S-expression: Difference between revisions

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~~In mathematics~~, a '''~~square~~ [[~~superparticular~~]] ~~ratio~~''', ~~also called a~~ '''~~square~~-~~particular ratio~~''', is the ~~ratio of two consecutive integer numbers whose higher one is a square number~~ (~~3:4~~, 8:9, 15:16...).

An '''S-expression''' is any product, or ratio of products, of the square superparticulars S''k'', where S''k'' is defined as the fraction of the form (''k''2)/(''k''2 - 1). Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.

== Quick rules of S-expressions ==

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 in terms of [[tempering out|tempering]]. The linked sections provide deeper information into each comma family.

* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', fractions of the form (''k''2)/(''k''2 - 1). Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).

* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).

* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')2 * S(''k''+1) and (S''k'')2 * S(''k''-1)'''. Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))2, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')2.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: '''S''k''/S(''k''+1)'''. Tempering this out equates (''k''+2)/(''k''-1) with ((''k''+1)/''k'')3.

* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: '''S''k''/S(''k''+2)'''. Tempering this out equates (''k''+3)/(''k''-1) with ((''k''+2)/''k'')2.

== Sk (square-particulars) ==

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

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

<math>\Large \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k}</math>

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Furthermore, defining another sequence of commas with [[semiparticular|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.

~~=== S-expressions ===~~

~~An S-expression is any product, or ratio of products, of square superparticulars S''k''.~~

== Sk*S(k + 1) (triangle-particulars) ==

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-In mathematics, a '''square [[superparticular]] ratio''', also called a '''square-particular ratio''', is the ratio of two consecutive integer numbers whose higher one is a square number (3:4, 8:9, 15:16...).
+An '''S-expression''' is any product, or ratio of products, of the square superparticulars S''k'', where S''k'' is defined as the fraction of the form (''k''<sup>2</sup>)/(''k''<sup>2</sup> - 1). Commas defined by S-expressions turn out to represent intuitive and wide-reaching families of tempered equivalences, and therefore present a very useful framework to learn for a good understanding of the [[commas]] that appear frequently in xen.
+== Quick rules of S-expressions ==
+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 in terms of [[tempering out|tempering]].  The linked sections provide deeper information into each comma family.
+* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', fractions of the form (''k''<sup>2</sup>)/(''k''<sup>2</sup> - 1). Tempering out S''k'' equates (''k''+1)/''k'' and ''k''/(''k''-1).
+* [[#Sk*S(k + 1) (triangle-particulars)|Triangle-particulars]]: '''S''k'' * S(''k''+1)''', fractions of the form (''k''(''k''+1)/2)/((''k''-1)(''k''+2)/2). Tempering out S''k'' * S(''k''+1) equates (''k''+2)/(''k''+1) and ''k''/(''k''-1), or (''k''+2)/''k'' and (''k''+1)/(''k''-1).
+* [[#Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas)|Lopsided commas]]: '''(S''k'')<sup>2</sup> * S(''k''+1) and (S''k'')<sup>2</sup> * S(''k''-1)'''. Tempering out the former equates (''k''+2)/''k'' with (''k''/(''k''-1))<sup>2</sup>, and tempering out the latter equates ''k''/(''k''-2) with ((''k''+1)/''k'')<sup>2</sup>.
+* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: '''S''k''/S(''k''+1)'''. Tempering this out equates (''k''+2)/(''k''-1) with ((''k''+1)/''k'')<sup>3</sup>.
+* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: '''S''k''/S(''k''+2)'''. Tempering this out equates (''k''+3)/(''k''-1) with ((''k''+2)/''k'')<sup>2</sup>.
 == Sk (square-particulars) ==
-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
+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
 <math>\Large \frac {k^2}{k^2 - 1} = \frac {k/(k - 1)}{(k + 1)/k}</math>
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 Furthermore, defining another sequence of commas with [[semiparticular|formula S''k''/S(''k'' + 2) leads to semiparticulars]] which inform many natural ways in which one might want to halve intervals with other intervals, and with their own more structural consequences, talked about there. These also arise from tempering consecutive ultraparticulars.
-=== S-expressions ===
-An S-expression is any product, or ratio of products, of square superparticulars S''k''.
 == Sk*S(k + 1) (triangle-particulars) ==