S-expression: Difference between revisions

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'''k2/(k2 - 1) = (k/(k-1)) / ((k+1)/k)'''

which is square-(super)particular '''k''' for a given integer '''k > 1'''. A suggested shorthand for this interval is '''Sk''' 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 ~~would be~~ 1/0.

which is square-(super)particular '''k''' for a given integer '''k > 1'''. A suggested shorthand for this interval is '''Sk''' 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 '''(n+1)'''th harmonic, as well as the potential representational sacrifices that must be made from that point onward.

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It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas Uk = Sk/S(k+1), we get [[#Sk/S(k+1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary consequence: Because (k+2)/(k+1) and k/(k-1) are equidistant from (k+1)/k (because of tempering Sk/S(k+1)), this means that another expression for Sk/S(k+1) is the following:

~~'''~~Sk/S(k+1) = (k+2)/(k-1) / ((k+1)/k)3~~'''~~

Sk/S(k+1) = (k+2)/(k-1) / ((k+1)/k)3

This means you can read the ~~'''~~k~~'''~~ and ~~'''~~k+1~~'''~~ from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in the table).

This means you can read the k and k+1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k+1) (ultraparticulars)|the table of ultraparticulars]]).

* In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as Uk = Sk/S(k-1) and Uk = S(k+1)/S(k+2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.

<nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as Uk = Sk/S(k-1) and Uk = S(k+1)/S(k+2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.

== Sk/S(k+1) (ultraparticulars) ==

@@ Line 4: / Line 4: @@
 '''k<sup>2</sup>/(k<sup>2</sup> - 1) = (k/(k-1)) / ((k+1)/k)'''
-which is square-(super)particular '''k''' for a given integer '''k > 1'''. A suggested shorthand for this interval is '''Sk''' 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 would be 1/0.
+which is square-(super)particular '''k''' for a given integer '''k > 1'''. A suggested shorthand for this interval is '''Sk''' 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 '''(n+1)'''th harmonic, as well as the potential representational sacrifices that must be made from that point onward.
@@ Line 10: / Line 10: @@
 It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy it can be more beneficial to instead temper differences between consecutive square superparticulars so that the corresponding consecutive superparticulars are tempered to have equal spacing between them. If we define a sequence of commas Uk = Sk/S(k+1), we get [[#Sk/S(k+1) (ultraparticulars)|ultraparticulars]]*. Ultraparticulars have a secondary consequence: Because (k+2)/(k+1) and k/(k-1) are equidistant from (k+1)/k (because of tempering Sk/S(k+1)), this means that another expression for Sk/S(k+1) is the following:
-'''Sk/S(k+1) = (k+2)/(k-1) / ((k+1)/k)<sup>3</sup>'''
+Sk/S(k+1) = (k+2)/(k-1) / ((k+1)/k)<sup>3</sup>
-This means you can read the '''k''' and '''k+1''' from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in the table).
+This means you can read the k and k+1 from the S-expression of an ultraparticular as being the interval involved in the cubing equivalence (abbreviated to "cube relation" in [[#Sk/S(k+1) (ultraparticulars)|the table of ultraparticulars]]).
-* In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as Uk = Sk/S(k-1) and Uk = S(k+1)/S(k+2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.
+<nowiki>*</nowiki> In analogy with the "super-", "ultra-" progression and because these would be differences between adjacent differences between adjacent superparticulars, which means a higher order of "particular", and as we will see, no longer [[superparticular]], hence the need for a new name. Also note that the choice of indexing is rather arbitrary and up to debate, as Uk = Sk/S(k-1) and Uk = S(k+1)/S(k+2) also make sense. Therefore it is advised to use the S-expression to refer to an ultraparticular unambiguously, or the comma itself.
 == Sk/S(k+1) (ultraparticulars) ==