41edo: Difference between revisions

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== 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]]. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 ~~is debatable~~. ~~(In comparison, [[31edo]] is only consistent up to~~ the ~~11-odd-limit, and the intervals 12\31 and 19\31 have no 11-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent up to [[21-odd~~-limit~~]]. Among these~~, it is 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 ~~have~~ the [[9-odd-limit]] ~~be [[consistent|~~distinctly consistent]].

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 13-limit [[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.

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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=== ~~Divisors~~ ===

=== Subsets and supersets ===

41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]].

== Intervals ==

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 == 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]]. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, [[31edo]] is only consistent up to the 11-odd-limit, and the intervals 12\31 and 19\31 have no 11-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent up to [[21-odd-limit]]. Among these, it is 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 have the [[9-odd-limit]] be [[consistent|distinctly consistent]].
+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 13-limit [[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.
 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]].
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 {{Harmonics in equal|41}}
-=== Divisors ===
+=== Subsets and supersets ===
 edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]].
 == Intervals ==