Non radical intervals with musical significance - Revision history

Lériendil: added zetave link

2025-03-26T13:13:07Z

added zetave link

← Older revision		Revision as of 13:13, 26 March 2025
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	\| \| <math>e^{2\pi} \approx 535.4917</math>		\| \| <math>e^{2\pi} \approx 535.4917</math>
	\| \| 10877.66		\| \| 10877.66
	\| \|		\| \| [[Zetave]]
	\| \| The zeta function has units that are given as divisions of the interval <math>e^{2\pi}</math>.		\| \| The [[zeta]] function has units that are given as divisions of the interval <math>e^{2\pi}</math>.
	\|}		\|}

	[[Category:Lists of intervals]]		[[Category:Lists of intervals]]
	[[Category:Math]]		[[Category:Math]]

BudjarnLambeth: Todo link, fix math formatting

2024-10-20T07:03:26Z

Todo link, fix math formatting

← Older revision		Revision as of 07:03, 20 October 2024
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			{{todo\|link\|cleanup\|comment=fix broken math formatting}}
	Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.		Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

Moremajorthanmajor at 01:22, 4 May 2023

2023-05-04T01:22:36Z

← Older revision		Revision as of 01:22, 4 May 2023
Line 1:		Line 1:
	Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.		Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

	There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>(a/b)^~~{1/~~c}</math>, where <math>a</math>, <math>b</math> and <math>c</math> ~~are integers~~. What follows is a list of musically significant non-radical intervals.		There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>(a/b)^c</math>, where <math>a</math> and <math>b</math> are integers and <math>c</math> is rational. What follows is a list of musically significant non-radical intervals.

	{\| class="wikitable"		{\| class="wikitable"

TallKite: fixed the definition of non-radical

2023-05-03T00:33:05Z

fixed the definition of non-radical

← Older revision		Revision as of 00:33, 3 May 2023
Line 1:		Line 1:
	Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.		Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

	There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>a^{1/b}</math>, where <math>a</math> and <math>b</math> are integers. What follows is a list of musically significant non-radical intervals.		There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>(a/b)^{1/c}</math>, where <math>a</math>, <math>b</math> and <math>c</math> are integers. What follows is a list of musically significant non-radical intervals.

	{\| class="wikitable"		{\| class="wikitable"

Fredg999: Fix math tags, add names, unify precision (cents), misc. edits

2023-05-02T05:23:27Z

Fix math tags, add names, unify precision (cents), misc. edits

← Older revision		Revision as of 05:23, 2 May 2023
Line 1:		Line 1:
	Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.		Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

	There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form [[math~~\|math]]~~ a^{1/b} [[math~~\|math]]~~, where a and b are integers. What follows is a list of musically significant non-radical intervals.		There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>a^{1/b}</math>, where <math>a</math> and <math>b</math> are integers. What follows is a list of musically significant non-radical intervals.

