40/39: Difference between revisions
m brevity & clarity & remove typo |
+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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'''40/39''' is the difference between the third octave of the third [[5/4]] (40 = 5 | 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| //| }}. | ||
Revision as of 09:15, 13 November 2024
| Interval information |
tridecimal minor diesis
reduced
In 13-limit just intonation, 40/39 is the difference between the third octave of the third 5/4 (40 = 5 × 23) 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 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, 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 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 .