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}}. 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'''.

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).

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== Approximations by EDOs ==

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"~~

|-

~~! [[EDO]]~~

~~! class="unsortable" | deg\~~edo

~~! Absolute <br> error ([[Cent~~|~~¢]])~~

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

~~! ↕~~

~~! 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 || ↓ ||~~

|-

~~| [[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 || ↑ ||~~

|-

|}

@@ Line 12: / Line 12: @@
 {{Wikipedia|Major fourth and minor fifth}}
-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'''.
+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).
@@ Line 29: / Line 29: @@
 == Approximations by EDOs ==
+{{Interval edo approximation|11/8}}
-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"
-|-
-! [[EDO]]
-! class="unsortable" | deg\edo
-! Absolute <br> error ([[Cent|¢]])
-! Relative <br> error ([[Relative cent|r¢]])
-! &#8597;
-! 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; ||
-|-
-|  [[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 group="note" />