S-expression: Difference between revisions

Line 1,269:

The generalisation of this method using commutative group theory is discussed in [[square superparticular#Abstraction|the abstraction section of this page]].

=== Using S-factorizations to show a useful equivalence (and hence redundancy) of S-expressions ===

Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:

==== The general S-expression equivalence ====

Consider:

<pre>

Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:

S(2k-1) * S(2k) * S(2k) * S(2k+1)

= [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(

[-1, 2, -1]

+ [-2, 4, -2]

+ [-1, 2, -1]

= [-1, 0, 2, 0, -1] )

</pre>

From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':

<math>

\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)

</math>

...where we use the notation S''k''''p'' to mean (S''k'')''p'' rather than S(''k''''p'') for convenience in the practical analysis of [[regular temperament]]s using [[S-expression]]s.

For tuning theory only integer ''k'' > 1 is of relevance. Technically, rational ''k'' other than 1 correspond to rational commas too; the most relevant case for tuning theory is that half-integer ''k'' work as an alternative notation for [[odd-particulars]], though for intuitively understanding the notation, the method described in [[#Abstraction]] may be recommendable as having (in a mathematical sense) exact analogues for every infinite family of commas defined in terms of an analogue of an S-expression, for which the most musically fruitful example is O''k'' = (''k'' / (''k'' - 2))/((''k'' + 2) / ''k'') for odd ''k'' as relevant to [[no-twos subgroup temperaments]].

== Sk2 * S(k + 1) and S(k - 1) * Sk2 (lopsided commas) ==

@@ Line 1,269: / Line 1,269: @@
 The generalisation of this method using commutative group theory is discussed in [[square superparticular#Abstraction|the abstraction section of this page]].
+=== Using S-factorizations to show a useful equivalence (and hence redundancy) of S-expressions ===
+Absent of restrictions on the form that an S-expression may take, there is no unique S-expression for any given rational number. This is in fact a huge advantage, because it allows one to understand the landscape of commas in a way that sees interconnectedness of subgroups and corresponding tempering opportunities. But then what S-expressions are equivalent, other than mathematical one-offs? The most important general rule can be derived quite simply using S-factorizations:
+==== The general S-expression equivalence ====
+Consider:
+<pre>
+Sk = [k-1, k, k+1]^[-1, 2, -1] versus what it is claimed to be equivalent to:
+S(2k-1) * S(2k) * S(2k) * S(2k+1)
+ = [2k-2, 2k-1, 2k, 2k+1, 2k+2]^(
+   [-1,    2,   -1]
+       + [-2,    4,   -2]
+             + [-1,    2,   -1]
+ = [-1,    0,    2,    0,   -1] )
+</pre>
+From here we can observe that the exponents are on even integers and that the factors of 2 involved cancel (we divide by 2 once for 2k-2 and 2k+2 having -1 as the power and we multiply by 2 twice for 2k having 2 as the power). Therefore the expressions are algebraically equivalent, which leads to the surprising fact that the following equivalence is true for all real and complex ''k'':
+<math>
+\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
+</math>
+...where we use the notation S''k''<sup>''p''</sup> to mean (S''k'')<sup>''p''</sup> rather than S(''k''<sup>''p''</sup>) for convenience in the practical analysis of [[regular temperament]]s using [[S-expression]]s.
+For tuning theory only integer ''k'' > 1 is of relevance. Technically, rational ''k'' other than 1 correspond to rational commas too; the most relevant case for tuning theory is that half-integer ''k'' work as an alternative notation for [[odd-particulars]], though for intuitively understanding the notation, the method described in [[#Abstraction]] may be recommendable as having (in a mathematical sense) exact analogues for every infinite family of commas defined in terms of an analogue of an S-expression, for which the most musically fruitful example is O''k'' = (''k'' / (''k'' - 2))/((''k'' + 2) / ''k'') for odd ''k'' as relevant to [[no-twos subgroup temperaments]].
 == Sk<sup>2</sup> * S(k + 1) and S(k - 1) * Sk<sup>2</sup> (lopsided commas) ==