14/11

Interval information

Ratio	14/11
Factorization	2 × 7 × 11-1
Monzo	[1 0 0 1 -1⟩
Size in cents	417.508¢
Names	undecimal major third, pentacircle major third
Color name	1uz4, luzo 4th
FJS name	[math]\displaystyle{ \text{P4}^{7}_{11} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	7.26679
Weil norm (log2 max(n, d))	7.61471
Wilson norm (sopfr(nd))	20
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In 11-limit just intonation, 14/11 is an undecimal major third, specifically the pentacircle major third, or neogothic major third, a major or supermajor third of about 417.5¢. It represents the difference between the 11th and 14th harmonics of the harmonic series.

In many notation systems based on the 3-limit with commatic alterations (e.g. FJS, HEJI), it is an imperfect fourth, as it is a perfect fourth (4/3) minus an instance of 22/21, which is a stack consisting of an undecimal quartertone (33/32) and a septimal comma (64/63), neither of which changes the scale degree or quality. However, it is only sharp of the Pythagorean (3-limit) major third of 81/64 (about 407.8¢) by a pentacircle comma (896/891), which makes it function more often as a major third, hence the names.

14/11 can function as a neo-Gothic major third, as it falls between 5/4 and 9/7. Indeed, it is the mediant ratio between those simpler intervals, as it is (5 + 9)/(4 + 7), and is 56/55 sharp of 5/4, 99/98 flat of 9/7. Other relatively simple thirds in this region can be generated by taking the mediant between 5/4 and 14/11 (which is (5 + 14)/(4 + 11) = 19/15, about 409.2¢) and between 14/11 and 9/7 (which is (14 + 9)/(11 + 7) = 23/18, about 424.4¢. The fact that this interval functions as a type of third is one of the reasons why 7/4, the octave reduced version of the 14th harmonic, can be argued to be a type of "sinth" – a cross between a sixth and a seventh- as opposed to merely a subminor seventh.

It also appears in chords such as 8:11:14, the principal triad of orgone temperament.

Approximation

Edo approximations for 14/11 (417.51 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
3	1\3	400.00	-17.51	-4.38
6	2\6	400.00	-17.51	-8.75
17	6\17	423.53	+6.02	+8.53
20	7\20	420.00	+2.49	+4.15
23	8\23	417.39	-0.12	-0.22
26	9\26	415.38	-2.12	-4.60
29	10\29	413.79	-3.71	-8.98
40	14\40	420.00	+2.49	+8.31
43	15\43	418.60	+1.10	+3.93
46	16\46	417.39	-0.12	-0.45
49	17\49	416.33	-1.18	-4.82
52	18\52	415.38	-2.12	-9.20
63	22\63	419.05	+1.54	+8.08
66	23\66	418.18	+0.67	+3.71
69	24\69	417.39	-0.12	-0.67
72	25\72	416.67	-0.84	-5.05
75	26\75	416.00	-1.51	-9.42

See also

• 11/7 – its octave complement
• 33/28 – its fifth complement
• Gallery of just intervals
• Gentle chords
• List of root-3rd-P5 triads in JI

External links

• The Noble Mediant by Margo Schulter and David Keenan