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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox MOS |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | Periods = 2 |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-05-23 16:08:30 UTC</tt>.<br>
| | | nLargeSteps = 2 |
| : The original revision id was <tt>551973612</tt>.<br>
| | | nSmallSteps = 8 |
| : The revision comment was: <tt></tt><br>
| | | Equalized = 1 |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | | Collapsed = 0 |
| <h4>Original Wikitext content:</h4>
| | | Pattern = ssLssssLss |
| <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">This is the MOS pattern of the [[decatonic]] scale of [[Paul Erlich]] and others.
| | | Name = jaric |
| | }} |
|
| |
|
| The only significant harmonic entropy minimum that is [[Rothenberg propriety|proper]] is the decatonic scale itself ([[Diaschismic family|pajara]][10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes [[4_3|4/3]], and another generator down makes [[5_4|5/4]]. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:
| | {{MOS intro}} |
| http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
| | 2L 8s is the mos pattern of the decatonic scale of [[Paul Erlich]] and others. |
| http://www.lumma.org/tuning/erlich/
| |
|
| |
|
| Improper harmonic entropy minima include [[Meantone family#Injera|injera]] (which is similar to pajara except that 5/4 is now four generators **up** and no periods) and [[Diaschismic family#Shrutar|shrutar]] (which is basically pajara with the generator divided in two).
| | The only significant harmonic entropy minimum that is [[Rothenberg propriety|Rothenberg proper]] is the decatonic scale itself ([[Pajara]][10]), in which the period is [[7/5]][[~]][[10/7]] (tempered to be the same interval), one generator down from that makes [[4/3]], and another generator down makes [[5/4]]. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it is the next step up from the [[5L 2s|diatonic]] scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found here: [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf ''Tuning, Tonality, and Twenty-Two-Tone Temperament''] ([http://www.lumma.org/tuning/erlich/ alt link]). |
|
| |
|
| In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the "symmetric" scale and LsssssLsss is called the "pentachordal" scale (because it has two identical "pentachords" in the same way that the diatonic scale has two identical tetrachords).
| | Improper harmonic entropy minima include [[injera]] (which is similar to pajara except that 5/4 is now four generators ''up'' and no periods) and [[shrutar]] (which is basically pajara with the generator divided in two). |
| ||||||||||||||||||~ Generator ||~ Cents ||~ Comments ||
| |
| || 0\2 || || || || || || || || || 0 ||= ||
| |
| || || || || || || || || || 1\26 || 46.15 ||= ||
| |
| || || || || || || || || 1\24 || || 50 ||= ||
| |
| || || || || || || || || || 2\46 || 52.17 ||= Shrutar is around here ||
| |
| || || || || || || || 1\22 || || || 54.55 ||= ||
| |
| || || || || || || 1\20 || || || || 60 ||= ||
| |
| || || || || || 1\18 || || || || || 66.67 ||= ||
| |
| || || || || 1\16 || || || || || || 75 ||= L/s = 4 ||
| |
| || || || || || || || || || || 600/(4+pi) || ||
| |
| || || || 1\14 || || || || || || || 85.71 ||= L/s = 3 ||
| |
| || || || || || || || || || || 600/(4+e) || ||
| |
| || || || || 2\26 || || || || || || 92.31 ||= ||
| |
| || || || || || 3\38 || || || || || 94.74 ||= ||
| |
| || || || || || || 4\50 || || || || 96 ||= Injera is around here ||
| |
| || || || || || || || 5\62 || || || 96.77 ||= ||
| |
| || || 1\12 || || || || || || || || 100 ||= Boundary of propriety (generators
| |
| larger than this are proper) ||
| |
| || || || || || 4\46 || || || || || 104.35 ||= ||
| |
| || || || || 3\34 || || || || || || 105.88 ||= ||
| |
| || || || || || || 8\90 || || || || 106.67 ||= ||
| |
| || || || || || || || || 21\236 || || 106.78 ||= ||
| |
| || || || || || || || || || 34\382 || 106.81 ||= Golden decatonic (Srutal/pajara) ||
| |
