S-expression: Difference between revisions

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

Note that ~~while a lot~~ of ~~these have pages~~, ~~not all~~ of ~~them do~~.

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. An arithmetic of rather interesting commas whose elements are square-particulars 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.)

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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.)

Note from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3k+1)/S(3k+2) for a positive integer k, because then the superparticular can be expressed as:

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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.)

== Glossary ==

'''Superparticular''': The interval/comma between two consecutive harmonics. See [[superparticular]].

These are of the form (k+1)/k.

'''Square-particular''': A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.

These are of the form k<sup>2</sup>/(k<sup>2</sup> - 1) = Sk.

'''Odd-particular''': An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.

These are of the form (2k+1)/(2k-1).

'''Throdd-particular''': An interval/comma between two harmonics 3 apart which is not superparticular.

These are of the form (3k+1)/(3k) or (3k+2)/(3k+1).

'''Quodd-particular''': An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.

These are of the form (4k+1)/(4k-1) and (4k+3)/(4k+1).

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

These are of the form Sk/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 Sk/S(k+2).

'''S-expression''': An expression using the Sk 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.

'''Metaparticulars''': A suggested name for the general class of commas describable by S-expressions.

== Mathematical derivation ==

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:: = ( 2m + 2(3a) + 1 )/( 2m + 2(3a) - 1 )

:: ...which is of the form (x + 1)/(x - 1) meaning it is odd-particular.

:: ...which is of the form (2x + 1)/(2x - 1) meaning it is odd-particular.

In conclusion: Sk/S(k+2) is superparticular for k =/= 3 (mod 4) and is odd-particular when k = 3 (mod 4).

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 == Sk/S(k+1) (ultraparticulars) ==
-Note that while a lot of these have pages, not all of them do.
+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. An arithmetic of rather interesting commas whose elements are square-particulars 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.)
 {| class="wikitable center-all
@@ Line 85: / Line 86: @@
 | [[24576/24565]]
 |}
+(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't.)
 Note from this table how the shorthand becomes increasingly convenient higher up the series, where (preferably [[consistent]]) temperaments that temper out the ultraparticular but neither of the superparticulars which it is a difference between are of increasing precision. Note also how every three superparticulars the interval divided into three equal parts simplifies to a superparticular. This happens for S(3k+1)/S(3k+2) for a positive integer k, because then the superparticular can be expressed as:
@@ Line 198: / Line 202: @@
 | [[4375/4374]]
 |}
+(Note that while a lot of these have pages, not all of them do, although that doesn't mean they shouldn't.)
+== Glossary ==
+'''Superparticular''': The interval/comma between two consecutive harmonics. See [[superparticular]].
+These are of the form (k+1)/k.
+'''Square-particular''': A superparticular interval/comma whose numerator is a square number. A shorthand (nick)name for square superparticular.
+These are of the form k<sup>2</sup>/(k<sup>2</sup> - 1) = Sk.
+'''Odd-particular''': An interval/comma between two consecutive odd harmonics. The odd analogue of superparticular.
+These are of the form (2k+1)/(2k-1).
+'''Throdd-particular''': An interval/comma between two harmonics 3 apart which is not superparticular.
+These are of the form (3k+1)/(3k) or (3k+2)/(3k+1).
+'''Quodd-particular''': An interval/comma between two harmonics 4 apart which is not superparticular or odd-particular.
+These are of the form (4k+1)/(4k-1) and (4k+3)/(4k+1).
+'''Ultraparticular''': An interval/comma which is the ratio of two consecutive square-particulars.
+These are of the form Sk/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 Sk/S(k+2).
+'''S-expression''': An expression using the Sk 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.
+'''Metaparticulars''': A suggested name for the general class of commas describable by S-expressions.
 == Mathematical derivation ==
@@ Line 271: / Line 311: @@
 :: = ( 2m + 2(3a) + 1 )/( 2m + 2(3a) - 1 )
-:: ...which is of the form (x + 1)/(x - 1) meaning it is odd-particular.
+:: ...which is of the form (2x + 1)/(2x - 1) meaning it is odd-particular.
 In conclusion: Sk/S(k+2) is superparticular for k =/= 3 (mod 4) and is odd-particular when k = 3 (mod 4).