	{\| class="wikitable"		{\| class="wikitable"
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	\| \| '''Ratio'''		\| \| '''Ratio'''
	\| \| '''Cents'''		\| \| '''Cents'''
	\| \| '''Name'''		\| \| '''Name(s)'''
	\| \| '''Musical ~~Significance~~'''		\| \| '''Musical significance'''
	\|-		\|-
	\| \| <math>2^{1/\phi}		\| \| <math>2^{1/\phi} \approx 1.5348</math>
	\approx		\| \| 741.641
	1.5348</math>		\| \| [[Logarithmic phi]]
	\| \| 741.64		\| \| Divides the octave into two parts, one being phi times larger than the other in cents.
	\| \|
	\| \| ~~"Logarithmic phi" which divides~~ the octave into two parts, one being ~~Phi~~ times larger than the other in cents.
	\|-		\|-
	\| \| <math>\dfrac{\sqrt{5}+1}{2}		\| \| <math>\phi = \dfrac{\sqrt{5}+1}{2} \approx 1.6180</math>
	\approx		\| \| 833.090
	1.6180</math>		\| \| [[Acoustic phi]]<br>[[Golden ratio]]<br>Linear phi
	\| \| 833.09		\| \| "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [[Wikipedia:Diophantine_approximation#General_upper_bound\|Dirichlet's approximation theorem]].
	\| \| <~~math~~>~~\text{Phi } (\phi)~~<~~/math~~>
	\| \| "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [~~https~~:~~//en.wikipedia.org/wiki/~~Diophantine_approximation#General_upper_bound Dirichlet's ~~Approximation Theorem~~].
	\|-		\|-
	\| \| <math>e \approx 2.7183</math>		\| \| <math>e \approx 2.7183</math>
	\| \| 1731.23		\| \| 1731.23
	\| \| "e-tave"		\| \| [[Natave]]<br>[[Neper]]<br>"e-tave"
	\| \| In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave.		\| \| In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave.
	\|-		\|-
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	\| \| 10877.66		\| \| 10877.66
	\| \|		\| \|
	\| \| The zeta function has units that are given as divisions of the interval		\| \| The zeta function has units that are given as divisions of the interval <math>e^{2\pi}</math>.

	<math>e^{2\pi}</math>
	\|}		\|}

	[[Category:Lists of intervals]]		[[Category:Lists of intervals]]
	[[Category:Math]]		[[Category:Math]]

BudjarnLambeth: Categorised this uncategorised page

2023-04-26T01:19:53Z

Categorised this uncategorised page

← Older revision		Revision as of 01:19, 26 April 2023
Line 36:		Line 36:
	<math>e^{2\pi}</math>		<math>e^{2\pi}</math>
	\|}		\|}

			[[Category:Lists of intervals]]
			[[Category:Math]]

BudjarnLambeth: Undo revision 109412 by BudjarnLambeth (talk)

2023-04-26T01:19:19Z

Undo revision 109412 by BudjarnLambeth (talk)

← Older revision		Revision as of 01:19, 26 April 2023
Line 36:		Line 36:
	<math>e^{2\pi}</math>		<math>e^{2\pi}</math>
	\|}		\|}
	~~[[Category:Lists of intervals]]~~
	~~[[Category:Math]]~~

BudjarnLambeth: Categorised this uncategorised page

2023-04-26T01:19:07Z

Categorised this uncategorised page

← Older revision		Revision as of 01:19, 26 April 2023
Line 36:		Line 36:
	<math>e^{2\pi}</math>		<math>e^{2\pi}</math>
	\|}		\|}
			[[Category:Lists of intervals]]
			[[Category:Math]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>Sarzadoce: Imported revision 586762957 - Original comment:

2016-07-10T20:01:53Z

**Imported revision 586762957 - Original comment: **

← Older revision		Revision as of 20:01, 10 July 2016
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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:Sarzadoce\|Sarzadoce]] and made on <tt>2016-07-10 19:55:26 UTC</tt>.<br>		: This revision was by author [[User:Sarzadoce\|Sarzadoce]] and made on <tt>2016-07-10 20:01:53 UTC</tt>.<br>
	: The original revision id was <tt>~~586762787~~</tt>.<br>		: The original revision id was <tt>586762957</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>
Line 8:		Line 8:
	<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">Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.		<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">Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

