29th-octave temperaments

Template:Fractional-octave navigation 29edo is notable for being the first equal division to have a more precise 3/2 than 12edo, and the first tuning to be consistent in the 15-odd-limit. 29th-octave temperaments occur naturally when temperament-merging edos whose greatest common divisor is 29.

Mystery (5-limit)

Main article: Mystery and for higher-limit versions see Hemifamity temperaments #Mystery

The mystery temperament in the 5-limit is described by tempering out the comma [46 -29⟩, where a circle of 29 fifths closes on 17 octaves, and it is supported by small multiples of 29edo.

Subgroup: 2.3.5

Comma: [46 -29⟩

Mapping: [⟨29 46 0], ⟨0 0 1]]

Mapping generators: ~531441/524288, ~5

POTE generator: ~5/4 = 387.408

Optimal ET sequence: 29, 58, 87, 232, 319

Badness: 1.020556

Copper

Copper temperament is derived from a 5-limit comma called copper comma, because it is constructed the same way towards 29edo as Kirnberger's atom is towards 12edo. A fifth of each of these tunings is modified by a tiny amount, then a circle of these fifths is set to close eventually at the octave.

Surprisingly, despite 29edo's fifth being closer to 3/2 than 12edo's, copper has a higher TE error than atomic and is not a very high accuracy temperament.

Subgroup: 2.3.5

Comma list: [-481 261 29⟩

Mapping: [⟨29 0 481], ⟨0 1 -9]]

Mapping generators: ~[-199 12 108⟩ = 1\29, ~3/2 = 701.905

Optimal tuning (CTE): ~3/2 = 701.905

Supporting ETs: 29, 754, 783, 812, 1566, 1537, 2320, 3103, 3132, ...