S-expression: Difference between revisions

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As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.

* [[#Sk (square-particulars)|Square ~~superparticulars~~]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''2|''k''2 − 1}}. Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.

* [[#Sk (square-particulars)|Square-particulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''2|''k''2 − 1}}. Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.

* [[#Sk⋅S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k''⋅S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. Tempering out {{nowrap|S''k''⋅S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.

* [[#Sk2⋅S(k + 1) and S(k − 1)⋅Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')2⋅S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')2⋅S(''k'' − 1)'''}}. Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}2 and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}3, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}2 and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

* [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}3.

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

* [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''3|''k''3 − 1}} and {{sfrac|''k''3 + 1|''k''3}}, respectively.

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

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| [[256000/255879]]

|}

== 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, 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.

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.

=== Significance ===

# Cube-particulars are related to ultraparticulars via the cube-relation identity: S''k''/S(''k'' + 1) = C''k''⋅Cp(''k'' + 1), which suggests structurally induced extensions for temperaments that temper out ultraparticulars.

# Cube-particulars are very sensitive to the tuning of the harmonic in question, so they can be used to keep track of how the harmonic is tuned across systems.

=== Properties ===

# C''k''/Cp''k'' = C(''k''2). Note S''k''/Sp''k'' = S(k2) holds too if Sp''k'' notation is used for (''k''2 + 1)/k2.

# C(''k''2) = S(''k''3). In other words, C''k'' is a square-particular if ''k'' is a perfect square, and conversely S''k'' is a cube-particular if ''k'' is a perfect cube.

== Using S-factorizations to understand the significance of S-expressions ==

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: For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.

: These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk⋅S(k + 1)⋅…⋅S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].

; Ultraparticular

: An interval/comma which is the ratio of two consecutive square-particulars.

: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.

; Semiparticular

: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:

: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.

; Cube-particular

: A superparticular interval/comma whose numerator or denominator is a cube number. A shorthand (nick)name for cube superparticular.

: These are of the form {{nowrap|{{sfrac|''k''3|''k''3 − 1}} {{=}} C''k''}} or {{nowrap|{{sfrac|''k''3 + 1|''k''3}} {{=}} Cp''k''}}.

; Odd-particular

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: If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.

: If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.

~~; Ultraparticular~~

~~: An interval/comma which is the ratio of two consecutive square-particulars.~~

~~: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.~~

~~; Semiparticular~~

~~: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:~~

~~: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.~~

; S-expression

: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.

: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only arbitrary square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars. (By extension, the expression could also include cube-particulars, C''k'' and Cp''k''.)

; S-factorization

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; S-comma

: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).

: Any comma within one of the infinite families of commas discussed here, excluding 1/''n''-square-particulars for ''n'' > 5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than ''n'' > 3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).

; Indirect S-comma

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 As S-expressions are deployed widely on the wiki and in the broader xen community, below is a list of what the most common S-expression categories imply when they are [[tempering out|tempered out]]. The linked sections provide deeper information into each comma family.
-* [[#Sk (square-particulars)|Square superparticulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.
+* [[#Sk (square-particulars)|Square-particulars]]: '''S''k''''', superparticular fractions of the form {{sfrac|''k''<sup>2</sup>|''k''<sup>2</sup> − 1}}. <br>Tempering out S''k'' equates {{sfrac|''k'' + 1|''k''}} with {{sfrac|''k''|''k'' − 1}} and splits {{sfrac|''k'' + 1|''k'' − 1}} in two.
 * [[#Sk⋅S(k + 1) (triangle-particulars)|Triangle-particulars]]: {{nowrap|'''S''k''⋅S(''k'' + 1)'''}}, superparticular fractions of the form {{sfrac|''k''(''k'' + 1)/2|(''k'' − 1)(''k'' + 2)/2}}. <br>Tempering out {{nowrap|S''k''⋅S(''k'' + 1)}} equates {{sfrac|''k'' + 2|''k'' + 1}} with {{sfrac|''k''|''k'' − 1}}, and {{sfrac|''k'' + 2|''k''}} with {{sfrac|''k'' + 1|''k'' − 1}}.
 * [[#Sk2⋅S(k + 1) and S(k − 1)⋅Sk2 (lopsided commas)|Lopsided commas]]: {{nowrap|'''(S''k'')<sup>2</sup>⋅S(''k'' + 1)'''}} and {{nowrap|'''(S''k'')<sup>2</sup>⋅S(''k'' − 1)'''}}. <br>Tempering out the former equates {{sfrac|''k'' + 2|''k''}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>2</sup> and {{sfrac|''k'' + 2|''k'' − 1}} with {{pars|{{sfrac|''k''|''k'' − 1}}}}<sup>3</sup>, and tempering out the latter equates {{sfrac|''k''|''k'' − 2}} with  {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>2</sup> and {{sfrac|''k'' + 1|''k'' − 2}} with {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
 * [[#Sk/S(k + 1) (ultraparticulars)|Ultraparticulars]]: {{nowrap|'''S''k''/S(''k'' + 1)'''}}. Tempering this out splits {{sfrac|''k'' + 2|''k'' − 1}} into {{pars|{{sfrac|''k'' + 1|''k''}}}}<sup>3</sup>.
 * [[#Sk/S(k + 2) (semiparticulars)|Semiparticulars]]: {{nowrap|'''S''k''/S(''k'' + 2)'''}}. Tempering this out splits {{sfrac|''k'' + 3|''k'' − 1}} into {{pars|{{sfrac|''k'' + 2|''k''}}}}<sup>2</sup>.
+* [[#Ck and Cpk (cube-particulars)|Cube-particulars]]: '''C''k''''' and '''Cp''k''''', superparticular fractions of the form {{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} and {{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}}, respectively.
 == S''k'' (square-particulars) ==
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 | [[256000/255879]]
 |}
+== 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, 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.
+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.
+=== Significance ===
+# Cube-particulars are related to ultraparticulars via the cube-relation identity: S''k''/S(''k'' + 1) = C''k''⋅Cp(''k'' + 1), which suggests structurally induced extensions for temperaments that temper out ultraparticulars.
+# Cube-particulars are very sensitive to the tuning of the harmonic in question, so they can be used to keep track of how the harmonic is tuned across systems.
+=== Properties ===
+# C''k''/Cp''k'' = C(''k''<sup>2</sup>). Note S''k''/Sp''k'' = S(k<sup>2</sup>) holds too if Sp''k'' notation is used for (''k''<sup>2</sup> + 1)/k<sup>2</sup>.
+# C(''k''<sup>2</sup>) = S(''k''<sup>3</sup>). In other words, C''k'' is a square-particular if ''k'' is a perfect square, and conversely S''k'' is a cube-particular if ''k'' is a perfect cube.
 == Using S-factorizations to understand the significance of S-expressions ==
@@ Line 2,827: / Line 2,841: @@
 : For {{nowrap|''b'' {{=}} ''a'' + 1}} these can also be called triangle-particulars, in which case they are always superparticular.
 : These have implications for whether consistency in the {{nowrap|(''n'' + ''k'') {{=}} (''b'' + 1)}}-[[odd-limit]] is ''potentially'' possible in a given temperament; see the [[#Sk⋅S(k + 1)⋅…⋅S(k + n - 1) (1/n-square-particulars)|section on 1/''n''-square-particulars]].
+; Ultraparticular
+: An interval/comma which is the ratio of two consecutive square-particulars.
+: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.
+; Semiparticular
+: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:
+: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.
+; Cube-particular
+: A superparticular interval/comma whose numerator or denominator is a cube number. A shorthand (nick)name for cube superparticular.
+: These are of the form {{nowrap|{{sfrac|''k''<sup>3</sup>|''k''<sup>3</sup> − 1}} {{=}} C''k''}} or {{nowrap|{{sfrac|''k''<sup>3</sup> + 1|''k''<sup>3</sup>}} {{=}} Cp''k''}}.
 ; Odd-particular
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 : If ''n'' is a prime, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is not superparticular. For example, 5-odd-particular intervals are of the form {{sfrac|5''k'' + 1|5''k'' − 4}}, {{sfrac|5''k'' + 2|5''k'' − 3}}, {{sfrac|5''k'' + 3|5''k'' − 2}}, or {{sfrac|5''k'' + 4|5''k'' − 1}}.
 : If ''n'' is a composite, an ''n''-odd-particular interval is between two harmonics ''n'' apart which is neither superparticular nor of ''m''-odd-particular intervals where ''m'' is any other divisor of ''n''. For example, 6-odd-particular intervals are of the form {{sfrac|6''k'' + 1|6''k'' − 5}} or {{sfrac|6''k'' + 5|6''k'' − 1}}.
-; Ultraparticular
-: An interval/comma which is the ratio of two consecutive square-particulars.
-: These are of the form {{sfrac|S''k''|S(''k'' + 1)}}.
-; Semiparticular
-: A superparticular or odd-particular interval/comma which is the ratio between two adjacent-to-adjacent square-particulars, which is to say:
-: These are of the form {{sfrac|S''k''|S(''k'' + 2)}}.
 ; S-expression
-: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only (arbitrary) square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars.
+: An expression using the S''k'' shorthand notation corresponding strictly to multiplying and dividing only arbitrary square-particulars. S-expressions include singular square superparticulars and expressions for other superparticulars in terms of square superparticulars. (By extension, the expression could also include cube-particulars, C''k'' and Cp''k''.)
 ; S-factorization
@@ Line 2,861: / Line 2,879: @@
 ; S-comma
-: Any comma within one of the infinite families of commas discussed here, excluding 1/n-square-particulars for n>5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than n>3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).
+: Any comma within one of the infinite families of commas discussed here, excluding 1/''n''-square-particulars for ''n'' > 5 (so square-particulars, triangle-particulars, 1/3-square-particulars, 1/4-square-particulars and 1/5-square-particulars are included, but not anything beyond; this bound is used for exclusion (rather than ''n'' > 3) to allow the utility of 1/5-square-particulars in avoiding twin primes by equating superparticular intervals on either side of the twin primes).
 ; Indirect S-comma

S-expression: Difference between revisions

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