S-expression: Difference between revisions

Line 3,094:

== C''k'' and Cp''k'' (cube-particulars) ==

This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''3|''k''3 − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''3 + 1|''k''3}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''3 + 1 and ''k''3 − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''2 + 1 can be prime. The term ''S-expression'' ~~applies to these despite~~ not ~~using the letter~~ ''S'', in ~~avoidance~~ of ~~introducing additional terms~~.

This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''3|''k''3 − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''3 + 1|''k''3}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''3 + 1 and ''k''3 − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''2 + 1 can be prime. The term ''S-expression'' arguably does not apply here, but they are noted on this page nonetheless because they appear as an extension of [[ultraparticular]]s, as {{nowrap| S''k''/S(''k'' + 1) {{=}} C''k''⋅Cp(''k'' + 1) }}, therefore every temperament tempering out an ultraparticular can potentially be extended in a non-obvious way to tempering a pair of cube-particulars.

Note that as ''k'' increases, the maximal prime limit of a cube-particular grows more quickly than that of a square-particular; cube-particulars essentially rely on the factorizability of the term (''k''2 + ''k'' + 1) for C''k'' or (''k''2 − ''k'' + 1) for Cp''k'' to get to a reasonable prime limit.

@@ Line 3,094: / Line 3,094: @@
 == C''k'' and Cp''k'' (cube-particulars) ==
-This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''<sup>3</sup> + 1 and ''k''<sup>3</sup> − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''<sup>2</sup> + 1 can be prime. The term ''S-expression'' applies to these despite not using the letter ''S'', in avoidance of introducing additional terms.
+This family of superparticular interval is of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} and {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}} (for ''cube-particular complement''). Both C''k'' and Cp''k'' are notable because ''k''<sup>3</sup> + 1 and ''k''<sup>3</sup> − 1 are always composite for ''k'' ≥ 2, unlike with square-particulars, where ''k''<sup>2</sup> + 1 can be prime. The term ''S-expression'' arguably does not apply here, but they are noted on this page nonetheless because they appear as an extension of [[ultraparticular]]s, as {{nowrap| S''k''/S(''k'' + 1) {{=}} C''k''⋅Cp(''k'' + 1) }}, therefore every temperament tempering out an ultraparticular can potentially be extended in a non-obvious way to tempering a pair of cube-particulars.
 Note that as ''k'' increases, the maximal prime limit of a cube-particular grows more quickly than that of a square-particular; cube-particulars essentially rely on the factorizability of the term (''k''<sup>2</sup> + ''k'' + 1) for C''k'' or (''k''<sup>2</sup> − ''k'' + 1) for Cp''k'' to get to a reasonable prime limit.

S-expression: Difference between revisions

Godtone (talk | contribs)