Xenharmonic Wiki

11/8: Difference between revisions

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{{Infobox Interval
{{Infobox Interval
| Ratio = 11/8
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, <br>major fourth, <br>Axirabian paramajor fourth, <br>just paramajor fourth
| Color name = 1o4, ilo 4th
| Color name = 1o4, ilo 4th
| FJS name = P4<sup>11</sup>
| Sound = jid_11_8_pluck_adu_dr220.mp3
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
}}
{{Wikipedia|Major fourth and minor fifth}}


In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.
In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth|semiaugmented fourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented fourth'''.  


This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, 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).
This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, 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 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. Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths. Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].
== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts.
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.<ref>Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']</ref>
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.


== Potential usage ==


== Approximations by EDOs ==
This interval 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.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


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;).
In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (basically, a type of [[Wikipedia: Neapolitan chord|Neapolitan chord]]), and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence. This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".


{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
== Approximations by EDOs ==
|-
{{Interval edo approximation|11/8}}
! [[EDO]]
<references group="note" />
! 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; ||
|-
|}
 
<references />


== See also ==
== See also ==
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* [[Gallery of just intervals]]
* [[Gallery of just intervals]]


[[Category:11-limit]]
== References ==
<references />
 
[[Category:Fourth]]
[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]
[[Category:Alpharabian]]
[[Category:Octave-reduced harmonics]]
[[Category:Pages with internal sound examples]]
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