Syntonic–diatonic equivalence continuum: Difference between revisions

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The '''syntonic-diatonic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]].

All temperaments in the continuum satisfy (81/80)''n'' ~ 256/243. 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 [[5edo]] (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 4.1952…, and temperaments near this tend to be the most accurate ones.

All temperaments in the continuum satisfy (81/80)''n'' ~ 256/243. 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 [[5edo]] 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 4.1952…, and temperaments near this tend to be the most accurate ones.

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)''k'' = 16/15, which might be a preferred way of conceptualising it because:

256/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In 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)''k'' = 16/15, which might be a preferred way of conceptualising it because:

* 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.

* 16/15 is the classic 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.

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 The '''syntonic-diatonic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]].
-All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 256/243. 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 [[5edo]] (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 4.1952…, and temperaments near this tend to be the most accurate ones.
+All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 256/243. 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 [[5edo]] 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 4.1952…, and temperaments near this tend to be the most accurate ones.
-/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:
+/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In 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:
-* 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.
+* 16/15 is the classic 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)<sup>0</sup> ~ 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.