41edo: Difference between revisions

[[de:41edo]]

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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 represents a frequency ratio of 29.268 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63|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|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|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10|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 the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]].

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All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:

For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].  

A red-note/blue-note system, similar to the one proposed for [[36edo|36edo]], is one option for notating 41edo. (This is separate from and not compatible with Kite's [[Kite's_color_notation|color notation]].) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:

If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups_and_Downs_Notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.

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Whereas 12-edo has a circle of twelve 5ths, 41-edo has a spiral of twelve 5ths (since 24\41 is on the 7\12 kite in the scale tree). This spiral of 5th shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with microtonal music. There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo. The two innermost and two outermost intervals on the spiral are duplicates.

[[File:41-edo spiral with notes.png|730x730px]]

41 EDO tempers out the following commas using its patent val, < 41 65 95 115 142 152 168 174 185 199 203 |.

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[[List_of_edo-distinct_41et_rank_two_temperaments|List of edo-distinct 41et rank two temperaments]]

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A list of [[41edo_modes|41edo modes]] (MOS and others).

41edo is the first edo to do some justice to Mode 8 of the [[OverToneSeries|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).

The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.

Taking every third degree of 41edo produces a scale extremely close to [[88cET|88cET]] or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span style="">[[BP|Bohlen-Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See chart:

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[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]

A possible system to tune keyboards in 41EDO is discussed in http://launch.groups.yahoo.com/group/tuning/message/74155.

[http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro EveningHorizon] [http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 play] by Cameron Bobro

<ul><li>[http://en.wikipedia.org/wiki/41_equal_temperament Wikipedia article on 41edo]</li><li>[[Magic22_as_srutis#magic22assrutis|Magic22 as srutis]] describes a possible use of 41edo for [[Indian|indian]] music.</li><li>see also [[Magic_family|Magic family]]</li><li>Sword, Ron. [http://www.ronsword.com "Tetracontamonophonic Scales for Guitar"]</li><li>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.</li></ul>

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