41edo - Revision history

TallKite: /* Instruments */ added the lavta

2026-09-04T07:16:47Z

Instruments: added the lavta

← Older revision		Revision as of 07:16, 4 September 2026
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	[[File:Caleb's Kite guitar.jpg\|none\|thumb\|200px\|Kite guitar]]		[[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}}		{{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 ===		=== Metallophones ===

TallKite: /* Intervals */ added interval names to the main table

2026-08-11T02:54:56Z

Intervals: added interval names to the main table

Show changes

Squib: /* Subsets and supersets */

2026-07-30T02:45:49Z

Subsets and supersets

← Older revision		Revision as of 02:45, 30 July 2026
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	=== Subsets and supersets ===		=== 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''.		[[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''.

Fredg999: Interwiki

2026-07-19T05:27:26Z

Interwiki

← Older revision		Revision as of 05:27, 19 July 2026
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	\| de = 41-EDO		\| de = 41-EDO
	\| en = 41edo		\| en = 41edo
	\| es =		\| es = 41edo
	\| ja =		\| ja =
	}}		}}

Dave Keenan: /* Sagittal notation */ Corrected multiple errors in the table. It did not agree with the staff examples below it.

2026-06-02T01:36:30Z

Sagittal notation: Corrected multiple errors in the table. It did not agree with the staff examples below it.

← Older revision		Revision as of 01:36, 2 June 2026
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	=== Sagittal notation ===		=== 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 Stein–Zimmermann 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;"		{\| class="wikitable" style="text-align: center;"
Line 883:		Line 883:
	\|-		\|-
	! rowspan="3" \|Symbol		! 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		! Evo
	\| rowspan="2" \| <big>{{sagittal\|/\|\}}</big>		\| rowspan="2" \| <big>{{sagittal\| /\|\ }}</big>
	~~\| <small>{{sagittal\|#}}{{sagittal\|!/}}</small~~>
	\|-		\|-
	! Revo		! 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):		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.		See [[Sagittal notation #Revo\|apotome complements]] for equivalent accidental pairs.

FloraC: /* Notations */ improve description

2026-05-12T14:15:15Z

Notations: improve description

← Older revision		Revision as of 14:15, 12 May 2026
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	== Notations ==		== Notations ==
	=== Stein–Zimmermann–Gould notation ===		=== Stein–Zimmermann–Gould notation ===
	[[Stein–Zimmermann–Gould notation]] uses sharps and flats with arrows:		[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
	{{Sharpness-sharp4-szg}}		{{Sharpness-sharp4-szg}}

FloraC: /* Notations */ SZG notation

2026-05-12T13:13:43Z

Notations: SZG notation

← Older revision		Revision as of 13:13, 12 May 2026
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	== Notations ==		== 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 with arrows:
	{{~~Ups and downs sharpness~~}}		{{Sharpness-sharp4-szg}}

	~~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

	{{~~Sharpness-sharp4~~}}		=== 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.
			{{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 ===		=== 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}}		{{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.		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 ===		=== 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 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 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;"		{\| class="wikitable" style="text-align: center;"

FloraC: Undo random addition

2026-04-22T09:23:56Z

Undo random addition

← Older revision		Revision as of 09:23, 22 April 2026
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	\| 38		\| 38
	\| 1112.2		\| 1112.2
	\| [[19/10~~]], [[243/128~~]], [[21/11]], [[36/19]], [[40/21]]		\| [[19/10]], [[21/11]], [[36/19]], [[40/21]]
	\| {{UDnote\|step=38}}		\| {{UDnote\|step=38}}
	\|-		\|-

AleX-1337: Added 243/128 to approximations

2026-04-22T05:32:31Z

Added 243/128 to approximations

← Older revision		Revision as of 05:32, 22 April 2026
Line 236:		Line 236:
	\| 38		\| 38
	\| 1112.2		\| 1112.2
	\| [[19/10]], [[21/11]], [[36/19]], [[40/21]]		\| [[19/10]], [[243/128]], [[21/11]], [[36/19]], [[40/21]]
	\| {{UDnote\|step=38}}		\| {{UDnote\|step=38}}
	\|-		\|-

FloraC: /* Regular temperament properties */ correction and note the absolute accuracy in the 7-, 11-, and 17-limit.

2026-03-31T10:26:58Z

Regular temperament properties: correction and note the absolute accuracy in the 7-, 11-, and 17-limit.

Show changes

← Older revision		Revision as of 01:36, 2 June 2026
Line 872:		Line 872:

	=== Sagittal notation ===		=== 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 Stein–Zimmermann 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;"		{\| class="wikitable" style="text-align: center;"
Line 883:		Line 883:
	\|-		\|-
	! rowspan="3" \|Symbol		! 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		! Evo
	\| rowspan="2" \| <big>{{sagittal\|/\|\}}</big>		\| rowspan="2" \| <big>{{sagittal\| /\|\ }}</big>
	~~\| <small>{{sagittal\|#}}{{sagittal\|!/}}</small~~>
	\|-		\|-
	! Revo		! 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):		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.		See [[Sagittal notation #Revo\|apotome complements]] for equivalent accidental pairs.