Overtone scale: Difference between revisions
Wikispaces>Andrew_Heathwaite **Imported revision 265202148 - Original comment: ** |
Wikispaces>Andrew_Heathwaite **Imported revision 265239442 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | 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>2011-10-16 | : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-10-16 14:24:41 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>265239442</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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|| overtone || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20 || | || overtone || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20 || | ||
|| JI ratio || 1/1 || [[11_10|11/10]] || 6/5 || [[13_10|13/10]] || 7/5 || [[ | || JI ratio || 1/1 || [[11_10|11/10]] || 6/5 || [[13_10|13/10]] || 7/5 || [[3_2]] || 8/5 || [[17_10|17/10]] || 9/5 || [[19_10|19/10]] || 2/1 || | ||
Notice that the 15th harmonic is a 3/2 above 10. Although this may look like it breaks the Over-5 rule, it's just a reduced form of 15/10, which has a 2<span style="vertical-align: super;">n</span>*5 in the denominator. 10 may be too many notes for a particular purpose; we could take a subset of Mode 10 -- for instance 10:11:13:15:17:20, and it would also be an Over-5 scale. Below are some of the simplest Over-n scales as Modes of the Harmonic Series. All of them are ripe for the taking of subsets. | Notice that the 15th harmonic is a 3/2 above 10. Although this may look like it breaks the Over-5 rule, it's just a reduced form of 15/10, which has a 2<span style="vertical-align: super;">n</span>*5 in the denominator. 10 may be too many notes for a particular purpose; we could take a subset of Mode 10 -- for instance 10:11:13:15:17:20, and it would also be an Over-5 scale. Below are some of the simplest Over-n scales as Modes of the Harmonic Series. All of them are ripe for the taking of subsets. | ||
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Mode 3 -- 3:4:5:6 -- a major triad in 2nd inversion -- that is, with the perfect fifth in the bass. | Mode 3 -- 3:4:5:6 -- a major triad in 2nd inversion -- that is, with the perfect fifth in the bass. | ||
Mode 6 -- 6:7:8:9:10:11:12 -- an effective 6-tone scale. 9 is 3/2 above 3, so there is a perfect fifth above the bass. A septimal subminor triad -- 6:7:9 -- is available, as well as an undecimal 6:7:9:11 tetrad. | Mode 6 -- 6:7:8:9:10:11:12 -- an effective 6-tone scale. 9 is 3/2 above 3, so there is a perfect fifth above the bass. A septimal subminor triad -- 6:7:9 -- is available, as well as an undecimal 6:7:9:11 tetrad, which adds a neutral seventh of 11/6 to the septimal subminor triad. | ||
Mode 12 -- 12:13:14:15:16:17:18:19:20:21:22:23:24 -- as this scale has 12 tones, it fits nicely onto a traditional keyboard instrument, such as piano, melodica, organ, accordion, etc. It allows a 4:5:6:7 septimal tetrad above the bass (a reduced form of 12:15:18:21) as well as the subminor triad and undecimal tetrad given above. The fundamental is 4/3 above the bass, making 4/3 a strong attractor in the system. Andrew Heathwaite has composed with a 12:13:14:16:18:20:22:24 subset, and [[Jacob Barton]] retuned an electric organ to this scale. | Mode 12 -- 12:13:14:15:16:17:18:19:20:21:22:23:24 -- as this scale has 12 tones, it fits nicely onto a traditional keyboard instrument, such as piano, melodica, organ, accordion, etc. It allows a 4:5:6:7 septimal tetrad above the bass (a reduced form of 12:15:18:21) as well as the subminor triad and undecimal tetrad given above. The fundamental is 4/3 above the bass, making 4/3 a strong attractor in the system. Andrew Heathwaite has composed with a 12:13:14:16:18:20:22:24 subset, and [[Jacob Barton]] retuned an electric organ to this scale. | ||
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===Over-5 Scales=== | ===Over-5 Scales=== | ||
Mode 5 -- 5:6:7:8:9:10 -- This is essentially a [[7-limit]] fully-diminished seventh chord. | |||
Mode 10 -- 10:11:12:13:14:15:16:17:18:19:20 -- from 10 to 15 is a 3/2 perfect fifth. We have access to a 10:12:15 classic minor triad, as well as a number of other nice chords like the 10:13:15 barbados triad. 11/10 makes a strange second (or ninth), while 9/5 makes a very nice minor seventh (and an alternative to the 7/4 bluesy seventh of Over-1 scales and neutral seventh of Over-3 scales. | |||
Mode 20 -- 20:21:22:23:24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40 -- this has a lot of variety as it great for making subsets. In addition to the chords above, there's a 4:5:6:7 tetrad on 20:25:30:35. There's also an inframinor triad on 20:23:30 and a variety of sevenths. | |||
===Over-7 Scales=== | ===Over-7 Scales=== | ||
Mode 7 -- 7:8:9:10:11:12:13:14 -- with no 3/2 perfect fifth, it may be difficult to make 7 sound like tonic here. | |||
Mode 14 -- 14:15:16:17:18:19:20:21:22:23:24:25:26:27:28 -- 21 is 3/2 above 14, so we can get some root-3rd-P5 triads, such as 14:18:21, a septimal supermajor triad, which also sounds good with 27/14 -- a supermajor seventh; 14:17:21, a septendecimal ([[17-limit]]) supraminor triad, which works well with a 13/7 low major seventh. [[19_14|19/14]] is notable here as a wide and complex perfect fourth. | |||
===Over-9 Scales=== | ===Over-9 Scales=== | ||
Mode 9 -- 9:10:11:12:13:14:15:16:17:18 -- again, lacking a 3/2 above the bass, it's hard to make 9 sound like tonic. | |||
Mode 18 -- 18:19:20:21:22:23:24:25:26:27:28:29:30:31:32:33:34:35:36 -- now we have 27, a 3/2 above with bass, which allows 18:22:27:33, an undecimal neutral seventh chord; and 18:23:27, a [[23-limit]] supermajor triad (close to [[17edo]]). It's also worth noting that the entirety of Mode 6 is available here starting on 18 -- 18:21:24:27:30:33:36. | |||
===Over-11 Scales=== | ===Over-11 Scales=== | ||
Mode 11 -- 11:12:13:14:15:16:17:18:19:20:21:22 | |||
Mode 22 -- 22:23:24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44 -- with 33, we have a perfect fifth above the bass and can make such root-3rd-P5 triads as 22:26:33, a middle "Gothic" tridecimal minor triad; 22:27:33, an undecimal neutral triad; 22:28:23, a "Gothic" undecimal supermajor triad. The sevenths are all complex, ranging from an [[interseptimal]] [[19_11|19/11]]; to a neutral seventh [[20_11|20/11]]; to a wide major seventh at [[21_11|21/11]]. | |||
===Over-13 Scales=== | ===Over-13 Scales=== | ||
Mode 13 -- 13:14:15:16:17:18:19:20:21:22:23:24:25:26 | |||
Mode 26 -- 26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44:45:46:47:48:49:50:51:52 -- 39/26 is a 3/2 perfect fifth. Root-3rd-P5 chords include the tridecimal inframinor 26:30:39; a 31-limit minor triad at 26:31:39; a tridecimal neutral triad at 26:32:39; and a wide tridecimal major at 26:33:39. As odd harmonics go up to 51, a great variety is possible here. | |||
==A Solfege System== | ==A Solfege System== | ||
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Mode 3 -- 3:4:5:6 -- a major triad in 2nd inversion -- that is, with the perfect fifth in the bass.<br /> | Mode 3 -- 3:4:5:6 -- a major triad in 2nd inversion -- that is, with the perfect fifth in the bass.<br /> | ||
<br /> | <br /> | ||
