Mike Battaglia's Scratch Pad for 5L3s and 3L5s Temperaments
I discovered so many useful subgroup temperaments tonight that my feeble mind is in danger of just snapping. I have no idea how to organize them all and so I'm just making a scratch page on the wiki for now until I can figure out what temperaments these are supposed to go on.
Cliffnotes; there are so many useful temperaments generating 5L3s MOS's that I don't know what to do with them all.
For 5L3s temperaments, the generator doesn't necessarily have to be taken as 3/2 as in Father, although it can. It can also be 32/21, or 22/14, or even 11/7 if you're adventurous. The latter is actually not that adventurous, because if you manage to find a temperament in which you have a lot of 4:7:9:11 chords, it will work even if the 11/7's are 730 cents or so, which is 50 cents flat. This is probably because more complex intervals are less sensitive to mistuning, which is nice, because these temperaments sound awesome.
The basic concept that sparked all this - look at the circle of 738 cent generators in 13-equal (which we'll call 20/13 instead of 3/2 to appease Gene). It goes through, in order (octave equivalent) - 20/13, 7/3, 9/5, 11/8, 17/16, 13/8, and finally 369 cents. Inasmuch as it's a circle of "fifths," it's the most xenharmonic circle of fifths I've ever seen - screw mavila. It hits so many higher-limit intervals that it's almost insane.
[In the chain 20/13, 7/3, 9/5, 11/8 you can stop that point, avoiding the 17-limit and the 50/49 problem. You get a rank three temperament tempering out 351/350, 144/143 and 99/98, with generators 2, 3 and 20/13. It has a badness figure a little higher than the ones I listed, but not too bad; 88edo supplies the optimal patent val. Adding 81/80 or 105/104 to the list of commas gives 13-limit mothra, also with 88edo as optimal patent val. You could use one-third the Lucy fifth as a generator if you felt so inspired. Two mothra generators give you 13/10, which is a generator for your chain. It seems kind of a pity to leave out 3 and 7 by focusing just on every other note in the mothra generator chain, but you could concoct a subgroup temperament in this way. - GWS]
There are endless variations on this - if you make the generator just a little sharp, you can hit 386 cents for that 369 cent interval instead of 369 cents. If you make it just a little flat, you can hit 6/5 instead. And then, aside from that, if you make the generator flatter, you can hit 7/4 instead of 9/5, although this changes the rest of the chain. So there's lots of variations there.
Then there's even more variations in the mappings. If I call the generator 3/2, that's one mapping, if I call it 32/21, that's another mapping, if I call it 14/9, that's another. Equating 14/9 and 32/21 seems like a good idea, but that means that 50/49 vanishes, and the period splits in half. So that's even more stuff to work out. Then you can split the generator in half, and there's even more stuff.
The most interesting development is that sometimes, something like 0-400-333 cents in 18-equal sounds like 7:9:11, at least to my ears and from a virtual pitch standpoint. I started poking around in Scala randomly and found this 13-note MOS which somehow worked as a diatonic scale for 4:7:9:11, and I mean diatonic in the sense that the 7/9 and 9/11 share an interval class and hence contain utonal and otonal equivalents. It gave me this visual of there being "icicles," like everything was blue. We need to do more research into the 22.214.171.124 subgroup.
Lastly, I note that, for 4:7:9:11, 8-equal is the cutoff much like 7-equal is for meantone: sharp of 8-equal, things are much better intoned, and flat of it they're not and "backwards" like mavila. But in this case, all of these awesome and interesting temperaments I keep finding seem to be flat of 8-equal.
I don't know what to make of it all. For now I'm going to just write notes and come back tomorrow. The above notes will help me remember and organize this. I might change some of these names.
Commas: 28/27, 65/64, 88/81, 78/77
Turned out to be Sensi with a dumber mapping
Too tired to finish but should be able to rederive all the temperaments from the above.