Overtone scale: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~Andrew_Heathwaite~~|~~Andrew_Heathwaite~~]] and made on <tt>2010-02~~-04 12~~:51:28 UTC</tt>.

: This revision was by author [[User:guest|guest]] and made on <tt>2010-11-02 15:52:30 UTC</tt>.

: The original revision id was <tt>~~118101069~~</tt>.

: The original revision id was <tt>175806833</tt>.

: The revision comment was: <tt></tt>

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

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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;white-space: pre-wrap ! important" class="old-revision-html">==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.

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.

~~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:

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.

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 ||

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|| 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** || **te** || **tu** || **ti** || **da** || **do** ||

|| 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 te do re mi**

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. "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==

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|| 2-note || || **do** || **sol** || **do** || || || || || || || || || || || || || || || || || || || || ||

|| 3-note || || || **sol** || **do** || **mi** || **sol** || || || || || || || || || || || || || || || || || || ||

|| 4-note || || || || **do** || **mi** || **sol** || **te** || **do** || || || || || || || || || || || || || || || || ||

|| 4-note || || || || **do** || **mi** || **sol** || **ta** || **do** || || || || || || || || || || || || || || || || ||

|| 5-note || || || || || **mi** || **sol** || **te** || **do** || **re** || **mi** || || || || || || || || || || || || || || ||

|| 5-note || || || || || **mi** || **sol** || **ta** || **do** || **re** || **mi** || || || || || || || || || || || || || || ||

|| 6-note || || || || || || **sol** || **te** || **do** || **re** || **mi** || **fu** || **sol** || || || || || || || || || || || || ||

|| 6-note || || || || || || **sol** || **ta** || **do** || **re** || **mi** || **fu** || **sol** || || || || || || || || || || || || ||

|| 7-note || || || || || || || **te** || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **te** || || || || || || || || || || ||

|| 7-note || || || || || || || **ta** || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || || || || || || || || || || ||

|| 8-note || || || || || || || || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **te** || **ti** || **do** || || || || || || || || ||

|| 8-note || || || || || || || || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || || || || || || || || ||

|| 9-note || || || || || || || || || **re** || **mi** || **fu** || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || || || || || || ||

|| 9-note || || || || || || || || || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || || || || || || ||

|| 10-note || || || || || || || || || || **mi** || **fu** || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || **me** || **mi** || || || || ||

|| 10-note || || || || || || || || || || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || || || || ||

|| 11-note || || || || || || || || || || || **fu** || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || || ||

|| 11-note || || || || || || || || || || || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || || ||

|| 12-note || || || || || || || || || || || || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || **su** || **sol** ||

|| 12-note || || || || || || || || || || || || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || **su** || **sol** ||

==next steps==

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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 te do re mi fu sol**, which could be taken as the 7-note scale starting on **sol**).

* 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 te do**).

* 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.</pre></div>

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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"><html><head><title>overtone scales</title></head><body><h2 id="toc0"><a name="x-introduction"></a>introduction</h2>

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.

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.

~~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:

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:

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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.

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).

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<td>la

</td>

<td>te

<td>ta

</td>

<td>tu

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

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

Note: I have started using &quot;ta&quot; for 7/4 and &quot;te&quot; for 16/9. &quot;Te&quot; is the traditional name for a flat seventh in the solfege system. &quot;Ta&quot; 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>, &quot;te&quot; would be appropriate. &quot;Ta&quot; indicates a distinctly septimal interval. Also, it's a perfect fifth up from 7/6, which I call &quot;ma.&quot; 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....

<h2 id="toc2"><a name="x-twelve scales to learn"></a>twelve scales to learn</h2>

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<td>sol

</td>

<td>te

<td>ta

</td>

<td>do

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<td>sol

</td>

<td>te

<td>ta

</td>

<td>do

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<td>sol

</td>

<td>te

<td>ta

</td>

<td>do

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<td>

</td>

<td>te

<td>ta

</td>

<td>do

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<td>lu

</td>

<td>te

<td>ta

</td>

<td>

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<td>lu

</td>

<td>te

<td>ta

</td>

<td>ti

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<td>lu

</td>

<td>te

<td>ta

</td>

<td>ti

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<td>lu

</td>

<td>te

<td>ta

</td>

<td>ti

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<td>lu

</td>

<td>te

<td>ta

</td>

<td>ti

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<td>lu

</td>

<td>te

<td>ta

</td>

<td>ti

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Here are some next steps:

<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. sol lu te do re mi fu sol, which could be taken as the 7-note scale starting on sol).</li><li>Take subsets of larger scales, which are not strict adjacent overtone scales (eg. do re fe sol te do).</li><li>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.)</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></pre></div>

