14/11: Difference between revisions

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In [[11-limit]] [[just intonation]], '''14/11''' is the '''undecimal major third''' or '''pentacircle major third''', a supermajor third of about 417.5¢. It represents the difference between the 11th and 14th harmonics of the [[harmonic series]].

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fourth, as it is ~~the~~ [[4/3|perfect fourth (4/3)]] minus 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¢) 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 (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¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a major third, hence the names.

14/11 can 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¢. 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.

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 In [[11-limit]] [[just intonation]], '''14/11''' is the '''undecimal major third''' or '''pentacircle major third''', a supermajor third of about 417.5¢. It represents the difference between the 11th and 14th harmonics of the [[harmonic series]].
-In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fourth, as it is the [[4/3|perfect fourth (4/3)]] minus 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¢) 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 (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¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a major third, hence the names.
 /11 can 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¢. 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.