4096/3993: Difference between revisions

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{{Infobox Interval

~~| Icon =~~

| Name = Alpharabian paralimma, Alpharabian paradiatonic semilimma, Alpharabian semilimmic inframinor second

~~| Ratio = 4096/3993~~

| Color name = s1u<sup>3</sup>2, satrilu 2nd

~~| Monzo = 12 -1 0 0 -3~~

| Comma = yes

~~| Cents = 44.09117~~

| Name = Alpharabian paralimma, ~~<br>~~ Alpharabian inframinor second

| Color name = ~~Satrilu 2nd~~

~~| FJS name = M2~~<~~sub~~>~~11,11,11~~</~~sub~~>

| ~~Sound~~ =

}}

'''4096/3993''', the '''Alpharabian ~~inframinor second~~''', is ~~notable for~~ being ~~one~~ of ~~only two [[11-limit]] quartertone intervals needed in order to add up to a familiar~~ [[9/8]] ~~whole tone. Specifically, it is~~ the ~~quartertone that forms the difference between~~ the ~~whole tone and a stack of three~~ [[33/32]] ~~quartertones, and can thus be regarded as a sort of subminor second. Remarkably~~, ~~it is currently~~ the ~~simplest~~ interval ~~in terms~~ of ~~odd-limit that is known to result from stacking three identical quartertones with rational intervals and subtracting said stack from a 9/8 whole tone. Furthermore, although [[38/37]], [[35/34]]~~, [[~~32/31~~]] and [[28/27]] are all simpler intervals that can be called "quarter tones" and can safely be regarded as some kind of second, subtracting any one of these intervals from 9/8 yields an interval that has a ratio lacking a cubed number in the numerator and or the denominator, and such an interval cannot be split into three equal quartertones with rational intervals.

'''4096/3993''', the '''Alpharabian paralimma''' or '''Alpharabian paradiatonic semilimma''', is only just shy of being half of [[256/243]]- the Pythagorean limma- being separated from the nearby [[1331/1296]], the interval forming the other part of the Pythagorean limma, by the [[nexus comma]].

It is also notable for being one of only two [[quartertone]] intervals in the [[11-limit]]- specifically the 2.3.11 [[subgroup]]- needed in order to add up to a familiar [[9/8]] whole tone. Specifically, it is the quartertone that forms the difference between the whole tone and a stack of three [[33/32]] quartertones, and can thus be regarded as being some sort of second- specifically, the '''Alpharabian semilimmic inframinor second''', not to be confused with [[8192/8019]], the Alpharabian inframinor second, as the two intervals are only equated when [[243/242]], the rastma, is tempered out.

Remarkably, 4096/3993 is currently the simplest interval in terms of [[odd-limit]] that is known to result from stacking three identical quartertones with rational intervals and subtracting said stack from a 9/8 whole tone. Furthermore, although [[38/37]], [[35/34]], [[32/31]] and [[28/27]] are all simpler intervals that can be called "quarter tones" and can safely be regarded as some kind of second, subtracting any one of these intervals from 9/8 yields an interval that has a ratio lacking a cubed number in the numerator and or the denominator, and such an interval cannot be split into three equal quartertones with rational intervals.

== Temperaments ==

[[Tempering out]] the paralimma in the 2.3.11 subgroup results in [[No-fives subgroup temperaments#Paralimmal|paralimmal]] temperament, where [[3/1]] is divided into 3 flat [[16/11]] generators.

== See also ==

* [[3993/2048]] – its [[octave complement]]

* [[1331/1024]] – its [[fourth complement]]

* [[Gallery of just intervals]]

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

[[Category:Quartertone]]

[[Category:Alpharabian]]

[[Category:~~Interval~~]]

[[Category:Commas named after polymaths]]

[[Category:Commas named after their interval size]]

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 {{Infobox Interval
-| Icon =
+| Name = Alpharabian paralimma, Alpharabian paradiatonic semilimma, Alpharabian semilimmic inframinor second
-| Ratio = 4096/3993
+| Color name = s1u<sup>3</sup>2, satrilu 2nd
-| Monzo = 12 -1 0 0 -3
+| Comma = yes
-| Cents = 44.09117
-| Name = Alpharabian paralimma, <br> Alpharabian inframinor second
-| Color name = Satrilu 2nd
-| FJS name = M2<sub>11,11,11</sub>
-| Sound =
 }}
-'''4096/3993''', the '''Alpharabian inframinor second''', is notable for being one of only two [[11-limit]] quartertone intervals needed in order to add up to a familiar [[9/8]] whole tone. Specifically, it is the quartertone that forms the difference between the whole tone and a stack of three [[33/32]] quartertones, and can thus be regarded as a sort of subminor second.  Remarkably, it is currently the simplest interval in terms of odd-limit that is known to result from stacking three identical quartertones with rational intervals and subtracting said stack from a 9/8 whole tone.  Furthermore, although [[38/37]], [[35/34]], [[32/31]] and [[28/27]] are all simpler intervals that can be called "quarter tones" and can safely be regarded as some kind of second, subtracting any one of these intervals from 9/8 yields an interval that has a ratio lacking a cubed number in the numerator and or the denominator, and such an interval cannot be split into three equal quartertones with rational intervals.
+'''4096/3993''', the '''Alpharabian paralimma''' or '''Alpharabian paradiatonic semilimma''', is only just shy of being half of [[256/243]]- the Pythagorean limma- being separated from the nearby [[1331/1296]], the interval forming the other part of the Pythagorean limma, by the [[nexus comma]].
+It is also notable for being one of only two [[quartertone]] intervals in the [[11-limit]]- specifically the 2.3.11 [[subgroup]]- needed in order to add up to a familiar [[9/8]] whole tone.  Specifically, it is the quartertone that forms the difference between the whole tone and a stack of three [[33/32]] quartertones, and can thus be regarded as being some sort of second- specifically, the '''Alpharabian semilimmic inframinor second''', not to be confused with [[8192/8019]], the Alpharabian inframinor second, as the two intervals are only equated when [[243/242]], the rastma, is tempered out.
+Remarkably, 4096/3993 is currently the simplest interval in terms of [[odd-limit]] that is known to result from stacking three identical quartertones with rational intervals and subtracting said stack from a 9/8 whole tone.  Furthermore, although [[38/37]], [[35/34]], [[32/31]] and [[28/27]] are all simpler intervals that can be called "quarter tones" and can safely be regarded as some kind of second, subtracting any one of these intervals from 9/8 yields an interval that has a ratio lacking a cubed number in the numerator and or the denominator, and such an interval cannot be split into three equal quartertones with rational intervals.
+== Temperaments ==
+[[Tempering out]] the paralimma in the 2.3.11 subgroup results in [[No-fives subgroup temperaments#Paralimmal|paralimmal]] temperament, where [[3/1]] is divided into 3 flat [[16/11]] generators.
+== See also ==
+* [[3993/2048]] – its [[octave complement]]
+* [[1331/1024]] – its [[fourth complement]]
+* [[Gallery of just intervals]]
-[[Category:11-limit]]
 [[Category:Quartertone]]
 [[Category:Alpharabian]]
-[[Category:Interval]]
+[[Category:Commas named after polymaths]]
+[[Category:Commas named after their interval size]]