Re: retuning by Factor re-mapping, and hexany symmetries (was SCALA

Hi Gene,

> Are you or is anyone thinking of working on retuning software which
> will allow a general sort of retuning? One reason I've been asking
> about it is that I'd like to create some midi examples of
> automorphism groups acting on music to illustrate some things I would
> like to discuss on the tuning-math group, and your hexany pieces
> would be ideal for some purposes. If you would be interested in what
> happens when the group of the octahedron is applied to it (24
> elements) or even the full orthogonal group (48 different versions of
> each of your hexany pieces) perhaps you could send something in an
> ascii version which would allow a perspicuous editing and alteration
> of the tuning.

Sounds fun. Yes, I'd like to hear it.

This is independent of your factor remapping isn't it?

re-arranging 2, 3, 5, 7, in all poss. ways gives 24 possiblities.

Swapping two numbers corresponds to a reflection of the octahedron
in plane through centre normal to one of the opposite pairs of edges.

Easy to get the third turns. (e.g. 2 <-> 3, 3 <-> 5)

However, I don't think that method gets the quarter turns about one line
joining two opposite vertices.

So I suppose then to get those, one replaces 2 by 1/2, 3 by 1/3, 5 by 1/5
and 7 by 1/7, so that 2 * 3 -> 1/(2*3), then on multiplying all the ratios
by 2*3*5*7 one gets 2*3 -> 5*7, i.e. reflection in the equatorial square
of the octahedron. That would complete the group.

FTS now has an option to log ratios played (thanks to suggestion by
Mary Ackerley :-)). However that is meant for one to read, so that one can
see which intervals were played. It doesn't log the times, or channels, or say
which notes were switched on / off at any moment (e.g. two successive
notes show up as the same as a single sustained note with change of
accompaniment).

Do you have any suggestions for a suitable ascii format?

I can add a record to text as a third option (at present has
record to WAVE, and to MIDI). Could be a useful thing to have.

..............................................

After one of your earlier posts, I thought of idea of adding a remapping
channel to FTS.

Here is how it might work:

I have option to select a scale and mode into a midi channel.

Suppose scale shows the hexany:

Then have another scale which shows
2*3 2*5 2*7 3*5 3*7 5*7

(in FTS this would be as mode of scale also including 1/1 and 2/1
- so the 2*3 is relative to the 1/1)

which is just for factor remapping use, and not actually sounded.

"Play" a note in this scale's channel, and if you play a 3*5 it
swaps all 5s and 3s in all the playing channels, etc.

(which one could then octave reduce into a single octave
if doing it like that)

It would be a 2-cycle, so playing it again would get you back to the original scale.

1/3 would map 3 to 1/3

1/(3*5) would map 3 to 1/3 and 5 to 1/5

Not sure what 3*7/5 would do, maybe same as 3*7 combined with 1/5.

So to get all the remappings of the hexany, it would be sufficient
to use the scale

2*3 2*5 2*7 3*5 3*7 5*7 1/(2*3*5*7)

This might be another way to do it.

Could just save the .midi file prefixing the piece with all 48 combinations for
the factor remapping channel, and then play each of those in turn
and record them as one does so, though a bit awkward finding out the
combinations of reflections to use for each one (maybe I'm missing some
simplification here). Plus there's the matter of keeping track of it all
and not introducing errors, if doing it by hand like this.

Perhaps though, if I programmed it in, you could generate the midi
files from the hexany original in automatic fashion with extra
notes in the factor remapping channel, and I could then run
them all through FTS to retune them.

So that might be another approach to explore,...
..............................................

> I wrote 3-part hexany canon in the late 70's in order to do this very
> thing, but it's long gone and I like what you've done better. :)

Glad you like it :-)

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> Sounds fun. Yes, I'd like to hear it.

48 variations on a piece 44 seconds long would give something lasting
35 minutes and 12 seconds. Now, if only I can get this retuning thing
to work we'll be in business!

> This is independent of your factor remapping isn't it?

I have some ideas which are independent of factor remapping but this
isn't one of them. I am thinking of automorphisms of the 7-limit
which are isometries on note classes under a suitable metric, but I
think the other group would be the place for the details.

> re-arranging 2, 3, 5, 7, in all poss. ways gives 24 possibilities.

That is the symmetric group on four elements, S4; or the group of
isometries of the tetrahedron. Its rotation group is A4.

> Swapping two numbers corresponds to a reflection of the octahedron
> in plane through centre normal to one of the opposite pairs of
edges.

> Easy to get the third turns. (e.g. 2 <-> 3, 3 <-> 5)
>
> However, I don't think that method gets the quarter turns about one
line
> joining two opposite vertices.

The *rotations* of the octahedron are also abstractly S4, and permute
the six vertices of the octahedron by one of the permutation
representations of S4 on six elements.

> So I suppose then to get those, one replaces 2 by 1/2, 3 by 1/3, 5
by 1/5
> and 7 by 1/7, so that 2 * 3 -> 1/(2*3), then on multiplying all the
ratios
> by 2*3*5*7 one gets 2*3 -> 5*7, i.e. reflection in the equatorial
square
> of the octahedron. That would complete the group.

That's how you would make the group go from 24 to 48 elements, the
full group of isometries of an octahedron.

> Do you have any suggestions for a suitable ascii format?

I would suggest a note number, followed by a value in cents, in front
of the rest of the score. If a program which converts an ascii score
file to midi knows to always output note x as y cents from the
reference frequency, you can remap the notes to other notes.

> I can add a record to text as a third option (at present has
> record to WAVE, and to MIDI). Could be a useful thing to have.

I would say definitely yes.

> After one of your earlier posts, I thought of idea of adding a
remapping
> channel to FTS.

I would say whatever you do, it should allow you to at least take the
set of all notes which appear in a given piece, and remap them to
other notes; it doesn't seem to me what you are talking about below
does that. There are other kinds of remappings which this won't
support but I think getting at least that far would be grand.

> Could just save the .midi file prefixing the piece with all 48
combinations for
> the factor remapping channel, and then play each of those in turn
> and record them as one does so, though a bit awkward finding out the
> combinations of reflections to use for each one (maybe I'm missing
some
> simplification here).

The whole thing can be done by matrix products as S4 factors nicely,
unlike some less well-behaved groups.

When I did this stuff 20-25 years ago, I had an HP workstation (not
as good as a PC of today, but it made up for it by being larger and
far more expensive) which my brother would borrow from work over the
holidays and a black box he made which controlled four channels
outputting square waves and tuned by a frequency divider. It wasn't
very musical by today's standards, but it *was* in tune and readily
controllable by simple programs which also could do matrix
operations, etc. What a Christmas it was when the whole family was
forced to listen to beloved Christmas carols transformed in weird
ways! I'd like to get up to that speed at least.