Xenharmonic Wiki

S-expression: Difference between revisions

← Older edit
VisualWikitext
+ cube-particulars
 
(2 intermediate revisions by the same user not shown)
Line 1,951: Line 1,951:
| ([[3/1]])/([[5/3]])<sup>2</sup>
| ([[3/1]])/([[5/3]])<sup>2</sup>
| [[27/25]]
| [[27/25]]
| 2.3.5
| 3.5
|-
|-
| S4/S6 = ([[16/15]])/([[36/35]])
| S4/S6 = ([[16/15]])/([[36/35]])
Line 2,071: Line 2,071:
| ([[15/13]])/([[29/27]])<sup>2</sup>
| ([[15/13]])/([[29/27]])<sup>2</sup>
| [[10935/10933]]
| [[10935/10933]]
| 2.3.5.13.29
| 3.5.13.29
|-
|-
| S28/S30 = ([[784/783]])/([[900/899]])
| S28/S30 = ([[784/783]])/([[900/899]])
Line 2,091: Line 2,091:
| ([[17/15]])/([[33/31]])<sup>2</sup>
| ([[17/15]])/([[33/31]])<sup>2</sup>
| [[16337/16335]]
| [[16337/16335]]
| 2.3.5.11.17.31
| 3.5.11.17.31
|-
|-
| S32/S34 = ([[1024/1023]])/([[1156/1155]])
| S32/S34 = ([[1024/1023]])/([[1156/1155]])
Line 2,644: Line 2,644:


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


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.  
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.  
Line 2,653: Line 2,653:


=== Properties ===
=== Properties ===
# Like square-particulars, all cube-particulars involve prime 2, but unlike square-particulars, some cube-particulars are no-3. These are Cp(3''k'' + 1) and C(3''k'' + 2) for any integer ''k''.
# 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''/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.
# 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.
=== Table of cube-particulars ===
<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-all left-3 left-6"
|+ style="font-size: 105%;" | 31-limit cube-particulars – C''k''
|-
! S-expression
! Ratio
! Subgroup
|-
| –
| –
| –
|-
| C2
| [[8/7]]
| 2.7
|-
| C3
| [[27/26]]
| 2.3.13
|-
| C4
| [[64/63]]
| 2.3.7
|-
| C5
| [[125/124]]
| 2.5.31
|-
| C7
| [[343/342]]
| 2.3.7.19
|-
| C9
| [[729/728]]
| 2.3.7.13
|-
| C11
| [[1331/1330]]
| 2.5.7.11.19
|-
| C16
| [[4096/4095]]
| 2.3.5.7.13
|-
| C18
| [[5832/5831]]
| 2.3.7.17
|-
| C22
| [[10648/10647]]
| 2.3.7.11.13
|-
| C25
| [[15625/15624]]
| 2.3.5.7.31
|-
| C30
| [[27000/26999]]
| 2.3.5.7.19.29
|-
| –
| –
| –
|-
| –
| –
| –
|}
</div>
<div style="display: inline-grid;">
{| class="wikitable center-all left-3 left-6"
|+ style="font-size: 105%;" | 31-limit cube-particulars – Cp''k''
|-
! S-expression
! Ratio
! Subgroup
|-
| Cp2
| [[9/8]]
| 2.3
|-
| Cp3
| [[28/27]]
| 2.3.7
|-
| Cp4
| [[65/64]]
| 2.5.13
|-
| Cp5
| [[126/125]]
| 2.3.5.7
|-
| Cp6
| [[217/216]]
| 2.3.7.31
|-
| Cp8
| [[513/512]]
| 2.3.19
|-
| Cp10
| [[1001/1000]]
| 2.5.7.11.13
|-
| Cp12
| [[1729/1728]]
| 2.3.7.13.19
|-
| Cp17
| [[4914/4913]]
| 2.3.7.13.17
|-
| Cp19
| [[6860/6859]]
| 2.5.7.19
|-
| Cp23
| [[12168/12167]]
| 2.3.13.23
|-
| Cp26
| [[17577/17576]]
| 2.3.7.13.31
|-
| Cp31
| [[29792/29791]]
| 2.7.19.31
|-
| Cp68
| <small>[[314433/314432]]</small>
| 2.3.7.17.23.31
|-
| Cp69
| <small>[[328510/328509]]</small>
| 2.3.5.7.13.19.23
|}
</div></div>
Note that C''k'' and Cp(''k'' + 1) tend to share the same highest prime, and as we set the prime limit to 31, each C''k'' almost perfectly matches a Cp(''k'' + 1) – except for Cp68 and Cp69 at the bottom.


