27/26: Difference between revisions
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{{Infobox Interval | |||
| Name = small tridecimal third tone | |||
| Color name = 3u1, thu unison | |||
| Sound = jid_27_26_pluck_adu_dr220.mp3 | |||
| Comma = yes | |||
}} | |||
In [[13-limit]] [[just intonation]], '''27/26''', the '''small tridecimal third tone''', appears as the interval between the Pythagorean major sixth ([[27/16]]) and the [[octave reduction|octave-reduced]] thirteenth [[harmonic]] ([[13/8]]). It measures about 65.3{{cent}}. It is close in size to another 13-limit microtone – [[26/25]]. These intervals differ by the [[superparticular]] ratio [[676/675]], about 2.6{{cent}}, the island comma; tempering out this comma produces temperaments associated with [[The Archipelago]]. | |||
== Approximation == | |||
{{Interval edo approximation}} | |||
== Notation == | |||
See | 27/26 is significant in [[Helmholtz-Ellis notation]] as the tridecimal formal comma which translates a Pythagorean interval to a nearby tridecimal interval, analogous to [[64/63]] and [[33/32]] for septimal and undecimal, respectively. However, in the [[Functional Just System]], that role is taken by [[1053/1024]]. It is worth noting that [[13/8]] is closer to minor than it is to major (some might argue 13/8 to be [[supraminor]] instead of neutral), so [[1053/1024]] is significantly smaller. | ||
=== 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 large diesis''', or '''13L''' for short, because the simplest interval it notates is 13/1 (equiv. 13/8), as for example in C-A{{nbhsp}}{{sagittal | (!/ }}. The primary role of {{ sagittal | (!/ }} is [[8505/8192#Sagittal notation | 8192/8505]] (35L down). The upward version is called '''1/13L''' or '''13L up''' and is represented (in a secondary role) by {{sagittal| (|\ }}. | |||
== Temperaments == | |||
If treated as a [[comma]] to be [[tempering out|tempered out]], it leads to the [[ultraflat]] temperament of the [[2.3.13 subgroup]]. It is tempered out in the [[patent val]]s for edos 2, 5, 7, 9, 14, 16, 21, 23, 28 & 35. | |||
== See also == | |||
* [[52/27]] – its [[octave complement]] | |||
* [[13/9]] – its [[fifth complement]] | |||
* [[26/25]] - the large tridecimal third tone | |||
* [[Gallery of just intervals]] | |||
* [[List of superparticular intervals]] | |||
[[Category:Chroma]] | |||
[[Category:Third tone]] | |||
[[Category:Commas named after their interval size]] | |||
Latest revision as of 22:34, 20 October 2026
| Interval information |
reduced
[sound info]
In 13-limit just intonation, 27/26, the small tridecimal third tone, appears as the interval between the Pythagorean major sixth (27/16) and the octave-reduced thirteenth harmonic (13/8). It measures about 65.3 ¢. It is close in size to another 13-limit microtone – 26/25. These intervals differ by the superparticular ratio 676/675, about 2.6 ¢, the island comma; tempering out this comma produces temperaments associated with The Archipelago.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 17 | 1\17 | 70.59 | +5.25 | +7.44 |
| 18 | 1\18 | 66.67 | +1.33 | +1.99 |
| 19 | 1\19 | 63.16 | -2.18 | -3.45 |
| 20 | 1\20 | 60.00 | -5.34 | -8.90 |
| 35 | 2\35 | 68.57 | +3.23 | +9.43 |
| 36 | 2\36 | 66.67 | +1.33 | +3.99 |
| 37 | 2\37 | 64.86 | -0.47 | -1.46 |
| 38 | 2\38 | 63.16 | -2.18 | -6.90 |
| 54 | 3\54 | 66.67 | +1.33 | +5.98 |
| 55 | 3\55 | 65.45 | +0.12 | +0.54 |
| 56 | 3\56 | 64.29 | -1.05 | -4.91 |
| 72 | 4\72 | 66.67 | +1.33 | +7.98 |
| 73 | 4\73 | 65.75 | +0.42 | +2.53 |
| 74 | 4\74 | 64.86 | -0.47 | -2.91 |
| 75 | 4\75 | 64.00 | -1.34 | -8.36 |
Notation
27/26 is significant in Helmholtz-Ellis notation as the tridecimal formal comma which translates a Pythagorean interval to a nearby tridecimal interval, analogous to 64/63 and 33/32 for septimal and undecimal, respectively. However, in the Functional Just System, that role is taken by 1053/1024. It is worth noting that 13/8 is closer to minor than it is to major (some might argue 13/8 to be supraminor instead of neutral), so 1053/1024 is significantly smaller.
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 large diesis, or 13L for short, because the simplest interval it notates is 13/1 (equiv. 13/8), as for example in C-A . The primary role of is 8192/8505 (35L down). The upward version is called 1/13L or 13L up and is represented (in a secondary role) by .
Temperaments
If treated as a comma to be tempered out, it leads to the ultraflat temperament of the 2.3.13 subgroup. It is tempered out in the patent vals for edos 2, 5, 7, 9, 14, 16, 21, 23, 28 & 35.
See also
- 52/27 – its octave complement
- 13/9 – its fifth complement
- 26/25 - the large tridecimal third tone
- Gallery of just intervals
- List of superparticular intervals