Tetrachord
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Related pages: [[22edo tetrachords]], [[17edo tetrachords]], [[Tricesimoprimal Tetrachordal Tesseract]] The word "tetrachord" usually refers to the interval of a perfect fourth (just or not) divided into three subintervals by the interposition of two additional notes. John Chalmers, in [[http://eamusic.dartmouth.edu/%7Elarry/published_articles/divisions_of_the_tetrachord/index.html|Divisions of the Tetrachord]], tells us: //Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Medditterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.// =Ancient Greek Genera= The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the characteristic interval, or CI-- the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic. ===hyperenharmonic genus=== The CI is larger than 425 cents. ===enharmonic genus=== The CI approximates a major third, falling between 425 cents and 375 cents. ===chromatic genus=== The CI approximates a minor or neutral third, falling between 375 cents and 250 cents. ===diatonic genus=== The CI (and the other intervals) approximates a "tone," measuring less than 250 cents. ==Ptolomy's Catalog== In the //Harmonics//, Ptolomy catalogs several historical tetrachords and attributes them to particular theorists. ||||||~ Archytas's Genera || || 28/27, 36/35, 5/4 || 63 + 49 + 386 || enharmonic || || 28/27, 243/224, 32/27 || 63 + 141 + 294 || chromatic || || 28/27, 8/7, 9/8 || 63 + 231 + 204 || diatonic || ||||||~ Eratosthenes's Genera || || 40/39, 39/38, 19/15 || 44 + 45 + 409 || enharmonic || || 20/19, 19/18, 6/5 || 89 + 94 + 316 || chromatic || || 256/243, 9/8, 9/8 || 90 + 204 + 204 || diatonic || ||||||~ Didymos's Genera || || 32/31, 31/30, 5/4 || 55 + 57 + 386 || enharmonic || || 16/15, 25/24, 6/5 || 112 + 74 + 316 || chromatic || || 16/15, 10/9, 9/8 || 112 + 182 + 204 || diatonic || ||||||~ Ptolemy's Tunings || || 46/45, 24/23, 5/4 || 38 + 75 + 386 || enharmonic || || 28/27, 15/14, 6/5 || 63 + 119 + 316 || soft chromatic || || 22/21, 12/11, 7/6 || 81 + 151 + 267 || intense chromatic || || 21/20, 10/9, 8/7 || 85 + 182 + 231 || soft diatonic || || 28/27, 8/7, 9/8 || 63 + 231 + 204 || diatonon toniaion || || 256/243, 9/8, 9/8 || 90 + 204 + 204 || diatonon ditoniaion || || 16/15, 9/8, 10/9 || 112 + 182 + 204 || intense diatonic || || 12/11, 11/10, 10/9 || 151 + 165 + 182 || equable diatonic || ==Superparticular Intervals== In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular-- meaning of the form n/n-1 (eg. 5/4, 6/5, 11/10, 39/38...). =Tetrachords Generalized= All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals //a// & //b//, & then write our generalized tetrachord like this: 1/1, a, b, 4/3 We can build a heptatonic scale by copying this tetrachord at the perfect fifth. Thus: 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1 Between 3/2 and 4/3, we have 9/8, so another way to write it would be: [tetrachord], 9/8, [tetrachord] When a tetrachord is paired with its copy, in this way, I call it a "heptatonic mirror." Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord (eg. 1/1, c, d, 4/3): 1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1 ==Modes of a heptatonic mirror== Going back to our generalized heptatonic mirror, let's take a look at what modes we get by starting on different scale degrees. || mode 1 || 1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1 || || mode 2 || 1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1 || || mode 3 || 1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1 || || mode 4 || 1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1 || || mode 5 || 1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1 || || mode 6 || 1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1 || || mode 7 || 1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1 || A heptatonic mirror thus contains more than a single kind of tetrachord. In addition to 1/1, a, b, 4/3, it would also have: 1/1, b/a, 4/3a, 4/3 (mode 6) 1/1, 4/3b, 4a/3b, 4/3 (mode 7)
