S-expression: Difference between revisions

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== S''k''/S(''k'' + 2) (semiparticulars) ==

For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or "odd-particular", meaning an interval of the form (2''n'' + 1)/(2''n'' - 1) for some positive integer ''n''. (This terminology also suggests "throdd-particular" for intervals of the form (3''n'' + 2)/(3''n'') and (3''n'' + 1)/(3''n'' - 1) and maybe "quodd-particular" (sounding like "quad-particular") for (4''n'' + 3)/(4''n'' - 1) and (4''n'' + 1)/(4''n'' - 3).) Furthermore, S(''k'' + 1)/S(''k'' + 3), when tempered, implies that (''k'' + 4)/''k'' is divisible exactly into two halves of (''k'' + 3)/(''k'' + 1) which is equated with ((''k'' + 1)/''k'')((''k'' + 4)/(''k'' + 3)). It is for this reason that the indexing choice of S(''k'' + 1)/S(''k'' + 3) was chosen, as two (''k'' + 3)/(''k'' + 1)'s make a (''k'' + 4)/''k''. This form of comma does not yet have an official name, but a proposed name is "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular.

For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or "odd-particular", meaning an interval of the form (2''n'' + 1)/(2''n'' - 1) for some positive integer ''n''. (This terminology also suggests "throdd-particular" for intervals of the form (3''n'' + 1)/(3''n'' - 2) and (3''n'' + 2)/(3''n'' - 1) and maybe "quodd-particular" (sounding like "quad-particular") for (4''n'' + 3)/(4''n'' - 1) and (4''n'' + 1)/(4''n'' - 3).) Furthermore, S(''k'' + 1)/S(''k'' + 3), when tempered, implies that (''k'' + 4)/''k'' is divisible exactly into two halves of (''k'' + 3)/(''k'' + 1) which is equated with ((''k'' + 1)/''k'')((''k'' + 4)/(''k'' + 3)). It is for this reason that the indexing choice of S(''k'' + 1)/S(''k'' + 3) was chosen, as two (''k'' + 3)/(''k'' + 1)'s make a (''k'' + 4)/''k''. This form of comma does not yet have an official name, but a proposed name is "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular.

Specifically, S''k''/S(''k'' + 2) is superparticular when ''k'' is not a multiple of 4, and odd-particular otherwise. See [[#Mathematical derivation]] for details on this and other facts stated here.

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 == S''k''/S(''k'' + 2) (semiparticulars) ==
-For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or "odd-particular", meaning an interval of the form (2''n'' + 1)/(2''n'' - 1) for some positive integer ''n''. (This terminology also suggests "throdd-particular" for intervals of the form (3''n'' + 2)/(3''n'') and (3''n'' + 1)/(3''n'' - 1) and maybe "quodd-particular" (sounding like "quad-particular") for (4''n'' + 3)/(4''n'' - 1) and (4''n'' + 1)/(4''n'' - 3).) Furthermore, S(''k'' + 1)/S(''k'' + 3), when tempered, implies that (''k'' + 4)/''k'' is divisible exactly into two halves of (''k'' + 3)/(''k'' + 1) which is equated with ((''k'' + 1)/''k'')((''k'' + 4)/(''k'' + 3)). It is for this reason that the indexing choice of S(''k'' + 1)/S(''k'' + 3) was chosen, as two (''k'' + 3)/(''k'' + 1)'s make a (''k'' + 4)/''k''. This form of comma does not yet have an official name, but a proposed name is "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular.
+For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or "odd-particular", meaning an interval of the form (2''n'' + 1)/(2''n'' - 1) for some positive integer ''n''. (This terminology also suggests "throdd-particular" for intervals of the form (3''n'' + 1)/(3''n'' - 2) and (3''n'' + 2)/(3''n'' - 1) and maybe "quodd-particular" (sounding like "quad-particular") for (4''n'' + 3)/(4''n'' - 1) and (4''n'' + 1)/(4''n'' - 3).) Furthermore, S(''k'' + 1)/S(''k'' + 3), when tempered, implies that (''k'' + 4)/''k'' is divisible exactly into two halves of (''k'' + 3)/(''k'' + 1) which is equated with ((''k'' + 1)/''k'')((''k'' + 4)/(''k'' + 3)). It is for this reason that the indexing choice of S(''k'' + 1)/S(''k'' + 3) was chosen, as two (''k'' + 3)/(''k'' + 1)'s make a (''k'' + 4)/''k''. This form of comma does not yet have an official name, but a proposed name is "semiparticular", because most of the time it is superparticular but less often it is odd-particular, and because when tempered they all cause an interval to be divided into two equal parts where each part is a tempered version of a superparticular or odd-particular, and the interval being divided in half is sometimes quodd-particular, sometimes odd-particular and sometimes superparticular.
 Specifically, S''k''/S(''k'' + 2) is superparticular when ''k'' is not a multiple of 4, and odd-particular otherwise. See [[#Mathematical derivation]] for details on this and other facts stated here.