S-expression: Difference between revisions

Line 1,198:

This implies that s-ultraparticulars, s-semiparticulars, etc. are now about ratios between ultraparticulars, and that s-1/n-square-superparticulars are about ratios between square-particulars.

While the utility of the latter should be clear to those familiar with using square-particulars for analysis, it is not clear how useful a concept a ratio between ultraparticulars is, thus we should investigate why we might want such:

When we equate square-particulars, we lose accuracy the larger streak of them we equate, so eventually we are forced to break the chain.

When this happens, we have an ultraparticular that should, in any accurate temperament, be mapped positively, but as a result, we will have multiple such ultraparticulars.

Therefore, it is of interest to describe when these ultraparticulars are equated.

Now consider that as temperaments get more accurate, the streaks of equated square-particulars will either get higher/smaller or shorter, thus in the limit, we do not want to equate them.

Rather, we want to equate the ultraparticulars between them in order that we have a smooth growing of distances.

While the question of where it is most appropriate and accurate to equate ultraparticulars is beyond the scope of this section, it nonetheless shows that the 2D comma family Ss''a''/Ss''b'' has utility.

[[Category:Elementary math]]

[[Category:superparticular ratios]]

[[Category:Terms]]

@@ Line 1,198: / Line 1,198: @@
 This implies that s-ultraparticulars, s-semiparticulars, etc. are now about ratios between ultraparticulars, and that s-1/n-square-superparticulars are about ratios between square-particulars.
+While the utility of the latter should be clear to those familiar with using square-particulars for analysis, it is not clear how useful a concept a ratio between ultraparticulars is, thus we should investigate why we might want such:
+When we equate square-particulars, we lose accuracy the larger streak of them we equate, so eventually we are forced to break the chain.
+When this happens, we have an ultraparticular that should, in any accurate temperament, be mapped positively, but as a result, we will have multiple such ultraparticulars.
+Therefore, it is of interest to describe when these ultraparticulars are equated.
+Now consider that as temperaments get more accurate, the streaks of equated square-particulars will either get higher/smaller or shorter, thus in the limit, we do not want to equate them.
+Rather, we want to equate the ultraparticulars between them in order that we have a smooth growing of distances.
+While the question of where it is most appropriate and accurate to equate ultraparticulars is beyond the scope of this section, it nonetheless shows that the 2D comma family Ss''a''/Ss''b'' has utility.
 [[Category:Elementary math]]
 [[Category:superparticular ratios]]
 [[Category:Terms]]

S-expression: Difference between revisions

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