Just intonation subgroup: Difference between revisions
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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">[[toc|flat]] | <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">[[toc|flat]] | ||
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=Definition= | =Definition= | ||
A just intonation //subgroup// is a [[http://en.wikipedia.org/wiki/Free_abelian_group|group]] generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a [[Harmonic Limit|p-limit]] group for some minimal choice of prime p, which is the prime limit of the subgroup. | |||
It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite [[http://en.wikipedia.org/wiki/Index_of_a_subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full 7-limit group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzos]] of the generators. | It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite [[http://en.wikipedia.org/wiki/Index_of_a_subgroup|index]] and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full [[3-limit]] (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full 7-limit group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the [[monzos]] of the generators. | ||
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A canonical naming system for just intonation subgroups is to give a [[Normal lists|normal interval list]] for the generators of the group, which will also show the [[http://en.wikipedia.org/wiki/Rank_of_an_abelian_group|rank]] of the group by the number of generators in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below. | A canonical naming system for just intonation subgroups is to give a [[Normal lists|normal interval list]] for the generators of the group, which will also show the [[http://en.wikipedia.org/wiki/Rank_of_an_abelian_group|rank]] of the group by the number of generators in the list. Below we give some of the more interesting subgroup systems. If a scale is given with the system, it means the subgroup is generated by the notes of the scale. Just intonation subgroups can be described by listing their generators with dots between them; the purpose of using dots is to flag the fact that it is a subgroup which is being referred to. This naming convention is employed below. | ||
=7-limit subgroups= | =7-limit subgroups= | ||
2.3.7 | 2.3.7 | ||
Ets: 5, 31, 36, 135, 571 | Ets: 5, 31, 36, 135, 571 | ||
Archytas Diatonic | Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1] | ||
Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1] | Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1] | ||
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The [[Chromatic pairs|Terrain temperament]] subgroup. | The [[Chromatic pairs|Terrain temperament]] subgroup. | ||
=11-limit subgroups= | =11-limit subgroups= | ||
2.3.11 | 2.3.11 | ||
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The [[Chromatic pairs|Indium temperament]] subgroup. | The [[Chromatic pairs|Indium temperament]] subgroup. | ||
=13-limit subgroups= | =13-limit subgroups= | ||
2.3.13 | 2.3.13 | ||
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<!-- ws:end:WikiTextTocRule:13 --><hr /> | <!-- ws:end:WikiTextTocRule:13 --><hr /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Definition"></a><!-- ws:end:WikiTextHeadingRule:0 -->Definition</h1> | ||
A just intonation <em>subgroup</em> is a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">group</a> generated by a finite set of positive rational numbers via arbitrary multiplications and divisions. Any such group will be contained in a <a class="wiki_link" href="/Harmonic%20Limit">p-limit</a> group for some minimal choice of prime p, which is the prime limit of the subgroup.<br /> | |||
<br /> | <br /> | ||
It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Index_of_a_subgroup" rel="nofollow">index</a> and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full <a class="wiki_link" href="/3-limit">3-limit</a> (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full 7-limit group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the <a class="wiki_link" href="/monzos">monzos</a> of the generators.<br /> | It is only when the group in question is not the entire p-limit group that we have a just intonation subgroup in the strict sense. Such subgroups come in two flavors: finite <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Index_of_a_subgroup" rel="nofollow">index</a> and infinite index, where intuitively speaking the index measures the relative size of the subgroup within the entire p-limit group. For example, the subgroups generated by 4 and 3, by 2 and 9, and by 4 and 6 all have index 2 in the full <a class="wiki_link" href="/3-limit">3-limit</a> (Pythagorean) group. Half of the 3-limit intervals will belong to any one of them, and half will not, and all three groups are distinct. On the other hand, the group generated by 2, 3, and 7 is of infinite index in the full 7-limit group, which is generated by 2, 3, 5 and 7. The index can be computed by taking the determinant of the matrix whose rows are the <a class="wiki_link" href="/monzos">monzos</a> of the generators.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x7-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit subgroups</h1> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="x7-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->7-limit subgroups</h1> | ||
<br /> | <br /> | ||
2.3.7<br /> | 2.3.7<br /> | ||
Ets: 5, 31, 36, 135, 571<br /> | Ets: 5, 31, 36, 135, 571<br /> | ||
<br /> | <br /> | ||
Archytas Diatonic | Archytas Diatonic [8/7, 32/27, 4/3, 3/2, 12/7, 16/9, 2/1]<br /> | ||
Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]<br /> | Safi al-Din Septimal [8/7, 9/7, 4/3, 32/21, 12/7, 16/9, 2/1]<br /> | ||
<br /> | <br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit subgroups</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="x11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:4 -->11-limit subgroups</h1> | ||
<br /> | <br /> | ||
2.3.11<br /> | 2.3.11<br /> | ||
Ets: 7, 15, 17, 24, 159, 494, 518, 653<br /> | Ets: 7, 15, 17, 24, 159, 494, 518, 653<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x13-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit subgroups</h1> | <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x13-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:6 -->13-limit subgroups</h1> | ||
<br /> | <br /> | ||
2.3.13<br /> | 2.3.13<br /> | ||
Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424<br /> | Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424<br /> | ||