11/8: Difference between revisions

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

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

|-

! [[EDO]]

! class="unsortable" | deg\edo

! Absolute <br> error ([[Cent|¢]])

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

! ↕

! 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 || ↓ ||

|-

| [[13edo|13]] || 6\13 || 2.5282 || 2.7389 || ↑ || [[26edo|12\26]]

|-

| [[24edo|24]] || 11\24 || 1.3179 || 2.6359 || ↓ || [[48edo|22\48]]

|-

| [[37edo|37]] || 17\37 || 0.0334 || 0.1030 || ↑ || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]

|-

| [[50edo|50]] || 23\50 || 0.6821 || 2.8419 || ↑ || [[100edo|46\100]]

|-

| [[61edo|61]] || 28\61 || 0.4983 || 2.5329 || ↓ || [[122edo|56\122]]

|-

| [[63edo|63]] || 29\63 || 1.0630 || 5.5808 || ↑ ||

|-

| [[85edo|85]] || 39\85 || 0.7297 || 5.1688 || ↓ ||

|-

| [[87edo|87]] || 40\87 || 0.4062 || 2.9449 || ↑ || [[174edo|80\174]]

|-

| [[98edo|98]] || 45\98 || 0.2975 || 2.4299 || ↓ || [[196edo|90\196]]

|-

| [[124edo|124]] || 57\124 || 0.2950 || 3.0479 || ↑ ||

|-

| [[135edo|135]] || 62\135 || 0.2068 || 2.3269 || ↓ ||

|-

| [[137edo|137]] || 63\137 || 0.5069 || 5.7868 || ↑ ||

|-

| [[159edo|159]] || 73\159 || 0.3745 || 4.9627 || ↓ ||

|-

| [[161edo|161]] || 74\161 || 0.2349 || 3.1509 || ↑ ||

|-

| [[172edo|172]] || 79\172 || 0.1552 || 2.2238 || ↓ ||

|-

| [[198edo|198]] || 91\198 || 0.1972 || 3.2540 || ↑ ||

|-

|}

== See also ==

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

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

@@ Line 17: / Line 17: @@
 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.
+== 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;).
+{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
+|-
+! [[EDO]]
+! 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; ||
+|-
+|}
 == See also ==
 * [[16/11]] – its [[octave complement]]
 * [[12/11]] – its [[fifth complement]]