11/8: Difference between revisions

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| Monzo = -3 0 0 0 1
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Cents = 551.31794
| Name = undecimal superfourth, <br>major fourth
| Name = undecimal superfourth, <br>major fourth, <br>Alpharabian paramajor fourth, <br>just paramajor fourth
| Color name = 1o4, ilo 4th
| Color name = 1o4, ilo 4th
| FJS name = P4<sup>11</sup>
| FJS name = P4<sup>11</sup>
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}}
}}


In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&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|&cent;]]. This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the "'''Alpharabian paramajor fourth'''" or even the "'''just paramajor fourth'''".
 
This interval is the simplest superfourth in JI, and, falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. 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 ==
== Approximations by EDOs ==
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[[Category:Superfourth]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Overtone]]
[[Category:Over-2]]
[[Category:Over-2]]

Revision as of 06:03, 27 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,
Alpharabian paramajor fourth,
just paramajor 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¢. This interval has also been referred to as the major fourth. Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality), it can also be somewhat similarly dubbed the "Alpharabian paramajor fourth" or even the "just paramajor fourth".

This interval is the simplest superfourth in JI, and, falling about halfway between 12edo's perfect fourth and tritone, it is very xenharmonic. 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 ↑
  1. ↑ error magnitude below 7, both, absolute (in ¢) and relative (in r¢)
  2. ↑ Super EDOs up to 200 within the same error tolerance

See also

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