	There are many non-radical intervals which have musical significance. By non-radical is meant a number that cannot be written in the form		There are many non-radical intervals which have musical significance. By non-radical is meant a number that cannot be written in the form [[math]] a^{1/b} [[math]], where a and b are integers. What follows is a list of musically significant non-radical intervals.
	~~<span style="display: block; text-align: center;"><span class="MathJax"><span class="~~math" style="display: inline-block; width: 1.94em;"><span style="clip: rect(1.04em,1000em,2.38em,-0.54em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.17em; width: 1.6em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 1.61em;"><span style="clip: rect(1.86em,1000em,2.7em,-0.54em); left: 0em; position: absolute; top: -2.5em;"><span class="mi" style="font-family: MathJax_Math;">//a//</span></span><span style="left: 0.51em; position: absolute; top: -2.72em;"><span class="texatom"><span class="mrow"><span class="mn" style="font-family: MathJax_Main; font-size: 70.7%;">1</span><span class="texatom"><span class="mrow"><span class="mo" style="font-family: MathJax_Main; font-size: 70.7%;">/</span></span></span><span class="mi" style="font-family: MathJax_Math; font-size: 70.7%;">//b~~//</span></span></span></span></span></span></span></span></span></span></span>~~
	, where a and b are integers. What follows is a list of musically significant non-radical intervals.

	\|\| Ratio \|\| Cents \|\| Name \|\| Musical Significance \|\|		\|\| Ratio \|\| Cents \|\| Name \|\| Musical Significance \|\|
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	[[math]] \|\| 833.09 \|\| [[math]]		[[math]] \|\| 833.09 \|\| [[math]]
	\text{Phi } (\phi)		\text{Phi } (\phi)
	[[math]] \|\| "Linear phi," the unique interval whose continued fraction ~~convergents~~ \|\|		[[math]] \|\| "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [[https://en.wikipedia.org/wiki/Diophantine_approximation#General_upper_bound\|Dirichlet's Approximation Theorem]]. \|\|
	\|\| [[math]]		\|\| [[math]]
	e \approx 2.7183		e \approx 2.7183
Line 37:		Line 35:
	<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>Non radical intervals with musical significance</title></head><body>Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.<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>Non radical intervals with musical significance</title></head><body>Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.<br />
	<br />		<br />
	There are many non-radical intervals which have musical significance. By <strong>non-radical</strong> is meant a number that cannot be written in the form<~~br />~~		There are many non-radical intervals which have musical significance. By <strong>non-radical</strong> is meant a number that cannot be written in the form <a class="wiki_link" href="/math">math</a> a^{1/b} <a class="wiki_link" href="/math">math</a>, where a and b are integers. What follows is a list of musically significant non-radical intervals.<br />
	~~<span style="display: block; text-align: center;"><span~~ class="~~MathJax"><span style="display: inline-block; width: 1.94em;~~" ~~class~~="math"><span style="clip: rect(1.04em,1000em,2.38em,-0.54em); display: inline-block; font-size: 120%; height: 0px; left: 0em; position: absolute; top: -2.17em; width: 1.6em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; height: 0px; position: relative; width: 1.61em;"><span style="clip: rect(1.86em,1000em,2.7em,-0.54em); left: 0em; position: absolute; top: -2.5em;"><span style="font-family: MathJax_Math;" class="mi"><em>a</em></span></span><span style="left: 0.51em; position: absolute; top: -2.72em;"><span class="texatom"><span class="mrow"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mn">1~~<~~/~~span>~~<~~span~~ class="~~texatom~~"~~><span class~~="~~mrow"><span style="font-family: MathJax_Main; font-size: 70.7%;" class="mo">~~/~~</span></span></span><span style="font-family: MathJax_Math; font-size: 70.7%;" class="mi~~">~~<em>b~~</~~em></span></span></span></span></span></span></span></span></span></span></span><br /~~>
	, where a and b are integers. What follows is a list of musically significant non-radical intervals.<br />
	<br />		<br />

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	--><script type="math/tex">\text{Phi } (\phi)</script><!-- ws:end:WikiTextMathRule:2 --><br />		--><script type="math/tex">\text{Phi } (\phi)</script><!-- ws:end:WikiTextMathRule:2 --><br />
	</td>		</td>
	<td>&quot;Linear phi,&quot; the unique interval whose continued fraction ~~convergents~~<br />		<td>&quot;Linear phi,&quot; the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Diophantine_approximation#General_upper_bound" rel="nofollow">Dirichlet's Approximation Theorem</a>.<br />
	</td>		</td>
	</tr>		</tr>

← Older revision		Revision as of 05:23, 2 May 2023
Line 1:		Line 1:
	Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.		Musicians are typically interested in musical intervals that are rational (i.e. justly intoned ratios), or equal divisions of these intervals. However, not all intervals fit into this box.

