11/8: Difference between revisions

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{{Infobox Interval

| Name = undecimal superfourth, undecimal ~~neutral fourth, harmonic neutral~~ fourth, harmonic semiaugmented ~~fourth, harmonic semiperfect~~ fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth

| Name = undecimal superfourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth

| Color name = 1o4, ilo 4th

| Sound = jid_11_8_pluck_adu_dr220.mp3

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In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' 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'' ~~(a.k.a. ''imperfect'')~~ version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the ~~'''harmonic semiaugmented/semiperfect fourth''', or~~ '''harmonic/undecimal neutral fourth''' if you prefer to generalise the naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave-complements (which is also rigorous). 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'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. 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. 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.

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' 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/undecimal 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).

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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''. 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''' – see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. 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. 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.

== Approximations by EDOs ==

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 }}
 {{Infobox Interval
-| Name = undecimal superfourth, undecimal neutral fourth, harmonic neutral fourth, harmonic semiaugmented fourth, harmonic semiperfect fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
+| Name = undecimal superfourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
 | Color name = 1o4, ilo 4th
 | Sound = jid_11_8_pluck_adu_dr220.mp3
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 {{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 is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' (a.k.a. ''imperfect'') version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented/semiperfect fourth''', or '''harmonic/undecimal neutral fourth''' if you prefer to generalise the naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave-complements (which is also rigorous). 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'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. 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. 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.
+In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' 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/undecimal 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).
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 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''. 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''' – see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. 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. 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.
 == Approximations by EDOs ==