27/26

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
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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-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

• 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