Efficicency and ambiguity

If I'm understanding Eytan Agmon's paper "Numbers and the Western
Tone-System", he is interested in MOS in an equal temperament which
are efficent and have at most one ambiguous interval. It seems to me that
a 7 or 8 note MOS in Blackwood/15 (with 2/15 as a generator) or a 9 or
10 note MOS of Negri/19 (with a 2/19 generator) would qualify. Eytan
claims to have a proof this is not so, but I don't see how he can in
the face of such examples. Is there anyone following this business who
thinks they could explain this?

I suppose I should look the proof up before the library here closes
for moving.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> If I'm understanding Eytan Agmon's paper "Numbers and the Western
> Tone-System", he is interested in MOS in an equal temperament which
> are efficent and have at most one ambiguous interval. It seems to
me that
> a 7 or 8 note MOS in Blackwood/15 (with 2/15 as a generator) or a 9
or
> 10 note MOS of Negri/19 (with a 2/19 generator) would qualify.

Here are more examples:

15 or 16 notes with a 2/31 generator (Quartaminorthirds/31)

26 or 27 notes with a 2/53 generator

49 or 50 notes with a 2/99 generator

But 2/odd is not the only possibility:

20 or 21 notes of Miracle/41 (4/41 generator.)

51 or 52 notes of Miracle/103

Then there are some nice ones fitting Eytan's conditions:

35 or 37 notes with 35/72 generator

41 or 43 notes with 41/84 generator

>If I'm understanding Eytan Agmon's paper "Numbers and the Western
>Tone-System", he is interested in MOS in an equal temperament which
>are efficent and have at most one ambiguous interval.

Not at most, exactly one. So your examples don't qualify.

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
> >If I'm understanding Eytan Agmon's paper "Numbers and the Western
> >Tone-System", he is interested in MOS in an equal temperament which
> >are efficent and have at most one ambiguous interval.
>
> Not at most, exactly one. So your examples don't qualify.
>
> Manuel

why on earth should there be exactly one ambiguous interval? it seems
to me that musical academia has been staring for too long at its 7-
out-of-12-equal navel.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:

> why on earth should there be exactly one ambiguous interval? it seems
> to me that musical academia has been staring for too long at its 7-
> out-of-12-equal navel.

Good question. However, as we've seen, Eytan actually exterminates
these with an extra assumption that the generator is to be an
approximate fifth. If we solve 1/2+1/(4*n) = log2(3/2) for n,
we get n = 2.94; the nearest interger solution and clearly by far the
best is n = 3, leading to the 12-et. Needless to say, I am not
convinced by all these assumptions, and wish Eytan had presented this
as "here are some interesting conditions on scales, which lead to 12
as their solution".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
> > <manuel.op.de.coul@e...> wrote:
>
> > why on earth should there be exactly one ambiguous interval? it
seems
> > to me that musical academia has been staring for too long at its
7-
> > out-of-12-equal navel.
>
> Good question. However, as we've seen, Eytan actually exterminates
> these

these? what are these?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> > Good question. However, as we've seen, Eytan actually exterminates
> > these
>
> these? what are these?

MOS of size (n+1)/2 within an n-et, such that n is odd and 2/n is the
generator.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> > Good question. However, as we've seen, Eytan actually exterminates
> > these
>
> these? what are these?

Systems with an odd et n and 2/n as generator.

>> why on earth should there be exactly one ambiguous interval? it seems
>> to me that musical academia has been staring for too long at its 7-
>> out-of-12-equal navel.

>> these? what are these?

>MOS of size (n+1)/2 within an n-et, such that n is odd and 2/n is the
>generator.

Yeah, those have no ambiguous interval, leaving only the even n with
exactly one ambiguous interval.

Manuel