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)]].

The '''syntonic-diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].

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 [[tempering out|tempers out]] both commas and thus tempers out 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 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 generator chain.

256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. For example:

* Superpyth (''n'' = 1) is generated by a fifth;

* Immunity (''n'' = 2) splits its twelfth in two;

* Rodan (''n'' = 3) splits its fifth in three;

* Etc.

At ''n'' = 5, the corresponding temperament splits the ''octave'' into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again.

~~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. Some prefer this way of conceptualising it because:

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. Some prefer this way of conceptualising it because:

* 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.)

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We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''superpyth-diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130…

We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''superpyth-diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130… The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.

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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)]].
+The '''syntonic-diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed by [[5edo]].
-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 [[tempering out|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is 4.1952…, and temperaments near this tend to be the most accurate ones.
-/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 generator chain.
+/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. For example:
+* Superpyth (''n'' = 1) is generated by a fifth;
+* Immunity (''n'' = 2) splits its twelfth in two;
+* Rodan (''n'' = 3) splits its fifth in three;
+* Etc.
+At ''n'' = 5, the corresponding temperament splits the ''octave'' into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again.
-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. Some prefer this way of conceptualising it because:
+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. Some prefer this way of conceptualising it because:
 * 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.)
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 |}
-We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''superpyth-diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130…
+We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''superpyth-diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130… The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.
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