Question about MOS's (completeness)

Hi list,

I want to know if there are moments of symmetry (MOS's) that cannot be derived
from a maximally even scale by doing the following:

Replace the smaller step with s, and the larger step with L, where 0 < s < L.

I'll call such a hypothetical MOS an irregular MOS. Also, I assume that an
irregular MOS can be a MODMOS, which is contrary to the usual definition,
which dictates that a MODMOS cannot be an MOS.

Obviously, an MOS is a linear tuning, since both steps can be used as generators
for the scale, and a change of basis can be applied s.th. one of the generators is
either the period of the scale, or a fraction of it.
Since there exists an MOS for each number of small and large steps, an irregular
MOS has to be a permutation of its steps, which is a special kind of MODMOS.

Does someone know more? I never saw such an irregular MOS, and from what I'm
reading they don't seem to exist, but I'd like to know for sure.

You're basically asking if every MOS can be generated by starting with a
maximally even scale and changing the ratio of L/s, right? Then the answer
is yes, assuming you take the proper care in defining your terms so that
your pathological corner cases are dealt with (such as the 6-note ME scale
in 12-EDO).

Mike

On Mon, Sep 1, 2014 at 3:55 PM, gedankenwelt94@... [tuning-math] <
tuning-math@yahoogroups.com> wrote:

>
>
> Hi list,
>
>
> I want to know if there are moments of symmetry (MOS's) that cannot be
> derived
>
> from a maximally even scale by doing the following:
>
>
> Replace the smaller step with s, and the larger step with L, where 0 < s <
> L.
>
>
> I'll call such a hypothetical MOS an irregular MOS. Also, I assume that an
>
> irregular MOS can be a MODMOS, which is contrary to the usual definition,
>
> which dictates that a MODMOS cannot be an MOS.
>
>
> Obviously, an MOS is a linear tuning, since both steps can be used as
> generators
>
> for the scale, and a change of basis can be applied s.th. one of the
> generators is
>
> either the period of the scale, or a fraction of it.
>
> Since there exists an MOS for each number of small and large steps, an
> irregular
>
> MOS has to be a permutation of its steps, which is a special kind of
> MODMOS.
>
>
> Does someone know more? I never saw such an irregular MOS, and from what
> I'm
>
> reading they don't seem to exist, but I'd like to know for sure.
>
>
>

Hi Mike, thanks for your answer! :)

Yes, I assume that EDOs like 6-EDO are not ME scales.

Is there a simple proof that this is true? I.e. that rearranging the steps of
an MOS cannot result in a different MOS, translational symmetry assumed?

Rearranging the steps? I thought you were referring to changing the L/s
ratio...

Mike

On Monday, September 1, 2014, gedankenwelt94@... [tuning-math] <
tuning-math@yahoogroups.com> wrote:

>
>
> Hi Mike, thanks for your answer! :)
>
>
> Yes, I assume that EDOs like 6-EDO are not ME scales.
>
>
> Is there a simple proof that this is true? I.e. that rearranging the steps
> of
>
> an MOS cannot result in a different MOS, translational symmetry assumed?
>
>
>

--
Mike

Sorry for the confusing wording, and for omitting some intermediate steps...

What I wanted to know is if there is a simple proof that there is no MOS which can't be constructed by changing the L/s ratio of an ME scale.

Since such a hypothetical "irregular MOS" would have to be a linear tuning, and would have to have two step sizes, we should be able to construct it by rearranging the steps of an existing MOS; just like the MODMOS melodic minor scale can be constructed by rearranging the steps of the diatonic scale.

If one can show that rearranging the steps of an MOS never results in a different MOS (root and mode ignored), this suffices to proof that no such "irregular MOS" exists.

On 04/09/14 20:35, gedankenwelt94@... [tuning-math] wrote:

> What I wanted to know is if there is a simple proof that there is no MOS which can't be constructed by changing the L/s ratio of an ME scale.

It depends how strictly you want obvious things to be proved. I've shown that every node on the scale tree can be associated with a maximally even scale. If you're happy that every MOS is on the scale tree somewhere (for the correct period) then you can count it proven.

The Carey & Clampitt paper "Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogues" has the proof of equivalence between Myhill's property and well formedness. What you probably want is to tie distributional evenness to well formedness, which may be Carey and Clampitt's "Aspects of Well-formed Scales". I haven't checked.

Graham

Graham Breed wrote:
> It depends how strictly you want obvious things to be proved.

I'm happy with proofs for the non-obvious things. I guess I should
write what is obvious to me, then, and what is not. ^^

> I've shown that every node on the scale tree can be associated with a
> maximally even scale.

This is clear to me.

> If you're happy that every MOS is on the scale
> tree somewhere (for the correct period) then you can count it proven.

I can see that every MOS that can be derived from an ME scale can be
found on the scale tree. But it's not obvious to me yet that there aren't
other scales with the MOS (or max-variety-2 / DE) property.

> The Carey & Clampitt paper "Self-Similar Pitch Structures, Their Duals,
> and Rhythmic Analogues" has the proof of equivalence between Myhill's
> property and well formedness. What you probably want is to tie
> distributional evenness to well formedness, which may be Carey and
> Clampitt's "Aspects of Well-formed Scales". I haven't checked.

Does the definition of (non-degenerate) well formedness include that it is
an ME scale with possibly modified L/s ratio, and period = formal octave?

And does "tie to" mean to show that DE scales are exclusively repetitions of a scale pattern that is well formed / has Myhill's property?
If so, then those are the proofs I'm looking for (especially the second one).

- Gedankenwelt