Mode 6 -- 6:7:8:9:10:11:12 -- an effective 6-tone scale. 9 is 3/2 above 3, so there is a perfect fifth above the bass. A septimal subminor triad -- 6:7:9 -- is available, as well as an undecimal 6:7:9:11 tetrad.<br /> | Mode 6 -- 6:7:8:9:10:11:12 -- an effective 6-tone scale. 9 is 3/2 above 3, so there is a perfect fifth above the bass. A septimal subminor triad -- 6:7:9 -- is available, as well as an undecimal 6:7:9:11 tetrad, which adds a neutral seventh of 11/6 to the septimal subminor triad.<br /> | ||
<br /> | <br /> | ||
Mode 12 -- 12:13:14:15:16:17:18:19:20:21:22:23:24 -- as this scale has 12 tones, it fits nicely onto a traditional keyboard instrument, such as piano, melodica, organ, accordion, etc. It allows a 4:5:6:7 septimal tetrad above the bass (a reduced form of 12:15:18:21) as well as the subminor triad and undecimal tetrad given above. The fundamental is 4/3 above the bass, making 4/3 a strong attractor in the system. Andrew Heathwaite has composed with a 12:13:14:16:18:20:22:24 subset, and <a class="wiki_link" href="/Jacob%20Barton">Jacob Barton</a> retuned an electric organ to this scale.<br /> | Mode 12 -- 12:13:14:15:16:17:18:19:20:21:22:23:24 -- as this scale has 12 tones, it fits nicely onto a traditional keyboard instrument, such as piano, melodica, organ, accordion, etc. It allows a 4:5:6:7 septimal tetrad above the bass (a reduced form of 12:15:18:21) as well as the subminor triad and undecimal tetrad given above. The fundamental is 4/3 above the bass, making 4/3 a strong attractor in the system. Andrew Heathwaite has composed with a 12:13:14:16:18:20:22:24 subset, and <a class="wiki_link" href="/Jacob%20Barton">Jacob Barton</a> retuned an electric organ to this scale.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Over-n Scales-Over-5 Scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->Over-5 Scales</h3> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Over-n Scales-Over-5 Scales"></a><!-- ws:end:WikiTextHeadingRule:10 -->Over-5 Scales</h3> | ||
<br /> | <br /> | ||
Mode 5 -- 5:6:7:8:9:10 -- This is essentially a <a class="wiki_link" href="/7-limit">7-limit</a> fully-diminished seventh chord.<br /> | |||
< | <br /> | ||
Mode 10 -- 10:11:12:13:14:15:16:17:18:19:20 -- from 10 to 15 is a 3/2 perfect fifth. We have access to a 10:12:15 classic minor triad, as well as a number of other nice chords like the 10:13:15 barbados triad. 11/10 makes a strange second (or ninth), while 9/5 makes a very nice minor seventh (and an alternative to the 7/4 bluesy seventh of Over-1 scales and neutral seventh of Over-3 scales.<br /> | |||
<!-- ws:start:WikiTextHeadingRule: | <br /> | ||
< | Mode 20 -- 20:21:22:23:24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40 -- this has a lot of variety as it great for making subsets. In addition to the chords above, there's a 4:5:6:7 tetrad on 20:25:30:35. There's also an inframinor triad on 20:23:30 and a variety of sevenths.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Over-n Scales-Over-7 Scales"></a><!-- ws:end:WikiTextHeadingRule:12 -->Over-7 Scales</h3> | |||
<br /> | |||
Mode 7 -- 7:8:9:10:11:12:13:14 -- with no 3/2 perfect fifth, it may be difficult to make 7 sound like tonic here.<br /> | |||
<br /> | |||
Mode 14 -- 14:15:16:17:18:19:20:21:22:23:24:25:26:27:28 -- 21 is 3/2 above 14, so we can get some root-3rd-P5 triads, such as 14:18:21, a septimal supermajor triad, which also sounds good with 27/14 -- a supermajor seventh; 14:17:21, a septendecimal (<a class="wiki_link" href="/17-limit">17-limit</a>) supraminor triad, which works well with a 13/7 low major seventh. <a class="wiki_link" href="/19_14">19/14</a> is notable here as a wide and complex perfect fourth.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Over-n Scales-Over-9 Scales"></a><!-- ws:end:WikiTextHeadingRule:14 -->Over-9 Scales</h3> | |||
<br /> | |||
