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Wikispaces>Andrew_Heathwaite
**Imported revision 23287421 - Original comment: I set some limits w/i which I intend to find every possible scale. ... .. .**
Wikispaces>spt3125
**Imported revision 504166090 - Original comment: **
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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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2008-05-01 23:31:07 UTC</tt>.<br>
: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-04-23 19:02:31 UTC</tt>.<br>
: The original revision id was <tt>23287421</tt>.<br>
: The original revision id was <tt>504166090</tt>.<br>
: The revision comment was: <tt>I set some limits w/i which I intend to find every possible scale. ... .. .</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">=Superparticular-Nonoctave-MOS=  
<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">=Superparticular-Nonoctave-MOS=  


NOTE: I haven't completed th list of scales on this page. Consider that part under construction. You can check th intro &amp; th few scales I have in th meantime, tho!
NOTE: I haven't completed the list of scales on this page. Consider that part under construction. You can check the intro &amp; the few scales I have in the meantime, though!


...
...


A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on th nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:
A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on the nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:


# [[Superparticular]] - meaning that th steps of th scale represent th intervals between adjacent notes in th harmonic series. You can identify these intervals easily, because they appear in th form //n/n-1.// Examples: 5:4, 7:6, 13:12, 41:40, etc.
# [[Superparticular]] - meaning that the steps of the scale represent the intervals between adjacent notes in the harmonic series. You can identify these intervals easily, because they appear in the form //n/n-1.// Examples: 5:4, 7:6, 13:12, 41:40, etc.
# [[Nonoctave]] - meaning that th scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, &amp; so on. These intervals can sound very harsh, but they can also sound incredibly rich &amp; dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.
# [[Nonoctave]] - meaning that the scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, &amp; so on. These intervals can sound very harsh, but they can also sound incredibly rich &amp; dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.
# [[MOSScales|Moment of Symmetry]] - meaning that th scale contain exactly two step sizes, spaced out as evenly as possible w/i th scale. Normally, you build MOS scales by continuously adding notes a given interval, called th generator, away from one another in one long chain until th resulting scale has only two step sizes. Pythagorean scales use 3/2 (th perfect fifth) as th generator. In 12edo, we can identify th standard pentatonic scale &amp; th various diatonic scales as MOS scales because you can build them using a chain of fifths.
# [[MOSScales|Moment of Symmetry]] - meaning that the scale contains exactly two step sizes, spaced out as evenly as possible within the scale. Normally, you build MOS scales by continuously adding notes a given interval, called the generator, away from one another in one long chain until the resulting scale has only two step sizes. Pythagorean scales use 3/2 (the perfect fifth) as the generator. In 12edo, we can identify the standard pentatonic scale &amp; the various diatonic scales as MOS scales because you can build them using a chain of fifths.


Jacky Ligon's scale meets these three criteria. I will use it as an example:
Jacky Ligon's scale meets these three criteria. I will use it as an example:
Line 22: Line 22:
//Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11//
//Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11//


# It fits as Superparticular because its intervals, 9/8 &amp; 12/11, both fit th form //n/n-1//.
# It is superparticular because its intervals, 9/8 &amp; 12/11, both fit the form //n/n-1//.
# It fits as nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.
# It is nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.
# It fits as MOS because it contains exactly two step sizes, spaced out as evenly as possible w/i th scale.
# It is MOS because it contains exactly two step sizes, spaced out as evenly as possible within the scale.


Inspired by th peculiar musical qualities of this scale, I set about looking for others, &amp; found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type w/i these (admittedly arbitrary) limits:
Inspired by the peculiar musical qualities of this scale, I set about looking for others, &amp; found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type within these (admittedly arbitrary) limits:


# Superparticular Limit: smallest interval: 41/40 = 42.8 cents.
# Superparticular Limit: smallest interval: 41/40 = 42.8 cents.
Line 32: Line 32:
# Moment of Symmetry Limit: greatest number of notes in a scale = 10.
# Moment of Symmetry Limit: greatest number of notes in a scale = 10.


Even w/ these limits in place, this produces a multitude of fascinating scales for our enjoyment &amp; fascination. I invite you to play &amp; share yr results!
Even with these limits in place, this produces a multitude of fascinating scales for our enjoyment &amp; fascination. I invite you to play &amp; share your results!


==Pentatonic (5-note) Scales:==  
==Pentatonic (5-note) Scales:==  
Line 41: Line 41:
SNM320614 : //6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents//
SNM320614 : //6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents//
SNM320615 : //6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents//
SNM320615 : //6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents//
==Heptatonic (7-note) Scales:==  
==Heptatonic (7-note) Scales:==  
===MOS 2+5 : sLsssLs===  
===MOS 2+5 : sLsssLs===  
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<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Superparticular-Nonoctave-MOS&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Superparticular-Nonoctave-MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Superparticular-Nonoctave-MOS&lt;/h1&gt;
<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">&lt;html&gt;&lt;head&gt;&lt;title&gt;Superparticular-Nonoctave-MOS&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Superparticular-Nonoctave-MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Superparticular-Nonoctave-MOS&lt;/h1&gt;
  &lt;br /&gt;
  &lt;br /&gt;
NOTE: I haven't completed th list of scales on this page. Consider that part under construction. You can check th intro &amp;amp; th few scales I have in th meantime, tho!&lt;br /&gt;
NOTE: I haven't completed the list of scales on this page. Consider that part under construction. You can check the intro &amp;amp; the few scales I have in the meantime, though!&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
...&lt;br /&gt;
...&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on th nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:&lt;br /&gt;
A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on the nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Superparticular"&gt;Superparticular&lt;/a&gt; - meaning that th steps of th scale represent th intervals between adjacent notes in th harmonic series. You can identify these intervals easily, because they appear in th form &lt;em&gt;n/n-1.&lt;/em&gt; Examples: 5:4, 7:6, 13:12, 41:40, etc.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Nonoctave"&gt;Nonoctave&lt;/a&gt; - meaning that th scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, &amp;amp; so on. These intervals can sound very harsh, but they can also sound incredibly rich &amp;amp; dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; - meaning that th scale contain exactly two step sizes, spaced out as evenly as possible w/i th scale. Normally, you build MOS scales by continuously adding notes a given interval, called th generator, away from one another in one long chain until th resulting scale has only two step sizes. Pythagorean scales use 3/2 (th perfect fifth) as th generator. In 12edo, we can identify th standard pentatonic scale &amp;amp; th various diatonic scales as MOS scales because you can build them using a chain of fifths.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Superparticular"&gt;Superparticular&lt;/a&gt; - meaning that the steps of the scale represent the intervals between adjacent notes in the harmonic series. You can identify these intervals easily, because they appear in the form &lt;em&gt;n/n-1.&lt;/em&gt; Examples: 5:4, 7:6, 13:12, 41:40, etc.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/Nonoctave"&gt;Nonoctave&lt;/a&gt; - meaning that the scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, &amp;amp; so on. These intervals can sound very harsh, but they can also sound incredibly rich &amp;amp; dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.&lt;/li&gt;&lt;li&gt;&lt;a class="wiki_link" href="/MOSScales"&gt;Moment of Symmetry&lt;/a&gt; - meaning that the scale contains exactly two step sizes, spaced out as evenly as possible within the scale. Normally, you build MOS scales by continuously adding notes a given interval, called the generator, away from one another in one long chain until the resulting scale has only two step sizes. Pythagorean scales use 3/2 (the perfect fifth) as the generator. In 12edo, we can identify the standard pentatonic scale &amp;amp; the various diatonic scales as MOS scales because you can build them using a chain of fifths.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
Jacky Ligon's scale meets these three criteria. I will use it as an example:&lt;br /&gt;
Jacky Ligon's scale meets these three criteria. I will use it as an example:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;em&gt;Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11&lt;/em&gt;&lt;br /&gt;
&lt;em&gt;Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11&lt;/em&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;It fits as Superparticular because its intervals, 9/8 &amp;amp; 12/11, both fit th form &lt;em&gt;n/n-1&lt;/em&gt;.&lt;/li&gt;&lt;li&gt;It fits as nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.&lt;/li&gt;&lt;li&gt;It fits as MOS because it contains exactly two step sizes, spaced out as evenly as possible w/i th scale.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;It is superparticular because its intervals, 9/8 &amp;amp; 12/11, both fit the form &lt;em&gt;n/n-1&lt;/em&gt;.&lt;/li&gt;&lt;li&gt;It is nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.&lt;/li&gt;&lt;li&gt;It is MOS because it contains exactly two step sizes, spaced out as evenly as possible within the scale.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
Inspired by th peculiar musical qualities of this scale, I set about looking for others, &amp;amp; found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type w/i these (admittedly arbitrary) limits:&lt;br /&gt;
Inspired by the peculiar musical qualities of this scale, I set about looking for others, &amp;amp; found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type within these (admittedly arbitrary) limits:&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;Superparticular Limit: smallest interval: 41/40 = 42.8 cents.&lt;/li&gt;&lt;li&gt;Nonoctave Limit: greatest deviation from octave allowed = 25 cents.&lt;/li&gt;&lt;li&gt;Moment of Symmetry Limit: greatest number of notes in a scale = 10.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
&lt;ol&gt;&lt;li&gt;Superparticular Limit: smallest interval: 41/40 = 42.8 cents.&lt;/li&gt;&lt;li&gt;Nonoctave Limit: greatest deviation from octave allowed = 25 cents.&lt;/li&gt;&lt;li&gt;Moment of Symmetry Limit: greatest number of notes in a scale = 10.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
Even w/ these limits in place, this produces a multitude of fascinating scales for our enjoyment &amp;amp; fascination. I invite you to play &amp;amp; share yr results!&lt;br /&gt;
Even with these limits in place, this produces a multitude of fascinating scales for our enjoyment &amp;amp; fascination. I invite you to play &amp;amp; share your results!&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Superparticular-Nonoctave-MOS-Pentatonic (5-note) Scales:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Pentatonic (5-note) Scales:&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Superparticular-Nonoctave-MOS-Pentatonic (5-note) Scales:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Pentatonic (5-note) Scales:&lt;/h2&gt;
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  SNM320614 : &lt;em&gt;6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents&lt;/em&gt;&lt;br /&gt;
  SNM320614 : &lt;em&gt;6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents&lt;/em&gt;&lt;br /&gt;
SNM320615 : &lt;em&gt;6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents&lt;/em&gt;&lt;br /&gt;
SNM320615 : &lt;em&gt;6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents&lt;/em&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Heptatonic (7-note) Scales:&lt;/h2&gt;
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Heptatonic (7-note) Scales:&lt;/h2&gt;
  &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:-MOS 2+5 : sLsssLs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;MOS 2+5 : sLsssLs&lt;/h3&gt;
  &lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc5"&gt;&lt;a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:-MOS 2+5 : sLsssLs"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;MOS 2+5 : sLsssLs&lt;/h3&gt;

Revision as of 19:02, 23 April 2014

IMPORTED REVISION FROM WIKISPACES

This is an imported revision from Wikispaces. The revision metadata is included below for reference:

This revision was by author spt3125 and made on 2014-04-23 19:02:31 UTC.
The original revision id was 504166090.
The revision comment was:

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

Original Wikitext content:

=Superparticular-Nonoctave-MOS= 

NOTE: I haven't completed the list of scales on this page. Consider that part under construction. You can check the intro & the few scales I have in the meantime, though!

...

A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on the nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:

# [[Superparticular]] - meaning that the steps of the scale represent the intervals between adjacent notes in the harmonic series. You can identify these intervals easily, because they appear in the form //n/n-1.// Examples: 5:4, 7:6, 13:12, 41:40, etc.
# [[Nonoctave]] - meaning that the scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, & so on. These intervals can sound very harsh, but they can also sound incredibly rich & dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.
# [[MOSScales|Moment of Symmetry]] - meaning that the scale contains exactly two step sizes, spaced out as evenly as possible within the scale. Normally, you build MOS scales by continuously adding notes a given interval, called the generator, away from one another in one long chain until the resulting scale has only two step sizes. Pythagorean scales use 3/2 (the perfect fifth) as the generator. In 12edo, we can identify the standard pentatonic scale & the various diatonic scales as MOS scales because you can build them using a chain of fifths.

Jacky Ligon's scale meets these three criteria. I will use it as an example:

//Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11//

# It is superparticular because its intervals, 9/8 & 12/11, both fit the form //n/n-1//.
# It is nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.
# It is MOS because it contains exactly two step sizes, spaced out as evenly as possible within the scale.

Inspired by the peculiar musical qualities of this scale, I set about looking for others, & found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type within these (admittedly arbitrary) limits:

# Superparticular Limit: smallest interval: 41/40 = 42.8 cents.
# Nonoctave Limit: greatest deviation from octave allowed = 25 cents.
# Moment of Symmetry Limit: greatest number of notes in a scale = 10.

Even with these limits in place, this produces a multitude of fascinating scales for our enjoyment & fascination. I invite you to play & share your results!

==Pentatonic (5-note) Scales:== 
===MOS 2+3 : sLsLs=== 
[[SNM230513]] : //13:12, 5:4, 13:12, 5:4, 13:12 = 1188.3 cents//
SNM230610 : //10:9, 6:5, 10:9, 6:5, 10:9 = 1178.5 cents//
===MOS 3+2 : LsLsL=== 
SNM320614 : //6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents//
SNM320615 : //6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents//

==Heptatonic (7-note) Scales:== 
===MOS 2+5 : sLsssLs=== 
[[SNM250520]] : //20:19, 5:4, 20:19, 20:19, 20:19, 5:4, 20:19 = 1216.6 cents//
[[SNM250521]] : //21:20, 5:4, 21:20, 21:20, 21:20, 5:4, 21:20 = 1195.0 cents (nickname: Mercury Sand)//
SNM250616 : //16:15, 6:5, 16:15, 16:15, 16:15, 6:5, 16:15 = 1189.9 cents//
===MOS 3+4 : sLsLsLs=== 
===MOS 4+3 : LsLsLsL=== 
===MOS 5+2 : LsLLLsL=== 

==Octatonic (8-note) Scales:== 
===MOS 3+5 : sLssLsLs=== 
===MOS 5+3 : LsLLsLsL=== 

==Nonatonic (9-note) Scales:== 
===MOS 2+7 : ssLsssLss=== 
[[SNM270528]] : //28:27, 28:27, 5:4, 28:27, 28:27, 28:27, 5:4, 28:27, 28:27 = 1213.4 cents//
[[SNM270529]] : //29:28, 29:28, 5:4, 29:28, 29:28, 29:28, 5:4, 29:28, 29:28 = 1197.9 cents//
[[SNM270530]] : //30:29, 30:29, 5:4, 30:29, 30:29, 30:29, 5:4, 30:29, 30:29 = 1183.5 cents//
SNM270622 : //22:21, 22:21, 6:5, 22:21, 22:21, 22:21, 6:5, 22:21, 22:21 = 1195.0 cents//
===MOS 4+5 : LsLsLsLsL=== 
===MOS 5+4 : sLsLsLsLs=== 
===MOS 7+2 : LLsLLLsLL=== 

==Dekatonic (10-note) Scales:== 
===MOS 3+7 : sLsssLssLs=== 
===MOS 7+3 : LsLLLsLLsL=== 

Original HTML content:

