124edo: Difference between revisions

Line 1:

~~'''124edo''' is the [[~~EDO|~~equal division of the octave]] into~~ 124 ~~parts of 9.6774 cents each. It~~ is closely related to [[31edo]], but the patent vals differ on the mapping for 3. It tempers out 2048/2025 (diaschisma) and 19073486328125/18075490334784 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 9th harmonic is especially accurate as a result, so it can be considered a dual-fifths system, in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 subgroup (AKA 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 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 vals differ on the mapping for 3. It tempers out 2048/2025 (diaschisma) and 19073486328125/18075490334784 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 9th harmonic is especially accurate as a result, so it can be considered a dual-fifths system, in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 subgroup (AKA 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 inconsistencies, so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.

==Harmonics==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''124edo''' is the [[EDO|equal division of the octave]] into 124 parts of 9.6774 cents each. It is closely related to [[31edo]], but the patent vals differ on the mapping for 3. It tempers out 2048/2025 (diaschisma) and 19073486328125/18075490334784 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 9th harmonic is especially accurate as a result, so it can be considered a dual-fifths system, in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 subgroup (AKA 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 inconsistencies, so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.
+{{EDO intro|124}}
+edo is closely related to [[31edo]], but the patent vals differ on the mapping for 3. It tempers out 2048/2025 (diaschisma) and 19073486328125/18075490334784 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 9th harmonic is especially accurate as a result, so it can be considered a dual-fifths system, in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 subgroup (AKA 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 inconsistencies, so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.
 ==Harmonics==