S-expression: Difference between revisions

As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply is equated when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.

* [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''³|''k''³ − 1}} and {{sfrac|''k''³ + 1|''k''³}}, respectively.

The reader is encouraged to familiarize themself with the structure of this argument, as it generalizes to arbitrary S''k'' (→ [[S-expression/Advanced_results#Mathematical_derivations]]); the algebraic proof is tedious, but the intuition is the same:

# Tempering out any two consecutive ultraparticulars will imply tempering out the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]], which is their 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 S''k'' notation to achieve that compactness.

# Tempering out the ultraparticular S''k''/S({{nowrap|''k'' + 1}}) along with either the corresponding 1/2-square-particular {{nowrap|S''k''⋅S(''k'' + 1)}} or one of the two corresponding lopsided commas {{nowrap|S''k''²⋅S(''k'' + 1)}} or {{nowrap|S''k''⋅S(''k'' + 1)²}} implies tempering both of S''k'' and S({{nowrap|''k'' + 1}}) individually, and vice versa, so that there is a total of ''five'' equivalences—corresponding to ''five'' infinite families of commas—for every such S''k'' and {{nowrap|S(''k'' + 1)}}. This only gets better if you temper out a third consecutive square-particular. This is an abundance of "at-a-glance" essential tempering information that is fully general, so it only needs to be learned once, and is another motivation of the use of S-expressions. For example, {{nowrap|{S16, S17} → {S16⋅S17, S16/S17, S16²⋅S17, S16⋅S17²} }}, and any of the two commas in the latter set imply all the other commas too.

… and notice that the latter expression is the one we have shown is equal to S(''k'' - 1)/S(''k'' + 1) up to an offset ''k'' (→ [[S-expression/Advanced results #Mathematical derivations]]). In other words, that tempering out S(''k'' - 1)/S(''k'' + 1) results in (''k'' + 1)/(''k'' - 1) being half of (''k'' + 2)/(''k'' - 2) is an implication that it makes (''k'' + 2)/''k'', (''k'' + 1)/(''k'' - 1), and ''k''/(''k'' - 2) equidistant.  

All S-expressions have other equivalent S-expressions; however, a comma might appear in more than one family of S-commas or arise as a specific product of them. This often means nontrivial, deep tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.

This is important to note because using this simple rule we can derive an infinite amount of trivial equivalent S-expressions. See [[S-expression/Advanced results]] for mathematical details.

| S351, S78/S80, S25/S27/S65