27/26: Difference between revisions

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Sectioning; note the resulting temp from this interval
 
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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]].
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]].


== Temperaments ==
== Approximation ==
27/26 is tempered out in the patent vals for edos 2, 5, 7, 9, 14, 16, 21, 23, 28 & 35.
{{Interval edo approximation}}
 
{{interval_edo_approximation}}


== Notation ==
== Notation ==
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=== 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 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| (|\ }}.
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 ==
== See also ==

Latest revision as of 08:04, 6 October 2026

Interval information
Ratio 27/26
Factorization 2-1 × 33 × 13-1
Monzo [-1 3 0 0 0 -1⟩
Size in cents 65.33734 ¢
Name small tridecimal third tone
Color name 3u1, thu unison
FJS name [math]\displaystyle{ \text{A1}_{13} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 9.45533
Weil norm (log2 max(n, d)) 9.50978
Wilson norm (sopfr(nd)) 24
Comma size medium

[sound info]
Open this interval in xen-calc

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 approximations for 27/26 (65.34 ¢)
≤ 80edo, relative error ≤ 10%
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.

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

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