Omnitetrachordal Scales

Hi,

The following is pure "stream of consciousness" thinking so don't
take it too seriously. :)

What if we started with melodic considerations instead of harmonic
when we search for interesting scales?

Is there any kind of general algorithm which will generate all
omnitetrachordal scales, possibly in an abstract form of patterns of
L (larger step) and s (smaller step)?

I'm speculating that there might be a way to investigate the possible
harmonic properties of such scales after giving all kinds of
different values to L and s.

Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi,
>
> The following is pure "stream of consciousness" thinking so don't
> take it too seriously. :)
>
> What if we started with melodic considerations instead of harmonic
> when we search for interesting scales?
>
> Is there any kind of general algorithm which will generate all
> omnitetrachordal scales, possibly in an abstract form of patterns
of
> L (larger step) and s (smaller step)?

I've posted omnitetrachordal scales with *three* step sizes. But
among those with two step sizes, there seem to be only two classes
that I've found so far:

1. MOSs of temperaments where the period is an octave and the
generator is the approximate fifth/fourth;

2. Non-MOSs of temperaments where the period is a half-octave and the
generator can be expressed as the approximate fifth/fourth.

In the first class we have the pentatonic and diatonic scales in
meantone and mavila, 17- and 29-tone schismic scales, 19-tone
meantone scales . . .

In the second class, the 'pentachordal' and 'hexachordal' pajara
scales, the 'hexachordal' and 'heptachordal' injera scales . . .