14/11
In 11-limit just intonation, 14/11 is an undecimal major third, specifically the pentacircle major third, a major or supermajor third of about 417.5 cents. It represents the difference between the 11th and 14th harmonics of the harmonic series.
| Interval information |
pentacircle major third
[sound info]
In many notation systems based on the diatonic chain-of-fifths notation 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 is the simplest neogothic major third. It falls between 5/4 and 9/7, and is the mediant ratio between those simpler intervals, as it is (5 + 9)/(4 + 7). It is 56/55 sharp of 5/4, and 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.
Despite being around a third in size, due to being notated as an imperfect fourth in many systems, many chords involving it are awkward to notate in terms of diatonic degrees. For example, in 11:14:16:20, if 16/11 is considered a type of fifth and 14/11 is considered a type of third, then 8/7 would be considered a type of third, which is awkward due to its size, but actually makes sense if its inversion 7/4 sometimes considered a sixth. It also appears in the chord 7:9:11:14, which consists of three thirds stacked to an octave, but in a diatonic system three thirds would stack to a type of seventh, so one of the intervals would have to be considered a type of fourth instead. In FJS and other systems, 14/11 is considered an imperfect fourth, but 9/7 is wider than it, so it may be considered a diminished fourth in this context instead. In short, this interval shows that diatonic interval classification is far from perfect, and such ambiguity also occurs with 13/10 and 7/5.
It also appears in chords such as 8:11:14, the principal triad of orgone temperament.
Approximation
| 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
External links
- The Noble Mediant by Margo Schulter and David Keenan