re: is 19-et j.i.

Hi Kraig,

> It appears we have all type of infinite continuums ranging for JI generators to ET generators as
> well as irrational generators. There is no reason to not map it out in an way one can think of. If
> fact the more approaches beside stand ET and JI i think the better.

Yes, and the idea was to come up with another approach, and to make it so that it can
stand on its own so that you don't need to keep translating everything into the
standard approach and back again all the time (which is the way the
infinitesimals are dealt with at present), but deal with them on their own terms.

As Gene said, it prob. won't have much influence on actual tuning theory.

1.0371550444461919
itself is a ratio, if you write it as
10371550444461919/1000000000000000

and the standard approach would be to treat it as an
infinite expansion
1.0371550444461919.......

I'd treat it as
1.0371550444461919.........12341378/1000000000000....000000
where I've added a tail to each one to make it into a ratio.
In this ratio, only the head matters, and you can have
anything in the tail - all of them will be treated as
identical. So the practical implications are the same,
the difference is in how one thinks about things,
so philosophical, and also a certain elegance you get in the
proofs of mathematical theorems using infinitesimals
instead of limits.

The classical approach to infinitesimals
gives exactly the same results as you get without them,
so you just set up the maths, and you have all the
classical results automatically.

With this approach you have to prove everything from
scratch as you can't transfer results back and forth
which is kind of interesting as one could get new
results though in practice you seem to get all the
same theorems as before. The idea originates in
the work of a Czech mathematician Vopenka.

I still hope to go back to it some day and finish it for publication.

> if you are asking how i would see this , I would say (being of a gestalt leaning psychology) I
> would say you have a 19 ET scale if you were superimposing this interval. My understanding is the
> mind will attempt to place any perception into its simplest form or combinations there of

I agree.

> One interesting feature i have found going the other direction with an ET being associated with an
> irrational (phi). If one forms an 13 MOS of this interval you end up with a close interval with
> the 12 ET semitone. Iy has made me wonder if all that circus music might be drawing upon this
> interpretation of this interval. What could be more turning our world upside down that giving a
> different meaning to what is already here.

That's interesting. I've just had a go at making this MOS and the semitone is very close to 100 cents.

> I think this type of difference might be of interest to Mayan Astronomers. In fact isn't this
> close to one revolution of the Milky way? or am i about 400 million short?

Sounds about right. I'll check it up.

http://seds.lpl.arizona.edu/messier/more/mw.html
Yes, 220 million years for the rotation period.

> One thing that has not been mentioned in there are assumption to JI that have not been touched
> upon and that is that the ratios will be used in certain ways and that lattices and this has been
> the practice so far. So while it is possible for some JI user to use 419914/404871 such an
> interval would be reached by the simpler intervals in between.
> Lou Harrison mentioned that JI has only examined half of it universe in that it has been scale
> oriented as opposed to interval oriented. In the latter we would have a scale of intervals at hand
> regardless of weather it forms a cohesive whole or not. This for me is out of my interest in that
> i find expression in such structures that the intervals outline out.
> It seems possible to have an array of different ET scales or intervals or a matrix of any
> Irrationals from which to proceed through the continuum

Yes, I've come across quite a few ideas that can't be expressed in terms of a scale.
In FTS, the fibonacci tonescapes are like that - they use pure ratios at present
but one could develop the same idea using irrationals too. Anyway the music
you make in the tonescapes couldn't be notated in any single scale with a small fintie
number of intervals.

Robert

Hi Jon,

Yes it was slightly dead pan humour. Not sure
what to use as an emoticon, maybe :-|
perhaps.

Robert