An improv with a fiveless intro

If we temper out 118098/117649 by itself, we get a fiveless half-octave temperament. Its generator is ~9/7 and 5 of them make ~7/2.

Interestingly, this temperament has some recognizable approximations to 5-limit imtervals as well, which makes it essentially hedgehog but with a completely different view on it than usual.

This improv begins with a fiveless hedgehog pump:
http://dl.dropbox.com/u/8497979/pp_hedgehog_improv.ogg

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:
>
> If we temper out 118098/117649 by itself, we get a fiveless half-octave
> temperament. Its generator is ~9/7 and 5 of them make ~7/2.
>
> Interestingly, this temperament has some recognizable approximations to
> 5-limit imtervals as well, which makes it essentially hedgehog but with a
> completely different view on it than usual.

Essentially hedgehog? The generator is pretty flat for a 10/9; it's much closer to 11/10. In fact, if you temper out 9801/9800 then you can equate 2/(9/7)^2 with (11/10)^2, and hence sqrt(2)/(9/7) with 11/10.

All of this leaves me wondering where, if I was to link this piece as a musical example on the xenwiki, what I should consider it an example of. That's the regular mapping paradigm for you. But I think your no-fives and/or rank three temperament extends very naturally to the 11-limit by tossing 9801/9800 into the mix.

Gene wrote:

> Essentially hedgehog? The generator is pretty flat for a 10/9; it's much > closer to 11/10. In fact, if you temper out
> 9801/9800 then you can equate 2/(9/7)^2 with (11/10)^2, and hence > sqrt(2)/(9/7) with 11/10.

If I add 250/243 to the unison vectors, I get hedgehog, and if I add your suggestion of 9801/9800, I get the 11-limit extension which is common to both hedgehog and porcupine:
http://x31eq.com/cgi-bin/rt.cgi?ets=8d_14c&limit=11

My original idea was to temper out 118098/117649 without involving 5, which allows to tune intervals like 7/4 or 7/6 with a mistuning smaller than a cent.

When I then realized that there were recognizable 5-limit approximations in that temperament, I started playing around with it and understood that this "enhanced" mapping was actually described as hedgehog in various listings, the only difference being that the stress seemed to be usually given on 5-limit intervals, quite understandably.

A couple of years ago, before recording another improv, I found the same mapping by completely different means (by taking a 9-limit scale and tempering out 245/243 and 2430/2401 within the system). At that time, I chose a generator equal to the 5th root of 8/5.

Even though the mapping is the same in both cases, each improv is in a slightly different mood, I think that's because of the slightly different generators and also the stress on 7-limit intervals in the newer one.

The older one is here:
http://dl.dropbox.com/u/8497979/PPImprovX.mp3

Petr