Automatically turning one rank-1 temperament into another one?

I've had the idea for a long time now to automatically take a piece
that's in meantone and map it to mavila, such that the fifth-based
mapping is preserved, major thirds turn into minor thirds, major
seconds turn into minor seconds, etc. The basic idea was that that's a
good way to take a piece from one linear temperament (meantone) and
turn it into another one (mavila). It works in rank-2, so that's
pretty simple.

One of the main selling points of this approach is that existing music
notation doesn't temper out 128/125, because it distinguishes between
C# and Db. The only problem is that MIDI itself does temper out
128/125, which is a huge pain. Furthermore, lots of existing music
also tempers out 128/125, which also causes some problems with this
approach.

One option might be to generalize the rank-2 transformations to rank-1
ones; to turn the temperament that eliminates both 81/80 and 128/125
into the temperament that eliminates both 135/128 and 128/125 - which
is to say to turn 12-EDO into 9-EDO. Or, you could turn the
temperament that eliminates both 81/80 and 648/625 into the one
eliminating 135/128 and 648/625, which is 16-equal. etc.

Has anyone worked out a method in which this might be done? What gets
me is that in one sense, it seems like the most mathematically "pure"
way to do it would be to just map 1\12 to 1\9 and say harmony be
damned, but if you do so then even the octaves get screwed up, so
that's certainly less than ideal - if nothing else, the concept of an
"EDO" at -least- suggests a mapping for 2/1. Might there be some
clever approach to this problem I'm not seeing?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Has anyone worked out a method in which this might be done? What gets
> me is that in one sense, it seems like the most mathematically "pure"
> way to do it would be to just map 1\12 to 1\9 and say harmony be
> damned, but if you do so then even the octaves get screwed up, so
> that's certainly less than ideal - if nothing else, the concept of an
> "EDO" at -least- suggests a mapping for 2/1. Might there be some
> clever approach to this problem I'm not seeing?
>
> -Mike
>

The best way I can see this happening is to convert the rank-1 temperament into a rank-2 temperament, and then follow the process you described for rank-2 conversions.

For example, in the song Heart and Soul (C Am F G) you could convert 12-equal into 1/4 comma meantone. Then if you wanted to turn it into mavila, you flatten every fifth by W76;24 cents. Your final step is to turn this regular temperament into an equal temperament, like 9-equal for example.

I agree with you though that there must be a more clever solution to this problem that we aren't seeing at the moment.

--- In tuning@yahoogroups.com, "domeofatonement" <domeofatonement@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > Has anyone worked out a method in which this might be done? What gets
> > me is that in one sense, it seems like the most mathematically "pure"
> > way to do it would be to just map 1\12 to 1\9 and say harmony be
> > damned, but if you do so then even the octaves get screwed up, so
> > that's certainly less than ideal - if nothing else, the concept of an
> > "EDO" at -least- suggests a mapping for 2/1. Might there be some
> > clever approach to this problem I'm not seeing?
> >
> > -Mike
> >
>
> The best way I can see this happening is to convert the rank-1 temperament into a rank-2 temperament, and then follow the process you described for rank-2 conversions.
>
> For example, in the song Heart and Soul (C Am F G) you could convert 12-equal into 1/4 comma meantone. Then if you wanted to turn it into mavila, you flatten every fifth by b 24 cents. Your final step is to turn this regular temperament into an equal temperament, like 9-equal for example.
>
> I agree with you though that there must be a more clever solution to this problem that we aren't seeing at the moment.
>

Actually shoot, I forgot that mavila and meantone have different mappings. Using the method above would turn major chords into minor chords, and vice versa. This is a bit of a predicament.

-Ryan

On Wed, Jun 15, 2011 at 4:45 PM, domeofatonement
<domeofatonement@...> wrote:
>
> Actually shoot, I forgot that mavila and meantone have different mappings. Using the method above would turn major chords into minor chords, and vice versa. This is a bit of a predicament.

That was actually part of my goal with this. But the more I think
about it the more I realize that the whole concept doesn't make any
sense. If I'm treating everything as being on the spiral of fifths,
then it doesn't matter whether or not a dyad comes closer to 5/4 or
6/5, so it doesn't matter whether 128/125 vanishes or not. The
equivalent would be to make 108/125 vanish, so that three -minor-
thirds gives you the octave. As you might expect, 108/125 vanishing
and 135/128 vanishing leads to the 12c mapping, where 5/1 is mapped to
27\12 instead of 28\12 (the <12 19 27| val). So much for that.

I know that Gene has somehow managed to work out a full translation of
every temperament to every other one, so I was hoping he could perhaps
give some insight into the rank-1 case; if the approach is basically
just to map 1\x to 1\y, or if there's a more elegant solution I'm not
thinking of.

-Mike

On Wed, Jun 15, 2011 at 11:10 PM, Mike Battaglia <battaglia01@...> wrote:
>
> I know that Gene has somehow managed to work out a full translation of
> every temperament to every other one, so I was hoping he could perhaps
> give some insight into the rank-1 case; if the approach is basically
> just to map 1\x to 1\y, or if there's a more elegant solution I'm not
> thinking of.

Hey, here's something stupidly obvious that I didn't think of at all:
If both of them are augmented temperaments, then you can map
everything as an augmented temperament. I don't know why I didn't
consider that. The problem is that there's going to be an infinite
number of ways to map every interval - in 12-equal, C-D could be 0
period, 2 generators, or it could be 1 period, -2 generators, or it
could be 2 periods, -6 generators. This is the case because 12 is also
a meantone temperament. I'm not immediately sure how the translation
into mavila would work.

Perhaps we can obtain two mappings for each 12-tet interval - the
shortest path(s) to the interval via the meantone mapping, and then
the shortest path(s) to the interval by the augmented mapping, and use
the combination of that data somehow to work out the proper 9-TET
mapping when you change the first one to mavila.

-Mike