Overtone scale: Difference between revisions

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==Introduction== 

One way of using the [[OverToneSeries|overtone series]] to generate scalar material is to take an octave-long subset of the series and make it octave-repeating. So for instance, starting at the fifth overtone and continuing up the sequence to the tenth overtone (which is a doubling of five, and thus an octave higher) produces a pentatonic scale:

|| overtone || 5 || 6 || 7 || 8 || 9 || 10 ||
|| JI ratio || 1/1 || 6/5 || 7/5 || 8/5 || 9/5 || 2/1 ||

Such a scale has the overtonal characteristic of containing all [[superparticular]] steps ("superparticular" refers to ratios of the form n/(n-1)) that are decreasing in pitch size as one ascends the scale).

|| steps || 6:5 || 7:6 || 8:7 || 9:8 || 10:9 ||
|| common name || just minor third || septimal subminor third || septimal supermajor second || large major second || small major second ||

[[Denny Genovese]] would call the above scale "Mode 5 of the Harmonic Series," or "Mode 5" for short.

==A Solfege System== 

[[Andrew Heathwaite]] proposes a solfege system for overtones 16-32 (Mode 16):

|| overtone || 16 || 17 || 18 || 19 || 20 || 21 || 22 || 23 || 24 || 25 || 26 || 27 || 28 || 29 || 30 || 31 || 32 ||
|| JI ratio || 1/1 || 17/16 || 9/8 || 19/16 || 5/4 || 21/16 || 11/8 || 23/16 || 3/2 || 25/16 || 13/8 || 27/16 || 7/4 || 29/16 || 15/8 || 31/16 || 2/1 ||
|| solfege || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || **su** || **sol** || **le** || **lu** || **la** || **ta** || **tu** || **ti** || **da** || **do** ||

Thus, the pentatonic scale in the example above could be sung: **mi sol ta do re mi**

==Twelve Scales== 

For those interested in learning to sing and hear just intervals, here are twelve otonal scales to try. I leave it up to the curious learner to decide the value, beauty, or usefulness of these particular scales for their compositional purposes.

||   ||   || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20 || 21 || 22 || 23 || 24 ||
|| Mode 1 || 1-note || **do** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 2 || 2-note ||   || **do** || **sol** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 3 || 3-note ||   ||   || **sol** || **do** || **mi** || **sol** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 4 || 4-note ||   ||   ||   || **do** || **mi** || **sol** || **ta** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 5 || 5-note ||   ||   ||   ||   || **mi** || **sol** || **ta** || **do** || **re** || **mi** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 6 || 6-note ||   ||   ||   ||   ||   || **sol** || **ta** || **do** || **re** || **mi** || **fu** || **sol** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 7 || 7-note ||   ||   ||   ||   ||   ||   || **ta** || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 8 || 8-note ||   ||   ||   ||   ||   ||   ||   || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||
|| Mode 9 || 9-note ||   ||   ||   ||   ||   ||   ||   ||   || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** ||   ||   ||   ||   ||   ||   ||
|| Mode 10 || 10-note ||   ||   ||   ||   ||   ||   ||   ||   ||   || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** ||   ||   ||   ||   ||
|| Mode 11 || 11-note ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** ||   ||   ||
|| Mode 12 || 12-note ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || **su** || **sol** ||

==Next Steps== 

Here are some next steps:
* Go beyond the 24th overtone (eg. overtones 16-32 or higher).
* Experiment with using different pitches as the "tonic" of the scale (eg. **sol lu ta do re mi fu sol**, which could be taken as the 7-note scale starting on **sol**).
* Take subsets of larger scales, which are not strict adjacent overtone scales (eg. **do re fe sol ta do**).
* Learn the inversions of these scales, which would be **undertone** scales. (Undertone scales would have smaller steps at the bottom of the scale, which would get larger as one ascends.)
* Borrow overtones & undertones from the overtones & undertones of the fundamental -- this process can produce rich fields of interlocking harmonic series, and is often the sort of thing that composers do when they're composing in just intonation. Harry Partch's "Monophonic Fabric," which consists of 43 unequal tones per octave, is one famous example.

