41edo: Difference between revisions
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== Theory == | == Theory == | ||
41-ET can be seen as a tuning of the | 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|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]] 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. | 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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0-14-28 = D F# A# = Daug = D aug | 0-14-28 = D F# A# = Daug = D aug | ||
For a more complete list, see [[Ups and Downs Notation# | For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. | ||
== Notations == | == Notations == | ||
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|- | |- | ||
! rowspan="2" |Error | ! rowspan="2" |Error | ||
![[TE error|absolute]] (¢) | ! [[TE error|absolute]] (¢) | ||
| 0.153 | | 0.153 | ||
| 1.26 | | 1.26 | ||
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== Temperaments == | == Temperaments == | ||
[[List of edo-distinct 41et rank two temperaments]] | * [[List of edo-distinct 41et rank two temperaments]] | ||
{| class="wikitable right-1 right-2" | {| class="wikitable right-1 right-2" | ||
|+ Table of Temperaments by generator | |||
|- | |||
! Degree | ! Degree | ||
! Cents | ! Cents | ||
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=== Harmonic Scale === | === Harmonic Scale === | ||
41edo is the first edo to do some justice to Mode 8 of the [[ | 41edo is the first edo to do some justice to Mode 8 of the [[Overtone_series|harmonic series]], which Dante Rosati calls the "[[overtone_scales|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave). | ||
{| class="wikitable" style="text-align:center" | {| class="wikitable" style="text-align:center" | ||
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While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.) | While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.) | ||
7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match. | * 7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match. | ||
* 6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents). | |||
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents). | * 5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents). | ||
* 4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents). | |||
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents). | |||
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents). | |||
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4. | The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4. | ||
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== Links == | == Links == | ||
* [[Wikipedia:41_equal_temperament|41 Equal Temperament - Wikipedia]] | |||
* [[Magic22_as_srutis#magic22assrutis|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. | |||
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* [[#cite_ref-1|^]] [http://x31eq.com/schismic.htm "Schismic Temperaments"] at x31eq.com, the website of [[Graham Breed]] | |||
* [[#cite_ref-2|^]] [http://x31eq.com/decimal_lattice.htm "Lattices with Decimal Notation"] at x31eq.com | |||
* [[#cite_ref-3|^]] [[Wikipedia:Schismatic_temperament|Schismatic temperament - Wikipedia]] | |||
* [[#cite_ref-4|^]] [[Wikipedia:Magic_temperament|Magic temperament - Wikipedia]] | |||
[[Category:41edo]] | [[Category:41edo]] | ||