Just intonation subgroup: Difference between revisions
Wikispaces>genewardsmith **Imported revision 143678901 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 143688563 - 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:genewardsmith|genewardsmith]] and made on <tt>2010-05-21 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-05-21 04:15:13 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>143688563</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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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. | 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. | ||
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.</pre></div> | 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. | ||
====7-limit==== | |||
[2, 3, 7] | |||
Ets: 5, 31, 36, 135, 571 | |||
[2, 5, 7] | |||
Ets: 6, 25, 31, 171, 239, 379, 410, 789 | |||
[2, 3, 7/5] | |||
Ets: 10, 29, 31, 41, 70, 171, 241, 412 | |||
[2, 5/3, 7] | |||
Ets: 12, 15, 42, 57, 270, 327 | |||
[2, 5/3, 7/3] | |||
Ets: 27, 68, 72, 99, 171, 517 | |||
====11-limit==== | |||
[2, 3, 11] | |||
Ets: 7, 15, 17, 24, 159, 494, 518, 653 | |||
[2, 5, 11] | |||
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320 | |||
[2, 7, 11] | |||
Ets: 6, 9, 11, 20, 26, 135, 161, 296 | |||
</pre></div> | |||
<h4>Original HTML content:</h4> | <h4>Original HTML content:</h4> | ||
<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>Just intonation subgroups</title></head><body>By a just intonation subgroup is meant 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 /> | <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>Just intonation subgroups</title></head><body>By a just intonation subgroup is meant 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 /> | ||
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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 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.<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 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.<br /> | ||
<br /> | <br /> | ||
A canonical naming system for just intonation subgroups is to give a <a class="wiki_link" href="/Normal%20lists">normal interval list</a> for the generators of the group, which will also show the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rank_of_an_abelian_group" rel="nofollow">rank</a> of the group by the number of generators in the list.</body></html></pre></div> | A canonical naming system for just intonation subgroups is to give a <a class="wiki_link" href="/Normal%20lists">normal interval list</a> for the generators of the group, which will also show the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Rank_of_an_abelian_group" rel="nofollow">rank</a> 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.<br /> | ||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h4&gt; --><h4 id="toc0"><a name="x---7-limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->7-limit</h4> | |||
[2, 3, 7]<br /> | |||
Ets: 5, 31, 36, 135, 571<br /> | |||
<br /> | |||
[2, 5, 7]<br /> | |||
Ets: 6, 25, 31, 171, 239, 379, 410, 789<br /> | |||
<br /> | |||
[2, 3, 7/5]<br /> | |||
Ets: 10, 29, 31, 41, 70, 171, 241, 412<br /> | |||
<br /> | |||
[2, 5/3, 7]<br /> | |||
Ets: 12, 15, 42, 57, 270, 327<br /> | |||
<br /> | |||
[2, 5/3, 7/3]<br /> | |||
Ets: 27, 68, 72, 99, 171, 517<br /> | |||
<br /> | |||
<!-- ws:start:WikiTextHeadingRule:2:&lt;h4&gt; --><h4 id="toc1"><a name="x---11-limit"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit</h4> | |||
<br /> | |||
[2, 3, 11]<br /> | |||
Ets: 7, 15, 17, 24, 159, 494, 518, 653<br /> | |||
<br /> | |||
[2, 5, 11]<br /> | |||
Ets: 6, 7, 9, 13, 15, 22, 37, 87, 320<br /> | |||
<br /> | |||
[2, 7, 11]<br /> | |||
Ets: 6, 9, 11, 20, 26, 135, 161, 296</body></html></pre></div> | |||