k'' = 1 means ''n'' = 0, etc. then the continuum corresponds to (81/80)<sup>''k''</sup> = 16/15, which might be a preferred way of conceptualising it because:
-* 25/24 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.
+* 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''.