124edo: Difference between revisions

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== Theory ==

124edo is closely related to [[31edo]], but the [[patent val]]s differ on the [[mapping]] for [[3/1|3]]. The equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) and {{monzo| -6 -24 19 }} in the 5-limit. Using the patent val, it tempers out [[3136/3125]], [[4000/3969]], and 33614/32805 in the 7-limit; [[385/384]], 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; [[196/195]], [[364/363]], 572/567, [[625/624]], and [[1001/1000]] in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its [[9/1|9th harmonic]] is especially accurate as a result, so it can be considered a [[dual-fifth system]], in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 [[subgroup]] (the dual-fifth ~~no-31's~~ [[37-limit]]), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to [[consistency|inconsistencies]], so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.

124edo is closely related to [[31edo]], but the [[patent val]]s differ on the [[mapping]] for [[3/1|3]]. The equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) and {{monzo| -6 -24 19 }} in the 5-limit. Using the patent val, it tempers out [[3136/3125]], [[4000/3969]], and 33614/32805 in the 7-limit; [[385/384]], 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; [[196/195]], [[364/363]], 572/567, [[625/624]], and [[1001/1000]] in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its [[9/1|9th harmonic]] is especially accurate as a result, so it can be considered a [[dual-fifth system]], in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 [[subgroup]] (the dual-fifth [[23-limit]] add-37), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to [[consistency|inconsistencies]], so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.

=== Odd harmonics ===

{{Harmonics in equal|124|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 124edo (continued)}}

=== No-3 approach ===

If prime 3 is ignored, 124edo represents the no-3 28-integer-limit consistently. 124edo is distinctly consistent within the no-3 26-integer-limit.

== Intervals ==

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 == Theory ==
-edo is closely related to [[31edo]], but the [[patent val]]s differ on the [[mapping]] for [[3/1|3]]. The equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) and {{monzo| -6 -24 19 }} in the 5-limit. Using the patent val, it tempers out [[3136/3125]], [[4000/3969]], and 33614/32805 in the 7-limit; [[385/384]], 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; [[196/195]], [[364/363]], 572/567, [[625/624]], and [[1001/1000]] in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its [[9/1|9th harmonic]] is especially accurate as a result, so it can be considered a [[dual-fifth system]], in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 [[subgroup]] (the dual-fifth no-31's [[37-limit]]), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to [[consistency|inconsistencies]], so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.
+edo is closely related to [[31edo]], but the [[patent val]]s differ on the [[mapping]] for [[3/1|3]]. The equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]) and {{monzo| -6 -24 19 }} in the 5-limit. Using the patent val, it tempers out [[3136/3125]], [[4000/3969]], and 33614/32805 in the 7-limit; [[385/384]], 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; [[196/195]], [[364/363]], 572/567, [[625/624]], and [[1001/1000]] in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its [[9/1|9th harmonic]] is especially accurate as a result, so it can be considered a [[dual-fifth system]], in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 [[subgroup]] (the dual-fifth [[23-limit]] add-37), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to [[consistency|inconsistencies]], so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.
 === Odd harmonics ===
 {{Harmonics in equal|124|columns=12}}
 {{Harmonics in equal|124|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 124edo (continued)}}
+=== No-3 approach ===
+If prime 3 is ignored, 124edo represents the no-3 28-integer-limit consistently. 124edo is distinctly consistent within the no-3 26-integer-limit.
 == Intervals ==