Sqrtphi/Chords - Revision history

Lériendil: Lériendil moved page Chords of sqrtphi to Sqrtphi/Chords

2025-06-22T21:52:09Z

Lériendil moved page Chords of sqrtphi to Sqrtphi/Chords

← Older revision	Revision as of 21:52, 22 June 2025
(No difference)

BudjarnLambeth: Categorised this uncategorised page

2023-04-25T11:08:33Z

Categorised this uncategorised page

← Older revision		Revision as of 11:08, 25 April 2023
Line 1,492:		Line 1,492:
	\| \| otonal		\| \| otonal
	\|}		\|}
			[[Category:Lists of chords]]
			[[Category:Dyadic chords]]
			[[Category:15-odd-limit]]
			[[Category:Sqrtphi]]

Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>genewardsmith: Imported revision 305389616 - Original comment:

2012-02-27T03:51:56Z

**Imported revision 305389616 - Original comment: **

Wikispaces>genewardsmith: Imported revision 305350808 - Original comment:

2012-02-26T23:13:11Z

**Imported revision 305350808 - Original comment: **

← Older revision		Revision as of 23:13, 26 February 2012
Line 1:		Line 1:
	<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:genewardsmith\|genewardsmith]] and made on <tt>2012-02-26 23:01:42 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2012-02-26 23:13:11 UTC</tt>.<br>
	: The original revision id was <tt>~~305348348~~</tt>.<br>		: The original revision id was <tt>305350808</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">Below are listed the 15-limit [[Dyadic chord\|dyadic chords]] of [[Kleismic family#Sqrtphi\|sqrtphi temperament]], subject to the condition that no interval smaller than [[12_11\|12/11]], at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. [[Rank and codimension\|Codimension]] two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it.		<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">Below are listed the 15-limit [[Dyadic chord\|dyadic chords]] of [[Kleismic family#Sqrtphi\|sqrtphi temperament]], subject to the condition that no interval smaller than [[12_11\|12/11]], at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. [[Rank and codimension\|Codimension]] two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it.

	Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ, which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the "official" 13/8 of sqrtphi, φ is 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-1-2-~~9-10 or 0-1~~-8-9-10 ~~work a stack~~ of φs ~~into chords which~~ composers might find of interest.		Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ, which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the "official" 13/8 of sqrtphi, φ is 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.

	=Triads=		=Triads=
Line 315:		Line 315:
	<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>Chords of sqrtphi</title></head><body>Below are listed the 15-limit <a class="wiki_link" href="/Dyadic%20chord">dyadic chords</a> of <a class="wiki_link" href="/Kleismic%20family#Sqrtphi">sqrtphi temperament</a>, subject to the condition that no interval smaller than <a class="wiki_link" href="/12_11">12/11</a>, at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. <a class="wiki_link" href="/Rank%20and%20codimension">Codimension</a> two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it. <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>Chords of sqrtphi</title></head><body>Below are listed the 15-limit <a class="wiki_link" href="/Dyadic%20chord">dyadic chords</a> of <a class="wiki_link" href="/Kleismic%20family#Sqrtphi">sqrtphi temperament</a>, subject to the condition that no interval smaller than <a class="wiki_link" href="/12_11">12/11</a>, at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. <a class="wiki_link" href="/Rank%20and%20codimension">Codimension</a> two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it. <br />
	<br />		<br />
	Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ, which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the &quot;official&quot; 13/8 of sqrtphi, φ is 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-1-2-~~9-10 or 0-1~~-8-9-10 ~~work a stack~~ of φs ~~into chords which~~ composers might find of interest.<br />		Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ, which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the &quot;official&quot; 13/8 of sqrtphi, φ is 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.<br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Triads</h1>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Triads</h1>

Wikispaces>genewardsmith: Imported revision 305348348 - Original comment:

2012-02-26T23:01:42Z

**Imported revision 305348348 - Original comment: **

Show changes

Wikispaces>genewardsmith: Imported revision 305328792 - Original comment:

2012-02-26T21:38:58Z

**Imported revision 305328792 - Original comment: **

Show changes

← Older revision		Revision as of 03:51, 27 February 2012
Line 1:		Line 1:
	<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:genewardsmith\|genewardsmith]] and made on <tt>2012-02-~~26 23~~:13:11 UTC</tt>.<br>		: This revision was by author [[User:genewardsmith\|genewardsmith]] and made on <tt>2012-02-27 03:51:56 UTC</tt>.<br>
	: The original revision id was <tt>~~305350808~~</tt>.<br>		: The original revision id was <tt>305389616</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">Below are listed the 15-limit [[Dyadic chord\|dyadic chords]] of [[Kleismic family#Sqrtphi\|sqrtphi temperament]], subject to the condition that no interval smaller than [[12_11\|12/11]], at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. [[Rank and codimension\|Codimension]] two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it.		<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">Below are listed the 15-limit [[Dyadic chord\|dyadic chords]] of [[Kleismic family#Sqrtphi\|sqrtphi temperament]], subject to the condition that no interval smaller than [[12_11\|12/11]], at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. [[Rank and codimension\|Codimension]] two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it.

	Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ, which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the "official" 13/8 of sqrtphi, φ is 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.		Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ = √((1+√5)/2), which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the "official" 13/8 of sqrtphi, φ is only 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.

	=Triads=		=Triads=
Line 310:		Line 310:
	\|\| Number \|\| Chord \|\| Transversal \|\| Type \|\|		\|\| Number \|\| Chord \|\| Transversal \|\| Type \|\|
	\|\| 1 \|\| 0-21-22-30-35-60 \|\| 1-18/11-9/7-3/2-9/5-9/8 \|\|utonal \|\|		\|\| 1 \|\| 0-21-22-30-35-60 \|\| 1-18/11-9/7-3/2-9/5-9/8 \|\|utonal \|\|
	\|\| 2 \|\| 0-25-30-38-39-60 \|\| 1-5/4-3/2-7/4-11/8-9/8 \|\| otonal \|\|		\|\| 2 \|\| 0-25-30-38-39-60 \|\| 1-5/4-3/2-7/4-11/8-9/8 \|\| otonal \|\|</pre></div>
	</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>Chords of sqrtphi</title></head><body>Below are listed the 15-limit <a class="wiki_link" href="/Dyadic%20chord">dyadic chords</a> of <a class="wiki_link" href="/Kleismic%20family#Sqrtphi">sqrtphi temperament</a>, subject to the condition that no interval smaller than <a class="wiki_link" href="/12_11">12/11</a>, at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. <a class="wiki_link" href="/Rank%20and%20codimension">Codimension</a> two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it. <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>Chords of sqrtphi</title></head><body>Below are listed the 15-limit <a class="wiki_link" href="/Dyadic%20chord">dyadic chords</a> of <a class="wiki_link" href="/Kleismic%20family#Sqrtphi">sqrtphi temperament</a>, subject to the condition that no interval smaller than <a class="wiki_link" href="/12_11">12/11</a>, at 150 cents, appears. The essentially just chords are typed as otonal, utonal, or ambitonal. Those requiring tempering only by 325/324 are labeled marveltwin, by 364/363 gentle, by 540/539 swetismic, by 676/675 island, and by 1575/1573 nicolic. <a class="wiki_link" href="/Rank%20and%20codimension">Codimension</a> two temperaments are labeled swetismic-gentle, island-gentle and kleismic-marveltwin according to two commas which define it. <br />
	<br />		<br />
	Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ, which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the &quot;official&quot; 13/8 of sqrtphi, φ is 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.<br />		Sqrtphi has MOS of size 5, 8, 11, 14, 17, 20, 23, 26, 49 and 72. The largest chords on these lists have complexity 70, and so would require the 72 note MOS, but there are many chords of much lower complexity. Sqrtphi has a generator very close to √φ = √((1+√5)/2), which can be chosen to be exactly √φ. Either way, aside from the dyadic chords, chords with stacks of φ are interesting since φ² - φ = 1, resulting in a relationship between difference tones. Though it isn't the &quot;official&quot; 13/8 of sqrtphi, φ is only 7.44 cents flat of 13/8 which is close enough for government work. Chords such as 0-2-4-8-10 containing stacks of φs composers might find of interest.<br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Triads</h1>		<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Triads"></a><!-- ws:end:WikiTextHeadingRule:0 -->Triads</h1>

Sqrtphi/Chords - Revision history

Lériendil: Lériendil moved page Chords of sqrtphi to Sqrtphi/Chords

BudjarnLambeth: Categorised this uncategorised page

Wikispaces>FREEZE at 00:00, 17 July 2018

Wikispaces>genewardsmith: **Imported revision 305389616 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 305350808 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 305348348 - Original comment: **

Wikispaces>genewardsmith: **Imported revision 305328792 - Original comment: **

Wikispaces>genewardsmith: Imported revision 305389616 - Original comment:

Wikispaces>genewardsmith: Imported revision 305350808 - Original comment:

Wikispaces>genewardsmith: Imported revision 305348348 - Original comment:

Wikispaces>genewardsmith: Imported revision 305328792 - Original comment: