11/7: Difference between revisions
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Rework; +"pentacircle minor sixth" and reasons; -"undecimal augmented fifth" (its octave complement isn't "undecimal diminished fourth" yet) |
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In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782. | 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]]. | 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]]. | ||
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. | |||
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}} | |||
== 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. | |||
== See also == | == See also == | ||
* [[14/11]] – its octave complement | * [[14/11]] – its [[octave complement]] | ||
* [[21/11]] – its [[twelfth complement]] | |||
* [[Ed11/7]] | |||
* [[Gallery of just intervals]] | * [[Gallery of just intervals]] | ||
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example | * [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example | ||
| Line 76: | Line 31: | ||
[[Category:Subminor sixth]] | [[Category:Subminor sixth]] | ||
[[Category:Pentacircle]] | [[Category:Pentacircle]] | ||
[[Category:Taxicab-2 intervals]] | |||
Latest revision as of 00:34, 3 February 2026
| Interval information |
pentacircle minor sixth
[sound info]
In 11-limit just intonation, 11/7 is an undecimal minor sixth, specifically the pentacircle minor sixth, measuring about 782.5 cents. 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 perfect fifth (3/2) plus an instance of 22/21, which is a stack consisting of an undecimal quartertone (33/32) and a septimal comma (64/63), neither of which changes the scale degree or quality. It functions as such in voicings of the harmonic eleventh chord, 4:5:6:7:9:11.
However, it is only flat of the Pythagorean minor sixth (about 792.2 ¢) by a 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 ¢) above the perfect fifth, it can be resolved down by a step to the perfect fifth.
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.
Approximation
| Edo | Step size | Cents (¢) | Absolute error (¢) | Relative error (%) |
|---|---|---|---|---|
| 3 | 2\3 | 800.00 | +17.51 | +4.38 |
| 6 | 4\6 | 800.00 | +17.51 | +8.75 |
| 17 | 11\17 | 776.47 | -6.02 | -8.53 |
| 20 | 13\20 | 780.00 | -2.49 | -4.15 |
| 23 | 15\23 | 782.61 | +0.12 | +0.22 |
| 26 | 17\26 | 784.62 | +2.12 | +4.60 |
| 29 | 19\29 | 786.21 | +3.71 | +8.98 |
| 40 | 26\40 | 780.00 | -2.49 | -8.31 |
| 43 | 28\43 | 781.40 | -1.10 | -3.93 |
| 46 | 30\46 | 782.61 | +0.12 | +0.45 |
| 49 | 32\49 | 783.67 | +1.18 | +4.82 |
| 52 | 34\52 | 784.62 | +2.12 | +9.20 |
| 63 | 41\63 | 780.95 | -1.54 | -8.08 |
| 66 | 43\66 | 781.82 | -0.67 | -3.71 |
| 69 | 45\69 | 782.61 | +0.12 | +0.67 |
| 72 | 47\72 | 783.33 | +0.84 | +5.05 |
| 75 | 49\75 | 784.00 | +1.51 | +9.42 |
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.
See also
- 14/11 – its octave complement
- 21/11 – its twelfth complement
- Ed11/7
- Gallery of just intervals
- File:Ji-11-7-csound-foscil-220hz.mp3 – another sound example