Fifth-chroma temperaments: Difference between revisions

Line 18:

* 77 & 80 & 87 & 94 (omitting 84e)

* 80 & 84e & 87 & 94 (omitting 77)

That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]*, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].

That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]**, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].

<nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].

:: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].

:: <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.

:: <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.

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

@@ Line 18: / Line 18: @@
 * 77 & 80 & 87 & 94 (omitting 84e)
 * 80 & 84e & 87 & 94 (omitting 77)
-That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]*, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].
+That is, it represents the shared harmonic and spacing logic between at least* [[77edo]], [[80edo]], [[84edo]]**, [[87edo]] and [[94edo]], so that its join can be formed by any four of these five edos ''except'' for the one formed by omitting 80edo, because 77 & 84e & 87 & 94 reduces to something simpler (77 & 87 & 94), corresponding to 80edo being the only one that doesn't temper out [[385/384]].
-<nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].
+:: <nowiki>*</nowiki> There is a more idiosyncratic edo that fits into this scheme as well: [[70edo]] also obeys the intended 13-limit mapping using the patent val, but it's more dubious as a self-contained system due to being better conceptualized as a subset of [[140edo]], so that it's more of a dual-5's dual-7's 17-limit system with high-limit capabilities inherited from 140edo, which also performs well in a wide range of large odd-limits according to the same metric discussed in the section [[#Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness]].
-:: <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.
+:: <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.
 == Broad high-limit tuning results based on a hand-optimized measure of odd-limit approximation faithfulness ==