Original HTML content:

<html><head><title>overtone scales</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Introduction"></a><!-- ws:end:WikiTextHeadingRule:0 -->Introduction</h2>
 <br />
One way of using the <a class="wiki_link" href="/OverToneSeries">overtone series</a> to generate scalar material is to take an octave-long subset of the series and make it octave-repeating. So for instance, starting at the fifth overtone and continuing up the sequence to the tenth overtone (which is a doubling of five, and thus an octave higher) produces a pentatonic scale:<br />
<br />


<table class="wiki_table">
    <tr>
        <td>overtone<br />
</td>
        <td>5<br />
</td>
        <td>6<br />
</td>
        <td>7<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
    </tr>
    <tr>
        <td>JI ratio<br />
</td>
        <td>1/1<br />
</td>
        <td>6/5<br />
</td>
        <td>7/5<br />
</td>
        <td>8/5<br />
</td>
        <td>9/5<br />
</td>
        <td>2/1<br />
</td>
    </tr>
</table>

<br />
Such a scale has the overtonal characteristic of containing all <a class="wiki_link" href="/superparticular">superparticular</a> steps (&quot;superparticular&quot; refers to ratios of the form n/(n-1)) that are decreasing in pitch size as one ascends the scale).<br />
<br />


<table class="wiki_table">
    <tr>
        <td>steps<br />
</td>
        <td>6:5<br />
</td>
        <td>7:6<br />
</td>
        <td>8:7<br />
</td>
        <td>9:8<br />
</td>
        <td>10:9<br />
</td>
    </tr>
    <tr>
        <td>common name<br />
</td>
        <td>just minor third<br />
</td>
        <td>septimal subminor third<br />
</td>
        <td>septimal supermajor second<br />
</td>
        <td>large major second<br />
</td>
        <td>small major second<br />
</td>
    </tr>
</table>

<br />
<a class="wiki_link" href="/Denny%20Genovese">Denny Genovese</a> would call the above scale &quot;Mode 5 of the Harmonic Series,&quot; or &quot;Mode 5&quot; for short.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h2&gt; --><h2 id="toc1"><a name="x-A Solfege System"></a><!-- ws:end:WikiTextHeadingRule:2 -->A Solfege System</h2>
 <br />
<a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> proposes a solfege system for overtones 16-32 (Mode 16):<br />
<br />


<table class="wiki_table">
    <tr>
        <td>overtone<br />
</td>
        <td>16<br />
</td>
        <td>17<br />
</td>
        <td>18<br />
</td>
        <td>19<br />
</td>
        <td>20<br />
</td>
        <td>21<br />
</td>
        <td>22<br />
</td>
        <td>23<br />
</td>
        <td>24<br />
</td>
        <td>25<br />
</td>
        <td>26<br />
</td>
        <td>27<br />
</td>
        <td>28<br />
</td>
        <td>29<br />
</td>
        <td>30<br />
</td>
        <td>31<br />
</td>
        <td>32<br />
</td>
    </tr>
    <tr>
        <td>JI ratio<br />
</td>
        <td>1/1<br />
</td>
        <td>17/16<br />
</td>
        <td>9/8<br />
</td>
        <td>19/16<br />
</td>
        <td>5/4<br />
</td>
        <td>21/16<br />
</td>
        <td>11/8<br />
</td>
        <td>23/16<br />
</td>
        <td>3/2<br />
</td>
        <td>25/16<br />
</td>
        <td>13/8<br />
</td>
        <td>27/16<br />
</td>
        <td>7/4<br />
</td>
        <td>29/16<br />
</td>
        <td>15/8<br />
</td>
        <td>31/16<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>solfege<br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>ra</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>me</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fe</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>su</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>le</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>la</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>tu</strong><br />
</td>
        <td><strong>ti</strong><br />
</td>
        <td><strong>da</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
    </tr>
</table>

