Just intonation subgroup/Mike's tips
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On Thu, Nov 1, 2012 at 9:10 PM, Chris Vaisvil <[[@mailto/chrisvaisvil%40gmail.com|[email protected]]]> wrote: > > For certain, no. I could only guess that subgroups are actually harmonic > series prime limits. Subgroups expand the concept of a prime limit. For instance, say you want the 7-limit, but you don't care about prime 5; you just want primes 2, 3, and 7. So that's the 2.3.7 subgroup. Or say that you want the 7-limit, but you don't care about 3/1 but you do care about 9/1. Then that's the 2.5.7.9 subgroup. Or, say that you want primes 2 and 3, and the composite interval 7/5; that's the 2.3.7/5 subgroup. The rule for any subgroup is that if you multiply or divide intervals, that's also in the subgroup. So for the 2.3.7 subgroup, 7*3 = 21/1 is in the subgroup, as is 7/(2*3) = 7/6. And for the 2.3.7/5 subgroup, 2/(7/5 * 7/5) = 50/49 is in the subgroup, and so on. They're infinite lattices of intervals. 11-EDO happens to be a decent temperament for the 2.7.9.11 subgroup. -Mike Anyway, you asked about figuring out what steps in 11-EDO approximate what intervals. So if 11-EDO supports the 2.7.9.11 subgroup, I can just give you the mappings for 2/1, 7/1, 9/1, and 11/1, and you can mix and match them to get what intervals you want, right. So for instance, 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1 is 38 steps. So then 11/9 is 38-35 = 3 steps. That should be enough information for you to get all the intervals. -Mike OK, so rather than write all of that out in English, though, we can just use a simple notation. So if 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1 is 38 steps, then we can just condense all that as follows: <11 31 35 38| where it's understood in this particular case that the coefficients represent how many steps map to 2/1, 7/1, 9/1 and 11/1, respectively. This is called a val, and this is why we use them; so we can figure out how many steps every interval maps to. So 9/7 in the above case is 35-31 = 4 steps. -Mike <span style="color: #ffffff; display: block;">__._,_.___</span>
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<html><head><title>n00b page</title></head><body>On Thu, Nov 1, 2012 at 9:10 PM, Chris Vaisvil <<a class="wiki_link" href="http://mailto.wikispaces.com/chrisvaisvil%2540gmail.com" target="_blank">[email protected]</a>><br /> <br /> wrote:<br /> <br /> ><br /> <br /> <ul class="quotelist"><li>For certain, no. I could only guess that subgroups are actually harmonic</li><li>series prime limits.</li></ul>Subgroups expand the concept of a prime limit. For instance, say you<br /> want the 7-limit, but you don't care about prime 5; you just want<br /> primes 2, 3, and 7. So that's the 2.3.7 subgroup. Or say that you want<br /> the 7-limit, but you don't care about 3/1 but you do care about 9/1.<br /> Then that's the 2.5.7.9 subgroup. Or, say that you want primes 2 and<br /> 3, and the composite interval 7/5; that's the 2.3.7/5 subgroup.<br /> <br /> The rule for any subgroup is that if you multiply or divide intervals,<br /> that's also in the subgroup. So for the 2.3.7 subgroup, 7*3 = 21/1 is<br /> in the subgroup, as is 7/(2*3) = 7/6. And for the 2.3.7/5 subgroup,<br /> 2/(7/5 * 7/5) = 50/49 is in the subgroup, and so on. They're infinite<br /> lattices of intervals.<br /> <br /> 11-EDO happens to be a decent temperament for the 2.7.9.11 subgroup.<br /> <br /> -Mike<br /> <br /> Anyway, you asked about figuring out what steps in 11-EDO approximate<br /> what intervals. So if 11-EDO supports the 2.7.9.11 subgroup, I can<br /> just give you the mappings for 2/1, 7/1, 9/1, and 11/1, and you can<br /> mix and match them to get what intervals you want, right. So for<br /> instance, 2/1 is 11 steps, 7/1 is 31 steps, 9/1 is 35 steps, and 11/1<br /> is 38 steps. So then 11/9 is 38-35 = 3 steps. That should be enough<br /> information for you to get all the intervals.<br /> <br /> -Mike<br /> <br /> OK, so rather than write all of that out in English, though, we can<br /> just use a simple notation. So if 2/1 is 11 steps, 7/1 is 31 steps,<br /> 9/1 is 35 steps, and 11/1 is 38 steps, then we can just condense all<br /> that as follows:<br /> <br /> <11 31 35 38|<br /> <br /> where it's understood in this particular case that the coefficients<br /> represent how many steps map to 2/1, 7/1, 9/1 and 11/1, respectively.<br /> This is called a val, and this is why we use them; so we can figure<br /> out how many steps every interval maps to. So 9/7 in the above case is<br /> 35-31 = 4 steps.<br /> <br /> -Mike<br /> <span style="color: #ffffff; display: block;"><u>._,_.</u>_</span></body></html>