41edo: Difference between revisions

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| Augmented 1sn = 4\41 (117¢)

}}

The '''41 equal divisions of the octave''' ('''41edo'''), or '''41(-tone) equal temperament''' ('''41tet''', '''41et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 41 [[equal]]ly-sized steps. Each step is about 29.3 [[cent]]s, an [[interval]] close in size to [[64/63]], the [[septimal comma]].

== 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><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 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~~#Tuning_Spectra~~|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 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]] and the [[Superkleismic|superkleismic (26&41) temperament]]. 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]].

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.

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| 14

| 409.76

| [[Hocum]]<br>[[~~hocus~~]]

| [[Hocum]]<br>[[Hocus]]

| (P8, c<sup>3</sup>P4/10)

| 32-tone MOS

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Line 1,906:

== Links ==

* [[Wikipedia: 41 equal temperament]]

* [[Magic22 as srutis]] describes a possible use of 41edo for [[indian]] music.

* [[Magic family]]

* Sword, Ron. [http://www.ronsword.com "Tetracontamonophonic Scales for Guitar"]

* Taylor, Cam. [https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M Intervals, Scales and Chords in 41EDO], a work in progress using just intonation concepts and simplified Sagittal notation.

@@ Line 8: / Line 8: @@
 | Augmented 1sn = 4\41 (117¢)
 }}
+{{Wikipedia| 41 equal temperament }}
 The '''41 equal divisions of the octave''' ('''41edo'''), or '''41(-tone) equal temperament''' ('''41tet''', '''41et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 41 [[equal]]ly-sized steps. Each step is about 29.3 [[cent]]s, an [[interval]] close in size to [[64/63]], the [[septimal comma]].
 == 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><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 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#Tuning_Spectra|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 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]] and the [[Superkleismic|superkleismic (26&41) temperament]]. 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]].
 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.
@@ Line 1,284: / Line 1,285: @@
 | 14
 | 409.76
-| [[Hocum]]<br>[[hocus]]
+| [[Hocum]]<br>[[Hocus]]
 | (P8, c<sup>3</sup>P4/10)
 | 32-tone MOS
@@ Line 1,905: / Line 1,906: @@
 == Links ==
-* [[Wikipedia: 41 equal temperament]]
+* [[Magic22 as srutis]] describes a possible use of 41edo for [[indian]] music.
-* [[Magic22 as srutis]] describes a possible use of 41edo for [[indian]] music.
-* [[Magic family]]
 * Sword, Ron. [http://www.ronsword.com "Tetracontamonophonic Scales for Guitar"]
 * Taylor, Cam. [https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M Intervals, Scales and Chords in 41EDO], a work in progress using just intonation concepts and simplified Sagittal notation.