40/39: Difference between revisions

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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 ==

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=== 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

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[[Comma list]]: 40/39

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

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

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

[[Badness]] (Sintel): 0.227

== 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| //| }}.

@@ Line 4: / Line 4: @@
 }}
-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 ==
@@ Line 10: / Line 10: @@
 === 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
@@ Line 16: / Line 16: @@
 [[Comma list]]: 40/39
-{{Mapping|legend=2| 1 0 0 3 | 0 1 0 -1 | 0 0 1 1 }}
+{{Mapping|legend=3| 1 0 0 3 | 0 1 0 -1 | 0 0 1 1 }}
 : sval mapping generators: ~2, ~3, ~5
@@ Line 23: / Line 23: @@
 * [[CWE]]: ~2 = 1200{{c}}, ~3/2 = 710.153{{c}}, ~5/4 = 383.023{{c}}
-[[Badness]] (Dirichlet): 0.227
+[[Badness]] (Sintel): 0.227
 == 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| //| }}.

40/39: Difference between revisions

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