Complex number temperaments

Dearest members of the Alternative Tuning list,

I am trying to calculate out Western Equal musical temperaments using
imaginary and real numbers.

Most Western Equal temperaments are based on:

{12th root 2)^n, but I really want to do my own version of these using:

a) (2i^(1/12)^n
b) (2i^(1/12i)^n
c) (2^(1/12i)^n
d) (2i^(1/12)^in
e) (2i^(1/12i)^in
f) (2^(1/12i)^in

I have emailed a forum called "Dr.Math" to ask about how to calculate
imaginary roots, and imaginary powers, but a hell of a lot of convention
is assummed, and knowledge is necessecary to do these calculations, so I
will need
somebody with a good working knowledge of algebra and complex numbers to
"guide me through" these calculations, and point out any useful information,
and temperaments that could be made from these.

Would any of these prove interesting???

Any advice, links, resources and information as to aid me in my quest
would be appreciated...

Any comments?

Sincerely,

Sarn Richard Ursell.

Sarn Ursell wrote:

> I am trying to calculate out Western Equal musical temperaments using
> imaginary and real numbers.
>
> Most Western Equal temperaments are based on:
>
> {12th root 2)^n, but I really want to do my own version of these using:
>
> a) (2i^(1/12)^n
> b) (2i^(1/12i)^n
> c) (2^(1/12i)^n
> d) (2i^(1/12)^in
> e) (2i^(1/12i)^in
> f) (2^(1/12i)^in
>
> I have emailed a forum called "Dr.Math" to ask about how to calculate
> imaginary roots, and imaginary powers, but a hell of a lot of convention
> is assummed, and knowledge is necessecary to do these calculations, so I
> will need
> somebody with a good working knowledge of algebra and complex numbers to
> "guide me through" these calculations, and point out any useful
> information,
> and temperaments that could be made from these.
>
> Would any of these prove interesting???

I did think about this kind of thing a while back, with reference to
LucyTuning. You can find it somewhere on Charles Lucy's site, which was
http://www.harmonics.com/lucy/ last I remember. I can't remember the
details myself.

Charles had this idea that the pi which occurs in the LucyTuning formula
relates the circular nature of the sine function that underlies sine
waves, and so on. My first thought was that this couldn't be right,
because the dimensions were wrong. Then I thought a bit more and found
that it did sort of work if you used complex numbers, although it still
had nothing to do with Physics or Psychoacoustics. So the idea of writing
it up was to prove that it _doesn't_ mean anything important.

I don't know if Harrison was thinking this way: probably not, because all
he said in his explanation was that the major third related to the octave
as the diameter of a circle relates to its circumference, whith the whole
tone as the radius. Probably he had this picture in his head, and no
deeper theory behind it. If he knew the right people, he may have known
the relevant mathematics, so it's not *completely* out of the question.

Background information on LucyTuning, Harrison, etc is all at the site
mentioned above.

Anyway, that's the only example I can think of where complex numbers have
anything to do with tuning theory. If you really want to play
mathematical tricks, you could try using a complex function to choose
notes, and take the real part as key velocity and the imaginary part as
pitch. That's fairly harmless, could even be productive.

Graham

John F. Sprague wrote...

>It is due to the limitations of our sense organs that we perceive slower
>variations as rhythm, medium speed variations (about 20 to 8000 Hz) as
>pitch and faster variations (about 40 to 20,000 Hz) as tone color, with the
>interactions between simultaneous tones as harmony.

True or not, what kind of "limitation" is that?

>On the other hand, painting could be described as a two dimensional art
>form and sculpture as three dimensional.

All sorts of things can be described in all sorts of dimensions, as you
like. The important thing is the usefulness of the description. Just one
example of a multi-dimensional model being very useful in the field of
music would be Manfred Schroeder's compelling study of concert-hall acousitcs
(see his book, _Fractals, Chaos, and Power Laws_, pp. 74-79.

-Carl