Other equivalences like meantone -> mavila

It's possible to algorithmically turn any meantone composition into a
mavila composition, given a score, simply by retuning the chain of
fifths to be flatter than 4\7.

Does anyone know of any other simple equivalences like this that
exist, such that 1 generator + 4 generators yields a triad that's a
simple consonance? I don't care if the triad is 4:5:6 or 10:12:15 or
7:9:11 or whatever. I'm not seeing any simple solution to this but
perhaps there's something clever that I'm missing.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Does anyone know of any other simple equivalences like this that
> exist, such that 1 generator + 4 generators yields a triad that's a
> simple consonance? I don't care if the triad is 4:5:6 or 10:12:15 or
> 7:9:11 or whatever. I'm not seeing any simple solution to this but
> perhaps there's something clever that I'm missing.

You've obviously never looked at the 36edo page on the Xenwiki:

http://xenharmonic.wikispaces.com/36edo

"As a 5-limit temperament, the patent val for 36edo is contorted, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a transversal for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val."

http://xenharmonic.wikispaces.com/36edo#Music

On Sat, Sep 3, 2011 at 11:51 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Does anyone know of any other simple equivalences like this that
> > exist, such that 1 generator + 4 generators yields a triad that's a
> > simple consonance? I don't care if the triad is 4:5:6 or 10:12:15 or
> > 7:9:11 or whatever. I'm not seeing any simple solution to this but
> > perhaps there's something clever that I'm missing.
>
> You've obviously never looked at the 36edo page on the Xenwiki:
>
> http://xenharmonic.wikispaces.com/36edo
>
> "As a 5-limit temperament, the patent val for 36edo is contorted, meaning there are notes of it which cannot be reached from the unison using only 5-limit intervals. A curious alternative val for the 5-limit is <36 65 116| which is not contorted. It is also a meantone val, in the sense that 81/80 is tempered out. However, the "comma" |29 0 -9> is also tempered out, and the "fifth", 29\36, is actually approximately 7/4, whereas the "major third", 44\36, is actually approximately 7/3. Any 5-limit musical piece or scale which is a transversal for a meantone piece or scale will be converted to a no-fives piece tempering out 1029/1024 in place of 81/80 by applying this val."
>
> http://xenharmonic.wikispaces.com/36edo#Music

Right - I forgot about that one. So that maps 4:5:6 to 12:21:28,
right? Do you know of any ways to map arbitrary triads to other
triads, such as mapping 4:5:6 to 7:9:11 somehow? You can map 4:5:6 to
5:6:8 by turning a meantone 3/2 generator into a mavila 4/3 generator,
and melodies end up being upside down as a result too.

-Mike