4096/3993: Difference between revisions

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'''4096/3993''', the '''Alpharabian ~~Subminor 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 ~~both p-limit and~~ 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.

{{Infobox Interval

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

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

| Comma = yes

}}

'''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:Quartertone]]

[[Category:Alpharabian]]

[[Category:Commas named after polymaths]]

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

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-'''4096/3993''', the '''Alpharabian Subminor 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 both p-limit and 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.
+{{Infobox Interval
+| Name = Alpharabian paralimma, Alpharabian paradiatonic semilimma, Alpharabian semilimmic inframinor second
+| Color name = s1u<sup>3</sup>2, satrilu 2nd
+| Comma = yes
+}}
+'''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:Quartertone]]
+[[Category:Alpharabian]]
+[[Category:Commas named after polymaths]]
+[[Category:Commas named after their interval size]]