40/39: Difference between revisions
No edit summary |
|||
| (One intermediate revision by one other user not shown) | |||
| Line 10: | Line 10: | ||
=== Unintendo === | === Unintendo === | ||
This temperament was accidentally | 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 | ||
| Line 23: | Line 23: | ||
* [[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]] ( | [[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 .