Overtone scale
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author guest and made on 2010-11-02 15:58:34 UTC.
- The original revision id was 175809095.
- 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:
==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. I (Andrew Heathwaite) propose experimenting with this technique as a pathway into just intonation as well as a practice worthwhile in itself. I am not the first to do this, but perhaps the use of a modified solfege system is original to me.
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 ||
==a solfege system==
I propose a solfege system for overtones 16-32 which goes:
|| 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**
**Note: I have started using "ta" for 7/4 and "te" for 16/9 and 9/5. "Te" is the traditional name for a flat seventh in the solfege system. "Ta" is a new name. As the traditional solfege system does not admit subminor sevenths, it seemed appropriate to use a new name. In systems which do not distinguish between a minor and subminor seventh, such as [[12edo]] or [[22edo]], "te" would be appropriate. "Ta" indicates a distinctly septimal interval. Also, it's a perfect fifth up from 7/6, which I call "ma." That gives us the fifths ma-ta, me-te, mu-tu, mi-ti and mo-to. Of course, the system breaks down in systems where there are, for instance, multiple subminor sevenths and multiple neutral thirds....**
==twelve scales to learn==
I propose, for those interested in learning to sing and hear just intervals, 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 ||
|| 1-note || **do** || **do** || || || || || || || || || || || || || || || || || || || || || || ||
|| 2-note || || **do** || **sol** || **do** || || || || || || || || || || || || || || || || || || || || ||
|| 3-note || || || **sol** || **do** || **mi** || **sol** || || || || || || || || || || || || || || || || || || ||
|| 4-note || || || || **do** || **mi** || **sol** || **ta** || **do** || || || || || || || || || || || || || || || || ||
|| 5-note || || || || || **mi** || **sol** || **ta** || **do** || **re** || **mi** || || || || || || || || || || || || || || ||
|| 6-note || || || || || || **sol** || **ta** || **do** || **re** || **mi** || **fu** || **sol** || || || || || || || || || || || || ||
|| 7-note || || || || || || || **ta** || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || || || || || || || || || || ||
|| 8-note || || || || || || || || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || || || || || || || || ||
|| 9-note || || || || || || || || || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || || || || || || ||
|| 10-note || || || || || || || || || || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || || || || ||
|| 11-note || || || || || || || || || || || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || || ||
|| 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:<h2> --><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. I (Andrew Heathwaite) propose experimenting with this technique as a pathway into just intonation as well as a practice worthwhile in itself. I am not the first to do this, but perhaps the use of a modified solfege system is original to me.<br />
<br />
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 ("superparticular" 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 />
<!-- ws:start:WikiTextHeadingRule:2:<h2> --><h2 id="toc1"><a name="x-a solfege system"></a><!-- ws:end:WikiTextHeadingRule:2 -->a solfege system</h2>
<br />
I propose a solfege system for overtones 16-32 which goes:<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 />
<strong>Note: I have started using "ta" for 7/4 and "te" for 16/9 and 9/5. "Te" is the traditional name for a flat seventh in the solfege system. "Ta" is a new name. As the traditional solfege system does not admit subminor sevenths, it seemed appropriate to use a new name. In systems which do not distinguish between a minor and subminor seventh, such as <a class="wiki_link" href="/12edo">12edo</a> or <a class="wiki_link" href="/22edo">22edo</a>, "te" would be appropriate. "Ta" indicates a distinctly septimal interval. Also, it's a perfect fifth up from 7/6, which I call "ma." That gives us the fifths ma-ta, me-te, mu-tu, mi-ti and mo-to. Of course, the system breaks down in systems where there are, for instance, multiple subminor sevenths and multiple neutral thirds....</strong><br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h2> --><h2 id="toc2"><a name="x-twelve scales to learn"></a><!-- ws:end:WikiTextHeadingRule:4 -->twelve scales to learn</h2>
<br />
I propose, for those interested in learning to sing and hear just intervals, 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>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>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>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>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>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<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>
<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>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>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>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>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>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
</tr>
<tr>
<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>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>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>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:<h2> --><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 "tonic" 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 & 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.</li></ul></body></html>