S-expression: Difference between revisions

A '''square superparticular''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form

<nowiki>*</nowiki> 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 U''k'' = S''k''/S(''k'' - 1) and U''k'' = 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.

If we examine (''k'' + 1)/''k'' then we can notice that if we equate (''k'' + 2)/(''k'' + 1) with ''k''/(''k'' - 1), we have:

(Note: after 75, 76, 77, 78, streaks of four consecutive harmonics in the 23-limit become very sparse. The last few streaks are deeply related to the consistency and structure of [[311edo]], as [[311edo]] can be described as the unique 23-limit temperament that tempers all triangle-particulars from [[595/594]] up to [[21736/21735]]. It also tempers all the square-particulars composing those triangle-particulars with the exception of S169 and S170. It also maps the corresponding intervals of the 77-odd-limit consistently. 170/169 is the only place where the logic seems to 'break' as it is mapped to 2 steps instead of 3 meaning the mapping of that superparticular is inconsistent.)

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:

<pre>

Note how there

1/n-square-particulars, which is to say, commas which can be written in the form of a product of ''n'' consecutive square-particulars (including S''k'' but not including S(''k'' + ''n'')) and which can therefore be written as the ratio between the two superparticulars ''k''/(''k'' - 1) and (''k'' + ''n'')/(''k'' + ''n'' - 1) have implications for the [[consistency]] of the (''k'' + ''n'')-[[odd-limit]] when tempered. Specifically:

Therefore any superparticular interval ''x''/(''x'' - 1) between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the (''k'' + ''n'')-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering S''k''*S(''k''+1)*...*S(''k''+''n''-1) but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the (''k'' - 1)-odd-limit.

Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.

Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.

For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]]. (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).)

(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't.)

If instead of working through things algebraically we look at square-particulars as describing a relationship between adjacent harmonics, we can use this to understand why certain simplifications and equivalences exist in a way that is equivalent to the sometimes harder-to-understand usual algebraic form: