Superpartient ratio: Difference between revisions

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<math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>

When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta'''(proposed by [[Kite Giedraitis]]). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.

== Examples ==

* Delta-2 (superbipartient) ratios: [[3/1]], [[5/3]], [[7/5]], [[9/7]], [[11/9]], [[13/11]], etc.

* Delta-3 (supertripartient) ratios: [[4/1]], [[5/2]], [[7/4]], [[8/5]], [[10/7]], [[11/8]], etc.

* Delta-4 (superquadripartient) ratios: [[5/1]], [[7/3]], [[9/5]], [[11/7]], [[13/9]], [[15/11]], etc.

== See also ==

@@ Line 19: / Line 19: @@
 <math>\displaystyle \prod_{i \mathop = 1}^{P \mathop - 1} \dfrac {i + 1} {i} = P</math>
-When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.
+When considering ratios, and particularly when they are ratios for [[comma]]s, it can be useful to introduce the notion of the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta'''(proposed by [[Kite Giedraitis]]). In terms of ''p''/''q'' reduced to lowest terms it is ''p'' - ''q''. An epimoric ratio has degree 1, the 7-limit comma 245/243 degree 2, the 5-limit comma 128/125 degree 3, and so forth. [[Wikipedia:Størmer's theorem|Størmer's theorem]] can be extended to show that for each prime limit ''p'' and each degree of epimericity ''n'', there are only finitely many ''p''-limit ratios with degree of epimoricity less than or equal to ''n''.
+== Examples ==
+* Delta-2 (superbipartient) ratios: [[3/1]], [[5/3]], [[7/5]], [[9/7]], [[11/9]], [[13/11]], etc.
+* Delta-3 (supertripartient) ratios: [[4/1]], [[5/2]], [[7/4]], [[8/5]], [[10/7]], [[11/8]], etc.
+* Delta-4 (superquadripartient) ratios: [[5/1]], [[7/3]], [[9/5]], [[11/7]], [[13/9]], [[15/11]], etc.
 == See also ==