Eufalesio: Birth (WIP)

2026-08-03T22:50:30Z

Birth (WIP)

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The '''2.3.5.7.11.13.19.29 subgroup''' (a.k.a. ''yazalathanatwena'' in [[color notation]], hereon referred to as ''the subgroup'') consists of [[just intonation]] [[Interval|intervals]] such that the highest [[prime factor]] in all [[Ratio|ratios]] is 29, but without 17 or 23. It is thus a subset of the [[29-limit]], or alternatively, it can be seen as the [[13-limit]] with extra primes [[19/1|19]] and [[29/1|29]], or the [[2.3.5.7.11.13.19 subgroup]] with 29 added in.

This subgroup is a [[Rank and codimension|rank-8]] system, and can be modeled in a 6-dimensional [[lattice]], with the primes [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], [[13/1|13]], [[19/1|19]] and [[29/1|29]]<nowiki/>represented by each dimension. The prime [[2/1|2]] does not appear in typical lattices because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a seventh dimension is needed.

This subgroup is significant because 29 mirrors [[27/1|27]] in the 24::32 [[harmonic series segment]], and 27 is already present in the [[3-limit]]; thus any fifth-bounded chord involving 27 over the [[root]] has a harmonic (i.e. frequency-scale) inverse involving 29 in this subgroup. 29/16 is close to 9/5, a kind of superminor seventh, which provides a special "utonal" flavor to [[otonal]] chords without clashing ''too'' much with the fundamental. Meanwhile, primes [[17/1|17]] and [[23/1|23]] may be considered to clash with the fundamental, being close to a semitone and a tritone when [[Octave reduction|octave reduced]], so people may wish to exclude them. Therefore the subgroup can be considered to complete the harmonic series up to the lower half of the fifth octave without the more difficult ones near the edges. The same reasons also give rise to the [[2.3.5.7.11.13.19 subgroup]] as a [[retraction]].

== Regular temperaments ==

=== Rank-1 temperaments (edos) ===
[[Edo|Edos]] which represents the subgroup better ([[monotonic]] in the subgroup and decreasing [[TE error]]): {{EDOs|'''41''', 53*, '''58h''', 72, 77, 80, 87* (94), 99ef, '''103h'''* (113, 118f, 118), 121, 130, 140, '''152fj''', 159, 183, 190, 198, 217, 224, 243e, '''270''', 422, 472, 494h, 552}}... and so on. Edos in parenthesis are inconsistent in the subgroup.

<nowiki>*</nowiki> Monotonicity considered for {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 27, 29} (omission of 25 is intentional)

2.3.5.7.11.13.19.29 subgroup - Revision history

Eufalesio: Birth (WIP)