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. It functions as such in voicings of the harmonic eleventh chord, 4:5:6:7:9:11.
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 sometimes as a major third, hence the names. Indeed, 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. As such, it is used to form the gentle major triad, 22:28:33[note 1]. Compare this to 22:28:32 (11:14:16), which has the much more dissonant 16/11 as the outside interval in place of 3/2; 11:14:16 can be voiced as 8:11:14 however, which is less dissonant. 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 14/11 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.
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
Notes
- ↑ This is a minor minthmic chord where 14/11 and 13/11 sum to a perfect fifth. Shown here is the simplest JI representation.