S-expression: Difference between revisions

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which is square-(super)particular ''k'' for a given integer ''k > 1''. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article. Note that this means S2 = [[4/3]] is the first musically meaningful square-particular, as S1 = 1/0.

Square-particulars are important structurally because they are the intervals between consecutive superparticular ~~intervals~~ while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''k'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward.

=== Significance/motivation ===

Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''k'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals.

=== Table of square-particulars ===

Below is a table of [[23-limit]] square-particulars:

{| class="wikitable center-all

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 which is square-(super)particular ''k'' for a given integer ''k > 1''. A suggested shorthand for this interval is '''S''k''''' for the ''k''-th square superparticular, where the ''S'' stands for "(Shorthand for) Second-order/Square Superparticular". This will be used later in this article. Note that this means S2 = [[4/3]] is the first musically meaningful square-particular, as S1 = 1/0.
-Square-particulars are important structurally because they are the intervals between consecutive superparticular intervals while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''k'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward.
+=== Significance/motivation ===
+Square-particulars are important structurally because they are the intervals between consecutive [[superparticular]] [[interval]]s while simultaneously being superparticular themselves, which means that whether and how they are tempered tells us information about how well a temperament can represent the harmonic series up to the (''k'' + 1)th harmonic, as well as the potential representational sacrifices that must be made from that point onward. In other words, understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticular intervals.
+=== Table of square-particulars ===
 Below is a table of [[23-limit]] square-particulars:
 {| class="wikitable center-all