| || || || || || || || 13\146 || || || 106.85 ||= ||
| |
| || || || || || 5\56 || || || || || 107.14 ||= Srutal/pajara decatonic is around here ||
| |
| || || || 2\22 || || || || || || || 109.09 ||= Optimum rank range (L/s=3/2) decatonic ||
| |
| || || || || 3\32 || || || || || || 112.5 ||= ||
| |
| || || || || || 4\42 || || || || || 114.29 ||= ||
| |
| || 1\10 || || || || || || || || || 120 ||= ||</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>2L 8s</title></head><body>This is the MOS pattern of the <a class="wiki_link" href="/decatonic">decatonic</a> scale of <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> and others.<br />
| |
| <br />
| |
| The only significant harmonic entropy minimum that is <a class="wiki_link" href="/Rothenberg%20propriety">proper</a> is the decatonic scale itself (<a class="wiki_link" href="/Diaschismic%20family">pajara</a>[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes <a class="wiki_link" href="/4_3">4/3</a>, and another generator down makes <a class="wiki_link" href="/5_4">5/4</a>. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it's the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found at either of these links:<br />
| |
| <!-- ws:start:WikiTextUrlRule:976:http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf --><a class="wiki_link_ext" href="http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf" rel="nofollow">http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf</a><!-- ws:end:WikiTextUrlRule:976 --><br />
| |
| <!-- ws:start:WikiTextUrlRule:977:http://www.lumma.org/tuning/erlich/ --><a class="wiki_link_ext" href="http://www.lumma.org/tuning/erlich/" rel="nofollow">http://www.lumma.org/tuning/erlich/</a><!-- ws:end:WikiTextUrlRule:977 --><br />
| |
| <br />
| |
| Improper harmonic entropy minima include <a class="wiki_link" href="/Meantone%20family#Injera">injera</a> (which is similar to pajara except that 5/4 is now four generators <strong>up</strong> and no periods) and <a class="wiki_link" href="/Diaschismic%20family#Shrutar">shrutar</a> (which is basically pajara with the generator divided in two).<br /> | |
| <br />
| |
| In addition to the true MOS form, LssssLssss, these scales also exist in a near-MOS form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the &quot;symmetric&quot; scale and LsssssLsss is called the &quot;pentachordal&quot; scale (because it has two identical &quot;pentachords&quot; in the same way that the diatonic scale has two identical tetrachords).<br />
| |
|
| |
|
| | In addition to the true mos form, LssssLssss, these scales also exist in a near-mos form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the ''symmetric'' scale and LsssssLsss is called the ''pentachordal'' scale (because it has two identical [[pentachord]]s in the same way that the diatonic scale has two identical [[tetrachord]]s). |
|
| |
|
| <table class="wiki_table">
| | == Scale properties == |
| <tr>
| | {{TAMNAMS use}} |
| <th colspan="9">Generator<br />
| |
| </th>
| |
| <th>Cents<br />
| |
| </th>
| |
| <th>Comments<br />
| |
| </th>
| |
| </tr>
| |
| <tr>
| |
| <td>0\2<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>0<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\26<br />
| |
| </td>
| |
| <td>46.15<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\24<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>50<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>2\46<br />
| |
| </td>
| |
| <td>52.17<br />
| |
| </td>
| |
| <td style="text-align: center;">Shrutar is around here<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\22<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>54.55<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\20<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>60<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\18<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>66.67<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\16<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>75<br />
| |
| </td>
| |
