Fractal scales vs. Fractal tunings

Donning his fire-proof jacket the new-comer asks:

Is there not a difference between a fractal scale and a fractal
tuning/temperament? I agree that there can be no equal-tempered fractal
tuning, but is it not possible to have a scale that is derived from a
self-similar, recursive process, even within an equal temperament? For
example:

0.......5..7.......12(octave)

divide the interval of 5 (0 to 5 in a similar way):

0..2.3..5 (using uneven halves that fall within the tuning)

divide the interval of 5 (7 to 12 in a similar way):

7..9.10..12 (again using uneven halves that fall within the tuning)

put the two halves together and get:

0..2.3..5..7..9.10..12 (or the so-mis-called "dorian" mode)

Not very exotic, I'll grant you, but self-similar nonetheless. Similar
processes could be carried out with other equal temperaments.

Am I defining either "fractal" or "scale" too narrowly (or too broadly)
hear?

Hoping that the fire-proff jacket isn't made of asbestos,

Jon Southwood
noj@cedar-rapids.net

Jon Southwood wrote,

>0..2.3..5..7..9.10..12 (or the so-mis-called "dorian" mode)

>Not very exotic, I'll grant you, but self-similar nonetheless. Similar
>processes could be carried out with other equal temperaments.

> Am I defining either "fractal" or "scale" too narrowly (or too
broadly)
>hear?

If you (a) used golden meantone tuning and (b) continued dividing the large
intervals into smaller ones ad infinitum, you'd have a truly self-similar
scale. Your example is only stage two in the construction so would hardly
qualify as a fractal -- there isn't really a part of the scale that's
similar to the _whole_ scale. Many natural fractals, such as organs found in
the human body, have ten or more levels of self-similarity.

Replying to Jon Southwood, I wrote,

>If you (a) used golden meantone tuning

oops -- I think that should have been Paul Hahn's positive fractal tuning.