40/39: Difference between revisions
+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. |
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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 | ||
}} | }} | ||
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 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 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 | |||
[[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]] (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| //| }}. | ||
Latest revision as of 16:39, 6 June 2025
| Interval information |
tridecimal minor diesis,
unintendo comma
reduced
In 13-limit just intonation, 40/39 is the difference between the third octave of the third 5/4 (5 × 23) and the fifth of the thirteenth partial of the same root (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 perfect fourth (4/3) and the tridecimal naiadic (13/10), and between the Pythagorean whole tone (9/8) and the tridecimal semifourth (15/13).
Temperaments
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 discovered when 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 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
Sval mapping: [⟨1 0 0 3], ⟨0 1 0 -1], ⟨0 0 1 1]]
- sval mapping generators: ~2, ~3, ~5
- CWE: ~2 = 1200 ¢, ~3/2 = 710.153 ¢, ~5/4 = 383.023 ¢
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 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 . The primary role of is 6400/6561 (25S). The upward version is called 5/13S or 13/5S up and is represented (in a secondary role) by .