S-expression: Difference between revisions

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== 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 ===

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$$

~~\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 ===

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|-

| [[64/63]]

| S8, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3

| S8, S6/S9, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3

|-

| [[81/80]]

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|-

| [[243/242]]

| S9/S11, S15/(~~[[3025/3024|~~S22/S24 ~~= S55]]~~)

| S9/S11, S15/S55, S15/(S22/S24)

|-

| [[325/324]]

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| [[1225/1224]]

| S35, S49⋅S50

|-

| [[2601/2600]]

| S51, S17/(S25⋅S26)

|-

| [[3025/3024]]

| S55, S22/S24, (S25/S27)⋅S99

|-

~~| [[2601/2600]]~~

~~| S51, S17/(S25⋅S26)~~

|-

| [[9801/9800]]

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| S161, S46/S48

|-

| [[123201/123200]]

| <small>[[123201/123200]]</small>

| 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| 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. }}

@@ Line 2,876: / Line 2,876: @@
 == 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 ===
@@ Line 2,883: / 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 ===
@@ Line 2,903: / Line 2,902: @@
 |-
 | [[64/63]]
-| S8, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
+| S8, S6/S9, S4/(S6⋅S7), (S4⋅S5⋅S6)/S3
 |-
 | [[81/80]]
@@ Line 2,912: / Line 2,911: @@
 |-
 | [[243/242]]
-| S9/S11, S15/([[3025/3024|S22/S24 = S55]])
+| S9/S11, S15/S55, S15/(S22/S24)
 |-
 | [[325/324]]
@@ Line 2,925: / Line 2,924: @@
 | [[1225/1224]]
 | S35, S49⋅S50
+|-
+| [[2601/2600]]
+| S51, S17/(S25⋅S26)
 |-
 | [[3025/3024]]
 | S55, S22/S24, (S25/S27)⋅S99
-|-
-| [[2601/2600]]
-| S51, S17/(S25⋅S26)
 |-
 | [[9801/9800]]
@@ Line 2,938: / Line 2,937: @@
 | S161, S46/S48
 |-
-| [[123201/123200]]
+| <small>[[123201/123200]]</small>
 | 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| 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. }}