13/11
| Ratio | 13/11 |
| Subgroup monzo | 11.13 [-1 1⟩ |
| Size in cents | 289.20972¢ |
| Names | tridecimal minor third, neo-Gothic minor third, major minthmic minor third |
| Color name | 3o1u3, tholu 3rd |
| FJS name | [math]\text{m3}^{13}_{11}[/math] |
| Special properties | reduced |
| Tenney height (log2 nd) | 7.15987 |
| Weil height (log2 max(n, d)) | 7.40088 |
| Wilson height (sopfr (nd)) | 24 |
| Harmonic entropy (Shannon, [math]\sqrt{n\cdot d}[/math]) |
~4.55235 bits |
[sound info] | |
| open this interval in xen-calc | |
In 13-limit just intonation, 13/11 is the tridecimal minor third, neo-Gothic minor third or major minthmic minor third, measuring about 289.2¢. It is the difference between the 11th and 13th harmonics. The (octave-reduced) 11th harmonic (11/8, about 551.3¢) and 13th harmonic (13/8, about 840.5¢) are both quite xenharmonic and demand new interval categories, while 13/11 can be likened unto some kind of relatively complex minor third – it is a major minthma (352/351) narrower than the Pythagorean minor third (32/27). It can even function as such in a 13-limit neo-Gothic minor triad of 22:26:33, with a 3/2 perfect fifth between 33 and 22. Compare this to 22:26:32 (11:13:16), which has the much more dissonant 16/11 as the outside interval in place of 3/2. The latter triad sounds more like a xenharmonic version of a diminished triad, and could not be confused with simpler diminished triads such as 5:6:7.
13/11 is the classic mediant between the simpler and more familiar ratios 6/5 and 7/6, as it can be given as (6+7)/(5+6). This puts in between the latter ratios, slightly closer to 7/6. More complex minor thirds can be generated by taking the mediant between 13/11 and 7/6 (which yields (13+7)/(11+6) = 20/17, the septendecimal subminor third, about 281.4¢) and between 13/11 and 6/5 (which yields (13+6)/(11+5) = 19/16, the overtone minor third of 19-limit JI, about 297.5¢). (See the diagram below.)
| subminor and minor third | 7/6 266.9¢ |
6/5 315.6¢ | |||||||
|---|---|---|---|---|---|---|---|---|---|
| interval in between | << | 36:35 48.7¢ |
>> | ||||||
| add mediant (13/11) | 7/6 266.9¢ |
13/11 289.2¢ |
6/5 315.6¢ | ||||||
| intervals in between | << | 78:77 22.3¢ |
>> | << | 66:65 26.4¢ |
>> | |||
| add mediants (20/17 and 19/16) | 7/6 266.9¢ |
20/17 281.4¢ |
13/11 289.2¢ |
19/16 297.5¢ |
6/5 315.6¢ | ||||
| intervals in between | << 120:119 >> 14.5¢ |
<< 221:220 >> 7.9¢ |
<< 209:208 >> 8.3¢ |
<< 96:95 >> 18.1¢ |
|||||
Approximation
This interval is well approximated by 4\17 (282.353 cents), and even better, by 7\29 (289.655 cents).
See also
- 22/13 – its octave complement
- 33/26 – its fifth complement
- 44/39 – its fourth complement
- Gallery of just intervals
- Gentle chords
- List of root-3rd-P5 triads in JI
- File:Ji-13-11-csound-foscil-220hz.mp3 – another sound example
External links
- The Noble Mediant by Margo Schulter and David Keenan, the earliest description of 13/11 as the "neo-Gothic" minor third