41edo: Difference between revisions

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[[de:41edo]]

~~__FORCETOC__~~

The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[septimal comma]].

== Theory ==

The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], 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_extensions|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-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], 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_extensions|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 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-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.

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 [[de:41edo]]
-__FORCETOC__
+The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[septimal comma]].
 == Theory ==
-The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&amp;26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], 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_extensions|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.
+-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&amp;26) temperament. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], 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_extensions|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.
 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-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.