S-expression: Difference between revisions

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== Equivalent S-expressions ==

=== Significance and meaning ===

All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this ~~basically always~~ means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. 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 as you allow higher n, but are still quite rare for small n.

All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. 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 as you allow higher n, but are still quite rare for small n.

=== A useful general rule ===

While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:

<math>

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

</math>

This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.

=== Examples ===

Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable):

Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).

'''Importantly:''' examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are considered invalid/trivial.

{| class="wikitable center-all

@@ Line 2,575: / Line 2,575: @@
 == Equivalent S-expressions ==
 === Significance and meaning ===
-All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this basically always means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. 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 as you allow higher n, but are still quite rare for small n.
+All S-expressions have other equivalent S-expressions, however when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending how you temper further. 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 as you allow higher n, but are still quite rare for small n.
+=== A useful general rule ===
+While there are likely arbitrarily many ways of rewriting S-expressions due to the redundancy in representation, the following equivalence is, due to its simplicity and elegance, arguably most likely to be useful:
+<math>
+\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
+</math>
+This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.
 === Examples ===
-Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable):
+Here is an incomplete list of examples (feel free to expand with any equivalences you find that you think are valuable).
+'''Importantly:''' examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are considered invalid/trivial.
 {| class="wikitable center-all