Original HTML content:
<html><head><title>tetrachord</title></head><body>Related pages: <a class="wiki_link" href="/22edo%20tetrachords">22edo tetrachords</a>, <a class="wiki_link" href="/17edo%20tetrachords">17edo tetrachords</a>, <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a><br />
<br />
<br />
The word "tetrachord" usually refers to the interval of a perfect fourth (just or not) divided into three subintervals by the interposition of two additional notes.<br />
<br />
John Chalmers, in <a class="wiki_link_ext" href="http://eamusic.dartmouth.edu/%7Elarry/published_articles/divisions_of_the_tetrachord/index.html" rel="nofollow">Divisions of the Tetrachord</a>, tells us:<br />
<br />
<em>Tetrachords are modules from which more complex scalar and harmonic structures may be built. These structures range from the simple heptatonic scales known to the classical civilizations of the eastern Medditterranean to experimental gamuts with many tones. Furthermore, the traditional scales of much of the world's music, including that of Europe, the Near East, the Catholic and Orthodox churches, Iran and India, are still based on tetrachords. Tetrachords are thus basic to an understanding of much of the world's music.</em><br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="Ancient Greek Genera"></a><!-- ws:end:WikiTextHeadingRule:0 -->Ancient Greek Genera</h1>
<br />
The ancient Greeks distinguished between three primary genera depending on the size of the largest interval, the characteristic interval, or CI-- the enharmonic, chromatic, and diatonic. Modern theorists have added a fourth genera, called hyperenharmonic.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="Ancient Greek Genera--hyperenharmonic genus"></a><!-- ws:end:WikiTextHeadingRule:2 -->hyperenharmonic genus</h3>
The CI is larger than 425 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="Ancient Greek Genera--enharmonic genus"></a><!-- ws:end:WikiTextHeadingRule:4 -->enharmonic genus</h3>
The CI approximates a major third, falling between 425 cents and 375 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="Ancient Greek Genera--chromatic genus"></a><!-- ws:end:WikiTextHeadingRule:6 -->chromatic genus</h3>
The CI approximates a minor or neutral third, falling between 375 cents and 250 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="Ancient Greek Genera--diatonic genus"></a><!-- ws:end:WikiTextHeadingRule:8 -->diatonic genus</h3>
The CI (and the other intervals) approximates a "tone," measuring less than 250 cents.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:10:<h2> --><h2 id="toc5"><a name="Ancient Greek Genera-Ptolomy's Catalog"></a><!-- ws:end:WikiTextHeadingRule:10 -->Ptolomy's Catalog</h2>
<br />
In the <em>Harmonics</em>, Ptolomy catalogs several historical tetrachords and attributes them to particular theorists.<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="3">Archytas's Genera<br />
</th>
</tr>
<tr>
<td>28/27, 36/35, 5/4<br />
</td>
<td>63 + 49 + 386<br />
</td>
<td>enharmonic<br />
</td>
</tr>
<tr>
<td>28/27, 243/224, 32/27<br />
</td>
<td>63 + 141 + 294<br />
</td>
<td>chromatic<br />
</td>
</tr>
<tr>
<td>28/27, 8/7, 9/8<br />
</td>
<td>63 + 231 + 204<br />
</td>
<td>diatonic<br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<th colspan="3">Eratosthenes's Genera<br />
</th>
</tr>
<tr>
<td>40/39, 39/38, 19/15<br />
</td>
<td>44 + 45 + 409<br />
</td>
<td>enharmonic<br />
</td>
</tr>
<tr>
<td>20/19, 19/18, 6/5<br />
</td>
<td>89 + 94 + 316<br />
</td>
<td>chromatic<br />
</td>
</tr>
<tr>
<td>256/243, 9/8, 9/8<br />
</td>
<td>90 + 204 + 204<br />
</td>
<td>diatonic<br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<th colspan="3">Didymos's Genera<br />
</th>
</tr>
<tr>
<td>32/31, 31/30, 5/4<br />
</td>
