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. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the [[11-odd-limit]] except for [[11/10]] and [[20/11]] is represented with more than 10 cents of error in it. 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 edo with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest one 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.

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 perhaps the [[13-limit]], 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. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the [[11-odd-limit]] except for [[11/10]] and [[20/11]] is represented with more than 10 cents of error in it. 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 edo with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest one 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.

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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. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the [[11-odd-limit]] except for [[11/10]] and [[20/11]] is represented with more than 10 cents of error in it. 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 edo with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest one 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.
+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 perhaps the [[13-limit]], 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. It is also the first edo to either match or improve on 12edo's accuracy of every harmonic up to the 16th, and no interval from the [[11-odd-limit]] except for [[11/10]] and [[20/11]] is represented with more than 10 cents of error in it. 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 edo with a satisfactory model of the [[9-odd-limit]], not only because it is the smallest one 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.