29edo: Difference between revisions

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29 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.

=== ~~Divisors~~ ===

Due to 29edo's tone-efficient mapping of 2.3.7/5.11/5.13/5, it makes sense to collapse this subgroup to 29edo. One may then expand the subgroup to the full 13-limit, adding an independent generator to reach primes 5, 7, 11, and 13 in one generator. This is [[mystery]] temperament, which has very low badness despite so many periods per octave. The 58-note MOS gives scope for harmony, with 29 15-odd-limit otonal chords and 29 utonal chords.

29edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]].

=== Interval Flavors ===

29edo has inframinor, neogothic minor, supraminor, submajor, neogothic major, and supermajor seconds and thirds. This is in contrast to systems like [[31edo]], where there are subminor, minor, neutral, major, and supermajor seconds and thirds. This is due to it representing 2.3.7/5.11/5.13/5 well, and ratios between two primes greater than 3 tend to land between interval categories of intervals in a 2.3.p subgroup. For example, 2.3.5 intervals are major/minor, 2.3.7 intervals are supermajor/subminor, and 2.3.11 and 2.3.13 intervals are neutral. 31edo, on the other hand, represents 2.3.5.7.11 well, and therefore has interval categories resembling 2.3.5, 2.3.7, and 2.3.11. It can also be seen from the fact that the 29&31 temperament, [[tritonic]], maps seconds and thirds to large numbers of generators, so they differ more in tuning between the system.

=== Subsets and Supersets ===

29edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]]. Its supersets [[58edo]] and [[87edo]] correct many of the higher primes.

== Intervals ==

@@ Line 33: / Line 33: @@
 tempers out 352/351, 676/675 and 4000/3993 from the 2.3.11/5.13/5 subgroup, and in addition 196/195 and 364/363 from the 2.3.7/5.11/5.13/5 subgroup, so we have various relationships from the tempering, such as the fact that the 1-13/11-3/2 chord and the 1-14/11-3/2 chord are inverses of each other, a major-minor pairing. A larger subgroup containing both of these subgroups is the [[k*N subgroups|3*29 subgroup]] 2.3.125.175.275.325; on this subgroup 29 tunes the same as 87, and the commas of 29 on this subgroup are the same as the 13-limit commas of 87. Still another subgroup of interest is the [[k*N subgroups|2*29 subgroup]] 2.3.25.35.55.65.85; on this subgroup 29 tunes the same as 58 and has the same 17-limit commas.
-=== Divisors ===
+Due to 29edo's tone-efficient mapping of 2.3.7/5.11/5.13/5, it makes sense to collapse this subgroup to 29edo. One may then expand the subgroup to the full 13-limit, adding an independent generator to reach primes 5, 7, 11, and 13 in one generator. This is [[mystery]] temperament, which has very low badness despite so many periods per octave. The 58-note MOS gives scope for harmony, with 29 15-odd-limit otonal chords and 29 utonal chords.
-edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]].
+=== Interval Flavors ===
+edo has inframinor, neogothic minor, supraminor, submajor, neogothic major, and supermajor seconds and thirds. This is in contrast to systems like [[31edo]], where there are subminor, minor, neutral, major, and supermajor seconds and thirds. This is due to it representing 2.3.7/5.11/5.13/5 well, and ratios between two primes greater than 3 tend to land between interval categories of intervals in a 2.3.p subgroup. For example, 2.3.5 intervals are major/minor, 2.3.7 intervals are supermajor/subminor, and 2.3.11 and 2.3.13 intervals are neutral. 31edo, on the other hand, represents 2.3.5.7.11 well, and therefore has interval categories resembling 2.3.5, 2.3.7, and 2.3.11. It can also be seen from the fact that the 29&31 temperament, [[tritonic]], maps seconds and thirds to large numbers of generators, so they differ more in tuning between the system.
+=== Subsets and Supersets ===
+edo is the 10th [[prime edo]], following [[23edo]] and coming before [[31edo]]. Its supersets [[58edo]] and [[87edo]] correct many of the higher primes.
 == Intervals ==