Superparticular-Nonoctave-MOS
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- This revision was by author Andrew_Heathwaite and made on 2008-05-01 23:31:07 UTC.
- The original revision id was 23287421.
- The revision comment was: I set some limits w/i which I intend to find every possible scale. ... .. .
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=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 & th few scales I have in th meantime, tho! ... 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: # [[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. # [[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, & 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 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 & th 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 fits as Superparticular because its intervals, 9/8 & 12/11, both fit th form //n/n-1//. # It fits as 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. Inspired by th 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 w/i 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 w/ these limits in place, this produces a multitude of fascinating scales for our enjoyment & fascination. I invite you to play & share yr 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:<h1> --><h1 id="toc0"><a name="Superparticular-Nonoctave-MOS"></a><!-- ws:end:WikiTextHeadingRule:0 -->Superparticular-Nonoctave-MOS</h1> <br /> NOTE: I haven't completed th list of scales on this page. Consider that part under construction. You can check th intro & th few scales I have in th meantime, tho!<br /> <br /> ...<br /> <br /> 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:<br /> <br /> <ol><li><a class="wiki_link" href="/Superparticular">Superparticular</a> - 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 <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 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, & 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.</li><li><a class="wiki_link" href="/MOSScales">Moment of Symmetry</a> - 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 & th 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 fits as Superparticular because its intervals, 9/8 & 12/11, both fit th form <em>n/n-1</em>.</li><li>It fits as nonoctave (more accurately, near-octave) because it repeats at 1214.2 cents.</li><li>It fits as MOS because it contains exactly two step sizes, spaced out as evenly as possible w/i th scale.</li></ol><br /> Inspired by th 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 w/i 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 w/ these limits in place, this produces a multitude of fascinating scales for our enjoyment & fascination. I invite you to play & share yr results!<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h2> --><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:<h3> --><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:<h3> --><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 /> <!-- ws:start:WikiTextHeadingRule:8:<h2> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h2> --><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:<h3> --><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:<h3> --><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:<h2> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h3> --><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:<h2> --><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:<h3> --><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:<h3> --><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> </body></html>