<html><head><title>Superparticular-Nonoctave-MOS</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Superparticular-Nonoctave-MOS"></a><!-- ws:end:WikiTextHeadingRule:0 -->Superparticular-Nonoctave-MOS</h1>
 <br />
NOTE: I haven't completed the list of scales on this page. Consider that part under construction. You can check the intro &amp; the few scales I have in the meantime, though!<br />
<br />
...<br />
<br />
A few years ago, inspired by a fantastic scale revealed by Jacky Ligon on the nonoctave forum, I (Andrew Heathwaite) embarked on a quest to discover new scales that meet these three criteria:<br />
<br />
<ol><li><a class="wiki_link" href="/Superparticular">Superparticular</a> - meaning that the steps of the scale represent the intervals between adjacent notes in the harmonic series. You can identify these intervals easily, because they appear in the form <em>n/n-1.</em> Examples: 5:4, 7:6, 13:12, 41:40, etc.</li><li><a class="wiki_link" href="/Nonoctave">Nonoctave</a> - meaning that the scale repeats at an interval other than an octave. In fact, for this project I wanted near-octaves, intervals like 1193 cents, 1221 cents, &amp; so on. These intervals can sound very harsh, but they can also sound incredibly rich &amp; dynamic. Timbre plays an important role here in making these near-octave intervals function as octaves.</li><li><a class="wiki_link" href="/MOSScales">Moment of Symmetry</a> - meaning that the scale contains exactly two step sizes, spaced out as evenly as possible within the scale. Normally, you build MOS scales by continuously adding notes a given interval, called the generator, away from one another in one long chain until the resulting scale has only two step sizes. Pythagorean scales use 3/2 (the perfect fifth) as the generator. In 12edo, we can identify the standard pentatonic scale &amp; the various diatonic scales as MOS scales because you can build them using a chain of fifths.</li></ol><br />
Jacky Ligon's scale meets these three criteria. I will use it as an example:<br />
<br />
<em>Steps: 9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11</em><br />
<br />
<ol><li>It is superparticular because its intervals, 9/8 &amp; 12/11, both fit the form <em>n/n-1</em>.</li><li>It is nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.</li><li>It is MOS because it contains exactly two step sizes, spaced out as evenly as possible within the scale.</li></ol><br />
Inspired by the peculiar musical qualities of this scale, I set about looking for others, &amp; found quite a few. I gave some of them quirky nicknames. I have since then embarked on a search for all scales of this type within these (admittedly arbitrary) limits:<br />
<br />
<ol><li>Superparticular Limit: smallest interval: 41/40 = 42.8 cents.</li><li>Nonoctave Limit: greatest deviation from octave allowed = 25 cents.</li><li>Moment of Symmetry Limit: greatest number of notes in a scale = 10.</li></ol><br />
Even with these limits in place, this produces a multitude of fascinating scales for our enjoyment &amp; fascination. I invite you to play &amp; share your results!<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="Superparticular-Nonoctave-MOS-Pentatonic (5-note) Scales:"></a><!-- ws:end:WikiTextHeadingRule:2 -->Pentatonic (5-note) Scales:</h2>
 <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="Superparticular-Nonoctave-MOS-Pentatonic (5-note) Scales:-MOS 2+3 : sLsLs"></a><!-- ws:end:WikiTextHeadingRule:4 -->MOS 2+3 : sLsLs</h3>
 <a class="wiki_link" href="/SNM230513">SNM230513</a> : <em>13:12, 5:4, 13:12, 5:4, 13:12 = 1188.3 cents</em><br />
SNM230610 : <em>10:9, 6:5, 10:9, 6:5, 10:9 = 1178.5 cents</em><br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h3&gt; --><h3 id="toc3"><a name="Superparticular-Nonoctave-MOS-Pentatonic (5-note) Scales:-MOS 3+2 : LsLsL"></a><!-- ws:end:WikiTextHeadingRule:6 -->MOS 3+2 : LsLsL</h3>
 SNM320614 : <em>6:5, 14:13, 6:5, 14:13, 6:5 = 1203.5 cents</em><br />
