S-expression: Difference between revisions

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Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)2 is significant in that it has a special relationship: specifically, consider tempering (a/b)/(c/d)2 so therefore the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of the two superparticulars composing a superparticular or odd-particular with the two superparticulars directly adjacent to them. For example, as 9/8 = 18/17 * 17/16 we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering S16/S18 = (19/15)/(9/8)2 because we can multiply 9/8 by the tempered comma (19/15)/(9/8)2 to get (19/15)/(9/8) = (19/18)(16/15) (because 9/8 = 18/16), or as 13/11 = 13/12 * 12/11 we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering S11/S13 = (7/5)/(13/11)2 because we can multiply 13/11 by the tempered comma (7/5)/(13/11)2 to get (7/5)/(13/11) = (14/13)(11/10) (because 7/5 = 14/10). Note we have to replace ''both'' intervals ''simultaneously'' as this is lower error, and note that if we want to be able to replace them individually we must pick the higher error route of tempering S16 and S18 or S11 and S13 individually (for which tempering the semiparticular is then an implied consequence). (The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.)

Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]]. The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:

Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:

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 Also, the interval in the denominator of an expression of a semiparticular of the form (a/b)/(c/d)<sup>2</sup> is significant in that it has a special relationship: specifically, consider tempering (a/b)/(c/d)<sup>2</sup> so therefore the interval c/d is equal to the interval (a/b)/(c/d). This is significant because it allows the intuitive replacement of the two superparticulars composing a superparticular or odd-particular with the two superparticulars directly adjacent to them. For example, as 9/8 = 18/17 * 17/16 we can replace 18/17 with 19/18 and 17/16 with 16/15 by tempering S16/S18 = (19/15)/(9/8)<sup>2</sup> because we can multiply 9/8 by the tempered comma (19/15)/(9/8)<sup>2</sup> to get (19/15)/(9/8) = (19/18)(16/15) (because 9/8 = 18/16), or as 13/11 = 13/12 * 12/11 we can replace 13/12 with 14/13 and 12/11 with 11/10 by tempering S11/S13 = (7/5)/(13/11)<sup>2</sup> because we can multiply 13/11 by the tempered comma (7/5)/(13/11)<sup>2</sup> to get (7/5)/(13/11) = (14/13)(11/10) (because 7/5 = 14/10). Note we have to replace ''both'' intervals ''simultaneously'' as this is lower error, and note that if we want to be able to replace them individually we must pick the higher error route of tempering S16 and S18 or S11 and S13 individually (for which tempering the semiparticular is then an implied consequence). (The broader lesson is that you can rewrite exact JI equivalences with the commas you are tempering to find new interesting consequences of those commas.)
-Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]]. The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:
+Here follows a table of [[23-limit]] semiparticulars corresponding to square-particulars S''k'' for ''k'' < 96, plus all semiparticulars up to [[9801/9800|S33/S35 = S99]], an exceptional [[11-limit]] comma, plus all semiparticulars dividing [[superparticular interval]]s up to [[13/12]] (corresponding to the [[17-limit]] semiparticular [[31213/31212|S49/S51]]) for completeness. This table also shows all semiparticulars corresponding to splitting an [[#Glossary|odd-particular]] in two up to [[17/15]] (although a common strategy is to temper the square-particular that is the difference between the two superparticular intervals the odd-particular is composed of instead). The bound ''k'' < 96 was chosen as it corresponds to another remarkable semiparticular [[123201/123200|S78/S80 = S351]]. Perhaps many of the patterns will become clearer if you examine this table:
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