S-expression: Difference between revisions

In mathematics, a '''square [[superparticular]] ratio''', also called a ''square-particular ratio'', is the ratio of two consecutive integer numbers whose higher one is a square number (3:4, 8:9, 15:16...).

This section concerns commas of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) = ( (k-1)/(k-2) )/( (k+2)/(k+1) ) which therefore do not (directly) involve the kth harmonic. We can check their general algebraic expression for any potential simplifications:

In other words, what this shows is all 1/3-square-particulars of the form S(''k'' - 1) * S''k'' * S(''k'' + 1) are superparticular iff ''k'' is throdd (not a multiple of 3), and all 1/3-square-particulars of the form S(3''k'' - 1) * S(3''k'') * S(3''k'' + 1) are throdd-particular with the numerator and denominator always being one less than a multiple of 3 (which is to say, commas of this form are throdd-particular iff k is threven and superparticular iff k is throdd).

This table has been expanded following every ultraparticular from S2/S3 to S16/S17 having its own page, with S12/S13 and S13/S14 expected to have their own pages soon. Note that ultraparticulars are, in general, extremely precise commas so that usually one wouldn't consider tempering them directly rather than through tempering the square-particulars S''k'' which they are composed of. As an example of this, notice that [[4000/3993|S10/S11]] is the largest ultraparticular categorised as an [[unnoticeable comma]], which means not unnoticeable in the absolute sense but rather in the sense of being smaller than the melodic just-noticeable difference, despite only dividing a superparticular as simple and unremarkable as [[4/3]]. For this reason, a [[cent]]s column has been included to aid an appreciation of their precision. The cent value of a [[semiparticular]] is roughly double that of any of the two ultraparticulars it is composed of; this becomes more true the higher you go.