== Using S-factorizations to understand the significance of S-expressions ==
== Using S-factorizations to understand the significance of S-expressions ==
Line 2,733: Line 2,876:


== Equivalent S-expressions ==
== Equivalent S-expressions ==
=== Significance and meaning ===
All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed above, or otherwise makes it equal to a product or ratio between two such commas, this often means nontrivial, deep tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.
All S-expressions have other equivalent S-expressions; however, when the equivalence makes one comma a member of two of the infinite families discussed on this page, or otherwise makes it equal to a product or ratio between two such commas, this often means exceptional and nontrivial ("deep") tempering opportunities, usually leading to multiple of the most elegant and efficient temperaments that we know of depending on how the tempering is further realized. Generally we exclude 1/''n''-square-particulars, only noting up to 1/3-square-particulars, because equivalent 1/''n''-square-particular expressions become very common for higher ''n'', but are still quite rare for small ''n''.


=== A useful general rule ===
=== A useful general rule ===
Line 2,740: Line 2,882:


$$
$$
\large {\rm S}k = \large {\rm S}(2k-1) \cdot \large {\rm S}(2k)^2 \cdot \large {\rm S}(2k+1)
{\rm S}k = {\rm S}(2k - 1) \cdot {\rm S}(2k)^2 \cdot {\rm S}(2k + 1)
$$
$$


This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions that should be discarded from [[#Examples]]. See [[S-expression/Advanced results]] for mathematical details.
This is important to note because using this simple rule we can derive an infinite amount of trivially and obviously equivalent S-expressions. See [[S-expression/Advanced results]] for mathematical details.


=== Examples ===
=== Examples ===
Line 2,760: Line 2,902:
|-
|-
| [[64/63]]
| [[64/63]]
| S8, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
| S8, S6/S9, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
|-
|-
| [[81/80]]
| [[81/80]]
Line 2,769: Line 2,911:
|-
|-
| [[243/242]]
| [[243/242]]
| S9/S11, S15/([[3025/3024|S22/S24 = S55]])
| S9/S11, S15/S55, S15/(S22/S24)
|-
|-
| [[325/324]]
| [[325/324]]
Line 2,782: Line 2,924:
| [[1225/1224]]
| [[1225/1224]]
| S35, S49⋅S50
| S35, S49⋅S50
|-
| [[2601/2600]]
| S51, S17/(S25⋅S26)
|-
|-
| [[3025/3024]]
| [[3025/3024]]
| S55, S22/S24, (S25/S27)⋅S99
| S55, S22/S24, (S25/S27)⋅S99
|-
| [[2601/2600]]
| S51, S17/(S25⋅S26)
|-
|-
| [[9801/9800]]
| [[9801/9800]]
Line 2,795: Line 2,937:
| S161, S46/S48
| S161, S46/S48
|-
|-
| [[123201/123200]]
| <small>[[123201/123200]]</small>
| S351, S78/S80
| S351, S78/S80
|}
|}


{{Note| Examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are not included. }}
{{Note| Examples that can ''easily'' (with one or two algebraic rewriting steps) be shown to result from the aforementioned [[#A useful general rule|useful general rule]] are not included. }}
{{Note| Where a comma written in the form ''a''/''b'' is used in an S-expression, this means to replace that comma with any equivalent S-expression. This is done in the case of [[3025/3024]] as there are many S-expressions for it so restating them each time it appears seems inconvenient. }}
{{Tip| Feel free to expand with any equivalences you find that you think are valuable. }}
{{Tip| Feel free to expand with any equivalences you find that you think are valuable. }}


Retrieved from "https://en.xen.wiki/w/S-expression"