11/8: Difference between revisions

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{{interwiki

| de = Alphorn-Fa

| en = 11/8

| es =

| ja =

}}

{{Infobox Interval

| ~~Icon~~ =

| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth

| ~~Ratio = 11/8~~

| Color name = 1o4, ilo 4th

~~| Monzo~~ = ~~-3 0 0 0 1~~

~~| Cents = 551.31794~~

~~| Name = undecimal superfourth~~

| Sound = jid_11_8_pluck_adu_dr220.mp3

}}

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 stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== 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 ==

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]].

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".

== Approximations by EDOs ==

In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[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 (5) and 12 (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.

== See also ==

~~:''See also~~ [[Gallery of ~~Just Intervals~~]]''

* [[16/11]] – its [[octave complement]]

* [[12/11]] – its [[fifth complement]]

* [[Gallery of just intervals]]

~~[[Category:11-limit]]~~

== References ==

~~[[Category:fourth]]~~

~~[[Category:interval]]~~

~~[[Category:ratio]]~~

~~[[Category:superfourth]]~~

~~[[Category:undecimal]]~~

~~[[Category:untwelve]]~~

~~~~

[[Category:Fourth]]

[[de:~~Alphorn-Fa~~]]

[[Category:Superfourth]]

[[Category:Alpharabian]]

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+{{interwiki
+| de = Alphorn-Fa
+| en = 11/8
+| es =
+| ja =
+}}
 {{Infobox Interval
-| Icon =
+| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
-| Ratio = 11/8
+| Color name = 1o4, ilo 4th
-| Monzo = -3 0 0 0 1
-| Cents = 551.31794
-| Name = undecimal superfourth
 | 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|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 stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).
+== 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 ==
+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]].
+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".
+== Approximations by EDOs ==
+{{Interval edo approximation|11/8}}
+<references group="note" />
-In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[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 (5) and 12 (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.
+== See also ==
-:''See also [[Gallery of Just Intervals]]''
+* [[16/11]] – its [[octave complement]]
+* [[12/11]] – its [[fifth complement]]
+* [[Gallery of just intervals]]
-[[Category:11-limit]]
+== References ==
-[[Category:fourth]]
+<references />
-[[Category:interval]]
-[[Category:ratio]]
-[[Category:superfourth]]
-[[Category:undecimal]]
-[[Category:untwelve]]
-<!-- interwiki -->
+[[Category:Fourth]]
-[[de:Alphorn-Fa]]
+[[Category:Superfourth]]
+[[Category:Alpharabian]]