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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]]. |
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| 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]]. |
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| 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. |
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| 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]]. |
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| == Approximations by edos == | | == Approximation == |
| | {{Interval edo approximation|11/7}} |
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| 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 (↑) or flat (↓).
| | == 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"
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| ! [[Edo]]
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| ! class="unsortable" | deg\edo
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| ! Absolute <br> error ([[Cent|¢]])
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| ! Relative <br> error ([[Relative cent|r¢]])
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| ! ↕
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| ! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
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| | [[20edo|20]] || 13\20 || 2.4920 || 4.1534 || ↓ ||
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| | [[23edo|23]] || 15\23 || 0.1167 || 0.2236 || ↑ || [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
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| | [[26edo|26]] || 17\26 || 2.1233 || 4.6006 || ↑ ||
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| | [[43edo|43]] || 28\43 || 1.0967 || 3.9298 || ↓ ||
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| | [[49edo|49]] || 32\49 || 1.1814 || 4.8242 || ↑ ||
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| | [[66edo|66]] || 43\66 || 0.6739 || 3.7062 || ↓ ||
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| | [[72edo|72]] || 47\72 || 0.8413 || 5.0478 || ↑ ||
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| | [[89edo|89]] || 58\89 || 0.4696 || 3.4826 || ↓ || [[178edo|116\178]]
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| | [[95edo|95]] || 62\95 || 0.6659 || 5.2714 || ↑ ||
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| | [[112edo|112]] || 73\112 || 0.3492 || 3.2590 || ↓ ||
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| | [[118edo|118]] || 77\118 || 0.5588 || 5.4950 || ↑ ||
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| | [[135edo|135]] || 88\135 || 0.2698 || 3.0354 || ↓ ||
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| | [[141edo|141]] || 92\141 || 0.4867 || 5.7186 || ↑ ||
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| | [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || ↓ ||
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| | [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || ↑ ||
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| | [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || ↓ ||
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| | [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || ↑ ||
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| |}
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| <references/>
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| == Proximity with π/2 ==
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| (11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only 0.7 cents.
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| == 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]] |