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.

Temperaments

27/26 is tempered out in the patent vals for edos 2, 5, 7, 9, 14, 16, 21, 23, 28 & 35.

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 .

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