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]]. 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.

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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]].  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.