S-expression: Difference between revisions

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For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]]. (This terminology also suggests "throdd-particular" for intervals of the form (3''n'' + 1)/(3''n'' - 2) and (3''n'' + 2)/(3''n'' - 1) and maybe "quodd-particular" (sounding like "quad-particular") for (4''n'' + 3)/(4''n'' - 1) and (4''n'' + 1)/(4''n'' - 3).)

Tempering S(''k'' - 1)/S(''k'' + 3) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' and ''k''/(''k'' - 2) are equidistant from (''k'' + 1)/(''k'' - 1) because to make them equidistant we need to temper:

Tempering S(''k'' - 1)/S(''k'' + 3) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (= small) and ''k''/(''k'' - 2) (= large) are equidistant from (''k'' + 1)/(''k'' - 1) (= medium) because to make them equidistant we need to temper:

( (''k''/(''k''-2))/((''k''+1)/(''k''-1)) )/( ((''k''+1)/(''k''-1))/((''k''+2)/''k'') )

( large/medium )/( medium/small )

= ( (''k''/(''k''-2))/((''k''+1)/(''k''-1)) )/( ((''k''+1)/(''k''-1))/((''k''+2)/''k'') )

= large * small / medium2

= (''k''+2)/''k'' * ''k''/(''k''-2) / ((''k''+1)/(''k''-1))2 = ((''k''+2)/(''k''-2))/((''k''+1)/(''k''-1))2

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 For differences between square-particulars of the form S(''k'' + 1)/S(''k'' + 3) the resulting comma is either [[superparticular]] or [[#Glossary|odd-particular]]. (This terminology also suggests "throdd-particular" for intervals of the form (3''n'' + 1)/(3''n'' - 2) and (3''n'' + 2)/(3''n'' - 1) and maybe "quodd-particular" (sounding like "quad-particular") for (4''n'' + 3)/(4''n'' - 1) and (4''n'' + 1)/(4''n'' - 3).)
-Tempering S(''k'' - 1)/S(''k'' + 3) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' and ''k''/(''k'' - 2) are equidistant from (''k'' + 1)/(''k'' - 1) because to make them equidistant we need to temper:
+Tempering S(''k'' - 1)/S(''k'' + 3) implies that (''k'' + 2)/(''k'' - 2) is divisible exactly into two halves of (''k'' + 1)/(''k'' - 1). It also implies that the intervals (''k'' + 2)/''k'' (= small) and ''k''/(''k'' - 2) (= large) are equidistant from (''k'' + 1)/(''k'' - 1) (= medium) because to make them equidistant we need to temper:
-( (''k''/(''k''-2))/((''k''+1)/(''k''-1)) )/( ((''k''+1)/(''k''-1))/((''k''+2)/''k'')  )
+( large/medium )/( medium/small )
+= ( (''k''/(''k''-2))/((''k''+1)/(''k''-1)) )/( ((''k''+1)/(''k''-1))/((''k''+2)/''k'')  )
+= large * small / medium<sup>2</sup>
 = (''k''+2)/''k'' * ''k''/(''k''-2) / ((''k''+1)/(''k''-1))<sup>2</sup> = ((''k''+2)/(''k''-2))/((''k''+1)/(''k''-1))<sup>2</sup>