S-expression: Difference between revisions

* In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as Uk = Sk/S(k-1) and Uk = S(k+1)/S(k+2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.

Note that while a lot of these have pages, not all of them do.

Also note that if you temper multiple adjacent ultraparticulars, you sometimes aren't required to use those ultraparticulars in the comma list due to an interesting case of [[Sk/S(k+2)]], discussed next.

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 (2n+1)/(2n-1) for some positive integer n. (This terminology also suggests "throdd-particular" for intervals of the form (3n+2)/(3n) and (3n+1)/(3n-1) and maybe "quodd-particular" (sounding like "quad-particular") for (4n+3)/(4n-1) and (4n+1)/(4n-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.

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(Note for readers: Familiarity with basic algebra is advised, especially the rules (a + b)(a - b) = a² - b² and 1/(a/b) = b/a. Often multiple steps are combined into a single step for brevity, so follow carefully.)