14/11: Difference between revisions
Expanded paragraph on diatonic classification |
"Neogothic" is used as an interval region these days. |
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In [[11-limit]] [[just intonation]], '''14/11''' is an '''undecimal major third''', specifically the '''pentacircle major third''', | 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 [[ | 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. | ||
14/11 | 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. | ||
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]]. | 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. | It also appears in chords such as 8:11:14, the principal triad of [[orgone]] temperament. | ||
== Approximation == | == Approximation == | ||
{{Interval edo approximation|14/11}} | {{Interval edo approximation|14/11}} | ||
== See also == | == See also == | ||
* [[11/7]] – its [[octave complement]] | * [[11/7]] – its [[octave complement]] | ||