| <td style="text-align: center;">L/s = 4<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>600/(4+pi)<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>1\14<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>85.71<br />
| |
| </td>
| |
| <td style="text-align: center;">L/s = 3<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>600/(4+e)<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>2\26<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>92.31<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>3\38<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>94.74<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>4\50<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>96<br />
| |
| </td>
| |
| <td style="text-align: center;">Injera is around here<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>5\62<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>96.77<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td>1\12<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>100<br />
| |
| </td>
| |
| <td style="text-align: center;">Boundary of propriety (generators<br />
| |
| larger than this are proper)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>4\46<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>104.35<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>3\34<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>105.88<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>8\90<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>106.67<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>21\236<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>106.78<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>34\382<br />
| |
| </td>
| |
| <td>106.81<br />
| |
| </td>
| |
| <td style="text-align: center;">Golden decatonic (Srutal/pajara)<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>13\146<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>106.85<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>5\56<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>107.14<br />
| |
| </td>
| |
| <td style="text-align: center;">Srutal/pajara decatonic is around here<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>2\22<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>109.09<br />
| |
| </td>
| |
| <td style="text-align: center;">Optimum rank range (L/s=3/2) decatonic<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>3\32<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>112.5<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>4\42<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>114.29<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1\10<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>120<br />
| |
| </td>
| |
| <td style="text-align: center;"><br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div> | | === Intervals === |
| | {{MOS intervals}} |
| | |
| | === Generator chain === |
| | {{MOS genchain}} |
| | |
| | === Modes === |
| | {{MOS mode degrees}} |
| | |
| | == Scale tree == |
| | {{MOS tuning spectrum |
| | | 6/5 = [[Semimiracle]] ↑ |
| | | 3/2 = [[Pajara]] |
| | | 8/5 = [[Keen]] |
| | | 12/7 = [[Srutal]] |
| | | 9/5 = [[Diaschismic]] |
| | | 9/4 = [[Bimeantone]] |
| | | 5/2 = [[Injera]] |
| | | 9/2 = [[Vishnu]] (incomplete) |
| | | 6/1 = [[Shrutar]], [[teff]]/[[pombe]] (incomplete) ↓ |
| | }} |
| | |
| | [[Category:jaric]] |
| | [[Category:10-tone scales]] |
2L 8s, named jaric in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 2 large steps and 8 small steps, with a period of 1 large step and 4 small steps that repeats every 600.0 ¢, or twice every octave. Generators that produce this scale range from 480 ¢ to 600 ¢, or from 0 ¢ to 120 ¢.
2L 8s is the mos pattern of the decatonic scale of Paul Erlich and others.
The only significant harmonic entropy minimum that is Rothenberg proper is the decatonic scale itself (Pajara[10]), in which the period is 7/5~10/7 (tempered to be the same interval), one generator down from that makes 4/3, and another generator down makes 5/4. More than a few people think this is a beautiful scale that deserves a lot of investigation and use, with some going so far as to say it is the next step up from the diatonic scale that preserves the most desirable features of diatonic melody and harmony. Paul Erlich's original paper on this scale can be found here: Tuning, Tonality, and Twenty-Two-Tone Temperament (alt link).
Improper harmonic entropy minima include injera (which is similar to pajara except that 5/4 is now four generators up and no periods) and shrutar (which is basically pajara with the generator divided in two).
In addition to the true mos form, LssssLssss, these scales also exist in a near-mos form, LsssssLsss, in which the period is the only interval class with more than two flavors. In the case of the decatonic scale, LssssLssss is called the symmetric scale and LsssssLsss is called the pentachordal scale (because it has two identical pentachords in the same way that the diatonic scale has two identical tetrachords).
Scale properties
- This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.
Intervals
Intervals of 2L 8s
| Intervals
|
Steps
subtended
|
Range in cents
|
| Generic
|
Specific
|
Abbrev.
|
| 0-jarastep
|
Perfect 0-jarastep
|
P0jas
|
0
|
0.0 ¢
|
| 1-jarastep
|
Perfect 1-jarastep
|
P1jas
|
s
|
0.0 ¢ to 120.0 ¢
|
| Augmented 1-jarastep
|
A1jas
|
L
|
120.0 ¢ to 600.0 ¢
|
| 2-jarastep
|
Minor 2-jarastep
|
m2jas
|
2s
|
0.0 ¢ to 240.0 ¢
|
| Major 2-jarastep
|
M2jas
|
L + s
|
240.0 ¢ to 600.0 ¢
|
| 3-jarastep
|
Minor 3-jarastep
|
m3jas
|
3s
|
0.0 ¢ to 360.0 ¢
|
| Major 3-jarastep
|
M3jas
|
L + 2s
|
360.0 ¢ to 600.0 ¢
|
| 4-jarastep
|
Diminished 4-jarastep
|
d4jas
|
4s
|
0.0 ¢ to 480.0 ¢
|
| Perfect 4-jarastep
|
P4jas
|
L + 3s
|
480.0 ¢ to 600.0 ¢
|
| 5-jarastep
|
Perfect 5-jarastep
|
P5jas
|
L + 4s
|
600.0 ¢
|
| 6-jarastep
|
Perfect 6-jarastep
|
P6jas
|
L + 5s
|
600.0 ¢ to 720.0 ¢
|
| Augmented 6-jarastep
|
A6jas
|
2L + 4s
|
720.0 ¢ to 1200.0 ¢
|
| 7-jarastep
|
Minor 7-jarastep
|
m7jas
|
L + 6s
|
600.0 ¢ to 840.0 ¢
|
| Major 7-jarastep
|
M7jas
|
2L + 5s
|
840.0 ¢ to 1200.0 ¢
|
| 8-jarastep
|
Minor 8-jarastep
|
m8jas
|
L + 7s
|
600.0 ¢ to 960.0 ¢
|
| Major 8-jarastep
|
M8jas
|
2L + 6s
|
960.0 ¢ to 1200.0 ¢
|
| 9-jarastep
|
Diminished 9-jarastep
|
d9jas
|
L + 8s
|
600.0 ¢ to 1080.0 ¢
|
| Perfect 9-jarastep
|
P9jas
|
2L + 7s
|
1080.0 ¢ to 1200.0 ¢
|
| 10-jarastep
|
Perfect 10-jarastep
|
P10jas
|
2L + 8s
|
1200.0 ¢
|
Generator chain
Generator chain of 2L 8s
| Bright gens |
Scale degree |
Abbrev. |
Scale degree |
Abbrev.
|
| 5 |
Augmented 0-jaradegree |
A0jad |
Augmented 5-jaradegree |
A5jad
|
| 4 |
Augmented 1-jaradegree |
A1jad |
Augmented 6-jaradegree |
A6jad
|
| 3 |
Major 2-jaradegree |
M2jad |
Major 7-jaradegree |
M7jad
|
| 2 |
Major 3-jaradegree |
M3jad |
Major 8-jaradegree |
M8jad
|
| 1 |
Perfect 4-jaradegree |
P4jad |
Perfect 9-jaradegree |
P9jad
|
| 0 |
Perfect 0-jaradegree
Perfect 5-jaradegree |
P0jad
P5jad |
Perfect 5-jaradegree
Perfect 10-jaradegree |
P5jad
P10jad
|
| −1 |
Perfect 1-jaradegree |
P1jad |
Perfect 6-jaradegree |
P6jad
|
| −2 |
Minor 2-jaradegree |
m2jad |
Minor 7-jaradegree |
m7jad
|
| −3 |
Minor 3-jaradegree |
m3jad |
Minor 8-jaradegree |
m8jad
|
| −4 |
Diminished 4-jaradegree |
d4jad |
Diminished 9-jaradegree |
d9jad
|
| −5 |
Diminished 5-jaradegree |
d5jad |
Diminished 10-jaradegree |
d10jad
|
Modes
Scale degrees of the modes of 2L 8s
| UDP
|
Cyclic
order
|
Step
pattern
|
Scale degree (jaradegree)
|
| 0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
| 8|0(2)
|
1
|
LssssLssss
|
Perf.
|
Aug.
|
Maj.
|
Maj.
|
Perf.
|
Perf.
|
Aug.
|
Maj.
|
Maj.
|
Perf.
|
Perf.
|
| 6|2(2)
|
5
|
sLssssLsss
|
Perf.
|
Perf.
|
Maj.
|
Maj.
|
Perf.
|
Perf.
|
Perf.
|
Maj.
|
Maj.
|
Perf.
|
Perf.
|
| 4|4(2)
|
4
|
ssLssssLss
|
Perf.
|
Perf.
|
Min.
|
Maj.
|
Perf.
|
Perf.
|
Perf.
|
Min.
|
Maj.
|
Perf.
|
Perf.
|
| 2|6(2)
|
3
|
sssLssssLs
|
Perf.
|
Perf.
|
Min.
|
Min.
|
Perf.
|
Perf.
|
Perf.
|
Min.
|
Min.
|
Perf.
|
Perf.
|
| 0|8(2)
|
2
|
ssssLssssL
|
Perf.
|
Perf.
|
Min.
|
Min.
|
Dim.
|
Perf.
|
Perf.
|
Min.
|
Min.
|
Dim.
|
Perf.
