14/11: Difference between revisions

Line 7:

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 ~~more often~~ as a major third, hence the names.

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 ~~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.

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.

Despite being around a third in size, due to being notated as an imperfect fourth in many systems, many chords involving it are awkward to notate in terms of diatonic degrees. For example, in [[11:14:16:20]], if [[16/11]] is considered a type of fifth and 14/11 is considered a type of third, then [[8/7]] would be considered a type of third, which is awkward due to its size, but actually makes sense if its inversion 7/4 sometimes considered a sixth. It also appears in the chord [[7:9:11:14]], which consists of three thirds stacked to an octave, but in a diatonic system three thirds would stack to a type of seventh, so one of the intervals would have to be considered a type of fourth instead. In FJS and other systems, 14/11 is considered an imperfect fourth, but 9/7 is wider than it, so it may be considered a diminished fourth in this context instead. In short, this interval shows that diatonic interval classification is far from perfect, and such ambiguity also occurs with [[13/10]] and [[7/5]].

~~It also appears in chords such as 8:11:14, the principal triad of [[orgone]] temperament~~.

== Approximation ==

Line 27:

Line 23:

== External links ==

* [http://dkeenan.com/Music/NobleMediant.txt ''The Noble Mediant''] by Margo Schulter and David Keenan

== Notes ==

[[Category:Third]]

@@ Line 7: / Line 7: @@
 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 more often as a major third, hence the names.
+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.
-/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 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.
+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.
-Despite being around a third in size, due to being notated as an imperfect fourth in many systems, many chords involving it are awkward to notate in terms of diatonic degrees. For example, in [[11:14:16:20]], if [[16/11]] is considered a type of fifth and 14/11 is considered a type of third, then [[8/7]] would be considered a type of third, which is awkward due to its size, but actually makes sense if its inversion 7/4 sometimes considered a sixth. It also appears in the chord [[7:9:11:14]], which consists of three thirds stacked to an octave, but in a diatonic system three thirds would stack to a type of seventh, so one of the intervals would have to be considered a type of fourth instead. In FJS and other systems, 14/11 is considered an imperfect fourth, but 9/7 is wider than it, so it may be considered a diminished fourth in this context instead. In short, this interval shows that diatonic interval classification is far from perfect, and such ambiguity also occurs with [[13/10]] and [[7/5]].
-It also appears in chords such as 8:11:14, the principal triad of [[orgone]] temperament.
 == Approximation ==
@@ Line 27: / Line 23: @@
 == External links ==
 * [http://dkeenan.com/Music/NobleMediant.txt ''The Noble Mediant''] by Margo Schulter and David Keenan
+== Notes ==
+<references group="note"/>
 [[Category:Third]]