Just intonation subgroup: Difference between revisions
Jump to navigation
Jump to search
Wikispaces>genewardsmith **Imported revision 143982061 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 215740780 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-03-31 03:45:58 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>215740780</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> | ||
| 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">By a just intonation subgroup is meant 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. | <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">By a just intonation subgroup is meant 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. | 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. 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. | 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. | ||
| Line 80: | Line 80: | ||
<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 /> | ||
<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 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 <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.<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. 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 /> | 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 /> | ||
Revision as of 03:45, 31 March 2011
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2011-03-31 03:45:58 UTC.
- The original revision id was 215740780.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
By a just intonation subgroup is meant 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. 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 subgroups=== [2, 3, 7] Ets: 5, 31, 36, 135, 571 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] [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, 7/3] Ets: 9, 31, 40, 50, 81, 90, 171, 261 [2, 5/3, 7/3] Ets: 27, 68, 72, 99, 171, 517 [2, 27/25, 7/3] Ets: 9 In effect, equivalent to 9EDO, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2] ===11-limit subgroups=== [2, 3, 11] Ets: 7, 15, 17, 24, 159, 494, 518, 653 Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1] [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 [2, 3, 5, 11] Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494 [2, 3, 7, 11] Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135 Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1] [2, 5, 7, 11] Ets: 6, 15, 31, 35, 37, 109, 618, 960 ===13-limit subgroups [2, 3, 13] Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424 Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1] [2, 3, 7, 13] Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234 Buzurg [14/13, 16/13, 4/3, 56/39, 3/2] Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1] Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]
Original HTML content:
<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 /> <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.<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. 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:<h3> --><h3 id="toc0"><a name="x--7-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:0 -->7-limit subgroups</h3> <br /> [2, 3, 7]<br /> Ets: 5, 31, 36, 135, 571<br /> <br /> 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 /> <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, 7/3]<br /> Ets: 9, 31, 40, 50, 81, 90, 171, 261<br /> <br /> [2, 5/3, 7/3]<br /> Ets: 27, 68, 72, 99, 171, 517<br /> <br /> [2, 27/25, 7/3]<br /> Ets: 9<br /> <br /> In effect, equivalent to 9EDO, which has a 7-limit version given by [27/25, 7/6, 63/50, 49/36, 72/49, 100/63, 12/7, 50/27, 2]<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x--11-limit subgroups"></a><!-- ws:end:WikiTextHeadingRule:2 -->11-limit subgroups</h3> <br /> [2, 3, 11]<br /> Ets: 7, 15, 17, 24, 159, 494, 518, 653<br /> <br /> Zalzal, al-Farabi's version [9/8, 27/22, 4/3, 3/2, 18/11, 16/9, 2/1]<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<br /> <br /> [2, 3, 5, 11]<br /> Ets: 7, 15, 22, 31, 65, 72, 87, 270, 342, 407, 494<br /> <br /> [2, 3, 7, 11]<br /> Ets: 9, 17, 26, 31, 41, 46, 63, 72, 135<br /> <br /> Ptolemy Intense Chromatic [22/21, 8/7, 4/3, 3/2, 11/7, 12/7, 2/1]<br /> <br /> [2, 5, 7, 11]<br /> Ets: 6, 15, 31, 35, 37, 109, 618, 960<br /> <br /> ===13-limit subgroups<br /> <br /> [2, 3, 13]<br /> Ets: 7, 10, 17, 60, 70, 130, 147, 277, 424<br /> <br /> Mustaqim mode, Ibn Sina [9/8, 39/32, 4/3, 3/2, 13/8, 16/9, 2/1]<br /> <br /> [2, 3, 7, 13]<br /> Ets: 10, 26, 27, 36, 77, 94, 104, 130, 234<br /> <br /> Buzurg [14/13, 16/13, 4/3, 56/39, 3/2]<br /> Safi al-Din tuning [8/7, 16/13, 4/3, 32/21, 64/39, 16/9, 2/1]<br /> Ibn Sina tuning [14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2]</body></html>