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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]]. 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]].


== Approximations by edos ==
== Approximation ==
{{Interval edo approximation|11/7}}


Following [[edo]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&uarr;) or flat (&darr;).
== 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.
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute <br> error ([[Cent|¢]])
! Relative <br> error ([[Relative cent|r¢]])
! &#8597;
! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &uarr; ||
|}
 
<references/>
 
== Proximity with π/2 ==
(11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only 0.7 cents.


== See also ==
== See also ==
* [[14/11]] – its octave complement
* [[14/11]] – its [[octave complement]]
* [[21/11]] – its [[twelfth complement]]
* [[Ed11/7]]
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]
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[[Category:Subminor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]
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