13/11: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = tridecimal minor third, major minthmic minor third | | Name = tridecimal minor third, major-minthmic minor third | ||
| Color name = 3o1u3, tholu 3rd | | Color name = 3o1u3, tholu 3rd | ||
| Sound = jid_13_11_pluck_adu_dr220.mp3 | | Sound = jid_13_11_pluck_adu_dr220.mp3 | ||
}} | }} | ||
In [[13-limit]] [[just intonation]], '''13/11''' is a '''tridecimal minor third''', specifically the '''major minthmic minor third''', measuring about 289.2 [[cent]]s. It is the difference between the [[11/1|11th]] and [[13/1|13th]] [[harmonic]]s. The octave-reduced 11th harmonic ([[11/8]], about 551.3{{c}}) and 13th harmonic ([[13/8]], about 840.5{{c}}) 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 [[352/351|major minthma (352/351)]] narrower than the [[32/27|Pythagorean minor third (32/27)]]. It is the simplest [[neogothic major and minor|neogothic minor third]], and can function as such in a 13-limit neogothic minor triad of [[22:26:33]], with a [[3/2]] perfect fifth between 33 and 22<ref group="note">This is a [[minor minthmic chords|minor minthmic chord]] where 13/11 and [[14/11]] sum to a perfect fifth. Shown here is the simplest JI representation. </ref>. 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]]. | In [[13-limit]] [[just intonation]], '''13/11''' is a '''tridecimal minor third''', specifically the '''major-minthmic minor third''', measuring about 289.2 [[cent]]s. It is the difference between the [[11/1|11th]] and [[13/1|13th]] [[harmonic]]s. The octave-reduced 11th harmonic ([[11/8]], about 551.3{{c}}) and 13th harmonic ([[13/8]], about 840.5{{c}}) 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 [[352/351|major minthma (352/351)]] narrower than the [[32/27|Pythagorean minor third (32/27)]]. It is the simplest [[neogothic major and minor|neogothic minor third]], and can function as such in a 13-limit neogothic minor triad of [[22:26:33]], with a [[3/2]] perfect fifth between 33 and 22<ref group="note">This is a [[minor minthmic chords|minor minthmic chord]] where 13/11 and [[14/11]] sum to a perfect fifth. Shown here is the simplest JI representation. </ref>. 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 | 13/11 is the [[mediant (operation)|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 it 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{{c}}) 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{{c}}). See the diagram below. | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
Latest revision as of 14:45, 22 July 2026
| Interval information |
major-minthmic minor third
[sound info]
In 13-limit just intonation, 13/11 is a tridecimal minor third, specifically the major-minthmic minor third, measuring about 289.2 cents. 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 is the simplest neogothic minor third, and can function as such in a 13-limit neogothic minor triad of 22:26:33, with a 3/2 perfect fifth between 33 and 22[note 1]. 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 it 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).
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 4 | 1\4 | 300.00 | +10.79 | +3.60 |
| 8 | 2\8 | 300.00 | +10.79 | +7.19 |
| 17 | 4\17 | 282.35 | -6.86 | -9.71 |
| 21 | 5\21 | 285.71 | -3.50 | -6.12 |
| 25 | 6\25 | 288.00 | -1.21 | -2.52 |
| 29 | 7\29 | 289.66 | +0.45 | +1.08 |
| 33 | 8\33 | 290.91 | +1.70 | +4.67 |
| 37 | 9\37 | 291.89 | +2.68 | +8.27 |
| 46 | 11\46 | 286.96 | -2.25 | -8.64 |
| 50 | 12\50 | 288.00 | -1.21 | -5.04 |
| 54 | 13\54 | 288.89 | -0.32 | -1.44 |
| 58 | 14\58 | 289.66 | +0.45 | +2.15 |
| 62 | 15\62 | 290.32 | +1.11 | +5.75 |
| 66 | 16\66 | 290.91 | +1.70 | +9.35 |
| 75 | 18\75 | 288.00 | -1.21 | -7.56 |
| 79 | 19\79 | 288.61 | -0.60 | -3.96 |
See also
- 22/13 – its octave complement
- 33/26 – its fifth complement
- 44/39 – its fourth complement
- Ed13/11
- 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
Notes
- ↑ This is a minor minthmic chord where 13/11 and 14/11 sum to a perfect fifth. Shown here is the simplest JI representation.