Superpartient ratio: Difference between revisions

Line 8:

== Etymology ==

In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".

[[Kite Giedraitis]] has proposed the term delta-1 (where [[delta]] means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.

== Definitions ==

In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.

== Superpartient subcategories ==

Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.

These subcategories are named as delta-2, delta-3, delta-4, etc., or as superbipartient, supertripartient, superquadripartient, etc. Superparticular or epimoric ratios can likewise be named delta-1.

=== 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.

== Properties ==

Line 19:

Line 27:

<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), 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''.

[[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 8: / Line 8: @@
 == Etymology ==
 In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
-[[Kite Giedraitis]] has proposed the term delta-1 (where [[delta]] means difference, here the difference between the numerator and the denominator) as a replacement for superparticular, delta-2 for ratios of the form <math>\frac{n+2}{n}</math>, likewise delta-3, delta-4, etc.
 == Definitions ==
 In ancient Greece, fractions like 3/1 and 5/1 were not considered to be epimeric ratios because of their additional restriction that [[Harmonic|multiples of the fundamental]] cannot be epimeric. Epimeric ratios were considered to be inferior to epimoric ratios.
+== Superpartient subcategories ==
+Superpartient ratios can be grouped into subcategories based on the exact difference between the numerator and the denominator. This is known as the '''degree of epimoricity''' (not to be confused with ''epimericity'' – see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.
+These subcategories are named as delta-2, delta-3, delta-4, etc., or as superbipartient, supertripartient, superquadripartient, etc. Superparticular or epimoric ratios can likewise be named delta-1.
+=== 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.
 == Properties ==
@@ Line 19: / Line 27: @@
 <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), 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''.
+[[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 ==