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 [[zeta integral edo]], after 31; it is not, however, a [[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 [[consonance]]s, 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 2.3.5.7.11.19.29.31 [[subgroup temperament]] for its size, and perhaps the smallest system with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest edo to tune the 9-odd-limit distinctly consistent, but it is also [[Consistency#Consistency to distance d|consistent in it to distance 2]] ~~(i.e~~. all intervals in the 9-odd-limit are more in-tune than out of tune).

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 [[zeta integral edo]], after 31; it is not, however, a [[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 [[consonance]]s, 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 2.3.5.7.11.19.29.31 [[subgroup temperament]] for its size, and perhaps the smallest system with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest edo to tune the 9-odd-limit distinctly consistent, but it is also [[Consistency #Consistency to distance d|consistent in it to distance 2]]. In other words, all intervals in the 9-odd-limit are more in-tune than out of tune.

A step of 41edo is close and consistently mapped to [[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]], providing proper approximations to the 7th and 11th harmonic (at the cost of the 13th), 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 out [[100/99]]. ~~Note that this~~ equivalence is especially useful in 41edo, wherein this comma-flat whole tone (a.k.a. the tetracot ~~2nd)~~ can also be more accurately interpreted as [[21/19]] — which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes 29 and 31 so that it is a uniquely good/versatile choice for (interpreting the harmony) of tetracot.

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]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, 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 out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of tetracot[7] can also be more accurately interpreted as [[21/19]] – which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.

41et is used by the [[Kite Guitar]], see below in [[#Instruments]].

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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 [[zeta integral edo]], after 31; it is not, however, a [[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 [[consonance]]s, 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 2.3.5.7.11.19.29.31 [[subgroup temperament]] for its size, and perhaps the smallest system with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest edo to tune the 9-odd-limit distinctly consistent, but it is also [[Consistency#Consistency to distance d|consistent in it to distance 2]] (i.e. all intervals in the 9-odd-limit are more in-tune than out of tune).
+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 [[zeta integral edo]], after 31; it is not, however, a [[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 [[consonance]]s, 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 2.3.5.7.11.19.29.31 [[subgroup temperament]] for its size, and perhaps the smallest system with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest edo to tune the 9-odd-limit distinctly consistent, but it is also [[Consistency #Consistency to distance d|consistent in it to distance 2]]. In other words, all intervals in the 9-odd-limit are more in-tune than out of tune.
 A step of 41edo is close and consistently mapped to [[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]], providing proper approximations to the 7th and 11th harmonic (at the cost of the 13th), 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 out [[100/99]]. Note that this equivalence is especially useful in 41edo, wherein this comma-flat whole tone (a.k.a. the tetracot 2nd) can also be more accurately interpreted as [[21/19]] — which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes 29 and 31 so that it is a uniquely good/versatile choice for (interpreting the harmony) of tetracot.
+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]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, 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 out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of tetracot[7] can also be more accurately interpreted as [[21/19]] – which is equated with [[32/29]] above [[31/28]] below (both very near) — providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.
 et is used by the [[Kite Guitar]], see below in [[#Instruments]].