41edo: Difference between revisions

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| de = 41-EDO

| en = 41edo

| es =

| es = 41edo

| ja =

}}

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=== Subsets and supersets ===

41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though ~~it contains [[41ed4]]. Although not technically subsets,~~ it essentially contains [[88cET]] as every third step and [[13edt]] as every fifth step.

41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it essentially contains [[88cET]] as every third step and [[13edt]] as every fifth step.

[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo. Its step of 1\205 is called a ''mem''.

[[2460edo]] has potential for a 41edo analog to [[Cent|cents]]. It divides the 41edo step into 60 equal parts, and 60 is a highly composite (a.k.a. antiprime) number, so it contains many other multiples of 41edo, including 205edo, and also contains [[12edo]] among other equal tunings. It also accurately represents [[14afdo|mode 14 of the harmonic series]], as it is consistent all the way up to the 27-odd-limit. This allows for precise detunings in a 41-tone framework to approximate pure just intonation more closely, especially for some higher harmonics. Its step of 1\2460 is called a ''mina''.

See also [[countercomp family]] and [[41st-octave temperaments]] for temperaments that augment 41edo.

== Intervals ==

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! Cents

! Approximate ratios*

! [[Kite's ups and downs notation|Ups and downs notation]]

! colspan="2" |[[Kite's ups and downs notation|Ups and downs notation]]

([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^d2)

|-

| 0

| 0.0

| [[1/1]]

|P1

| {{UDnote|step=0}}

|-

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| 29.3

| [[49/48]], [[50/49]], [[64/63]], [[81/80]]

|^1

| {{UDnote|step=1}}

|-

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| 58.5

| [[25/24]], [[28/27]], [[33/32]], [[36/35]]

|^^1, vm2

| {{UDnote|step=2}}

|-

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| 87.8

| [[19/18]], [[20/19]], [[21/20]], [[22/21]]

|vA1, m2

| {{UDnote|step=3}}

|-

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| 117.1

| [[14/13]], [[15/14]], [[16/15]]

|A1, ^m2

| {{UDnote|step=4}}

|-

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| 146.3

| [[12/11]], [[13/12]]

|~2

| {{UDnote|step=5}}

|-

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| 175.6

| [[10/9]], [[11/10]], [[21/19]]

|vM2

| {{UDnote|step=6}}

|-

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| 204.9

| [[9/8]]

|M2

| {{UDnote|step=7}}

|-

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| 234.1

| [[8/7]], [[15/13]]

|^M2

| {{UDnote|step=8}}

|-

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| 263.4

| [[7/6]], [[22/19]]

|vm3

| {{UDnote|step=9}}

|-

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| 292.7

| [[13/11]], [[19/16]], [[32/27]]

|m3

| {{UDnote|step=10}}

|-

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| 322.0

| [[6/5]]

|^m3

| {{UDnote|step=11}}

|-

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| 351.2

| [[11/9]], [[16/13]]

|~3

| {{UDnote|step=12}}

|-

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| 380.5

| [[5/4]], [[26/21]]

|vM3

| {{UDnote|step=13}}

|-

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| 409.8

| [[14/11]], [[19/15]], [[24/19]]

|M3

| {{UDnote|step=14}}

|-

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| 439.0

| [[9/7]], [[32/25]]

|^M3

| {{UDnote|step=15}}

|-

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| 468.3

| [[21/16]], [[13/10]]

|v4

| {{UDnote|step=16}}

|-

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| 497.6

| [[4/3]]

|P4

| {{UDnote|step=17}}

|-

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| 526.8

| [[15/11]], [[19/14]], [[27/20]]

|^4

| {{UDnote|step=18}}

|-

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| 556.1

| [[11/8]], [[18/13]], [[26/19]]

|~4, vd5

| {{UDnote|step=19}}

|-

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| 585.4

| [[7/5]], [[45/32]]