<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. sol lu ta do re mi fu sol, which could be taken as the 7-note scale starting on sol).</li><li>Take subsets of larger scales, which are not strict adjacent overtone scales (eg. do re fe sol ta do).</li><li>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.)</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></pre></div>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2010-02-04 12:51:28 UTC</tt>.<br>
+: This revision was by author [[User:guest|guest]] and made on <tt>2010-11-02 15:52:30 UTC</tt>.<br>
-: The original revision id was <tt>118101069</tt>.<br>
+: The original revision id was <tt>175806833</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>
@@ Line 8: / Line 8: @@
 <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">==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.
+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.
-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:
+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.
+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 ||
@@ Line 26: / Line 26: @@
 || 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** || **te** || **tu** || **ti** || **da** || **do** ||
+|| 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 te do re mi**
+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. "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==
@@ Line 38: / Line 40: @@
 || 2-note ||   || **do** || **sol** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
 || 3-note ||   ||   || **sol** || **do** || **mi** || **sol** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 4-note ||   ||   ||   || **do** || **mi** || **sol** || **te** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
+|| 4-note ||   ||   ||   || **do** || **mi** || **sol** || **ta** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 5-note ||   ||   ||   ||   || **mi** || **sol** || **te** || **do** || **re** || **mi** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
+|| 5-note ||   ||   ||   ||   || **mi** || **sol** || **ta** || **do** || **re** || **mi** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 6-note ||   ||   ||   ||   ||   || **sol** || **te** || **do** || **re** || **mi** || **fu** || **sol** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
+|| 6-note ||   ||   ||   ||   ||   || **sol** || **ta** || **do** || **re** || **mi** || **fu** || **sol** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 7-note ||   ||   ||   ||   ||   ||   || **te** || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **te** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
+|| 7-note ||   ||   ||   ||   ||   ||   || **ta** || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 8-note ||   ||   ||   ||   ||   ||   ||   || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **te** || **ti** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||
+|| 8-note ||   ||   ||   ||   ||   ||   ||   || **do** || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** ||   ||   ||   ||   ||   ||   ||   ||   ||
-|| 9-note ||   ||   ||   ||   ||   ||   ||   ||   || **re** || **mi** || **fu** || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** ||   ||   ||   ||   ||   ||   ||
+|| 9-note ||   ||   ||   ||   ||   ||   ||   ||   || **re** || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** ||   ||   ||   ||   ||   ||   ||
-|| 10-note ||   ||   ||   ||   ||   ||   ||   ||   ||   || **mi** || **fu** || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || **me** || **mi** ||   ||   ||   ||   ||
+|| 10-note ||   ||   ||   ||   ||   ||   ||   ||   ||   || **mi** || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** ||   ||   ||   ||   ||
-|| 11-note ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || **fu** || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** ||   ||   ||
+|| 11-note ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || **fu** || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** ||   ||   ||
-|| 12-note ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || **sol** || **lu** || **te** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || **su** || **sol** ||
+|| 12-note ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   || **sol** || **lu** || **ta** || **ti** || **do** || **ra** || **re** || **me** || **mi** || **fe** || **fu** || **su** || **sol** ||
 ==next steps==
@@ Line 52: / Line 54: @@
 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 te do re mi fu sol**, which could be taken as the 7-note scale starting on **sol**).
+* 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 te do**).
+* 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 &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 "Monophonic Fabric," which consists of 43 unequal tones per octave, is one famous example.</pre></div>
@@ Line 59: / Line 61: @@
 <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;overtone scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-introduction"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;introduction&lt;/h2&gt;
   &lt;br /&gt;
-One way of using the &lt;a class="wiki_link" href="/OverToneSeries"&gt;overtone series&lt;/a&gt; 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.&lt;br /&gt;
+One way of using the &lt;a class="wiki_link" href="/OverToneSeries"&gt;overtone series&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
-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:&lt;br /&gt;
+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:&lt;br /&gt;
 &lt;br /&gt;
@@ Line 101: / Line 103: @@
