11/7: Difference between revisions

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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.
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.


== 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;).
 
{| 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/>
<references/>

Revision as of 07:36, 3 November 2025

Interval information
Ratio 11/7
Factorization 7-1 × 11
Monzo [0 0 0 -1 1⟩
Size in cents 782.492¢
Names undecimal minor sixth,
pentacircle minor sixth
Color name 1or5, loru 5th
FJS name [math]\displaystyle{ \text{P5}^{11}_{7} }[/math]
Special properties reduced
Tenney norm (log2 nd) 6.26679
Weil norm (log2 max(n, d)) 6.91886
Wilson norm (sopfr(nd)) 18

[sound info]
Open this interval in xen-calc

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 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. However, it is only flat of the Pythagorean (3-limit) minor sixth of 128/81 (about 792.2¢) by a pentacircle comma (896/891), which makes it function more often as a minor sixth, hence the names.

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.

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.

Approximation

Edo approximations for 11/7 (782.49 ¢)
≤ 80edo, relative error ≤ 10%
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 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

Retrieved from "https://en.xen.wiki/index.php?title=11/7&oldid=215871"