4096/3993: Difference between revisions

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'''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.

'''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]].

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.

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#Paralimmic|paralimmic]] temperament, where [[3/1]] is divided into 3 flat [[16/11]] generators.

== See also ==

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

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

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-'''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.
+'''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]].
-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.
+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#Paralimmic|paralimmic]] temperament, where [[3/1]] is divided into 3 flat [[16/11]] generators.
 == See also ==
 * [[3993/2048]] – its [[octave complement]]
 * [[1331/1024]] – its [[fourth complement]]