Mode 9 -- 9:10:11:12:13:14:15:16:17:18 -- again, lacking a 3/2 above the bass, it's hard to make 9 sound like tonic.<br /> | |||
<br /> | |||
Mode 18 -- 18:19:20:21:22:23:24:25:26:27:28:29:30:31:32:33:34:35:36 -- now we have 27, a 3/2 above with bass, which allows 18:22:27:33, an undecimal neutral seventh chord; and 18:23:27, a <a class="wiki_link" href="/23-limit">23-limit</a> supermajor triad (close to <a class="wiki_link" href="/17edo">17edo</a>). It's also worth noting that the entirety of Mode 6 is available here starting on 18 -- 18:21:24:27:30:33:36.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Over-n Scales-Over-11 Scales"></a><!-- ws:end:WikiTextHeadingRule:16 -->Over-11 Scales</h3> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id=" | Mode 11 -- 11:12:13:14:15:16:17:18:19:20:21:22<br /> | ||
<br /> | |||
Mode 22 -- 22:23:24:25:26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44 -- with 33, we have a perfect fifth above the bass and can make such root-3rd-P5 triads as 22:26:33, a middle &quot;Gothic&quot; tridecimal minor triad; 22:27:33, an undecimal neutral triad; 22:28:23, a &quot;Gothic&quot; undecimal supermajor triad. The sevenths are all complex, ranging from an <a class="wiki_link" href="/interseptimal">interseptimal</a> <a class="wiki_link" href="/19_11">19/11</a>; to a neutral seventh <a class="wiki_link" href="/20_11">20/11</a>; to a wide major seventh at <a class="wiki_link" href="/21_11">21/11</a>.<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Over-n Scales-Over-13 Scales"></a><!-- ws:end:WikiTextHeadingRule:18 -->Over-13 Scales</h3> | |||
<br /> | |||
Mode 13 -- 13:14:15:16:17:18:19:20:21:22:23:24:25:26<br /> | |||
<br /> | |||
Mode 26 -- 26:27:28:29:30:31:32:33:34:35:36:37:38:39:40:41:42:43:44:45:46:47:48:49:50:51:52 -- 39/26 is a 3/2 perfect fifth. Root-3rd-P5 chords include the tridecimal inframinor 26:30:39; a 31-limit minor triad at 26:31:39; a tridecimal neutral triad at 26:32:39; and a wide tridecimal major at 26:33:39. As odd harmonics go up to 51, a great variety is possible here.<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="x-A Solfege System"></a><!-- ws:end:WikiTextHeadingRule:20 -->A Solfege System</h2> | |||
<br /> | <br /> | ||
<a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> proposes a solfege system for overtones 16-32 (Mode 16):<br /> | <a class="wiki_link" href="/Andrew%20Heathwaite">Andrew Heathwaite</a> proposes a solfege system for overtones 16-32 (Mode 16):<br /> | ||
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Thus, the pentatonic scale in the example above could be sung: <strong>mi sol ta do re mi</strong><br /> | Thus, the pentatonic scale in the example above could be sung: <strong>mi sol ta do re mi</strong><br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:22:&lt;h2&gt; --><h2 id="toc11"><a name="x-Twelve Scales"></a><!-- ws:end:WikiTextHeadingRule:22 -->Twelve Scales</h2> | ||
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For those interested in learning to sing and hear just intervals, here are twelve of the simplest 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 /> | For those interested in learning to sing and hear just intervals, here are twelve of the simplest 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 /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule: | <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="x-Next Steps"></a><!-- ws:end:WikiTextHeadingRule:24 -->Next Steps</h2> | ||
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Here are some next steps:<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></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. <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></pre></div> | ||