 &lt;br /&gt;
-Such a scale has the overtonal characteristic of containing all &lt;a class="wiki_link" href="/superparticular"&gt;superparticular&lt;/a&gt; steps (&amp;quot;superparticular&amp;quot; refers to ratios of the form n/(n-1)) that are decreasing in pitch size as one ascends the scale.&lt;br /&gt;
+Such a scale has the overtonal characteristic of containing all &lt;a class="wiki_link" href="/superparticular"&gt;superparticular&lt;/a&gt; steps (&amp;quot;superparticular&amp;quot; refers to ratios of the form n/(n-1)) that are decreasing in pitch size as one ascends the scale).&lt;br /&gt;
 &lt;br /&gt;
@@ Line 247: / Line 249: @@
          &lt;td&gt;&lt;strong&gt;la&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;tu&lt;/strong&gt;&lt;br /&gt;
@@ Line 261: / Line 263: @@
 &lt;br /&gt;
-Thus, the pentatonic scale in the example above could be sung: &lt;strong&gt;mi sol te do re mi&lt;/strong&gt;&lt;br /&gt;
+Thus, the pentatonic scale in the example above could be sung: &lt;strong&gt;mi sol ta do re mi&lt;/strong&gt;&lt;br /&gt;
+&lt;br /&gt;
+&lt;strong&gt;Note: I have started using &amp;quot;ta&amp;quot; for 7/4 and &amp;quot;te&amp;quot; for 16/9. &amp;quot;Te&amp;quot; is the traditional name for a flat seventh in the solfege system. &amp;quot;Ta&amp;quot; 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 &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt; or &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &amp;quot;te&amp;quot; would be appropriate. &amp;quot;Ta&amp;quot; indicates a distinctly septimal interval. Also, it's a perfect fifth up from 7/6, which I call &amp;quot;ma.&amp;quot; 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....&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x-twelve scales to learn"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;twelve scales to learn&lt;/h2&gt;
@@ Line 493: / Line 497: @@
          &lt;td&gt;&lt;strong&gt;sol&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;do&lt;/strong&gt;&lt;br /&gt;
@@ Line 545: / Line 549: @@
          &lt;td&gt;&lt;strong&gt;sol&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;do&lt;/strong&gt;&lt;br /&gt;
@@ Line 597: / Line 601: @@
          &lt;td&gt;&lt;strong&gt;sol&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;do&lt;/strong&gt;&lt;br /&gt;
@@ Line 649: / Line 653: @@
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;do&lt;/strong&gt;&lt;br /&gt;
@@ Line 663: / Line 667: @@
          &lt;td&gt;&lt;strong&gt;lu&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 715: / Line 719: @@
          &lt;td&gt;&lt;strong&gt;lu&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;ti&lt;/strong&gt;&lt;br /&gt;
@@ Line 767: / Line 771: @@
          &lt;td&gt;&lt;strong&gt;lu&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;ti&lt;/strong&gt;&lt;br /&gt;
@@ Line 819: / Line 823: @@
          &lt;td&gt;&lt;strong&gt;lu&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;ti&lt;/strong&gt;&lt;br /&gt;
@@ Line 871: / Line 875: @@
          &lt;td&gt;&lt;strong&gt;lu&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;ti&lt;/strong&gt;&lt;br /&gt;
@@ Line 923: / Line 927: @@
          &lt;td&gt;&lt;strong&gt;lu&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;strong&gt;te&lt;/strong&gt;&lt;br /&gt;
+         &lt;td&gt;&lt;strong&gt;ta&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;strong&gt;ti&lt;/strong&gt;&lt;br /&gt;
@@ Line 952: / Line 956: @@
   &lt;br /&gt;
 Here are some next steps:&lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;Go beyond the 24th overtone (eg. overtones 16-32 or higher).&lt;/li&gt;&lt;li&gt;Experiment with using different pitches as the &amp;quot;tonic&amp;quot; of the scale (eg. &lt;strong&gt;sol lu te do re mi fu sol&lt;/strong&gt;, which could be taken as the 7-note scale starting on &lt;strong&gt;sol&lt;/strong&gt;).&lt;/li&gt;&lt;li&gt;Take subsets of larger scales, which are not strict adjacent overtone scales (eg. &lt;strong&gt;do re fe sol te do&lt;/strong&gt;).&lt;/li&gt;&lt;li&gt;Learn the inversions of these scales, which would be &lt;strong&gt;undertone&lt;/strong&gt; scales. (Undertone scales would have smaller steps at the bottom of the scale, which would get larger as one ascends.)&lt;/li&gt;&lt;li&gt;Borrow overtones &amp;amp; undertones from the overtones &amp;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 &amp;quot;Monophonic Fabric,&amp;quot; which consists of 43 unequal tones per octave, is one famous example.&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;ul&gt;&lt;li&gt;Go beyond the 24th overtone (eg. overtones 16-32 or higher).&lt;/li&gt;&lt;li&gt;Experiment with using different pitches as the &amp;quot;tonic&amp;quot; of the scale (eg. &lt;strong&gt;sol lu ta do re mi fu sol&lt;/strong&gt;, which could be taken as the 7-note scale starting on &lt;strong&gt;sol&lt;/strong&gt;).&lt;/li&gt;&lt;li&gt;Take subsets of larger scales, which are not strict adjacent overtone scales (eg. &lt;strong&gt;do re fe sol ta do&lt;/strong&gt;).&lt;/li&gt;&lt;li&gt;Learn the inversions of these scales, which would be &lt;strong&gt;undertone&lt;/strong&gt; scales. (Undertone scales would have smaller steps at the bottom of the scale, which would get larger as one ascends.)&lt;/li&gt;&lt;li&gt;Borrow overtones &amp;amp; undertones from the overtones &amp;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 &amp;quot;Monophonic Fabric,&amp;quot; which consists of 43 unequal tones per octave, is one famous example.&lt;/li&gt;&lt;/ul&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>