14/11: Difference between revisions
"Neogothic" is used as an interval region these days. |
Rework as many chord examples here are substructures of the harmonic eleventh chord, where notating 14/11 as a fourth makes sense. The assumption that fourths must be greater than thirds is a loaded one. |
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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 [[cent]]s. It represents the difference between the 11th and 14th harmonics of the [[harmonic series]]. | 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 [[cent]]s. It represents the difference between the 11th and 14th harmonics of the [[harmonic series]]. | ||
In many notation systems based on the [[5L 2s|diatonic]] [[chain-of-fifths notation]] with commatic alterations (e.g. [[FJS]], [[HEJI]]), it is an imperfect fourth, as it is a [[4/3|perfect fourth (4/3)]] minus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only sharp of the Pythagorean ([[3-limit]]) major third of [[81/64]] (about 407.8{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function | In many notation systems based on the [[5L 2s|diatonic]] [[chain-of-fifths notation]] with commatic alterations (e.g. [[FJS]], [[HEJI]]), it is an imperfect fourth, as it is a [[4/3|perfect fourth (4/3)]] minus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|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{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function sometimes as a major third, hence the names. | ||
14/11 is the simplest [[neogothic major and minor|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{{c}}) and between 14/11 and 9/7 (which is (14 + 9)/(11 + 7) = [[23/18]], about 424.4{{c}}. The fact that | Indeed, 14/11 is the simplest [[neogothic major and minor|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]]<ref group="note">This is a [[minor minthmic chord]] where 14/11 and [[13/11]] sum to a perfect fifth. Shown here is the simplest JI representation. </ref>. 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{{c}}) and between 14/11 and 9/7 (which is (14 + 9)/(11 + 7) = [[23/18]], about 424.4{{c}}). 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 == | == Approximation == | ||
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== External links == | == External links == | ||
* [http://dkeenan.com/Music/NobleMediant.txt ''The Noble Mediant''] by Margo Schulter and David Keenan | * [http://dkeenan.com/Music/NobleMediant.txt ''The Noble Mediant''] by Margo Schulter and David Keenan | ||
== Notes == | |||
<references group="note"/> | |||
[[Category:Third]] | [[Category:Third]] | ||
Revision as of 09:01, 16 January 2026
| Interval information |
pentacircle major third
[sound info]
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.
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
- 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
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.