	There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form [[math~~\|math]]~~ a^{1/b} [[math~~\|math]]~~, where a and b are integers. What follows is a list of musically significant non-radical intervals.		There are many non-radical intervals which have musical significance. By '''non-radical''' is meant a number that cannot be written in the form <math>a^{1/b}</math>, where <math>a</math> and <math>b</math> are integers. What follows is a list of musically significant non-radical intervals.

	{\| class="wikitable"		{\| class="wikitable"
Line 7:		Line 7:
	\| \| '''Ratio'''		\| \| '''Ratio'''
	\| \| '''Cents'''		\| \| '''Cents'''
	\| \| '''Name'''		\| \| '''Name(s)'''
	\| \| '''Musical ~~Significance~~'''		\| \| '''Musical significance'''
	\|-		\|-
	\| \| <math>2^{1/\phi}		\| \| <math>2^{1/\phi} \approx 1.5348</math>
	\approx		\| \| 741.641
	1.5348</math>		\| \| [[Logarithmic phi]]
	\| \| 741.64		\| \| Divides the octave into two parts, one being phi times larger than the other in cents.
	\| \|
	\| \| ~~"Logarithmic phi" which divides~~ the octave into two parts, one being ~~Phi~~ times larger than the other in cents.
	\|-		\|-
	\| \| <math>\dfrac{\sqrt{5}+1}{2}		\| \| <math>\phi = \dfrac{\sqrt{5}+1}{2} \approx 1.6180</math>
	\approx		\| \| 833.090
	1.6180</math>		\| \| [[Acoustic phi]]<br>[[Golden ratio]]<br>Linear phi
	\| \| 833.09		\| \| "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [[Wikipedia:Diophantine_approximation#General_upper_bound\|Dirichlet's approximation theorem]].
	\| \| <~~math~~>~~\text{Phi } (\phi)~~<~~/math~~>
	\| \| "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of [~~https~~:~~//en.wikipedia.org/wiki/~~Diophantine_approximation#General_upper_bound Dirichlet's ~~Approximation Theorem~~].
	\|-		\|-
	\| \| <math>e \approx 2.7183</math>		\| \| <math>e \approx 2.7183</math>
	\| \| 1731.23		\| \| 1731.23
	\| \| "e-tave"		\| \| [[Natave]]<br>[[Neper]]<br>"e-tave"
	\| \| In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave.		\| \| In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave.
	\|-		\|-
Line 32:		Line 28:
	\| \| 10877.66		\| \| 10877.66
	\| \|		\| \|
	\| \| The zeta function has units that are given as divisions of the interval		\| \| The zeta function has units that are given as divisions of the interval <math>e^{2\pi}</math>.

	<math>e^{2\pi}</math>
	\|}		\|}

	[[Category:Lists of intervals]]		[[Category:Lists of intervals]]
	[[Category:Math]]		[[Category:Math]]

Non radical intervals with musical significance - Revision history

Lériendil: added zetave link

BudjarnLambeth: Todo link, fix math formatting

Moremajorthanmajor at 01:22, 4 May 2023

TallKite: fixed the definition of non-radical

Fredg999: Fix math tags, add names, unify precision (cents), misc. edits

BudjarnLambeth: Categorised this uncategorised page

BudjarnLambeth: Undo revision 109412 by BudjarnLambeth (talk)

BudjarnLambeth: Categorised this uncategorised page

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>Sarzadoce: **Imported revision 586762957 - Original comment: **

Wikispaces>Sarzadoce: Imported revision 586762957 - Original comment: