S-expression: Difference between revisions

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=== Significance/motivation ===

Square-superparticulars are the intervals between consecutive [[superparticular]] [[interval]]s, ~~such~~ that tempering Sk out makes the harmonic segment centered around k have equal steps; eg. tempering out {{nowrap|S9 {{=}} 81/80}} equalizes 8:9:10, as in [[meantone]]. ~~Understanding~~ the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive ~~superparticulars~~ and the way it represents (or tries to represent) the harmonic series.

Square-superparticulars 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 ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onwards.

'''For example:''' tempering S''k'' out makes the harmonic segment centered around ''k'' have equal steps; eg. tempering out {{nowrap|S9 {{=}} (9/8)/(10/9) {{=}} 81/80}} equalizes 8:9:10, as in [[meantone]].

In other words, understanding the mappings of S''k'' in a given temperament (especially an equal temperament like an [[edo]]) is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.

=== Table of square-particulars ===

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 === Significance/motivation ===
-Square-superparticulars are the intervals between consecutive [[superparticular]] [[interval]]s, such that tempering Sk out makes the harmonic segment centered around k have equal steps; eg. tempering out {{nowrap|S9 {{=}} 81/80}} equalizes 8:9:10, as in [[meantone]]. Understanding the mappings of S''k'' in a given temperament is equivalent to understanding the spacing of consecutive superparticulars and the way it represents (or tries to represent) the harmonic series.
+Square-superparticulars 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 ({{nowrap|''k'' + 1}})th harmonic, as well as the potential representational sacrifices that must be made from that point onwards.
+'''For example:''' tempering S''k'' out makes the harmonic segment centered around ''k'' have equal steps; eg. tempering out {{nowrap|S9 {{=}} (9/8)/(10/9) {{=}} 81/80}} equalizes 8:9:10, as in [[meantone]].
+In other words, understanding the mappings of S''k'' in a given temperament (especially an equal temperament like an [[edo]]) is equivalent to understanding the spacing of consecutive superparticular intervals, and thereby to understanding the way it represents (or tries to represent) the harmonic series.
 === Table of square-particulars ===

S-expression: Difference between revisions

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