MOS scales

I just looked at the Xenharmonic WIKI article on MOS scales. My understanding of the concept precludes scales such as the octatonic that divide the octave (or other interval of equivalence or repetition) evenly as they cannot be generated by cycles of intervals relatively prime to the octave or IE. There may be a confusion here between Well-Formed and Maximally Even scales-- MOS are both, but ME are not necessarily MOS or WF, IIRC.

--John

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> I just looked at the Xenharmonic WIKI article on MOS scales. My
> understanding of the concept precludes scales such as the octatonic that
> divide the octave (or other interval of equivalence or repetition)
> evenly as they cannot be generated by cycles of intervals relatively
> prime to the octave or IE. There may be a confusion here between
> Well-Formed and Maximally Even scales-- MOS are both, but ME are not
> necessarily MOS or WF, IIRC.

Greetings, Mr. Chalmers,
Well, since you got your understanding of MOS probably DIRECTLY from Erv Wilson himself, I'd probably defer to you on the concept, but my understanding was that MOS can apply to scales with non-octave periods; i.e. the octatonic would be an MOS scale with a period of 1/4 of an octave (300 cents) and a generator of somewhere around a 9/8 (200 cents in 12-tET). Scales like the Blackwood Decatonic use a period of 1/5 of an octave (240 cents), and there are plenty of scales with 1/2 octave periods.

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> I just looked at the Xenharmonic WIKI article on MOS scales. My
> understanding of the concept precludes scales such as the octatonic that
> divide the octave (or other interval of equivalence or repetition)
> evenly as they cannot be generated by cycles of intervals relatively
> prime to the octave or IE. There may be a confusion here between
> Well-Formed and Maximally Even scales-- MOS are both, but ME are not
> necessarily MOS or WF, IIRC.

Can you give an example of a scale you would call maximally even but not MOS/WF?

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> I just looked at the Xenharmonic WIKI article on MOS scales. My
> understanding of the concept precludes scales such as the
> octatonic that divide the octave (or other interval of equivalence
> or repetition) evenly as they cannot be generated by cycles of
> intervals relatively prime to the octave or IE.

Hi John,

There's been a great deal of discussion on this over the years.
Paul had Kraig going back and forth with Wilson, who apparently
gave ambiguous answers (as he is known to do!). Paul finally
decided to call everything DES (Distributionally Even Scales),
but then IIRC there was some problem with that, and anyway it
didn't stick. What seems to have happened is that MOS has become
the umbrella term. It seems fair to me. There's really no need
to have two terms, since the octatonic scale is MOS within a
1/4-octave period.

-Carl

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Can you give an example of a scale you would call maximally even
> but not MOS/WF?

So the reference for this is the Clough et al A Taxonomy paper,
now appearing in a files section near you (see the table on
the 3rd page). -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> It seems fair to me. There's really no need
> to have two terms, since the octatonic scale is MOS within a
> 1/4-octave period.

Not having two terms is how mathematicians would do it, in case anyone cares. Always make things abstract and therefore simple. The academic literature on scales similarly sometimes confounds itself by needlessly restricting the definitions of things.

Bangarang. I needed this paper, like, 3 years ago.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > Can you give an example of a scale you would call maximally even
> > but not MOS/WF?
>
> So the reference for this is the Clough et al A Taxonomy paper,
> now appearing in a files section near you (see the table on
> the 3rd page). -Carl
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > Can you give an example of a scale you would call maximally even
> > but not MOS/WF?
>
> So the reference for this is the Clough et al A Taxonomy paper,
> now appearing in a files section near you (see the table on
> the 3rd page). -Carl

And the answer is: a maximally even scale is a type of distributionally even scale, which is a scale with the bivalence property (or else equal), that every generic interval class contains at most two specific intervals. A maximally even scale only applies to scales constructed inside rank one tunings, basically meaning EDOs, and requires the specific intervals in the class to be consecutive integers in terms of the step size.

It strikes me as a gross misnomer for several reasons: only rank one is *truly* "maximally even", and there's nothing obviously maximal about the evenness of requiring consecutive integers. That entails that the scale notes are not too small a proportion of the total notes, which seems to have nothing much to do with evenness.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > It seems fair to me. There's really no need
> > to have two terms, since the octatonic scale is MOS within a
> > 1/4-octave period.
>
> Not having two terms is how mathematicians would do it, in case
> anyone cares. Always make things abstract and therefore simple.
> The academic literature on scales similarly sometimes confounds
> itself by needlessly restricting the definitions of things.

The thing that matters is that you're rank 2, and if you are
you'll have a generator/period pair that get you Myhill's
property. One can certainly see a purpose in having a term for
rank 2 scales, as distinct from rank 2 abstract temperaments, and
since Wilson seems to have propriety I suggest we call it MOS
and leave these other terms relegated to the obscure corner of
the liberal arts academy that... they're already relegated to.

-Carl