|
Scale tree
Scale tree and tuning spectrum of 2L 8s
| Generator(edo)
|
Cents
|
Step ratio
|
Comments
|
| Bright
|
Dark
|
L:s
|
Hardness
|
| 4\10
|
|
|
|
|
|
480.000
|
120.000
|
1:1
|
1.000
|
Equalized 2L 8s
|
|
|
|
|
|
|
21\52
|
484.615
|
115.385
|
6:5
|
1.200
|
Semimiracle ↑
|
|
|
|
|
|
17\42
|
|
485.714
|
114.286
|
5:4
|
1.250
|
|
|
|
|
|
|
|
30\74
|
486.486
|
113.514
|
9:7
|
1.286
|
|
|
|
|
|
13\32
|
|
|
487.500
|
112.500
|
4:3
|
1.333
|
Supersoft 2L 8s
|
|
|
|
|
|
|
35\86
|
488.372
|
111.628
|
11:8
|
1.375
|
|
|
|
|
|
|
22\54
|
|
488.889
|
111.111
|
7:5
|
1.400
|
|
|
|
|
|
|
|
31\76
|
489.474
|
110.526
|
10:7
|
1.429
|
|
|
|
|
9\22
|
|
|
|
490.909
|
109.091
|
3:2
|
1.500
|
Soft 2L 8s
Pajara
|
|
|
|
|
|
|
32\78
|
492.308
|
107.692
|
11:7
|
1.571
|
|
|
|
|
|
|
23\56
|
|
492.857
|
107.143
|
8:5
|
1.600
|
Keen
|
|
|
|
|
|
|
37\90
|
493.333
|
106.667
|
13:8
|
1.625
|
|
|
|
|
|
14\34
|
|
|
494.118
|
105.882
|
5:3
|
1.667
|
Semisoft 2L 8s
|
|
|
|
|
|
|
33\80
|
495.000
|
105.000
|
12:7
|
1.714
|
Srutal
|
|
|
|
|
|
19\46
|
|
495.652
|
104.348
|
7:4
|
1.750
|
|
|
|
|
|
|
|
24\58
|
496.552
|
103.448
|
9:5
|
1.800
|
Diaschismic
|
|
|
5\12
|
|
|
|
|
500.000
|
100.000
|
2:1
|
2.000
|
Basic 2L 8s
Scales with tunings softer than this are proper
|
|
|
|
|
|
|
21\50
|
504.000
|
96.000
|
9:4
|
2.250
|
Bimeantone
|
|
|
|
|
|
16\38
|
|
505.263
|
94.737
|
7:3
|
2.333
|
|
|
|
|
|
|
|
27\64
|
506.250
|
93.750
|
12:5
|
2.400
|
|
|
|
|
|
11\26
|
|
|
507.692
|
92.308
|
5:2
|
2.500
|
Semihard 2L 8s
Injera
|
|
|
|
|
|
|
28\66
|
509.091
|
90.909
|
13:5
|
2.600
|
|
|
|
|
|
|
17\40
|
|
510.000
|
90.000
|
8:3
|
2.667
|
|
|
|
|
|
|
|
23\54
|
511.111
|
88.889
|
11:4
|
2.750
|
|
|
|
|
6\14
|
|
|
|
514.286
|
85.714
|
3:1
|
3.000
|
Hard 2L 8s
|
|
|
|
|
|
|
19\44
|
518.182
|
81.818
|
10:3
|
3.333
|
|
|
|
|
|
|
13\30
|
|
520.000
|
80.000
|
7:2
|
3.500
|
|
|
|
|
|
|
|
20\46
|
521.739
|
78.261
|
11:3
|
3.667
|
|
|
|
|
|
7\16
|
|
|
525.000
|
75.000
|
4:1
|
4.000
|
Superhard 2L 8s
|
|
|
|
|
|
|
15\34
|
529.412
|
70.588
|
9:2
|
4.500
|
Vishnu (incomplete)
|
|
|
|
|
|
8\18
|
|
533.333
|
66.667
|
5:1
|
5.000
|
|
|
|
|
|
|
|
9\20
|
540.000
|
60.000
|
6:1
|
6.000
|
Shrutar, teff/pombe (incomplete) ↓
|
| 1\2
|
|
|
|
|
|
600.000
|
0.000
|
1:0
|
→ ∞
|
Collapsed 2L 8s
|
