Euler–Fokker genus - Revision history

ArrowHead294 at 15:29, 23 January 2025

2025-01-23T15:29:37Z

← Older revision		Revision as of 15:29, 23 January 2025
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	{{Wikipedia}}		{{Wikipedia}}
	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]].		As originally defined by {{w\|Leonhard Euler}}, an '''Euler genus''' is a [[scale]] that consists of all {{w\|divisor}}s 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 {{nowrap\|''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 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.		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.

ArrowHead294: ArrowHead294 moved page Euler-Fokker genus to Euler–Fokker genus

2025-01-23T15:28:47Z

ArrowHead294 moved page Euler-Fokker genus to Euler–Fokker genus

← Older revision	Revision as of 15:28, 23 January 2025
(No difference)

FloraC: +category

2023-12-13T16:59:08Z

+category

← Older revision		Revision as of 16:59, 13 December 2023
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	* [[Yer]]		* [[Yer]]

			[[Category:Euler-Fokker genus\| ]] <!-- main article -->
	[[Category:Math]]		[[Category:Math]]
	[[Category:Scale]]		[[Category:Scale]]

FloraC: Improve linking and categories

2023-12-13T16:56:11Z

Improve linking and categories

← Older revision		Revision as of 16:56, 13 December 2023
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	{{Wikipedia}}		{{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.		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 ~~sets~~\|combination product multisets]]. If we start from any ~~[[Wikipedia: Multiset~~\|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.		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:~~Euler-Fokker genera\|~~ ]] ~~<!-- main article -->~~		[[Category:Math]]
			[[Category:Scale]]

Fredg999: Wikipedia box, see also, categories

2023-04-28T04:35:58Z

Wikipedia box, see also, categories

← Older revision		Revision as of 04:35, 28 April 2023
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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.		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 ==		== See also ==
	* [[~~Wikipedia: Euler–Fokker genus~~]]		* [[Epimorphic Euler genera]]
			* [[Yantras]]
			* [[Yer]]

	[[Category:~~Math]]~~		[[Category:Euler-Fokker genera\| ]] <!-- main article -->
	~~[[Category:Scale~~]]

Fredg999 category edits: Removing from Category:Theory using Cat-a-lot

2023-02-26T07:54:55Z

Removing from Category:Theory using Cat-a-lot

← Older revision		Revision as of 07:54, 26 February 2023
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	[[Category:Math]]		[[Category:Math]]
	~~[[Category:Theory]]~~
	[[Category:Scale]]		[[Category:Scale]]

Fredg999 category edits: Moving from Category:Scale theory to Category:Scale using Cat-a-lot

2022-02-26T16:32:51Z

Moving from Category:Scale theory to Category:Scale using Cat-a-lot

← Older revision		Revision as of 16:32, 26 February 2022
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Theory]]		[[Category:Theory]]
	[[Category:Scale ~~theory~~]]		[[Category:Scale]]

FloraC: Cleanup; internalize wikipedia links; improve categories

2021-06-25T07:08:31Z

Cleanup; internalize wikipedia links; improve categories

← Older revision		Revision as of 07:08, 25 June 2021
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	As originally defined by [~~http~~:~~//en.wikipedia.org/wiki/Leonhard_Euler~~ Euler], an Euler genus consists of all [~~http~~:~~//en.wikipedia.org/wiki/~~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 [~~http~~:~~//en.wikipedia.org/wiki/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 \|~~e2 e3 e5 ... ep>~~ is the [[~~monzo\|~~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.		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_sets~~\|combination product multisets]]. If we start from any [~~http~~:~~//en.wikipedia.org/wiki/~~Multiset 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 ~~needn't~~ 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.		The Euler genus can be generalized in a natural way which brings out its relation to [[Combination product sets\|combination product multisets]]. If we start from any [[Wikipedia: Multiset\|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.
	[[Category:~~math~~]]
	[[Category:theory]]		== See also ==
			* [[Wikipedia: Euler–Fokker genus]]

			[[Category:Math]]
			[[Category:Theory]]
			[[Category:Scale theory]]

FloraC: FloraC moved page Euler genera to Euler-Fokker genus: Following Wikipedia and open for further expansions

2021-06-25T07:01:05Z

FloraC moved page Euler genera to Euler-Fokker genus: Following Wikipedia and open for further expansions

← Older revision	Revision as of 07:01, 25 June 2021
(No difference)

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

← Older revision		Revision as of 00:00, 17 July 2018
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	~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~		As originally defined by [http://en.wikipedia.org/wiki/Leonhard_Euler Euler], an Euler genus consists of all [http://en.wikipedia.org/wiki/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 [http://en.wikipedia.org/wiki/Adriaan_Fokker Adriaan Fokker] the 7-limit.
	~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2013-12-24 16:48:04 UTC</tt>.<br>~~
	~~: The original revision id was <tt>479290308</tt>.<br>~~
	~~: The revision comment was: <tt></tt><br>~~
	~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~
	~~<h4>Original Wikitext content:</h4>~~
	<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">As originally defined by [[http://en.wikipedia.org/wiki/Leonhard_Euler\|Euler]], an Euler genus consists of all [[http://en.wikipedia.org/wiki/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 [[http://en.wikipedia.org/wiki/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 \|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.		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\|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 sets~~\|combination product multisets]]. If we start from any [[http://en.wikipedia.org/wiki/Multiset\|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 needn't 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.~~</pre></div>~~		The Euler genus can be generalized in a natural way which brings out its relation to [[Combination_product_sets\|combination product multisets]]. If we start from any [http://en.wikipedia.org/wiki/Multiset 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 needn't 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.
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:math]]
	~~<div style="width~~:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Euler genera</title></head><body>As originally defined by <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Leonhard_Euler" rel="nofollow">Euler</a>, an Euler genus consists of all <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Divisor" rel="nofollow">divisors</a> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Adriaan_Fokker" rel="nofollow">Adriaan Fokker</a> the 7-limit.<br />		[[Category:theory]]
	~~<br />~~
	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&gt; is the <a class="wiki_link" href="/monzo">monzo</a> 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.<br />
	~~<br />~~
	The Euler genus can be generalized in a natural way which brings out its relation to <a class="wiki_link" href="/Combination%20product%20sets">combination product multisets</a>. If we start from any <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Multiset" rel="nofollow">multiset</a> 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 needn't 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.</body></html></pre></div>