11/8: Difference between revisions

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== 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.

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''' – 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.

== 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 (↑) or flat (↓).

Following [[EDO]]s (up to 200) contain good approximations<ref group="note">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 (↑) or flat (↓).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"

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! Relative <br> error ([[Relative cent|r¢]])

! ↕

! class="unsortable" | Equally acceptable multiples <ref>~~Super~~ EDOs up to 200 within the same error tolerance</ref>

! class="unsortable" | Equally acceptable multiples <ref group="note">EDOs up to 200 within the same error tolerance</ref>

|-

| [[11edo|11]] || 5\11 || 5.8634 || 5.3748 || ↓ ||

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|-

|}

== See also ==

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

* [[Gallery of just intervals]]

== References ==

[[Category:Fourth]]

[[Category:Superfourth]]

[[Category:Alpharabian]]

@@ Line 19: / Line 19: @@
 == 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.
+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''' – 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.
 == 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;).
+Following [[EDO]]s (up to 200) contain good approximations<ref group="note">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"
@@ Line 32: / Line 35: @@
 ! Relative <br> error ([[Relative cent|r¢]])
 ! &#8597;
-! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
+! class="unsortable" | Equally acceptable multiples <ref group="note">EDOs up to 200 within the same error tolerance</ref>
 |-
 |  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &darr; ||
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 |-
 |}
+<references group="note" />
-<references />
 == See also ==
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 * [[12/11]] – its [[fifth complement]]
 * [[Gallery of just intervals]]
+== References ==
+<references />
 [[Category:Fourth]]
 [[Category:Superfourth]]
 [[Category:Alpharabian]]