S-expression: Difference between revisions

Square-particulars are important structurally because they are the intervals between consecutive superparticular intervals while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''n'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward.

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas U''k'' = S''k''/S(''k'' + 1), we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because (''k'' + 2)/(''k'' + 1) and ''k''/(''k'' - 1) are equidistant from (''k'' + 1)/''k'' (because of tempering S''k''/S(''k'' + 1)), this means that another expression for S''k''/S(''k'' + 1) is the following: