Euler–Fokker genus: Difference between revisions

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As originally defined by [[Wikipedia: Leonhard Euler|Euler]], an Euler genus consists of all [[Wikipedia: Divisor|divisors]] of a given positive integer ''n'', reduced to an octave. Since we reduce to an octave, without loss of generality we can restrict ''n'' to be odd, in which case there is a one-to-one relationship between the Euler genus Euler (''n'') and the odd integers. However the real interest attaches to composite numbers of low prime limit; Euler himself considered mostly the 5-limit, and [[Wikipedia: Adriaan Fokker|Adriaan Fokker]] the 7-limit.

As originally defined by [[Wikipedia: Leonhard Euler|Euler]], an '''Euler genus''' consists of all [[Wikipedia: Divisor|divisors]] of a given positive integer ''n'', reduced to an octave. Since we reduce to an octave, without loss of generality we can restrict ''n'' to be odd, in which case there is a one-to-one relationship between the Euler genus Euler (''n'') and the odd integers. However the real interest attaches to composite numbers of low prime limit; Euler himself considered mostly the 5-limit, and [[Wikipedia: Adriaan Fokker|Adriaan Fokker]] the 7-limit.

Because of the way it is constructed, an Euler genus has chords related to the prime divisors of ''n'', with otonal and utonal chords appearing equally, and has scale size equal to ''d'' (''n''), the number of divisors of ''n''. If {{monzo| ''e''2 ''e''3 ''e''5 … e''p'' }} is the [[monzo]] for ''n'', then d (''n'') = (''e''2 + 1)(''e'3 + 1)…(''e''p + 1) and hence the size of the scale, ''d'' (''n''), is composite and tends to be highly composite.

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== See also ==

* [[~~Wikipedia: Euler–Fokker genus~~]]

* [[Epimorphic Euler genera]]

* [[Yantras]]

* [[Yer]]

[[Category:~~Math]]~~

[[Category:Euler-Fokker genera| ]]

~~[[Category:Scale~~]]

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-As originally defined by [[Wikipedia: Leonhard Euler|Euler]], an Euler genus consists of all [[Wikipedia: Divisor|divisors]] of a given positive integer ''n'', reduced to an octave. Since we reduce to an octave, without loss of generality we can restrict ''n'' to be odd, in which case there is a one-to-one relationship between the Euler genus Euler (''n'') and the odd integers. However the real interest attaches to composite numbers of low prime limit; Euler himself considered mostly the 5-limit, and [[Wikipedia: Adriaan Fokker|Adriaan Fokker]] the 7-limit.
+{{Wikipedia}}
+As originally defined by [[Wikipedia: Leonhard Euler|Euler]], an '''Euler genus''' consists of all [[Wikipedia: Divisor|divisors]] of a given positive integer ''n'', reduced to an octave. Since we reduce to an octave, without loss of generality we can restrict ''n'' to be odd, in which case there is a one-to-one relationship between the Euler genus Euler (''n'') and the odd integers. However the real interest attaches to composite numbers of low prime limit; Euler himself considered mostly the 5-limit, and [[Wikipedia: Adriaan Fokker|Adriaan Fokker]] the 7-limit.
 Because of the way it is constructed, an Euler genus has chords related to the prime divisors of ''n'', with otonal and utonal chords appearing equally, and has scale size equal to ''d'' (''n''), the number of divisors of ''n''. If {{monzo| ''e''<sub>2</sub> ''e''<sub>3</sub> ''e''<sub>5</sub> … e<sub>''p''</sub> }} is the [[monzo]] for ''n'', then d (''n'') = (''e''<sub>2</sub> + 1)(''e'<sub>3</sub> + 1)…(''e''<sub>p</sub> + 1) and hence the size of the scale, ''d'' (''n''), is composite and tends to be highly composite.
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 == See also ==
-* [[Wikipedia: Euler–Fokker genus]]
+* [[Epimorphic Euler genera]]
+* [[Yantras]]
+* [[Yer]]
-[[Category:Math]]
+[[Category:Euler-Fokker genera| ]] <!-- main article -->
-[[Category:Scale]]