14/11: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Icon =
| Name = undecimal major third, pentacircle major third
| Ratio = 14/11
| Monzo = 1 0 0 1 -1
| Cents = 417.50796
| Name = undecimal major third
| Color name = 1uz4, luzo 4th
| Color name = 1uz4, luzo 4th
| FJS name = P4<sup>7</sup><sub>11</sub>
| Sound = jid_14_11_pluck_adu_dr220.mp3
| Sound = jid_14_11_pluck_adu_dr220.mp3
}}
}}


In [[11-limit]] [[just intonation]], '''14/11''' is the '''undecimal major third''', a supermajor third of about 417.5¢. It represents the difference between the 11th and 14th overtones of the [[harmonic series]] and appears in chords such as 8:11:14, the principal triad of [[Orgonia|orgone]] temperament. 14/11 can also function as a [[neogothic|neo-Gothic]] major third, as it falls between [[5/4]] and [[9/7]]. Indeed, it is the [[mediant]] ratio between those simpler intervals, as it is (5+9)/(4+7), and is [[56/55]] sharp of [[5/4]], [[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¢. Also in this region is the Pythagorean ([[3-limit]]) major third of [[81/64]] (about 407.8¢), of which 14/11 is sharp by [[896/891]].  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.
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]]. 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. 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 chords|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 ==
{{Interval edo approximation|14/11}}


== See also ==
== See also ==
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* [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


[[Category:11-limit]]
== Notes ==
[[Category:Interval ratio]]
<references group="note"/>
 
[[Category:Third]]
[[Category:Major third]]
[[Category:Major third]]
[[Category:Supermajor third]]
[[Category:Supermajor third]]
[[Category:Third]]
[[Category:Over-11 intervals]]
[[Category:Over-11]]
[[Category:Pentacircle]]
[[Category:Pentacircle]]
[[Category:Gentle]]
[[Category:Gentle]]
[[Category:Neo-gothic]]
[[Category:Neo-gothic]]

Latest revision as of 09:03, 16 January 2026

Interval information
Ratio 14/11
Factorization 2 × 7 × 11-1
Monzo [1 0 0 1 -1⟩
Size in cents 417.508¢
Names undecimal major third,
pentacircle major third
Color name 1uz4, luzo 4th
FJS name [math]\displaystyle{ \text{P4}^{7}_{11} }[/math]
Special properties reduced
Tenney norm (log2 nd) 7.26679
Weil norm (log2 max(n, d)) 7.61471
Wilson norm (sopfr(nd)) 20

[sound info]
Open this interval in xen-calc

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 approximations for 14/11 (417.51 ¢)
≤ 80edo, relative error ≤ 10%
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

Notes

  1. ↑ This is a minor minthmic chord where 14/11 and 13/11 sum to a perfect fifth. Shown here is the simplest JI representation.
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