S-expression: Difference between revisions

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

~~Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that tempering any two~~ consecutive ~~ultraparticulars will imply tempering the~~ [[~~#Sk/S(k + 2) (semiparticulars)|semiparticular~~]] ~~which is their sum/product~~. ~~A rather-interesting arithmetic~~ of ~~square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions'''~~, which ~~is to say, expressions composed~~ of ~~square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness~~.

=== Motivational example ===

Often it is desirable to make consecutive [[superparticular]] intervals equidistant. This has a number of nice consequences, many of which not explained here — see the motivation section for each infinite family of commas defined on this page.

For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate ([[5/4]])/([[6/5]]) = [[25/24]] = S5 with ([[6/5]])/([[7/6]]) = [[36/35]] = S6, hence tempering S5/S6 = ([[25/24]])/([[36/35]]) = [[875/864]], but it's actually often not necessary to know the specific numbers, often familiarizing yourself with and understanding the "S''k''" notation will give you a lot of insight, as we'll see.

Back to our example: we know that S5 ~ S6 (because we're tempering S5/S6); from this we can deduce that the intervals must be arranged like this: 7/6 <— S5~S6 —> 6/5 <— S5~S6 —> 5/4.

From this you can deduce that ([[6/5]])<sup>3</sup> = [[7/4]], because you can lower one of the 6/5's to [[7/6]] (lowering it by S6) and raise another of the 6/5's to [[5/4]] (raising it by S5). Then because we've tempered S5 and S6 together, we've lowered and raised by the same amount, so the result of 7/6 * 6/5 * 5/4 = 7/4 must be the same as the result of 6/5 * 6/5 * 6/5 in this temperament.

Familiarize yourself with the structure of this argument, as [[Square superparticular#Mathematical derivation|it generalizes to arbitrary S''k'']]; the algebraic proof is tedious, but the intuition is the same:

(''k''+2)/(''k''+1) <— S(''k''+1)~S''k'' —> (''k''+1)/''k'' <— S(''k''+1)~S''k'' —> ''k''/(''k''-1)

...implies that three (''k''+1)/''k'''s is equal to a (''k''+2)/(''k''-1) iff we temper S''k''/S(''k''+1). {{qed}}

=== Significance ===

Note that tempering any two consecutive square-particulars S''k'' and S(''k''+1) will naturally imply tempering the ultraparticular between them (S''k''/S(''k''+1)) (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.

=== Table of ultraparticulars ===

{| class="wikitable center-all

|-

@@ Line 654: / Line 654: @@
 == Sk/S(k + 1) (ultraparticulars) ==
-Note that tempering any two consecutive square-particulars will naturally imply tempering the ultraparticular between them (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.
+=== Motivational example ===
+Often it is desirable to make consecutive [[superparticular]] intervals equidistant. This has a number of nice consequences, many of which not explained here — see the motivation section for each infinite family of commas defined on this page.
+For example, if you want 6/5 equidistant from 5/4 and 7/6, you must equate ([[5/4]])/([[6/5]]) = [[25/24]] = S5 with ([[6/5]])/([[7/6]]) = [[36/35]] = S6, hence tempering S5/S6 = ([[25/24]])/([[36/35]]) = [[875/864]], but it's actually often not necessary to know the specific numbers, often familiarizing yourself with and understanding the "S''k''" notation will give you a lot of insight, as we'll see.
+Back to our example: we know that S5 ~ S6 (because we're tempering S5/S6); from this we can deduce that the intervals must be arranged like this: 7/6 <— S5~S6 —> 6/5 <— S5~S6 —> 5/4.
+From this you can deduce that ([[6/5]])<sup>3</sup> = [[7/4]], because you can lower one of the 6/5's to [[7/6]] (lowering it by S6) and raise another of the 6/5's to [[5/4]] (raising it by S5). Then because we've tempered S5 and S6 together, we've lowered and raised by the same amount, so the result of 7/6 * 6/5 * 5/4 = 7/4 must be the same as the result of 6/5 * 6/5 * 6/5 in this temperament.
+Familiarize yourself with the structure of this argument, as [[Square superparticular#Mathematical derivation|it generalizes to arbitrary S''k'']]; the algebraic proof is tedious, but the intuition is the same:
+(''k''+2)/(''k''+1) <— S(''k''+1)~S''k'' —> (''k''+1)/''k'' <— S(''k''+1)~S''k'' —> ''k''/(''k''-1)
+...implies that three (''k''+1)/''k'''s is equal to a (''k''+2)/(''k''-1) iff we temper S''k''/S(''k''+1). {{qed}}
+=== Significance ===
+Note that tempering any two consecutive square-particulars S''k'' and S(''k''+1) will naturally imply tempering the ultraparticular between them (S''k''/S(''k''+1)) (meaning they are very common implicit commas), and that tempering any two consecutive ultraparticulars will imply tempering the [[#Sk/S(k + 2) (semiparticulars)|semiparticular]] which is their sum/product. A rather-interesting arithmetic of square-particular (and related) commas exists. This arithmetic can be described compactly with '''S-expressions''', which is to say, expressions composed of square superparticulars multiplied and divided together, using the Sk notation to achieve that compactness.
+=== Table of ultraparticulars ===
 {| class="wikitable center-all
 |-