11/8: Difference between revisions

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In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|¢]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.
In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|¢]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.
== Approximations by EDOs ==
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 11/8. 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>
|-
|  [[11edo|11]]  ||  5\11  || 5.8634 || 5.3748 || &darr; ||
|-
|  [[13edo|13]]  ||  6\13  || 2.5282 || 2.7389 || &uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &uarr; ||
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &darr; ||
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &uarr; ||
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &darr; ||
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &uarr; ||
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &darr; ||
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &uarr; ||
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &darr; ||
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &uarr; ||
|-
|}


== See also ==
== See also ==
* [[16/11]] – its [[octave complement]]
* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[12/11]] – its [[fifth complement]]

Revision as of 14:12, 25 October 2020

Interval information
Ratio 11/8
Factorization 2-3 × 11
Monzo [-3 0 0 0 1⟩
Size in cents 551.3179¢
Names undecimal superfourth,
major fourth
Color name 1o4, ilo 4th
FJS name [math]\displaystyle{ \text{P4}^{11} }[/math]
Special properties reduced,
reduced harmonic
Tenney norm (log2 nd) 6.45943
Weil norm (log2 max(n, d)) 6.91886
Wilson norm (sopfr(nd)) 17

[sound info]
Open this interval in xen-calc

In 11-limit just intonation, 11/8 is an undecimal superfourth of about 551.3¢. Falling about halfway between 12edo's perfect fourth and tritone, it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

Approximations by EDOs

Following EDOs (up to 200) contain good approximations[1] of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (↑) or flat (↓).

EDO deg\edo Absolute
error (¢) 		Relative
error (r¢) 		↕ Equally acceptable multiples [2]
11 5\11 5.8634 5.3748 ↓
13 6\13 2.5282 2.7389 ↑ 12\26
24 11\24 1.3179 2.6359 ↓ 22\48
37 17\37 0.0334 0.1030 ↑ 34\74, 51\111, 68\148, 85\185
50 23\50 0.6821 2.8419 ↑ 46\100
61 28\61 0.4983 2.5329 ↓ 56\122
63 29\63 1.0630 5.5808 ↑
85 39\85 0.7297 5.1688 ↓
87 40\87 0.4062 2.9449 ↑ 80\174
98 45\98 0.2975 2.4299 ↓ 90\196
124 57\124 0.2950 3.0479 ↑
135 62\135 0.2068 2.3269 ↓
137 63\137 0.5069 5.7868 ↑
159 73\159 0.3745 4.9627 ↓
161 74\161 0.2349 3.1509 ↑
172 79\172 0.1552 2.2238 ↓
198 91\198 0.1972 3.2540 ↑

See also

  1. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  2. ↑ Super EDOs up to 200 within the same error tolerance
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