41edo: Difference between revisions

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== Theory ==

41edo can be seen as a tuning of the [[Schismatic family #Garibaldi|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 family #Magic|magic temperament]], the [[Superkleismic|superkleismic (26&41) temperament]] and multiple temperaments in the [[tetracot family]]. It 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. ~~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]].~~ 41 ~~EDO~~ is ~~also a great~~ [[~~tetracot~~]] ~~tuning, and in some sense~~ is ~~an ideal tuning for it due~~ to ~~being~~ the ~~unique temperament to support [[Tetracot family#Monkey|Monkey]]~~, ~~[[Tetracot family#Bunya|Bunya]]~~ and ~~[[Tetracot family#Octacot|Octacot]] simultaneously, all of which being different ways to extend tetracot into~~ the ~~[[7~~-limit]]. ~~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 41 EDO due to also giving~~ a ~~more accurate interpretation of this comma-flat whole tone as being [[21/19]] in the~~ no-~~13's no-17's~~ [[21-odd-limit]]. ~~Overall~~, ~~41 EDO~~ is ~~an excellent~~ 2.3.5.7.11.19 subgroup ~~(AKA no-13's no-17's [[19-limit]])~~ 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 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]]. 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 distinctly consistent.

41edo ~~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 all the way up to~~ 21~~-odd-limit~~.

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]].

41et forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41et as the basic [[13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41et circle in [[205edo]]. 41et is also used by the [[Kite Guitar]], see below in [[#Instruments]].

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 == Theory ==
-edo can be seen as a tuning of the [[Schismatic family #Garibaldi|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 family #Magic|magic temperament]], the [[Superkleismic|superkleismic (26&41) temperament]] and multiple temperaments in the [[tetracot family]]. It 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. 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]]. 41 EDO is also a great [[tetracot]] tuning, and in some sense is an ideal tuning for it due to being the unique temperament to support [[Tetracot family#Monkey|Monkey]], [[Tetracot family#Bunya|Bunya]] and [[Tetracot family#Octacot|Octacot]] simultaneously, all of which being different ways to extend tetracot into the [[7-limit]]. 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 41 EDO due to also giving a more accurate interpretation of this comma-flat whole tone as being [[21/19]] in the no-13's no-17's [[21-odd-limit]]. Overall, 41 EDO is an excellent 2.3.5.7.11.19 subgroup (AKA no-13's no-17's [[19-limit]]) 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 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]]. 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 distinctly consistent.
-edo 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 all the way up to 21-odd-limit.
+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]].
 et forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41et as the basic [[13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41et circle in [[205edo]]. 41et is also used by the [[Kite Guitar]], see below in [[#Instruments]].