Re: Complex Number Temperments

When taking roots and powers of complex numbers it is easier to change them
to polar coordinates such as a*e^(i*theta). To take a root, simply divide
theta by the root you wish to have, and take the same root of a. For
example, the 2i can be expressed as 2*e^(i*pi/2), and its twelve root as
2^(1/12)*e^(i*pi/24). Not much help yet, huh? Well the first factor is
12TET. The second is cosine(pi/24)+i*sine(pi/24). For imaginary powers use
the hyperbolic cosines and sines. They can be layed out in a spreadsheet as
follows:

n 12TET 12TET*cos(n*pi/24)
12TET*sin(n*pi/24)
12TET*cosh(n*pi/24)
12TET*sinh(n*pi/24)
0 1.0000 1.0000 0.0000 1.0000 0.0000
1 1.0594 1.0504 0.1383 1.0686 0.1391
2 1.1225 1.0842 0.2905 1.1611 0.2972
3 1.1892 1.0987 0.4551 1.2821 0.4791
4 1.2599 1.0911 0.6300 1.4366 0.6903
5 1.3348 1.0590 0.8126 1.6311 0.9374
6 1.4142 1.0000 1.0000 1.8733 1.2285
7 1.4983 0.9121 1.1887 2.1726 1.5732
8 1.5874 0.7937 1.3747 2.5403 1.9832
9 1.6818 0.6436 1.5538 2.9903 2.4725
10 1.7817 0.4612 1.7211 3.5391 3.0579
11 1.8877 0.2464 1.8716 4.2070 3.7597
12 2.0000 0.0000 2.0000 5.0184 4.6026

Note that e) can be simplified to a), f to (2^(1/12)^n.

>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.
>