Fifth-chroma temperaments: Difference between revisions

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:: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.

If we also consider the aforediscussed idiosyncratic tuning 70edo, then ~~some~~ more edo joins are possible:

If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:

* 70 & 77 & 80 & 84e

* 70 & 77 & 80 & 94

* 70 & 77 & 84e & 87

* 70 & 77 & 84e & 94

* 70 & 77 & 87 & 94

* 70 & 80 & 84e & 87

* 70 & 80 & 87 & 94

* 70 & 84e & 87 & 94

Note that 70 & 77 & 80 & 87 simplifies to 77 & 80 & 87, which is notable as being the temperament that imposes another restriction on the representation of the 13-limit, by assuming finding exactly one interval between ~5/4 and ~14/11, distinct from both, by tempering (56/55)/(144/143)<sup>2</sup> = 13013/12960, where 56/55 = (14/11)/(5/4).

Another omission is 70 & 80 & 84e & 94 which reduces by tempering out [[540/539|S12/S14]]; though the significance of this is less clear, it can be observed that it is similar to the other omission by having the latter two edos being an offset of 10 from the former two edos.

=== Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===

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 :: <nowiki>**</nowiki> [[84edo]] is peculiar because though using the flat 11 makes some sense in lower limits, in higher limits using the patent val tends to be more performant, so that though one is likely mapping 11/9 as the subneutral third, this mapping may sometimes be inconsistent with the mapping of 11 used when building chords, corresponding to multiple possible tunings of ~8:9:11; the sharp tuning uses 28edo's sharp ~9/8 and ~11/8 while the flat tuning uses the flat ~11/8 and 12edo's major second; interestingly, both of these map 11/9 to the subneutral third, with the trick being that the 28edo rendition uses a sharp ~9/8 that can't be achieved from the 12edo circle of fifths.
-If we also consider the aforediscussed idiosyncratic tuning 70edo, then some more edo joins are possible:
+If we also consider the aforediscussed idiosyncratic tuning 70edo, then many more edo joins are possible for describing this rank 4 (= 4-dimensional) 13-limit temperament:
 * 70 & 77 & 80 & 84e
 * 70 & 77 & 80 & 94
+* 70 & 77 & 84e & 87
+* 70 & 77 & 84e & 94
 * 70 & 77 & 87 & 94
+* 70 & 80 & 84e & 87
 * 70 & 80 & 87 & 94
+* 70 & 84e & 87 & 94
 Note that 70 & 77 & 80 & 87 simplifies to 77 & 80 & 87, which is notable as being the temperament that imposes another restriction on the representation of the 13-limit, by assuming finding exactly one interval between ~5/4 and ~14/11, distinct from both, by tempering (56/55)/(144/143)<sup>2</sup> = 13013/12960, where 56/55 = (14/11)/(5/4).
+Another omission is 70 & 80 & 84e & 94 which reduces by tempering out [[540/539|S12/S14]]; though the significance of this is less clear, it can be observed that it is similar to the other omission by having the latter two edos being an offset of 10 from the former two edos.
 === Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ===