Goldonic series - Revision history

BudjarnLambeth: Category

2024-11-17T12:24:03Z

BudjarnLambeth: Added links to dead end pg

2024-01-09T05:39:16Z

Added links to dead end pg

← Older revision		Revision as of 05:39, 9 January 2024
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	A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [[Wikipedia:Geometric_progression\|geometric progression]] whose generating interval is the [[~~Wikipedia:Golden_ratio~~\|golden ratio]] (φ = 1.618...).		A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [[Wikipedia:Geometric_progression\|geometric progression]] whose generating interval is the [[phi\|golden ratio]] (φ = 1.618...).

	== Unique properties ==		== Unique properties ==
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	From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).		From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).

	Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.		Also, unlike the [[harmonic series]], the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.

	[[Category:Golden ratio]]		[[Category:Golden ratio]]
	[[Category:Nonoctave]]		[[Category:Nonoctave]]

BudjarnLambeth: Moved from category "golden" to "golden ratio"

2023-04-28T09:26:26Z

Moved from category "golden" to "golden ratio"

← Older revision		Revision as of 09:26, 28 April 2023
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	Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.		Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.

	[[Category:Golden]]		[[Category:Golden ratio]]
	[[Category:Nonoctave]]		[[Category:Nonoctave]]

Fredg999: Recategorization, improved markup, misc. edits

2021-08-05T15:43:54Z

Recategorization, improved markup, misc. edits

← Older revision		Revision as of 15:43, 5 August 2021
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	A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [~~https~~:~~//en.wikipedia.org/wiki/~~Geometric_progression geometric progression] whose generating interval is the [~~https~~:~~//en.wikipedia.org/wiki/~~Golden_ratio golden ratio] (1.~~61803.~~...).		A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [[Wikipedia:Geometric_progression\|geometric progression]] whose generating interval is the [[Wikipedia:Golden_ratio\|golden ratio]] (φ = 1.618...).

	==Unique properties==		== Unique properties ==

	The goldonic series is unique among geometric ~~sequencies~~ because only ''~~<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">~~φ~~</span>~~'' satisfies the equation ~~''x''~~<~~span style="vertical-align: super;"~~>n-1~~</span> ''~~+ x~~''<span style="vertical-align: super;">~~n~~</span> ''~~= x~~''<span style="vertical-align: super;">~~n+1</~~span~~>.		The goldonic series is unique among geometric sequences because only ''φ'' satisfies the equation <math>x^{n-1} + x^n = x^{n+1}</math>.

	From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).		From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).

	Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.		Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.
	[[Category:~~golden~~]]
	[[Category:~~golden_ratio~~]]		[[Category:Golden]]
			[[Category:Nonoctave]]

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>~~		A '''goldonic series''' or '''golden series''' is a series of frequencies that form a [https://en.wikipedia.org/wiki/Geometric_progression geometric progression] whose generating interval is the [https://en.wikipedia.org/wiki/Golden_ratio golden ratio] (1.61803....).
	~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~
	~~: This revision was by author [[User:xenwolf\|xenwolf]] and made on <tt>2015-12-01 03:28:17 UTC</tt>.<br>~~
	~~: The original revision id was <tt>568386057</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">A goldonic series or golden series is a series of frequencies that form a [[https://en.wikipedia.org/wiki/Geometric_progression\|geometric progression]] whose generating interval is the [[https://en.wikipedia.org/wiki/Golden_ratio\|golden ratio]] (1.61803....).

	==Unique properties==		==Unique properties==

	The goldonic series is unique among geometric sequencies because only ~~//<~~span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;"~~>~~φ~~<~~/span~~>//~~ satisfies the equation //x~~//<~~span style="vertical-align: super;"~~>~~n-1~~<~~/span~~> //~~+ x~~//<~~span style="vertical-align: super;"~~>~~n~~<~~/span~~> //~~= x~~//<~~span style="vertical-align: super;"~~>~~n+1~~<~~/span~~>~~.		The goldonic series is unique among geometric sequencies because only ''<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span>'' satisfies the equation ''x''<span style="vertical-align: super;">n-1</span> ''+ x''<span style="vertical-align: super;">n</span> ''= x''<span style="vertical-align: super;">n+1</span>.

	From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).		From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).

	Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in //both// directions and contains no fundamental.~~</pre></div>~~		Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in ''both'' directions and contains no fundamental.
	~~<h4>Original HTML content~~:~~</h4>~~		[[Category:golden]]
	<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>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow">golden ratio</a> (1.61803....).<br />		[[Category:golden_ratio]]
	~~<br />~~
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Unique properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Unique properties</h2>
	~~<br />~~
	~~The goldonic series is unique among geometric sequencies because only <em><span style="background-color~~: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span></em> satisfies the equation <em>x</em><span style="vertical-align: super;">n-1</span> <em>+ x</em><span style="vertical-align: super;">n</span> <em>= x</em><span style="vertical-align: super;">n+1</span>.<br />
	~~<br />~~
	From an acoustic standpoint, the goldonic series contains some &quot;harmonic-like&quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).<br />
	~~<br />~~
	~~Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in <em>both</em> directions and contains no fundamental.</body></html></pre></div>~~

Wikispaces>xenwolf: Imported revision 568386057 - Original comment:

2015-12-01T03:28:17Z

**Imported revision 568386057 - Original comment: **

← Older revision		Revision as of 03:28, 1 December 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:~~MasonGreen1~~\|~~MasonGreen1~~]] and made on <tt>2015-11-~~30 23~~:31:54 UTC</tt>.<br>		: This revision was by author [[User:xenwolf\|xenwolf]] and made on <tt>2015-12-01 03:28:17 UTC</tt>.<br>
	: The original revision id was <tt>~~568375387~~</tt>.<br>		: The original revision id was <tt>568386057</tt>.<br>
	: The revision comment was: <tt></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>		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>		<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">A goldonic series or golden series is a series of frequencies that form a [[https://en.wikipedia.org/wiki/Geometric_progression\|geometric progression]] whose generating interval is the golden ratio (1.61803....).		<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">A goldonic series or golden series is a series of frequencies that form a [[https://en.wikipedia.org/wiki/Geometric_progression\|geometric progression]] whose generating interval is the [[https://en.wikipedia.org/wiki/Golden_ratio\|golden ratio]] (1.61803....).

	==Unique properties==		==Unique properties==
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	Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in //both// directions and contains no fundamental.</pre></div>		Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in //both// directions and contains no fundamental.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the golden ratio (1.61803....).<br />		<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>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow">golden ratio</a> (1.61803....).<br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Unique properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Unique properties</h2>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Unique properties"></a><!-- ws:end:WikiTextHeadingRule:0 -->Unique properties</h2>

Wikispaces>MasonGreen1: Imported revision 568375387 - Original comment:

2015-11-30T23:31:54Z

**Imported revision 568375387 - Original comment: **

← Older revision		Revision as of 23:31, 30 November 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-30 23:31:11 UTC</tt>.<br>		: This revision was by author [[User:MasonGreen1\|MasonGreen1]] and made on <tt>2015-11-30 23:31:54 UTC</tt>.<br>
	: The original revision id was <tt>~~568375357~~</tt>.<br>		: The original revision id was <tt>568375387</tt>.<br>
	: The revision comment was: <tt></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>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	The goldonic series is unique among geometric sequencies because only //<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span>// satisfies the equation //x//<span style="vertical-align: super;">n-1</span> //+ x//<span style="vertical-align: super;">n</span> //= x//<span style="vertical-align: super;">n+1</span>.		The goldonic series is unique among geometric sequencies because only //<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span>// satisfies the equation //x//<span style="vertical-align: super;">n-1</span> //+ x//<span style="vertical-align: super;">n</span> //= x//<span style="vertical-align: super;">n+1</span>.

	From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors)..</pre></div>		From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).

			Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in //both// directions and contains no fundamental.</pre></div>
	<h4>Original HTML content:</h4>		<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the golden ratio (1.61803....).<br />		<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>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the golden ratio (1.61803....).<br />
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	The goldonic series is unique among geometric sequencies because only <em><span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span></em> satisfies the equation <em>x</em><span style="vertical-align: super;">n-1</span> <em>+ x</em><span style="vertical-align: super;">n</span> <em>= x</em><span style="vertical-align: super;">n+1</span>.<br />		The goldonic series is unique among geometric sequencies because only <em><span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span></em> satisfies the equation <em>x</em><span style="vertical-align: super;">n-1</span> <em>+ x</em><span style="vertical-align: super;">n</span> <em>= x</em><span style="vertical-align: super;">n+1</span>.<br />
	<br />		<br />
	From an acoustic standpoint, the goldonic series contains some &quot;harmonic-like&quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors)..</body></html></pre></div>		From an acoustic standpoint, the goldonic series contains some &quot;harmonic-like&quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors).<br />
			<br />
			Also, unlike the harmonic series, the goldonic series can be in theory extended infinitely in <em>both</em> directions and contains no fundamental.</body></html></pre></div>

Wikispaces>MasonGreen1: Imported revision 568375357 - Original comment:

2015-11-30T23:31:11Z

**Imported revision 568375357 - Original comment: **

New page

<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on <tt>2015-11-30 23:31:11 UTC</tt>.<br>
: The original revision id was <tt>568375357</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">A **goldonic series** or **golden series** is a series of frequencies that form a [[https://en.wikipedia.org/wiki/Geometric_progression|geometric progression]] whose generating interval is the golden ratio (1.61803....).

==Unique properties==

The goldonic series is unique among geometric sequencies because only //<span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span>// satisfies the equation //x//<span style="vertical-align: super;">n-1</span> //+ x//<span style="vertical-align: super;">n</span> //= x//<span style="vertical-align: super;">n+1</span>.

From an acoustic standpoint, the goldonic series contains some "harmonic-like" characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors)..</pre></div>
<h4>Original HTML 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;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Goldonic series</title></head><body>A <strong>goldonic series</strong> or <strong>golden series</strong> is a series of frequencies that form a <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Geometric_progression" rel="nofollow">geometric progression</a> whose generating interval is the golden ratio (1.61803....).<br />
<br />
<h2 id="toc0"><a name="x-Unique properties"></a>Unique properties</h2>
<br />
The goldonic series is unique among geometric sequencies because only <em><span style="background-color: #ffffff; color: #252525; font-family: sans-serif; font-size: 14px;">φ</span></em> satisfies the equation <em>x</em><span style="vertical-align: super;">n-1</span> <em>+ x</em><span style="vertical-align: super;">n</span> <em>= x</em><span style="vertical-align: super;">n+1</span>.<br />
<br />
From an acoustic standpoint, the goldonic series contains some &quot;harmonic-like&quot; characteristics despite being inharmonic. Because each term is equal the difference between the next highest and next lowest terms, a phenomenon similar to modelocking can occur (in which each oscillator becomes entrained to the difference tone generated by its nearest-neighbors)..</body></html></pre></div>

← Older revision		Revision as of 12:24, 17 November 2024
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	[[Category:Golden ratio]]		[[Category:Golden ratio]]
	[[Category:Nonoctave]]		[[Category:Nonoctave]]
			[[Category:Xenharmonic series]]