Delta-N ratio: Difference between revisions

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The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. (Delta is also known as degree of epimoricity.) A ratio with a delta of N is called a '''delta-N ratio'''.

The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. Delta is also known as '''degree of epimoricity'''. A ratio with a delta of ''N'' is called a '''delta-''N'' ratio'''.

{| class="wikitable" style="text-align:center;"

|+ Examples

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Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-N terminology was coined by [[Kite Giedraitis]].

Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-''N'' terminology was coined by [[Kite Giedraitis]].

More particularly, a superpartient ratio takes the form:

:<math>\~~frac~~{n + d}{n} = 1 + \~~frac~~{d}{n}</math>,

:<math>\dfrac{n + d}{n} = 1 + \dfrac{d}{n}</math>,

where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is ~~[[Wikipedia:Coprime~~|coprime]] to <math>n</math>.

where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is {{w|coprime}} to <math>n</math>.

== Etymology ==

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

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

== 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.

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.

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 superbipartient, supertripartient, superquadripartient, etc., or in ~~[[Delta-N|~~delta-N terminology]] as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.

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

=== Examples ===

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* 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''

* 4/1 = (16/13) (13/10) (10/7) (7/4).

Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.

Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors (including their order).

:''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''

~~[[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''.

{{W|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''.

[[Category:Terms]]

[[Category:Ratio]]

@@ Line 1: / Line 1: @@
+{{DISPLAYTITLE:Delta-''N'' ratio}}
 {{Beginner|Abc, high quality commas, and epimericity}}
 {{Wikipedia|Superpartient ratio}}
-The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. (Delta is also known as degree of epimoricity.) A ratio with a delta of N is called a '''delta-N ratio'''.
+The '''delta''' of a [[ratio]] is simply the difference between its numerator and its denominator. Delta is also known as '''degree of epimoricity'''. A ratio with a delta of ''N'' is called a '''delta-''N'' ratio'''.
 {| class="wikitable" style="text-align:center;"
 |+ Examples
@@ Line 42: / Line 44: @@
 |}
-Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-N terminology was coined by [[Kite Giedraitis]].
+Thus [[superparticular]] ratios are delta-1 ratios, and '''superpartient ratios''' are all ratios ''except'' delta-1 ratios. The delta-''N'' terminology was coined by [[Kite Giedraitis]].
 More particularly, a superpartient ratio takes the form:
-:<math>\frac{n + d}{n} = 1 + \frac{d}{n}</math>,
+:<math>\dfrac{n + d}{n} = 1 + \dfrac{d}{n}</math>,
-where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is [[Wikipedia:Coprime|coprime]] to <math>n</math>.
+where <math>n</math> and <math>d</math> are [[Wikipedia:Positive integer|positive integer]]s, <math>d > 1</math> and <math>d</math> is {{w|coprime}} to <math>n</math>.
 == Etymology ==
-In ancient Greece, they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
+In [[ancient Greek music|ancient Greece]], they were called epimeric (epimerēs) ratios, which is literally translated as "above a part".
 == 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.
+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''&mdash;see link below), or '''delta''' (proposed by [[Kite Giedraitis]]). This is particularly useful when considering ratios that are [[comma]]s.
+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 superbipartient, supertripartient, superquadripartient, etc., or in [[Delta-N|delta-N terminology]] as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.
+These subcategories are named as superbipartient, supertripartient, superquadripartient, etc., or in delta-''N'' terminology as delta-2, delta-3, delta-4, etc. Superparticular or epimoric ratios can likewise be named delta-1.
 === Examples ===
@@ Line 77: / Line 79: @@
 * 4/1 = (12/9) (9/6) (6/3) = '''(4/3) (3/2) (2/1)''' ''— now we can’t get delta-3 because there are 3 factors.''
 * 4/1 = (16/13) (13/10) (10/7) (7/4).
-Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors.
+Also, if you factorize like this into ''K'' factors, then each of them into ''L'' factors, you get the same as if you directly factored into ''K L'' factors (including their order).
 :''The general formula for this factorization is <math>\prod\limits_{i = 1}^K \frac {K A + i N} {K A + (i - 1) N} = \frac {A + N} A</math>. Here you can see more clearly that actual delta of factors will be <math>N / \operatorname{gcd}(K, N)</math>.''
-[[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''.
+{{W|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''.
 [[Category:Terms]]
 [[Category:Ratio]]