|vA4, d5

| {{UDnote|step=20}}

|-

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| 614.6

| [[10/7]], [[64/45]]

|A4, ^d5

| {{UDnote|step=21}}

|-

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| 643.9

| [[13/9]], [[16/11]], [[19/13]]

|~5, ^A4

| {{UDnote|step=22}}

|-

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| 673.2

| [[22/15]], [[28/19]], [[40/27]]

|v5

| {{UDnote|step=23}}

|-

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| 702.4

| [[3/2]]

|P5

| {{UDnote|step=24}}

|-

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| 731.7

| [[20/13]], [[32/21]]

|^5

| {{UDnote|step=25}}

|-

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| 761.0

| [[14/9]], [[25/16]]

|vm6

| {{UDnote|step=26}}

|-

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| 790.2

| [[11/7]], [[19/12]], [[30/19]]

|m6

| {{UDnote|step=27}}

|-

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| 819.5

| [[8/5]], [[21/13]]

|^m6

| {{UDnote|step=28}}

|-

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| 848.8

| [[13/8]], [[18/11]]

|~6

| {{UDnote|step=29}}

|-

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| 878.0

| [[5/3]]

|vM6

| {{UDnote|step=30}}

|-

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| 907.3

| [[22/13]], [[27/16]], [[32/19]]

|M6

| {{UDnote|step=31}}

|-

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| 936.6

| [[12/7]], [[19/11]]

|^M6

| {{UDnote|step=32}}

|-

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| 965.9

| [[7/4]], [[26/15]]

|vm7

| {{UDnote|step=33}}

|-

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| 995.1

| [[16/9]]

|m7

| {{UDnote|step=34}}

|-

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| 1024.4

| [[9/5]], [[20/11]], [[38/21]]

|^m7

| {{UDnote|step=35}}

|-

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| 1053.7

| [[11/6]], [[24/13]]

|~7

| {{UDnote|step=36}}

|-

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| 1082.9

| [[13/7]], [[15/8]], [[28/15]]

|vM7

| {{UDnote|step=37}}

|-

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| 1112.2

| [[19/10]], [[21/11]], [[36/19]], [[40/21]]

|M7

| {{UDnote|step=38}}

|-

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| 1141.5

| [[27/14]], [[35/18]], [[48/25]], [[64/33]]

|^M7

| {{UDnote|step=39}}

|-

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| 1170.7

| [[49/25]], [[63/32]], [[96/49]], [[160/81]]

|v8

| {{UDnote|step=40}}

|-

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| 1200.0

| [[2/1]]

|P8

| {{UDnote|step=41}}

|}

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== Notations ==

=== ~~Ups and downs~~ notation ===

=== Stein–Zimmermann–Gould notation ===

~~41edo can be notated with~~ [[~~ups and downs~~]]~~, spoken as up, dup, downsharp, sharp, upsharp etc.~~ and ~~down, dud, upflat etc. Note that dup is equivalent to dudsharp~~ and ~~dud is equivalent to dupflat.~~

[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:

~~41edo can also be notated with quarter-tone accidentals and [[Alternative symbols for ups and downs notation#Sharp-3|ups and downs]].~~

The notes within an octave from A are thus:

~~{{Ups and downs sharpness|41|true}}~~

~~Arrows borrowed from extended [[Helmholtz–Ellis notation]] can also be used:~~

A, B{{sesquiflat2}}, A{{demisharp2}}, B♭, A♯, B{{demiflat2}}, A{{sesquisharp2}}, B, C{{demiflat2}}, B{{demisharp2}}, C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D, E{{sesquiflat2}}, D{{demisharp2}}, E♭, D♯, E{{demiflat2}}, D{{sesquisharp2}}, E, F{{demiflat2}}, E{{demisharp2}}, F, G{{sesquiflat2}}, F{{demisharp2}}, G♭, F♯, G{{demiflat2}}, F{{sesquisharp2}}, G, A{{sesquiflat2}}, G{{demisharp2}}, A♭, G♯, A{{demiflat2}}, G{{sesquisharp2}}, A

=== Kite's ups and downs notation ===

41edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.

