11/7: Difference between revisions

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In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]].

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5 [[cent]]s. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]].

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus 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 flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names.

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus 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 flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. It functions as such in voicings of the harmonic eleventh chord, [[4:5:6:7:9:11]].

It is flat of the ~~5-limit minor sixth of~~ [[8/5]] (about ~~813~~.7¢) by [[56/~~55]]. It is sharp of the 7-limit subminor sixth of [[14~~/9]] ~~(about 764.9¢) by~~ a ~~mothwellsma~~, ~~[[99/98]]~~. ~~And finally~~, it is ~~sharp of the classic augmented fifth of [[25~~/~~16]]~~ (about ~~772~~.6¢) by a ~~valinorsma, [[176/175]]~~.

However, it is only flat of the [[128/81|Pythagorean minor sixth]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function sometimes as a minor sixth, hence the names. For one thing, as it is 22/21 (about 80.5{{c}}) above the perfect fifth, it can be resolved down by a step to the perfect fifth.

~~As 11~~/7 is 22/21 (about 80.5¢) ~~above~~ the ~~perfect~~ fifth~~, it can be resolved down~~ by a ~~step from 11/7 to 3~~/2.

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7{{c}}) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9{{c}}) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6{{c}}) by a valinorsma, [[176/175]].

== Approximation ==

~~<references/>~~

== Proximity with acoustic pi ==

[[22/7]], one octave higher, is a fraction convergent to the continued fraction of acoustic pi. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== Proximity with π ==

[[22/7]], one octave higher, is a fraction convergent to the continued fraction of π. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==

* [[14/11]] – its octave complement

* [[14/11]] – its [[octave complement]]

* [[21/11]] – its [[twelfth complement]]

* [[Ed11/7]]

* [[Gallery of just intervals]]

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 }}
-In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]].
+In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5 [[cent]]s. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]].
-In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus 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 flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names.
+In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus 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 flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. It functions as such in voicings of the harmonic eleventh chord, [[4:5:6:7:9:11]].
-It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]].
+However, it is only flat of the [[128/81|Pythagorean minor sixth]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function sometimes as a minor sixth, hence the names. For one thing, as it is 22/21 (about 80.5{{c}}) above the perfect fifth, it can be resolved down by a step to the perfect fifth.
-As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.
+It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7{{c}}) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9{{c}}) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6{{c}}) by a valinorsma, [[176/175]].
 == Approximation ==
 {{Interval edo approximation|11/7}}
-<references/>
+== Proximity with acoustic pi ==
+[[22/7]], one octave higher, is a fraction convergent to the continued fraction of acoustic pi. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.
-== Proximity with π ==
-[[22/7]], one octave higher, is a fraction convergent to the continued fraction of π. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.
 == See also ==
-* [[14/11]] – its octave complement
+* [[14/11]] – its [[octave complement]]
+* [[21/11]] – its [[twelfth complement]]
 * [[Ed11/7]]
 * [[Gallery of just intervals]]