40/39 - Revision history

VectorGraphics at 00:09, 22 March 2026

2026-03-22T00:09:55Z

← Older revision		Revision as of 00:09, 22 March 2026
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	}}		}}

	In [[13-limit]] [[just intonation]], '''40/39''' is the difference between the third octave of the third [[5/4]] ({{nowrap\| 40 = 5 × 2<sup>3</sup> }}) and the fifth of the thirteenth partial of the same root ({{nowrap\| 39 = 13 × 3 }}). Within an octave, it is the difference between [[39/32]] and [[5/4]] and thus between [[13/8]] and [[5/3]]. It is also the difference between the [[4/3\|perfect fourth (4/3)]] and the [[13/10\|tridecimal naiadic (13/10)]], and between the [[9/8\|Pythagorean whole tone (9/8)]] and the [[15/13\|tridecimal semifourth (15/13)]].		In [[13-limit]] [[just intonation]], '''40/39''' is the difference between [[39/32]] and [[5/4]] and thus between [[13/8]] and [[5/3]]. It is also the difference between the [[4/3\|perfect fourth (4/3)]] and the [[13/10\|tridecimal naiadic (13/10)]], and between the [[9/8\|Pythagorean whole tone (9/8)]] and the [[15/13\|tridecimal semifourth (15/13)]].

	== Temperaments ==		== Temperaments ==

FloraC: Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

2025-06-06T16:39:39Z

Text replacement - "Badness (Dirichlet): " to "Badness (Sintel): "

← Older revision		Revision as of 16:39, 6 June 2025
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	* [[CWE]]: ~2 = 1200{{c}}, ~3/2 = 710.153{{c}}, ~5/4 = 383.023{{c}}		* [[CWE]]: ~2 = 1200{{c}}, ~3/2 = 710.153{{c}}, ~5/4 = 383.023{{c}}

	[[Badness]] (~~Dirichlet~~): 0.227		[[Badness]] (Sintel): 0.227

	== Notation ==		== Notation ==
	=== Sagittal notation ===		=== Sagittal notation ===
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400 #Sagittal notation\|6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400 #Sagittal notation\|6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.

VectorGraphics at 00:42, 24 April 2025

2025-04-24T00:42:50Z

← Older revision		Revision as of 00:42, 24 April 2025
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	=== Unintendo ===		=== Unintendo ===
	This temperament was accidentally ~~named~~ when [[User:VectorGraphics\|Vector]] described [[fendo]] as tempering out 40/39 in the 2.3.5.13 subgroup. The mistake has since been corrected. Its generators are a sharp perfect fifth and a flat major third. It can be described as {{nowrap\| 7 & 15 & 10 }}. It equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.		This temperament was accidentally discovered when [[User:VectorGraphics\|Vector]] described [[fendo]] as tempering out 40/39 in the 2.3.5.13 subgroup. The mistake has since been corrected, and the 2.3.5.13 temperament was renamed to "unintendo" to reflect its unintended discovery. Its generators are a sharp perfect fifth and a flat major third. It can be described as {{nowrap\| 7 & 15 & 10 }}. It equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.

	[[Subgroup]]: 2.3.5.13		[[Subgroup]]: 2.3.5.13

VectorGraphics at 20:06, 5 March 2025

2025-03-05T20:06:20Z

← Older revision		Revision as of 20:06, 5 March 2025
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	{{Infobox Interval		{{Infobox Interval
	\| Name = tridecimal 1/5-tone, tridecimal minor diesis		\| Name = tridecimal 1/5-tone, tridecimal minor diesis, unintendo comma
	\| Color name = 3uy1, thuyo unison		\| Color name = 3uy1, thuyo unison
	}}		}}

FloraC: Move data of unintendo here

2025-03-05T15:05:29Z

Move data of unintendo here

← Older revision		Revision as of 15:05, 5 March 2025
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	}}		}}

	In [[13-limit]] [[just intonation]], '''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 × 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 × 3). Within an octave, it is the difference between [[39/32]] and [[5/4]] and thus between [[13/8]] and [[5/3]]. It is also the difference between the [[4/3\|perfect fourth (4/3)]] and the [[13/10\|tridecimal naiadic (13/10)]], and between the [[9/8\|Pythagorean whole tone (9/8)]] and the [[15/13\|tridecimal semifourth (15/13)]].		In [[13-limit]] [[just intonation]], '''40/39''' is the difference between the third octave of the third [[5/4]] ({{nowrap\| 40 = 5 × 2<sup>3</sup> }}) and the fifth of the thirteenth partial of the same root ({{nowrap\| 39 = 13 × 3 }}). Within an octave, it is the difference between [[39/32]] and [[5/4]] and thus between [[13/8]] and [[5/3]]. It is also the difference between the [[4/3\|perfect fourth (4/3)]] and the [[13/10\|tridecimal naiadic (13/10)]], and between the [[9/8\|Pythagorean whole tone (9/8)]] and the [[15/13\|tridecimal semifourth (15/13)]].

	== Temperaments ==		== Temperaments ==
	If treated as a comma to be tempered out, it leads to [[fendo]] ~~temperament, which~~ equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.		If treated as a comma to be tempered out in the 2.3.5.13 subgroup, it leads to '''unintendo''' temperament.

			=== Unintendo ===
			This temperament was accidentally named when [[User:VectorGraphics\|Vector]] described [[fendo]] as tempering out 40/39 in the 2.3.5.13 subgroup. The mistake has since been corrected. Its generators are a sharp perfect fifth and a flat major third. It can be described as {{nowrap\| 7 & 15 & 10 }}. It equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.