~~The notes within an octave from A are thus~~:

Half-sharps and half-flats can be used to avoid double arrows:

A, B{{sesquiflat2}}, A{{demisharp2}}, B♭, A♯, B{{demiflat2}}, A{{sesquisharp2}}, B, C{{demiflat2}}, B{{demisharp2}}, C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D, E{{sesquiflat2}}, D{{demisharp2}}, E♭, D♯, E{{demiflat2}}, D{{sesquisharp2}}, E, F{{demiflat2}}, E{{demisharp2}}, F, G{{sesquiflat2}}, F{{demisharp2}}, G♭, F♯, G{{demiflat2}}, F{{sesquisharp2}}, G, A{{sesquiflat2}}, G{{demisharp2}}, A♭, G♯, A{{demiflat2}}, G{{~~sesquisharp2~~}}~~, A~~

=== Red-Blue notation ===

A red-note/blue-note system, similar to the one proposed for [[36edo]], is another option for notating 41edo. This is a special case of Kite's [[color notation]], treating 41edo as a temperament of the 2.3.7 subgroup. 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:

A red-note/blue-note system, similar to the one proposed for [[36edo]], is another option for notating 41edo. This is a special case of [[Kite's color notation]], treating 41edo as a temperament of the 2.3.7 subgroup. 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:

{{colored note|A}}, {{colored note|red|A}}, {{colored note|blue|B♭}}, {{colored note|B♭}}, {{colored note|A♯}}, {{colored note|red|A♯}}, {{colored note|blue|B}}, {{colored note|B}}, {{colored note|red|B}}, {{colored note|blue|C}}, {{colored note|C}}, {{colored note|red|C}}, {{colored note|blue|D♭}}, {{colored note|D♭}}, {{colored note|C♯}}, {{colored note|red|C♯}}, {{colored note|blue|D}}, {{colored note|D}}, {{colored note|red|D}}, {{colored note|blue|E♭}}, {{colored note|E♭}}, {{colored note|D♯}}, {{colored note|red|D♯}}, {{colored note|blue|E}}, {{colored note|E}}, {{colored note|red|E}}, {{colored note|blue|F}}, {{colored note|F}}, {{colored note|red|F}}, {{colored note|blue|G♭}}, {{colored note|G♭}}, {{colored note|F♯}}, {{colored note|red|F♯}}, {{colored note|blue|G}}, {{colored note|G}}, {{colored note|red|G}}, {{colored note|blue|A♭}}, {{colored note|A♭}}, {{colored note|G♯}}, {{colored note|red|G♯}}, {{colored note|blue|A}}, {{colored note|A}}

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The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.

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.

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 Kite's ups and downs notation. The only difference is the use of minor tritone and major tritone.

=== Sagittal notation ===

41edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation #Spartan single-shaft|Spartan ~~extension~~]], with the apotome equal to 4 edosteps and the limma to 3 edosteps. Since the apotome can be split in two and the [[243/242|rastma]] is tempered out, a SZ half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:

41edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation #Spartan single-shaft|Spartan set]], with the apotome equal to 4 edosteps and the limma to 3 edosteps. Since the apotome can be split in two and the [[243/242|rastma]] is tempered out, a Stein–Zimmermann half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:

{| class="wikitable" style="text-align: center;"

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|-

! rowspan="3" |Symbol

! Evo+SZ

! Evo-SZ

| rowspan="3" | <big>{{sagittal||//|}}</big>

| rowspan="3" | <big>{{sagittal| |//| }}</big>

| rowspan="3" | <big>{{sagittal|/|}}</big>

| rowspan="3" | <big>{{sagittal| /| }}</big>

| <big>{{Sagittal|t}}</big>

| <big>{{Sagittal| t }}</big>

| <~~small~~>{{~~Sagittal~~|t}}{{sagittal||\}}</~~small~~>

| rowspan="2" | <big>{{sagittal| \! }}{{sagittal| # }}</big>

| rowspan="2" | <big>{{sagittal|#}}</big>

| rowspan="2" | <big>{{sagittal| # }}</big>

|-

! Evo

| rowspan="2" | <big>{{sagittal|/|\}}</big>

| rowspan="2" | <big>{{sagittal| /|\ }}</big>

~~| <small>{{sagittal|#}}{{sagittal|!/}}</small~~>

|-

! Revo

| <big>{{sagittal|(|~~3=)~~}}</big>

| <big>{{sagittal| ||\ }}</big>

| <big>{{sagittal|/||\}}</big>

| <big>{{sagittal| /||\ }}</big>

|}

The following enharmonics from the Spartan set are present (comma tempered out):

* {{Sagittal|//|}} = {{sagittal|/|)}} = {{sagittal|/|\}} ([[325/324]], [[352/351]])

* {{Sagittal| //| }} = {{sagittal| /|) }} = {{sagittal| /|\ }} ([[325/324]], [[352/351]])

* {{Sagittal|/|}} = {{sagittal||)}} ([[225/224]])

* {{Sagittal| /| }} = {{sagittal| |) }} ([[225/224]])

* {{Sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])

* {{Sagittal| |( }} = {{sagittal| |//| }} ([[5120/5103]])

See [[Sagittal notation #Revo|apotome complements]] for equivalent accidental pairs.

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[[File:Caleb's Kite guitar.jpg|none|thumb|200px|Kite guitar]]

For more photos of Kite guitars, see [[Kite Guitar Photographs]].

There exist Kite-fretted basses, Kite-fretted Chapman sticks, and a Kite-fretted cuatro. For more photos of Kite guitars, see [[Kite Guitar Photographs]].

=== Other string instruments ===

The Turkish [[wikipedia:Lavta|lavta]] is a lute-like instrument with movable frets. The strings are tuned D A D A or D A D G. The frets are mostly placed in clumps of 5, so that there are 5 seconds, 5 thirds, etc. The lavta is arguably fretted to a 27-note subset of 41edo. Each clump would correspond to 41edo's plainminor-upminor-mid-downmajor-plainmajor. The edosteps would be something like 0, 3-7, 10-14, 17, 20-24, 27-31, 34-38 and 41.

https://www.facebook.com/groups/497105067092502/permalink/3562297780573200/

=== Metallophones ===

@@ Line 2: / Line 2: @@
 | de = 41-EDO
 | en = 41edo
-| es =
+| es = 41edo
 | ja =
 }}
@@ Line 30: / Line 30: @@
 === Subsets and supersets ===
-edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it contains [[41ed4]]. Although not technically subsets, it essentially contains [[88cET]] as every third step and [[13edt]] as every fifth step.
+edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it essentially contains [[88cET]] as every third step and [[13edt]] as every fifth step.
 [[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo. Its step of 1\205 is called a ''mem''.
 [[2460edo]] has potential for a 41edo analog to [[Cent|cents]]. It divides the 41edo step into 60 equal parts, and 60 is a highly composite (a.k.a. antiprime) number, so it contains many other multiples of 41edo, including 205edo, and also contains [[12edo]] among other equal tunings. It also accurately represents [[14afdo|mode 14 of the harmonic series]], as it is consistent all the way up to the 27-odd-limit. This allows for precise detunings in a 41-tone framework to approximate pure just intonation more closely, especially for some higher harmonics. Its step of 1\2460 is called a ''mina''.
+See also [[countercomp family]] and [[41st-octave temperaments]] for temperaments that augment 41edo.
 == Intervals ==
@@ Line 42: / Line 44: @@
 ! Cents
 ! Approximate ratios*
-! [[Kite's ups and downs notation|Ups and downs notation]]
+! colspan="2" |[[Kite's ups and downs notation|Ups and downs notation]]
+([[Enharmonic unisons in ups and downs notation|EUs]]: v<sup>4</sup>A1 and ^d2)
 |-
 | 0
 | 0.0
 | [[1/1]]
+|P1
 | {{UDnote|step=0}}
 |-
@@ Line 52: / Line 56: @@
 | 29.3
 | [[49/48]], [[50/49]], [[64/63]], [[81/80]]
+|^1
 | {{UDnote|step=1}}
 |-
@@ Line 57: / Line 62: @@
 | 58.5
 | [[25/24]], [[28/27]], [[33/32]], [[36/35]]
+|^^1, vm2
 | {{UDnote|step=2}}
 |-
@@ Line 62: / Line 68: @@
 | 87.8
 | [[19/18]], [[20/19]], [[21/20]], [[22/21]]
+|vA1, m2
 | {{UDnote|step=3}}
 |-
@@ Line 67: / Line 74: @@
 | 117.1
 | [[14/13]], [[15/14]], [[16/15]]
+|A1, ^m2
 | {{UDnote|step=4}}
 |-
@@ Line 72: / Line 80: @@
 | 146.3
 | [[12/11]], [[13/12]]
+|~2
 | {{UDnote|step=5}}
 |-
@@ Line 77: / Line 86: @@
 | 175.6
 | [[10/9]], [[11/10]], [[21/19]]
+|vM2
 | {{UDnote|step=6}}
 |-
@@ Line 82: / Line 92: @@
 | 204.9
 | [[9/8]]
+|M2
 | {{UDnote|step=7}}
 |-
@@ Line 87: / Line 98: @@
 | 234.1
 | [[8/7]], [[15/13]]
+|^M2
 | {{UDnote|step=8}}
 |-
@@ Line 92: / Line 104: @@
 | 263.4
 | [[7/6]], [[22/19]]
+|vm3
 | {{UDnote|step=9}}
 |-
@@ Line 97: / Line 110: @@
 | 292.7
 | [[13/11]], [[19/16]], [[32/27]]
+|m3
 | {{UDnote|step=10}}
 |-
@@ Line 102: / Line 116: @@
 | 322.0
 | [[6/5]]
+|^m3
 | {{UDnote|step=11}}
 |-
@@ Line 107: / Line 122: @@
 | 351.2
 | [[11/9]], [[16/13]]
+|~3
 | {{UDnote|step=12}}
 |-
@@ Line 112: / Line 128: @@
 | 380.5
 | [[5/4]], [[26/21]]
+|vM3
 | {{UDnote|step=13}}
 |-
@@ Line 117: / Line 134: @@
 | 409.8
 | [[14/11]], [[19/15]], [[24/19]]
+|M3
 | {{UDnote|step=14}}
 |-
@@ Line 122: / Line 140: @@
 | 439.0
 | [[9/7]], [[32/25]]
+|^M3
 | {{UDnote|step=15}}
 |-
@@ Line 127: / Line 146: @@
 | 468.3
 | [[21/16]], [[13/10]]
+|v4
 | {{UDnote|step=16}}
 |-
@@ Line 132: / Line 152: @@
 | 497.6
 | [[4/3]]
+|P4
 | {{UDnote|step=17}}
 |-
@@ Line 137: / Line 158: @@
 | 526.8
 | [[15/11]], [[19/14]], [[27/20]]
+|^4
 | {{UDnote|step=18}}
 |-
@@ Line 142: / Line 164: @@
 | 556.1
 | [[11/8]], [[18/13]], [[26/19]]
+|~4, vd5
 | {{UDnote|step=19}}
 |-
@@ Line 147: / Line 170: @@
 | 585.4
 | [[7/5]], [[45/32]]
+|vA4, d5
 | {{UDnote|step=20}}
 |-
@@ Line 152: / Line 176: @@
 | 614.6
 | [[10/7]], [[64/45]]
+|A4, ^d5
 | {{UDnote|step=21}}
 |-
@@ Line 157: / Line 182: @@
 | 643.9
 | [[13/9]], [[16/11]], [[19/13]]
+|~5, ^A4
 | {{UDnote|step=22}}
 |-
@@ Line 162: / Line 188: @@
 | 673.2
 | [[22/15]], [[28/19]], [[40/27]]
+|v5
 | {{UDnote|step=23}}
 |-
@@ Line 167: / Line 194: @@
 | 702.4
 | [[3/2]]
+|P5
 | {{UDnote|step=24}}
 |-
@@ Line 172: / Line 200: @@
 | 731.7
 | [[20/13]], [[32/21]]
+|^5
 | {{UDnote|step=25}}
 |-
@@ Line 177: / Line 206: @@
 | 761.0
 | [[14/9]], [[25/16]]
+|vm6
 | {{UDnote|step=26}}
 |-
@@ Line 182: / Line 212: @@
 | 790.2
 | [[11/7]], [[19/12]], [[30/19]]
+|m6
 | {{UDnote|step=27}}
 |-
@@ Line 187: / Line 218: @@
 | 819.5
 | [[8/5]], [[21/13]]
+|^m6
 | {{UDnote|step=28}}
 |-
@@ Line 192: / Line 224: @@
 | 848.8
 | [[13/8]], [[18/11]]
+|~6
 | {{UDnote|step=29}}
 |-
@@ Line 197: / Line 230: @@
 | 878.0
 | [[5/3]]
+|vM6
 | {{UDnote|step=30}}
 |-
@@ Line 202: / Line 236: @@
 | 907.3
 | [[22/13]], [[27/16]], [[32/19]]
+|M6
 | {{UDnote|step=31}}
 |-
@@ Line 207: / Line 242: @@
 | 936.6
 | [[12/7]], [[19/11]]
+|^M6
 | {{UDnote|step=32}}
 |-
@@ Line 212: / Line 248: @@
 | 965.9
 | [[7/4]], [[26/15]]
+|vm7
 | {{UDnote|step=33}}
 |-
@@ Line 217: / Line 254: @@
 | 995.1
 | [[16/9]]
+|m7
 | {{UDnote|step=34}}
 |-
@@ Line 222: / Line 260: @@
 | 1024.4
 | [[9/5]], [[20/11]], [[38/21]]
+|^m7
 | {{UDnote|step=35}}
 |-
@@ Line 227: / Line 266: @@
 | 1053.7
 | [[11/6]], [[24/13]]
+|~7
 | {{UDnote|step=36}}
 |-
@@ Line 232: / Line 272: @@
 | 1082.9
 | [[13/7]], [[15/8]], [[28/15]]
+|vM7
 | {{UDnote|step=37}}
 |-
@@ Line 237: / Line 278: @@
 | 1112.2
 | [[19/10]], [[21/11]], [[36/19]], [[40/21]]
+|M7
 | {{UDnote|step=38}}
 |-
@@ Line 242: / Line 284: @@
 | 1141.5
 | [[27/14]], [[35/18]], [[48/25]], [[64/33]]
+|^M7
 | {{UDnote|step=39}}
 |-
@@ Line 247: / Line 290: @@
 | 1170.7
 | [[49/25]], [[63/32]], [[96/49]], [[160/81]]
+|v8
 | {{UDnote|step=40}}
 |-
@@ Line 252: / Line 296: @@
 | 1200.0
 | [[2/1]]
+|P8
 | {{UDnote|step=41}}
 |}
@@ Line 845: / Line 890: @@
 == Notations ==
-=== Ups and downs notation ===
+=== Stein–Zimmermann–Gould notation ===
-edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
-{{Ups and downs sharpness}}
+{{Sharpness-sharp4-szg}}
-edo can also be notated with quarter-tone accidentals and [[Alternative symbols for ups and downs notation#Sharp-3|ups and downs]].
+The notes within an octave from A are thus:
-{{Ups and downs sharpness|41|true}}
-Arrows borrowed from extended [[Helmholtz–Ellis notation]] can also be used:
+A, B{{sesquiflat2}}, A{{demisharp2}}, B♭, A♯, B{{demiflat2}}, A{{sesquisharp2}}, B, C{{demiflat2}}, B{{demisharp2}}, C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D, E{{sesquiflat2}}, D{{demisharp2}}, E♭, D♯, E{{demiflat2}}, D{{sesquisharp2}}, E, F{{demiflat2}}, E{{demisharp2}}, F, G{{sesquiflat2}}, F{{demisharp2}}, G♭, F♯, G{{demiflat2}}, F{{sesquisharp2}}, G, A{{sesquiflat2}}, G{{demisharp2}}, A♭, G♯, A{{demiflat2}}, G{{sesquisharp2}}, A
-{{Sharpness-sharp4}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
+{{Ups and downs sharpness}}
-The notes within an octave from A are thus:
+Half-sharps and half-flats can be used to avoid double arrows:
+{{Ups and downs sharpness|41|true}}
-A, B{{sesquiflat2}}, A{{demisharp2}}, B♭, A♯, B{{demiflat2}}, A{{sesquisharp2}}, B, C{{demiflat2}}, B{{demisharp2}}, C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D, E{{sesquiflat2}}, D{{demisharp2}}, E♭, D♯, E{{demiflat2}}, D{{sesquisharp2}}, E, F{{demiflat2}}, E{{demisharp2}}, F, G{{sesquiflat2}}, F{{demisharp2}}, G♭, F♯, G{{demiflat2}}, F{{sesquisharp2}}, G, A{{sesquiflat2}}, G{{demisharp2}}, A♭, G♯, A{{demiflat2}}, G{{sesquisharp2}}, A
 === Red-Blue notation ===
-A red-note/blue-note system, similar to the one proposed for [[36edo]], is another option for notating 41edo. This is a special case of Kite's [[color notation]], treating 41edo as a temperament of the 2.3.7 subgroup. 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:
+A red-note/blue-note system, similar to the one proposed for [[36edo]], is another option for notating 41edo. This is a special case of [[Kite's color notation]], treating 41edo as a temperament of the 2.3.7 subgroup. 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:
 {{colored note|A}}, {{colored note|red|A}}, {{colored note|blue|B♭}}, {{colored note|B♭}}, {{colored note|A♯}}, {{colored note|red|A♯}}, {{colored note|blue|B}}, {{colored note|B}}, {{colored note|red|B}}, {{colored note|blue|C}}, {{colored note|C}}, {{colored note|red|C}}, {{colored note|blue|D♭}}, {{colored note|D♭}}, {{colored note|C♯}}, {{colored note|red|C♯}}, {{colored note|blue|D}}, {{colored note|D}}, {{colored note|red|D}}, {{colored note|blue|E♭}}, {{colored note|E♭}}, {{colored note|D♯}}, {{colored note|red|D♯}}, {{colored note|blue|E}}, {{colored note|E}}, {{colored note|red|E}}, {{colored note|blue|F}}, {{colored note|F}}, {{colored note|red|F}}, {{colored note|blue|G♭}}, {{colored note|G♭}}, {{colored note|F♯}}, {{colored note|red|F♯}}, {{colored note|blue|G}}, {{colored note|G}}, {{colored note|red|G}}, {{colored note|blue|A♭}}, {{colored note|A♭}}, {{colored note|G♯}}, {{colored note|red|G♯}}, {{colored note|blue|A}}, {{colored note|A}}
@@ Line 869: / Line 914: @@
 The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.
-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.
+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 Kite's ups and downs notation. The only difference is the use of minor tritone and major tritone.
 === Sagittal notation ===
-edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation #Spartan single-shaft|Spartan extension]], with the apotome equal to 4 edosteps and the limma to 3 edosteps. Since the apotome can be split in two and the [[243/242|rastma]] is tempered out, a SZ half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:
+edo can be notated in [[Sagittal notation|Sagittal]] using the [[Sagittal notation #Spartan single-shaft|Spartan set]], with the apotome equal to 4 edosteps and the limma to 3 edosteps. Since the apotome can be split in two and the [[243/242|rastma]] is tempered out, a Stein–Zimmermann half-sharp and a half-flat may be used instead of pakai/pakao. Here is a simplified table:
 {| class="wikitable" style="text-align: center;"
@@ Line 883: / Line 928: @@
 |-
 ! rowspan="3" |Symbol
-! Evo+SZ
+! Evo-SZ
-| rowspan="3" | <big>{{sagittal||//|}}</big>
+| rowspan="3" | <big>{{sagittal| |//| }}</big>
-| rowspan="3" | <big>{{sagittal|/|}}</big>
+| rowspan="3" | <big>{{sagittal| /| }}</big>
-| <big>{{Sagittal|t}}</big>
+| <big>{{Sagittal| t }}</big>
-| <small>{{Sagittal|t}}{{sagittal||\}}</small>
+| rowspan="2" | <big>{{sagittal| \! }}{{sagittal| # }}</big>
-| rowspan="2" | <big>{{sagittal|#}}</big>
+| rowspan="2" | <big>{{sagittal| # }}</big>
 |-
 ! Evo
-| rowspan="2" | <big>{{sagittal|/|\}}</big>
+| rowspan="2" | <big>{{sagittal| /|\ }}</big>
-| <small>{{sagittal|#}}{{sagittal|!/}}</small>
 |-
 ! Revo
-| <big>{{sagittal|(|3=)}}</big>
+| <big>{{sagittal| ||\ }}</big>
-| <big>{{sagittal|/||\}}</big>
+| <big>{{sagittal| /||\ }}</big>
 |}
 The following enharmonics from the Spartan set are present (comma tempered out):
-* {{Sagittal|//|}} = {{sagittal|/|)}} = {{sagittal|/|\}} ([[325/324]], [[352/351]])
+* {{Sagittal| //| }} = {{sagittal| /|) }} = {{sagittal| /|\ }} ([[325/324]], [[352/351]])
-* {{Sagittal|/|}} = {{sagittal||)}} ([[225/224]])
+* {{Sagittal| /| }} = {{sagittal| |) }} ([[225/224]])
-* {{Sagittal||(}} = {{sagittal||//|}} ([[5120/5103]])
+* {{Sagittal| |( }} = {{sagittal| |//| }} ([[5120/5103]])
 See [[Sagittal notation #Revo|apotome complements]] for equivalent accidental pairs.
@@ Line 2,241: / Line 2,285: @@
 [[File:Caleb's Kite guitar.jpg|none|thumb|200px|Kite guitar]]
-For more photos of Kite guitars, see [[Kite Guitar Photographs]].
+There exist Kite-fretted basses, Kite-fretted Chapman sticks, and a Kite-fretted cuatro. For more photos of Kite guitars, see [[Kite Guitar Photographs]].
 {{clear}}
+=== Other string instruments ===
+The Turkish [[wikipedia:Lavta|lavta]] is a lute-like instrument with movable frets. The strings are tuned D A D A or D A D G. The frets are mostly placed in clumps of 5, so that there are 5 seconds, 5 thirds, etc. The lavta is arguably fretted to a 27-note subset of 41edo. Each clump would correspond to 41edo's plainminor-upminor-mid-downmajor-plainmajor. The edosteps would be something like 0, 3-7, 10-14, 17, 20-24, 27-31, 34-38 and 41.
+https://www.facebook.com/groups/497105067092502/permalink/3562297780573200/
 === Metallophones ===

41edo: Difference between revisions

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