SNM320615 : <em>6:5, 15:14, 6:5, 15:14, 6:5 = 1185.8 cents</em><br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:&lt;h2&gt; --><h2 id="toc4"><a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:"></a><!-- ws:end:WikiTextHeadingRule:8 -->Heptatonic (7-note) Scales:</h2>
 <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:-MOS 2+5 : sLsssLs"></a><!-- ws:end:WikiTextHeadingRule:10 -->MOS 2+5 : sLsssLs</h3>
 <a class="wiki_link" href="/SNM250520">SNM250520</a> : <em>20:19, 5:4, 20:19, 20:19, 20:19, 5:4, 20:19 = 1216.6 cents</em><br />
<a class="wiki_link" href="/SNM250521">SNM250521</a> : <em>21:20, 5:4, 21:20, 21:20, 21:20, 5:4, 21:20 = 1195.0 cents (nickname: Mercury Sand)</em><br />
SNM250616 : <em>16:15, 6:5, 16:15, 16:15, 16:15, 6:5, 16:15 = 1189.9 cents</em><br />
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:-MOS 3+4 : sLsLsLs"></a><!-- ws:end:WikiTextHeadingRule:12 -->MOS 3+4 : sLsLsLs</h3>
 <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:-MOS 4+3 : LsLsLsL"></a><!-- ws:end:WikiTextHeadingRule:14 -->MOS 4+3 : LsLsLsL</h3>
 <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Superparticular-Nonoctave-MOS-Heptatonic (7-note) Scales:-MOS 5+2 : LsLLLsL"></a><!-- ws:end:WikiTextHeadingRule:16 -->MOS 5+2 : LsLLLsL</h3>
 <br />
<!-- ws:start:WikiTextHeadingRule:18:&lt;h2&gt; --><h2 id="toc9"><a name="Superparticular-Nonoctave-MOS-Octatonic (8-note) Scales:"></a><!-- ws:end:WikiTextHeadingRule:18 -->Octatonic (8-note) Scales:</h2>
 <!-- ws:start:WikiTextHeadingRule:20:&lt;h3&gt; --><h3 id="toc10"><a name="Superparticular-Nonoctave-MOS-Octatonic (8-note) Scales:-MOS 3+5 : sLssLsLs"></a><!-- ws:end:WikiTextHeadingRule:20 -->MOS 3+5 : sLssLsLs</h3>
 <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="Superparticular-Nonoctave-MOS-Octatonic (8-note) Scales:-MOS 5+3 : LsLLsLsL"></a><!-- ws:end:WikiTextHeadingRule:22 -->MOS 5+3 : LsLLsLsL</h3>
 <br />
<!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Superparticular-Nonoctave-MOS-Nonatonic (9-note) Scales:"></a><!-- ws:end:WikiTextHeadingRule:24 -->Nonatonic (9-note) Scales:</h2>
 <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="Superparticular-Nonoctave-MOS-Nonatonic (9-note) Scales:-MOS 2+7 : ssLsssLss"></a><!-- ws:end:WikiTextHeadingRule:26 -->MOS 2+7 : ssLsssLss</h3>
 <a class="wiki_link" href="/SNM270528">SNM270528</a> : <em>28:27, 28:27, 5:4, 28:27, 28:27, 28:27, 5:4, 28:27, 28:27 = 1213.4 cents</em><br />
<a class="wiki_link" href="/SNM270529">SNM270529</a> : <em>29:28, 29:28, 5:4, 29:28, 29:28, 29:28, 5:4, 29:28, 29:28 = 1197.9 cents</em><br />
<a class="wiki_link" href="/SNM270530">SNM270530</a> : <em>30:29, 30:29, 5:4, 30:29, 30:29, 30:29, 5:4, 30:29, 30:29 = 1183.5 cents</em><br />
SNM270622 : <em>22:21, 22:21, 6:5, 22:21, 22:21, 22:21, 6:5, 22:21, 22:21 = 1195.0 cents</em><br />
<!-- ws:start:WikiTextHeadingRule:28:&lt;h3&gt; --><h3 id="toc14"><a name="Superparticular-Nonoctave-MOS-Nonatonic (9-note) Scales:-MOS 4+5 : LsLsLsLsL"></a><!-- ws:end:WikiTextHeadingRule:28 -->MOS 4+5 : LsLsLsLsL</h3>
 <!-- ws:start:WikiTextHeadingRule:30:&lt;h3&gt; --><h3 id="toc15"><a name="Superparticular-Nonoctave-MOS-Nonatonic (9-note) Scales:-MOS 5+4 : sLsLsLsLs"></a><!-- ws:end:WikiTextHeadingRule:30 -->MOS 5+4 : sLsLsLsLs</h3>
 <!-- ws:start:WikiTextHeadingRule:32:&lt;h3&gt; --><h3 id="toc16"><a name="Superparticular-Nonoctave-MOS-Nonatonic (9-note) Scales:-MOS 7+2 : LLsLLLsLL"></a><!-- ws:end:WikiTextHeadingRule:32 -->MOS 7+2 : LLsLLLsLL</h3>
 <br />
<!-- ws:start:WikiTextHeadingRule:34:&lt;h2&gt; --><h2 id="toc17"><a name="Superparticular-Nonoctave-MOS-Dekatonic (10-note) Scales:"></a><!-- ws:end:WikiTextHeadingRule:34 -->Dekatonic (10-note) Scales:</h2>
 <!-- ws:start:WikiTextHeadingRule:36:&lt;h3&gt; --><h3 id="toc18"><a name="Superparticular-Nonoctave-MOS-Dekatonic (10-note) Scales:-MOS 3+7 : sLsssLssLs"></a><!-- ws:end:WikiTextHeadingRule:36 -->MOS 3+7 : sLsssLssLs</h3>
 <!-- ws:start:WikiTextHeadingRule:38:&lt;h3&gt; --><h3 id="toc19"><a name="Superparticular-Nonoctave-MOS-Dekatonic (10-note) Scales:-MOS 7+3 : LsLLLsLLsL"></a><!-- ws:end:WikiTextHeadingRule:38 -->MOS 7+3 : LsLLLsLLsL</h3>
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