41edo: Difference between revisions

(7 intermediate revisions by 5 users not shown)

Line 2:

| de = 41-EDO

| en = 41edo

| es =

| es = 41edo

| ja =

}}

Line 30:

=== 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''.

Line 42:

! 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 54:

| 29.3

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

|^1

| {{UDnote|step=1}}

|-

Line 57:

Line 60:

| 58.5

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

|^^1, vm2

| {{UDnote|step=2}}

|-

Line 62:

Line 66:

| 87.8

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

|vA1, m2

| {{UDnote|step=3}}

|-

Line 67:

Line 72:

| 117.1

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

|A1, ^m2

| {{UDnote|step=4}}

|-

Line 72:

Line 78:

| 146.3

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

|~2

| {{UDnote|step=5}}

|-

Line 77:

Line 84:

| 175.6

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

|vM2

| {{UDnote|step=6}}

|-

Line 82:

Line 90:

| 204.9

| [[9/8]]

|M2

| {{UDnote|step=7}}

|-

Line 87:

Line 96:

| 234.1

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

|^M2

| {{UDnote|step=8}}

|-

Line 92:

Line 102:

| 263.4

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

|vm3

| {{UDnote|step=9}}

|-

Line 97:

Line 108:

| 292.7

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

|m3

| {{UDnote|step=10}}

|-

Line 102:

Line 114:

| 322.0

| [[6/5]]

|^m3

| {{UDnote|step=11}}

|-

Line 107:

Line 120:

| 351.2

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

|~3

| {{UDnote|step=12}}

|-

Line 112:

Line 126:

| 380.5

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

|vM3

| {{UDnote|step=13}}

|-

Line 117:

Line 132:

| 409.8

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

|M3

| {{UDnote|step=14}}

|-

Line 122:

Line 138:

| 439.0

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

|^M3

| {{UDnote|step=15}}

|-

Line 127:

Line 144:

| 468.3

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

|v4

| {{UDnote|step=16}}

|-

Line 132:

Line 150:

| 497.6

| [[4/3]]

|P4

| {{UDnote|step=17}}

|-

Line 137:

Line 156:

| 526.8

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

|^4

| {{UDnote|step=18}}

|-

Line 142:

Line 162:

| 556.1

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

|~4, vd5

| {{UDnote|step=19}}

|-

Line 147:

Line 168:

| 585.4

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

|vA4, d5

| {{UDnote|step=20}}

|-

Line 152:

Line 174:

| 614.6

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

|A4, ^d5

| {{UDnote|step=21}}

|-

Line 157:

Line 180:

| 643.9

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

|~5, ^A4

| {{UDnote|step=22}}

|-

Line 162:

Line 186:

| 673.2

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

|v5

| {{UDnote|step=23}}

|-

Line 167:

Line 192:

| 702.4

| [[3/2]]

|P5

| {{UDnote|step=24}}

|-

Line 172:

Line 198:

| 731.7

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

|^5

| {{UDnote|step=25}}

|-

Line 177:

Line 204:

| 761.0

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

|vm6

| {{UDnote|step=26}}

|-

Line 182:

Line 210:

| 790.2

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

|m6

| {{UDnote|step=27}}

|-

Line 187:

Line 216:

| 819.5

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

|^m6

| {{UDnote|step=28}}

|-

Line 192:

Line 222:

| 848.8

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

|~6

| {{UDnote|step=29}}

|-

Line 197:

Line 228:

| 878.0

| [[5/3]]

|vM6

| {{UDnote|step=30}}

|-

Line 202:

Line 234:

| 907.3

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

|M6

| {{UDnote|step=31}}

|-

Line 207:

Line 240:

| 936.6

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

|^M6

| {{UDnote|step=32}}

|-

Line 212:

Line 246:

| 965.9

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

|vm7

| {{UDnote|step=33}}

|-

Line 217:

Line 252:

| 995.1

| [[16/9]]

|m7

| {{UDnote|step=34}}

|-

Line 222:

Line 258:

| 1024.4

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

|^m7

| {{UDnote|step=35}}

|-

Line 227:

Line 264:

| 1053.7

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

|~7

| {{UDnote|step=36}}

|-

Line 232:

Line 270:

| 1082.9

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

|vM7

| {{UDnote|step=37}}

|-

Line 237:

Line 276:

| 1112.2

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

|M7

| {{UDnote|step=38}}

|-

Line 242:

Line 282:

| 1141.5

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

|^M7

| {{UDnote|step=39}}

|-

Line 247:

Line 288:

| 1170.7

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

|v8

| {{UDnote|step=40}}

|-

Line 252:

Line 294:

| 1200.0

| [[2/1]]

|P8

| {{UDnote|step=41}}

|}

Line 845:

Line 888:

== 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}}

Line 869:

Line 912:

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;"

Line 883:

Line 926:

|-

! 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: / 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''.
@@ Line 42: / Line 42: @@
 ! 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 54: @@
 | 29.3
 | [[49/48]], [[50/49]], [[64/63]], [[81/80]]
+|^1
 | {{UDnote|step=1}}
 |-
@@ Line 57: / Line 60: @@
 | 58.5
 | [[25/24]], [[28/27]], [[33/32]], [[36/35]]
+|^^1, vm2
 | {{UDnote|step=2}}
 |-
@@ Line 62: / Line 66: @@
 | 87.8
 | [[19/18]], [[20/19]], [[21/20]], [[22/21]]
+|vA1, m2
 | {{UDnote|step=3}}
 |-
@@ Line 67: / Line 72: @@
 | 117.1
 | [[14/13]], [[15/14]], [[16/15]]
+|A1, ^m2
 | {{UDnote|step=4}}
 |-
@@ Line 72: / Line 78: @@
 | 146.3
 | [[12/11]], [[13/12]]
+|~2
 | {{UDnote|step=5}}
 |-
@@ Line 77: / Line 84: @@
 | 175.6
 | [[10/9]], [[11/10]], [[21/19]]
+|vM2
 | {{UDnote|step=6}}
 |-
@@ Line 82: / Line 90: @@
 | 204.9
 | [[9/8]]
+|M2
 | {{UDnote|step=7}}
 |-
@@ Line 87: / Line 96: @@
 | 234.1
 | [[8/7]], [[15/13]]
+|^M2
 | {{UDnote|step=8}}
 |-
@@ Line 92: / Line 102: @@
 | 263.4
 | [[7/6]], [[22/19]]
+|vm3
 | {{UDnote|step=9}}
 |-
@@ Line 97: / Line 108: @@
 | 292.7
 | [[13/11]], [[19/16]], [[32/27]]
+|m3
 | {{UDnote|step=10}}
 |-
@@ Line 102: / Line 114: @@
 | 322.0
 | [[6/5]]
+|^m3
 | {{UDnote|step=11}}
 |-
@@ Line 107: / Line 120: @@
 | 351.2
 | [[11/9]], [[16/13]]
+|~3
 | {{UDnote|step=12}}
 |-
@@ Line 112: / Line 126: @@
 | 380.5
 | [[5/4]], [[26/21]]
+|vM3
 | {{UDnote|step=13}}
 |-
@@ Line 117: / Line 132: @@
 | 409.8
 | [[14/11]], [[19/15]], [[24/19]]
+|M3
 | {{UDnote|step=14}}
 |-
@@ Line 122: / Line 138: @@
 | 439.0
 | [[9/7]], [[32/25]]
+|^M3
 | {{UDnote|step=15}}
 |-
@@ Line 127: / Line 144: @@
 | 468.3
 | [[21/16]], [[13/10]]
+|v4
 | {{UDnote|step=16}}
 |-
@@ Line 132: / Line 150: @@
 | 497.6
 | [[4/3]]
+|P4
 | {{UDnote|step=17}}
 |-
@@ Line 137: / Line 156: @@
 | 526.8
 | [[15/11]], [[19/14]], [[27/20]]
+|^4
 | {{UDnote|step=18}}
 |-
@@ Line 142: / Line 162: @@
 | 556.1
 | [[11/8]], [[18/13]], [[26/19]]
+|~4, vd5
 | {{UDnote|step=19}}
 |-
@@ Line 147: / Line 168: @@
 | 585.4
 | [[7/5]], [[45/32]]
+|vA4, d5
 | {{UDnote|step=20}}
 |-
@@ Line 152: / Line 174: @@
 | 614.6
 | [[10/7]], [[64/45]]
+|A4, ^d5
 | {{UDnote|step=21}}
 |-
@@ Line 157: / Line 180: @@
 | 643.9
 | [[13/9]], [[16/11]], [[19/13]]
+|~5, ^A4
 | {{UDnote|step=22}}
 |-
@@ Line 162: / Line 186: @@
 | 673.2
 | [[22/15]], [[28/19]], [[40/27]]
+|v5
 | {{UDnote|step=23}}
 |-
@@ Line 167: / Line 192: @@
 | 702.4
 | [[3/2]]
+|P5
 | {{UDnote|step=24}}
 |-
@@ Line 172: / Line 198: @@
 | 731.7
 | [[20/13]], [[32/21]]
+|^5
 | {{UDnote|step=25}}
 |-
@@ Line 177: / Line 204: @@
 | 761.0
 | [[14/9]], [[25/16]]
+|vm6
 | {{UDnote|step=26}}
 |-
@@ Line 182: / Line 210: @@
 | 790.2
 | [[11/7]], [[19/12]], [[30/19]]
+|m6
 | {{UDnote|step=27}}
 |-
@@ Line 187: / Line 216: @@
 | 819.5
 | [[8/5]], [[21/13]]
+|^m6
 | {{UDnote|step=28}}
 |-
@@ Line 192: / Line 222: @@
 | 848.8
 | [[13/8]], [[18/11]]
+|~6
 | {{UDnote|step=29}}
 |-
@@ Line 197: / Line 228: @@
 | 878.0
 | [[5/3]]
+|vM6
 | {{UDnote|step=30}}
 |-
@@ Line 202: / Line 234: @@
 | 907.3
 | [[22/13]], [[27/16]], [[32/19]]
+|M6
 | {{UDnote|step=31}}
 |-
@@ Line 207: / Line 240: @@
 | 936.6
 | [[12/7]], [[19/11]]
+|^M6
 | {{UDnote|step=32}}
 |-
@@ Line 212: / Line 246: @@
 | 965.9
 | [[7/4]], [[26/15]]
+|vm7
 | {{UDnote|step=33}}
 |-
@@ Line 217: / Line 252: @@
 | 995.1
 | [[16/9]]
+|m7
 | {{UDnote|step=34}}
 |-
@@ Line 222: / Line 258: @@
 | 1024.4
 | [[9/5]], [[20/11]], [[38/21]]
+|^m7
 | {{UDnote|step=35}}
 |-
@@ Line 227: / Line 264: @@
 | 1053.7
 | [[11/6]], [[24/13]]
+|~7
 | {{UDnote|step=36}}
 |-
@@ Line 232: / Line 270: @@
 | 1082.9
 | [[13/7]], [[15/8]], [[28/15]]
+|vM7
 | {{UDnote|step=37}}
 |-
@@ Line 237: / Line 276: @@
 | 1112.2
 | [[19/10]], [[21/11]], [[36/19]], [[40/21]]
+|M7
 | {{UDnote|step=38}}
 |-
@@ Line 242: / Line 282: @@
 | 1141.5
 | [[27/14]], [[35/18]], [[48/25]], [[64/33]]
+|^M7
 | {{UDnote|step=39}}
 |-
@@ Line 247: / Line 288: @@
 | 1170.7
 | [[49/25]], [[63/32]], [[96/49]], [[160/81]]
+|v8
 | {{UDnote|step=40}}
 |-
@@ Line 252: / Line 294: @@
 | 1200.0
 | [[2/1]]
+|P8
 | {{UDnote|step=41}}
 |}
@@ Line 845: / Line 888: @@
 == 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 912: @@
 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 926: @@
 |-
 ! 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.