On Thu, Nov 1, 2012 at 9:10 PM, Chris Vaisvil <email@example.com>
> 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 22.214.171.124 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 126.96.36.199 subgroup.
Anyway, you asked about figuring out what steps in 11-EDO approximate
what intervals. So if 11-EDO supports the 188.8.131.52 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.
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.
Chris Vaisvil said:
> As someone pretty ignorant of tuning theory I wish there was a table that said
> If you play scale (tuning) steps 0:4:8:10 in 11 edo you will come close to
> approximating a harmonic series chord of 0:3:6:7
There's an easy way to do that in Scala.
Create a new 11-EDO scale by choosing menus File / New / Equal temperament (or press shift-alt-E).
Go to the Analyze menu and select Show Chord Presence (or press Shift-alt-W).
For our purposes, leave the default parameters and options, but make sure "Constrained" is checked. There's a way to change the default constraint, but I forget what it is, so mine says "0.0000". That will only detect JI chords, which clearly isn't so great for 11 EDO, so let's change your constraint.
Let's say you want only chords where no interval is more out of tune than the 5/4 is in 12 EDO: 400.0-386.3=13.69. To the right of the "Constrained" check box, there's a button labeled "Change". Click it, and another dialog box appears called "User Options". The top text field is "Maximum Difference in Cents", so enter 13.69 in that line. You can also change your prime limit there if you're only interested in, say, 13-limit chords. (The default is 29.) You can also play with the weighting factor, which I won't go into -- you can read the help and goof off with it to see what works for you.
Anyway, once you've set your constraints, click OK in the "User Options" dialog to get back to the "Show Chord Presence" dialog. You should see that it says "Maximum difference: 13.6900 cents".
Super. Click "OK", and you'll see the following on your main Scala window:
Scale is equal tempered, only showing first degree. 0-4-12: BP Wide Triad 2nd inversion 7:9:15 diff. 1.280,-10.352 0-5-9: NM Subfocal Tritonic Seventh 8:11:14 diff. -5.863, 12.992 0-1-3-4: Hüzzam 5-9-5 diff. -4.117, 10.292, 6.175 Total of 3 with 3 unique
The first line shows that, because this is an ET, talking about starting on scale step 1-10 would be redundant. If you had a scale with different step sizes, it would run through each chord available on each step.
The second line shows you that scale degrees 0, 4, and 12 approximate a 7:9:15 chord. You can also see that the 4th degree is only 1.28 cents sharp, but the 12th degree is about 10 cents flat.
The third line shows you that the scale degrees 0, 5, and 9 approximate an 8:11:14 chord. You can see that the 5th degree is about 6 cents flat, and the 9th degree is about 13 cents sharp.
I don't know what the fourth line means -- I'm sure someone here does, perhaps Dr. Oz -- but you're interested in approximations to JI chords, so I'm not sure whether it matters. The 5-9-5 represents steps of something, but I don't know what.
So there it is: Three unique chords, two of which are the type you appear to be looking for.
Two chords isn't much, so let's loosen our constraints and see what chords are available that are less out-of-tune than a 7/4 in 12 EDO: 1000.0 - 968.8 = 31.2 cents. Change the constraints as above, get the following:
0-4-7: Isoharmonic Augmented Triad 7:9:11 diff. 1.280,-18.856 0-4-12: BP Wide Triad 2nd inversion 7:9:15 diff. 1.280,-10.352 0-5-9: NM Subfocal Tritonic Seventh 8:11:14 diff. -5.863, 12.992 0-3-6: Isoharmonic Diminished Triad 9:11:13 diff. -20.135, 17.928 0-3-6-8: Isoharmonic Diminished Seventh 9:11:13:15 diff. -20.135, 17.928,-11.631 0-5-9: NM Trevicesimal Tritonic Seventh Triple 13:18:23 diff. -17.928,-5.929 0-3-6: Diminished 25:30:36 diff. 11.631, 23.263 0-2-4: Double Septimal Whole-Tone 49:56:64 diff. -12.992,-25.985 0-3-6: 7-Tone Neutral Triad 2-2 diff. -15.584,-31.169 0-1-2: Semitone Trichord 1-1 diff. 9.091, 18.182 0-1-3: Phrygian Trichord 1-2 diff. 9.091, 27.273 0-2-3: Minor Trichord 2-1 diff. 18.182, 27.273 0-3-8: Minor Trine 1st inversion 3-6 diff. 27.273,-27.273 0-3-8-11: Sixths Chord 3-6-3 diff. 27.273,-27.273, 0.000 0-3-9: Minor Quintal Triad 3-7 diff. 27.273,-18.182 0-4-6: 13-Tone NM Crunchy Pepper Triple 5-2 diff. -25.175, 8.392 0-4-11: 13-Tone NM Converse Trine 5-8 diff. -25.175, 0.000 0-7-11: 13-Tone NM Trine 8-5 diff. 25.175, 0.000 0-3-6-9: 16-Tone Diminished Seventh 4-5-4 diff. 27.273,-20.455, 6.818 0-5-9: 17-Tone NM Quasi-Trevicesimal Subfocal Tritonic Seventh 8-6 diff. -19.251,-6.417 0-2-9: 29-Tone NM Bivalent Ultra-Gothic Triple 6-17 diff. -30.094, 30.094 0-5-9: 36-Tone NM Smaller Tritone Subfocal Seventh 17-12 diff. -21.212, 15.152 0-5-9: 36-Tone NM Larger Tritone Subfocal Seventh 17-13 diff. -21.212,-18.182 0-5-9: 46-Tone NM Tritonic Subfocal Seventh 21-16 diff. -2.372, 16.601 0-1-3-4: Hüzzam 5-9-5 diff. -4.117, 10.292, 6.175 Total of 25 with 25 unique
Notice that the first eight lines are the same kind of thing that we had talked about before. More chords are now available, but the additions are more damaged.
The next line, "7-Tone Neutral Triad", says that this sounds a bit like playing a neutral triad in 7 EDO, and shows the difference from 7 EDO: 11 EDO's 327-cent interval differs from 7 EDO's 342-cent interval by about 16 cents. Similarly the 13-, 16-, 17-, 29-, 36-, and 46-Tone lines show you how similar you are to chords used in other ETs. (Looking at the errors: Not very.) These lines aren't about JI, then, but might help you relate one EDO to another -- "Hey, using 0-5-9 in 11 EDO is like using 0-8-14 in 17 EDO." (It's 0-8-14 in 17 EDO because that line says '8-6', which are the steps between each tone in the triad. The representations of the chords are different from what part of the program to the next. Not ideal, but good enough: Thanks, Manuel, I'll take it. :) )
Not sure about those guys in the middle: "0-3-8-11: Sixths Chord 3-6-3" doesn't mean anything to me. I believe the 3-6-3 again represents steps of something between tones, but steps of what? I don't know.
One thing to remember is that these chords are those that Manuel has defined and loaded up into Scala. There may be other chords you want to consider, and this may not give you those. He has a lot of chords in there, though, so it could be your first stop for any scale you want to look at, and certainly makes sense for most ETs.
Hope this helps,