More music and some thoughts on explaining commas

Hi again.

#1. So, after some hours of breaking 7-limit commas (I'm feeling almost like in a lab now), I've finally found two 2D temperaments with a period of 2/1. The generator for one of them is the 7th root of 14/5, the generator for the other is the 7th root of 12/5. That's because 3 generators of the former map to 14/9, while 2 generators of the latter map to 9/7. Interestingly enough, the latter also approximates other ratios very recognizably, like 7/2 or 16/11.

Of course, there are lots of other commas that I could temper out but I was doing it all "manually" and therefore I always tried to find only one step size to save time. For this reason, I often ended up with one approximant being not very recognizable. For example, there's a comma tempered out if you take 5 equal divisions of 5/1 and use 3 such steps instead of 21/8. But 21/8 is not very recognizable by ear and so I didn't try that one.

Okay, these are something like microtonal studies rather than seriously meant compositions.
http://dl.dropbox.com/u/8497979/pp_partial_7limit_1.ogg
http://dl.dropbox.com/u/8497979/pp_partial_7limit_2.ogg

#2. Recently, someone suggested that some less familiar commas might be worth more obvious illustration. Since I've been involved in tempering out small intervals for some time already, I was thinking about a similar idea as well. Interestingly enough, when I try to look them up on the web or temper them out using Graham's scripts, the "point of a particular comma" seems less obvious to me than when I turn the factors into exponent pairs "manually" using the procedure I mentioned some weeks ago. For example, 2430/2401 is, at least for me quite obviously, 15/8 compared to four steps of 7/6. Or 19683/19600 is, from my personal view again, quite evident 3/1 compared to stacking two 10/9s and two 14/9s. And what I haven't found mentioned anywhere and what I think is also important in the 7-limit context is the fact that 10976/10935 is an unmistakable 15/4 compared to three steps of 14/9. Therefore, I think there should be some mathematical "mechanism" that would allow to find these kinds of comparison for any particular comma -- at least for those which contain no more than 4 primes, AFAIC. I suspect there are great similarities with my recent idea of 3D temperaments having the lowest possible complexity, but I'm not sure.

Petr

PS: Eh, let's have a 3D one this time:
http://dl.dropbox.com/u/8497979/pp_partial_7limit_3.ogg

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:
>
> Hi again.
>
> #1. So, after some hours of breaking 7-limit commas (I'm feeling almost like
> in a lab now), I've finally found two 2D temperaments with a period of 2/1.
> The generator for one of them is the 7th root of 14/5, the generator for the
> other is the 7th root of 12/5.

The first seems to be godzilla, and the second I don't know about, but calling the generator 17/15 is interesting.