<td>55 + 57 + 386<br />
</td>
<td>enharmonic<br />
</td>
</tr>
<tr>
<td>16/15, 25/24, 6/5<br />
</td>
<td>112 + 74 + 316<br />
</td>
<td>chromatic<br />
</td>
</tr>
<tr>
<td>16/15, 10/9, 9/8<br />
</td>
<td>112 + 182 + 204<br />
</td>
<td>diatonic<br />
</td>
</tr>
</table>
<br />
<table class="wiki_table">
<tr>
<th colspan="3">Ptolemy's Tunings<br />
</th>
</tr>
<tr>
<td>46/45, 24/23, 5/4<br />
</td>
<td>38 + 75 + 386<br />
</td>
<td>enharmonic<br />
</td>
</tr>
<tr>
<td>28/27, 15/14, 6/5<br />
</td>
<td>63 + 119 + 316<br />
</td>
<td>soft chromatic<br />
</td>
</tr>
<tr>
<td>22/21, 12/11, 7/6<br />
</td>
<td>81 + 151 + 267<br />
</td>
<td>intense chromatic<br />
</td>
</tr>
<tr>
<td>21/20, 10/9, 8/7<br />
</td>
<td>85 + 182 + 231<br />
</td>
<td>soft diatonic<br />
</td>
</tr>
<tr>
<td>28/27, 8/7, 9/8<br />
</td>
<td>63 + 231 + 204<br />
</td>
<td>diatonon toniaion<br />
</td>
</tr>
<tr>
<td>256/243, 9/8, 9/8<br />
</td>
<td>90 + 204 + 204<br />
</td>
<td>diatonon ditoniaion<br />
</td>
</tr>
<tr>
<td>16/15, 9/8, 10/9<br />
</td>
<td>112 + 182 + 204<br />
</td>
<td>intense diatonic<br />
</td>
</tr>
<tr>
<td>12/11, 11/10, 10/9<br />
</td>
<td>151 + 165 + 182<br />
</td>
<td>equable diatonic<br />
</td>
</tr>
</table>
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:12:<h2> --><h2 id="toc6"><a name="Ancient Greek Genera-Superparticular Intervals"></a><!-- ws:end:WikiTextHeadingRule:12 -->Superparticular Intervals</h2>
<br />
In ancient Greek descriptions of tetrachords in use, we find a preference for tetrachordal steps that are superparticular-- meaning of the form n/n-1 (eg. 5/4, 6/5, 11/10, 39/38...).<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:14:<h1> --><h1 id="toc7"><a name="Tetrachords Generalized"></a><!-- ws:end:WikiTextHeadingRule:14 -->Tetrachords Generalized</h1>
<br />
All tetrachords share the interval of a perfect fourth, but they vary in the other two intervals. Assuming a just fourth, we can name the two variable intervals <em>a</em> & <em>b</em>, & then write our generalized tetrachord like this:<br />
<br />
1/1, a, b, 4/3<br />
<br />
We can build a heptatonic scale by copying this tetrachord at the perfect fifth. Thus:<br />
<br />
1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1<br />
<br />
Between 3/2 and 4/3, we have 9/8, so another way to write it would be:<br />
<br />
[tetrachord], 9/8, [tetrachord]<br />
<br />
When a tetrachord is paired with its copy, in this way, I call it a "heptatonic mirror." Of course, a tetrachord doesn't need to be paired with its copy. You might pair it with a dissimilar tetrachord (eg. 1/1, c, d, 4/3):<br />
<br />
1/1, a, b, 4/3, 3/2, 3c/2, 3d/2, 2/1<br />
<br />
<!-- ws:start:WikiTextHeadingRule:16:<h2> --><h2 id="toc8"><a name="Tetrachords Generalized-Modes of a heptatonic mirror"></a><!-- ws:end:WikiTextHeadingRule:16 -->Modes of a heptatonic mirror</h2>
<br />
Going back to our generalized heptatonic mirror, let's take a look at what modes we get by starting on different scale degrees.<br />
<br />
<table class="wiki_table">
<tr>
<td>mode 1<br />
</td>
<td>1/1, a, b, 4/3, 3/2, 3a/2, 3b/2, 2/1<br />
</td>
</tr>
<tr>
<td>mode 2<br />
</td>
<td>1/1, b/a, 4/3a, 3/2a, 3/2, 3b/2a, 2/a, 2/1<br />
</td>
</tr>
<tr>
<td>mode 3<br />
</td>
<td>1/1, 4b/3, 3b/2, 3a/2b, 3/2, 2/b, 2/b, 2/1<br />
</td>
</tr>
<tr>
<td>mode 4<br />
</td>
<td>1/1, 9/8, 9a/8, 9b/8, 3/2, 3a, 3b, 2/1<br />
</td>
</tr>
<tr>
<td>mode 5<br />
</td>
<td>1/1, a, b, 4/3, 4a/3, 4b/3, 16/9, 2/1<br />
</td>
</tr>
<tr>
<td>mode 6<br />
</td>
<td>1/1, b/a, 4/3a, 4/3, 4b/3a, 16/9a, 2/1<br />
</td>
</tr>
<tr>
<td>mode 7<br />
</td>
<td>1/1, 4/3b, 4a/3b, 4/3, 16/9b, 2/b, 2a/b, 2/1<br />
</td>
</tr>
</table>
<br />
A heptatonic mirror thus contains more than a single kind of tetrachord. In addition to 1/1, a, b, 4/3, it would also have:<br />
<br />
1/1, b/a, 4/3a, 4/3 (mode 6)<br />
1/1, 4/3b, 4a/3b, 4/3 (mode 7)</body></html>