S-expression: Difference between revisions

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* [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out equates {{sfrac|''k'' + 3|''k'' − 1}} with {{pars|{{sfrac|''k'' + 2|''k''}}}}2.

== Sk (square-particulars) ==

== S''k'' (square-particulars) ==

A '''square [[superparticular]]''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form

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=== Alternatives to tempering square-particulars ===

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons 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 {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">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 U''k'' = S''k''/S(''k'' ~~- 1)~~ and U''k'' = 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.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:

It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons 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 {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">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 U''k'' = S(''k'' - 1)/S''k'' and U''k'' = 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.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:

<math>\large {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3}</math>

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== {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ==

This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac| {{sfrac|''k'' − 1|''k'' − 2}} | {{sfrac|''k'' + 2|''k'' + 1}} }} which therefore do not (directly) involve the ''k''th harmonic.

This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac| {{sfrac|''k'' − 1|''k'' − 2}} | {{sfrac|''k'' + 2|''k'' + 1}} }}}} which therefore do not (directly) involve the ''k''th harmonic.

=== Significance ===

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Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.

== Sk/S(k + 2) (semiparticulars) ==

== {{nowrap|S''k''/S(''k'' + 2)}} (semiparticulars) ==

=== Motivational examples ===

If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:

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(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)

== Sk2 * S(k + 1) and S(k − 1) * Sk2 (lopsided commas) ==

== {{nowrap|S''k''2 * S(''k'' + 1)}} and {{nowrap|S(''k'' − 1) * S''k''2}} (lopsided commas) ==

=== Significance ===

1. Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.

@@ Line 10: / Line 10: @@
 * [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out equates {{sfrac|''k'' + 3|''k'' − 1}} with {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
-== Sk (square-particulars) ==
+== S''k'' (square-particulars) ==
 A '''square [[superparticular]]''', or ''square-particular'' for short, is a [[superparticular]] [[interval]] whose numerator is a square number, which is to say, a superparticular of the form
@@ Line 484: / Line 484: @@
 === Alternatives to tempering square-particulars ===
-It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons 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 {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">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 U''k'' = S''k''/S(''k'' - 1) and U''k'' = 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.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:
+It is common to temper square superparticulars, equating two adjacent superparticulars at some point in the harmonic series, but for higher accuracy or structural reasons 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 {{nowrap|U''k'' {{=}} {{sfrac|S''k''|S(''k'' + 1)}}}}, we get [[#Sk/S(k + 1) (ultraparticulars)|ultraparticulars]]<ref group="note">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 U''k'' = S(''k'' - 1)/S''k'' and U''k'' = 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.</ref>. Ultraparticulars have a secondary (and mathematically equivalent) consequence: Because {{sfrac|''k'' + 2|''k'' + 1}} and {{sfrac|''k''|''k'' − 1}} are equidistant from {{sfrac|''k'' + 1|''k''}} (because of tempering {{sfrac|S''k''|S(''k'' + 1)}}), this means that another expression for {{sfrac|S''k''|S(''k'' + 1)}} is the following:
 <math>\large {\rm S}k / {\rm S} (k + 1) = \frac{(k + 2) / (k - 1)}{((k + 1)/k)^3}</math>
@@ Line 783: / Line 783: @@
 == {{nowrap|S(''k'' − 1)*S''k''*S(''k'' + 1)}} (1/3-square-particulars) ==
-This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}} which therefore do not (directly) involve the ''k''th harmonic.
+This section concerns commas of the form {{nowrap|S(''k'' − 1) * S''k'' * S(''k'' + 1) {{=}} {{sfrac|&nbsp;{{sfrac|''k'' − 1|''k'' − 2}}&nbsp;|&nbsp;{{sfrac|''k'' + 2|''k'' + 1}}&nbsp;}}}} which therefore do not (directly) involve the ''k''th harmonic.
 === Significance ===
@@ Line 1,811: / Line 1,811: @@
 Also note that if you temper multiple adjacent ultraparticulars, you sometimes are not required to use those ultraparticulars in the comma list as description of (the bulk of) the tempering may be possible through [[#Sk/S(k + 2) (semiparticulars)|semiparticulars]], discussed next.
-== Sk/S(k + 2) (semiparticulars) ==
+== {{nowrap|S''k''/S(''k'' + 2)}} (semiparticulars) ==
 === Motivational examples ===
 If we want to halve one JI interval into two of another JI interval, there is a powerful and elegant pattern for doing so:
@@ Line 2,048: / Line 2,048: @@
 (Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't. A noticeable streak of commas currently without pages correspond to when dividing a superparticular interval implicates intervals from a higher [[prime limit]], as a surprising amount of 23-limit semiparticulars shown here already have pages.)
-== Sk<sup>2</sup> * S(k + 1) and S(k − 1) * Sk<sup>2</sup> (lopsided commas) ==
+== {{nowrap|S''k''<sup>2</sup> * S(''k'' + 1)}} and {{nowrap|S(''k'' − 1) * S''k''<sup>2</sup>}} (lopsided commas) ==
 === Significance ===
 . Tempering any two consecutive square-particulars, S''k'' and S(''k'' + 1), implies tempering the two associated lopsided commas as well as the associated [[triangle-particular]] and [[ultraparticular]], so the lopsided commas represent the general form of the highest-damage relations/consequences of doing so.