Euler–Fokker genus: Difference between revisions
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As originally defined by | As originally defined by {{w|Leonhard Euler}}, an '''Euler genus''' is a [[scale]] that consists of all {{w|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 {{w|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. | 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. | ||
The Euler genus can be generalized in a natural way which brings out its relation to [[Combination product | The Euler genus can be generalized in a natural way which brings out its relation to [[Combination product set|combination product multisets]]. If we start from any {{w|multiset}} S of positive real numbers, we may define the corresponding genus Euler (S) to be the set of products of all the combinations of elements of the multiset, reduced to an octave. When we start from a multiset of rational numbers, this very often this will be an Euler genus as defined by Euler, but it need not be. If we take the combination products 0 at a time, 1 at a time and so forth up to ''n'' at a time, we get the genus; combination product multisets are slices of a genus. | ||
== See also == | == See also == | ||
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* [[Yer]] | * [[Yer]] | ||
[[Category: | [[Category:Math]] | ||
[[Category:Scale]] | |||
Revision as of 16:56, 13 December 2023
As originally defined by Leonhard Euler, an Euler genus is a scale that consists of all 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 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 [e2 e3 e5 … ep⟩ is the monzo for n, then d (n) = (e2 + 1)(e3 + 1)…(ep + 1) and hence the size of the scale, d (n), is composite and tends to be highly composite.
The Euler genus can be generalized in a natural way which brings out its relation to combination product multisets. If we start from any multiset S of positive real numbers, we may define the corresponding genus Euler (S) to be the set of products of all the combinations of elements of the multiset, reduced to an octave. When we start from a multiset of rational numbers, this very often this will be an Euler genus as defined by Euler, but it need not be. If we take the combination products 0 at a time, 1 at a time and so forth up to n at a time, we get the genus; combination product multisets are slices of a genus.