41edo: Difference between revisions

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~~The '''~~41 equal divisions of the octave''' ('''41edo'''), or '''41(-tone) equal temperament''' ('''41tet''', '''41et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 41 [[equal]]ly-sized steps. Each step is about 29.3 [[cent]]s, an [[interval]] close in size to [[64/63]], the [[septimal comma]].

== Theory ==

41edo is the second smallest equal division (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo]], and is the seventh [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit]] fairly well, and the [[13-limit]] perhaps close enough for government work, though its [[~]][[13/10]] is 14 cents sharp. Anyway, it is [[consistent]] in the [[15-odd-limit]], or the no-17's [[21-odd-limit]]. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 arguably manifests itself as [[21/16]] rather than 13/10. Apart from the full 13-limit, it is even more prominent as a 2.3.5.7.11.19 subgroup temperament for its size, and perhaps the smallest system with a satisfactory model of the [[9-odd-limit]] because it is the smallest edo to tune the [[9-odd-limit]] distinctly consistent.

A step of 41edo is consistently mapped to the [[64/63]], the septimal comma.

41edo can be seen as a tuning of the [[Garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, the [[magic]] temperament, the [[superkleismic]] temperament and multiple temperaments in the [[tetracot family]]. Various 13-limit [[Magic family|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]]. 41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]] due to a much better approximation to the 7th harmonic, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering [[100/99]]. Note that this equivalence is especially nice in 41edo due to also giving a more accurate interpretation of this comma-flat whole tone as a [[21/19]].

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 {{Wikipedia| 41 equal temperament }}
-The '''41 equal divisions of the octave''' ('''41edo'''), or '''41(-tone) equal temperament''' ('''41tet''', '''41et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 41 [[equal]]ly-sized steps. Each step is about 29.3 [[cent]]s, an [[interval]] close in size to [[64/63]], the [[septimal comma]].
+{{EDO intro|41}}
 == Theory ==
 edo is the second smallest equal division (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo]], and is the seventh [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit]] fairly well, and the [[13-limit]] perhaps close enough for government work, though its [[~]][[13/10]] is 14 cents sharp. Anyway, it is [[consistent]] in the [[15-odd-limit]], or the no-17's [[21-odd-limit]]. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 arguably manifests itself as [[21/16]] rather than 13/10. Apart from the full 13-limit, it is even more prominent as a 2.3.5.7.11.19 subgroup temperament for its size, and perhaps the smallest system with a satisfactory model of the [[9-odd-limit]] because it is the smallest edo to tune the [[9-odd-limit]] distinctly consistent.
+A step of 41edo is consistently mapped to the [[64/63]], the septimal comma.
 edo can be seen as a tuning of the [[Garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, the [[magic]] temperament, the [[superkleismic]] temperament and multiple temperaments in the [[tetracot family]]. Various 13-limit [[Magic family|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]]. 41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]] due to a much better approximation to the 7th harmonic, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering [[100/99]]. Note that this equivalence is especially nice in 41edo due to also giving a more accurate interpretation of this comma-flat whole tone as a [[21/19]].