			[[Subgroup]]: 2.3.5.13

			[[Comma list]]: 40/39

			{{Mapping\|legend=2\| 1 0 0 3 \| 0 1 0 -1 \| 0 0 1 1 }}

			: sval mapping generators: ~2, ~3, ~5

			[[Optimal tuning]]s:
			* [[CWE]]: ~2 = 1200{{c}}, ~3/2 = 710.153{{c}}, ~5/4 = 383.023{{c}}

			[[Badness]] (Dirichlet): 0.227

	== Notation ==		== Notation ==
	=== Sagittal notation ===		=== Sagittal notation ===
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400 #Sagittal notation\|6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.

VectorGraphics: e

2025-02-02T10:46:43Z

← Older revision		Revision as of 10:46, 2 February 2025
Line 7:		Line 7:

	== Temperaments ==		== Temperaments ==
	If treated as a comma to be tempered out, it equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.		If treated as a comma to be tempered out, it leads to [[fendo]] temperament, which equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.

	== Notation ==		== Notation ==
	=== Sagittal notation ===		=== Sagittal notation ===
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.

FloraC: +as (4/3)/(13/10) and (15/13)/(9/8). 6/5 -> 5/3 and 16/13 -> 13/8 since both are simpler than their octave complements. Move the temp to a distinct section so that you don't have to think about the temp when you don't want to.

2024-11-13T09:15:52Z

+as (4/3)/(13/10) and (15/13)/(9/8). 6/5 -> 5/3 and 16/13 -> 13/8 since both are simpler than their octave complements. Move the temp to a distinct section so that you don't have to think about the temp when you don't want to.

← Older revision		Revision as of 09:15, 13 November 2024
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	}}		}}

	'''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 ⋅ 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 ⋅ 3). ~~If tempered out~~, it ~~equates~~ [[39/32]] ~~with~~ [[5/4]] and thus [[6/5]] ~~with~~ [[16/13]], ~~so that it does not assosciate major with greater neutral~~ and ~~minor with lesser neutral as one would expect~~ (~~see~~ [[65/64]]~~), but the other way around~~.		In [[13-limit]] [[just intonation]], '''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 × 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 × 3). Within an octave, it is the difference between [[39/32]] and [[5/4]] and thus between [[13/8]] and [[5/3]]. It is also the difference between the [[4/3\|perfect fourth (4/3)]] and the [[13/10\|tridecimal naiadic (13/10)]], and between the [[9/8\|Pythagorean whole tone (9/8)]] and the [[15/13\|tridecimal semifourth (15/13)]].

	== Sagittal notation ==		== Temperaments ==
			If treated as a comma to be tempered out, it equates 39/32 with 5/4 and equates 13/8 with 5/3, so it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.

			== Notation ==
			=== Sagittal notation ===
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.

Godtone: brevity & clarity & remove typo

2024-11-12T20:33:16Z

brevity & clarity & remove typo

← Older revision		Revision as of 20:33, 12 November 2024
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	}}		}}

	'''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 ⋅ 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 ⋅ 3). If tempered out, it ~~tempers together~~ [[5~~-limit~~]] and [[13~~-limit~~]]~~] intervals. However~~, it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around~~. In particular [[5/4]]~[[39/32]] and [[6/5]]~[[16/13]] are temperings~~.		'''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 ⋅ 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 ⋅ 3). If tempered out, it equates [[39/32]] with [[5/4]] and thus [[6/5]] with [[16/13]], so that it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around.

	== Sagittal notation ==		== Sagittal notation ==
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.

Cmloegcmluin: /* Sagittal notation */ indicating direction unnecessary

2024-10-21T03:32:48Z

Sagittal notation: indicating direction unnecessary

← Older revision		Revision as of 03:32, 21 October 2024
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	== Sagittal notation ==		== Sagittal notation ==
	In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S ~~down~~). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.		In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.

Cmloegcmluin: add Sagittal notation usage

2024-10-19T15:37:14Z

add Sagittal notation usage

← Older revision		Revision as of 15:37, 19 October 2024
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	'''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 ⋅ 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 ⋅ 3). If tempered out, it tempers together [[5-limit]] and [[13-limit]]] intervals. However, it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around. In particular [[5/4]]~[[39/32]] and [[6/5]]~[[16/13]] are temperings.		'''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 ⋅ 2<sup>3</sup>) and the fifth of the thirteenth partial of the same root (39 = 13 ⋅ 3). If tempered out, it tempers together [[5-limit]] and [[13-limit]]] intervals. However, it does not assosciate major with greater neutral and minor with lesser neutral as one would expect (see [[65/64]]), but the other way around. In particular [[5/4]]~[[39/32]] and [[6/5]]~[[16/13]] are temperings.

			== Sagittal notation ==
			In the [[Sagittal]] system, the downward version of this comma (possibly tempered) is represented (in a secondary role) by the sagittal {{sagittal \| \\! }} and is called the '''13/5 small diesis''', or '''13/5S''' for short, because the simplest interval it notates is 13/5 (equiv. 13/10), as for example in C-F{{nbhsp}}{{sagittal \| \\! }}. The primary role of {{ sagittal \| \\! }} is [[6561/6400#Sagittal notation \| 6400/6561]] (25S down). The upward version is called '''5/13S''' or '''13/5S up''' and is represented (in a secondary role) by {{sagittal\| //\| }}.