41edo: Difference between revisions

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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 is 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[septimal comma]].

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 is 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[septimal comma]].

== Theory ==

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41-ET 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><ref>[[Wikipedia: Schismatic temperament]]</ref>, the [[Magic family #Magic|~~Magic~~ temperament]]<ref>[[Wikipedia: Magic temperament]]</ref> and the [[Superkleismic|superkleismic (26&41) 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]] 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]]<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><ref>[[Wikipedia: Schismatic temperament]]</ref>, the [[Magic family #Magic|magic temperament]]<ref>[[Wikipedia: Magic temperament]]</ref> and the [[Superkleismic|superkleismic (26&41) 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]] 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-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.

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-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 is 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[septimal comma]].
+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 is 29.268 [[cents]], an [[interval]] close in size to [[64/63]], the [[septimal comma]].
 == Theory ==
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--ET 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><ref>[[Wikipedia: Schismatic temperament]]</ref>, the [[Magic family #Magic|Magic temperament]]<ref>[[Wikipedia: Magic temperament]]</ref> and the [[Superkleismic|superkleismic (26&41) 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]] 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]]<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><ref>[[Wikipedia: Schismatic temperament]]</ref>, the [[Magic family #Magic|magic temperament]]<ref>[[Wikipedia: Magic temperament]]</ref> and the [[Superkleismic|superkleismic (26&41) 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]] 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-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.