<br />
Thus, the pentatonic scale in the example above could be sung: <strong>mi sol ta do re mi</strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h2&gt; --><h2 id="toc2"><a name="x-Twelve Scales"></a><!-- ws:end:WikiTextHeadingRule:4 -->Twelve Scales</h2>
 <br />
For those interested in learning to sing and hear just intervals, here are twelve otonal scales to try. I leave it up to the curious learner to decide the value, beauty, or usefulness of these particular scales for their compositional purposes.<br />
<br />


<table class="wiki_table">
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>1<br />
</td>
        <td>2<br />
</td>
        <td>3<br />
</td>
        <td>4<br />
</td>
        <td>5<br />
</td>
        <td>6<br />
</td>
        <td>7<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
        <td>13<br />
</td>
        <td>14<br />
</td>
        <td>15<br />
</td>
        <td>16<br />
</td>
        <td>17<br />
</td>
        <td>18<br />
</td>
        <td>19<br />
</td>
        <td>20<br />
</td>
        <td>21<br />
</td>
        <td>22<br />
</td>
        <td>23<br />
</td>
        <td>24<br />
</td>
    </tr>
    <tr>
        <td>Mode 1<br />
</td>
        <td>1-note<br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>do</strong><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><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 2<br />
</td>
        <td>2-note<br />
</td>
        <td><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>do</strong><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><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 3<br />
</td>
        <td>3-note<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>sol</strong><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
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</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 4<br />
</td>
        <td>4-note<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>do</strong><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>
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</td>
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</td>
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</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 5<br />
</td>
        <td>5-note<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>mi</strong><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 6<br />
</td>
        <td>6-note<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>sol</strong><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 7<br />
</td>
        <td>7-note<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>ta</strong><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><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 8<br />
</td>
        <td>8-note<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>ti</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
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</td>
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</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 9<br />
</td>
        <td>9-note<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><strong>re</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>ti</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>ra</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 10<br />
</td>
        <td>10-note<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><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>ti</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>ra</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>me</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 11<br />
</td>
        <td>11-note<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><br />
</td>
        <td><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>ti</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>ra</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>me</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fe</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>Mode 12<br />
</td>
        <td>12-note<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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><strong>sol</strong><br />
</td>
        <td><strong>lu</strong><br />
</td>
        <td><strong>ta</strong><br />
</td>
        <td><strong>ti</strong><br />
</td>
        <td><strong>do</strong><br />
</td>
        <td><strong>ra</strong><br />
</td>
        <td><strong>re</strong><br />
</td>
        <td><strong>me</strong><br />
</td>
        <td><strong>mi</strong><br />
</td>
        <td><strong>fe</strong><br />
</td>
        <td><strong>fu</strong><br />
</td>
        <td><strong>su</strong><br />
</td>
        <td><strong>sol</strong><br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="x-Next Steps"></a><!-- ws:end:WikiTextHeadingRule:6 -->Next Steps</h2>
 <br />
Here are some next steps:<br />
<ul><li>Go beyond the 24th overtone (eg. overtones 16-32 or higher).</li><li>Experiment with using different pitches as the &quot;tonic&quot; of the scale (eg. <strong>sol lu ta do re mi fu sol</strong>, which could be taken as the 7-note scale starting on <strong>sol</strong>).</li><li>Take subsets of larger scales, which are not strict adjacent overtone scales (eg. <strong>do re fe sol ta do</strong>).</li><li>Learn the inversions of these scales, which would be <strong>undertone</strong> scales. (Undertone scales would have smaller steps at the bottom of the scale, which would get larger as one ascends.)</li><li>Borrow overtones &amp; undertones from the overtones &amp; undertones of the fundamental -- this process can produce rich fields of interlocking harmonic series, and is often the sort of thing that composers do when they're composing in just intonation. Harry Partch's &quot;Monophonic Fabric,&quot; which consists of 43 unequal tones per octave, is one famous example.</li></ul></body></html>

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