Re: A Property of MOS/DE Scales

Paul wrote on tuning-list:

"sorry i got confused on this matter!

but the main thrust of my reply was to suggest to you that we have
more sophisticated tools at our disposal now than i did when i wrote
that paper. we should not be restricted to an ET conception. each
MOS/DE scale can be defined independently in terms of its own
generator, in cents; and period of repetition, in 1/octaves. by
restricting ourselves to ETs and their best approximations to
consonant intervals, we may miss some interesting and wonderful
possibilities.

it might be a good idea to follow up to tuning-math, where
the "searchers" may be more willing to publically help."

I am aware of these more sophisticated (and also more elegant) tools
and possibilities. So if anyone is interested please read the thread
in tuning list and continue on tuning-math.

Kalle

Kalle wrote...
>I am aware of these more sophisticated (and also more elegant)
>tools and possibilities. So if anyone is interested please read
>the thread in tuning list and continue on tuning-math.

Anybody else interested in reviving this thread?

Relevant messages...

/tuning/topicId_41347.html#41371
/tuning/topicId_41347.html#41383

/tuning-math/message/5132
/tuning-math/message/5136

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <ekin@l...>"
<ekin@l...> wrote:
> Kalle wrote...
> >I am aware of these more sophisticated (and also more elegant)
> >tools and possibilities. So if anyone is interested please read
> >the thread in tuning list and continue on tuning-math.
>
> Anybody else interested in reviving this thread?
>
> Relevant messages...
>
> /tuning/topicId_41347.html#41371
> /tuning/topicId_41347.html#41383
>
> /tuning-math/message/5132
> /tuning-math/message/5136
>
> -Carl

yes, i am very interested!!

>> Anybody else interested in reviving this thread?
>>
>> Relevant messages...
>>
>> /tuning/topicId_41347.html#41371
>> /tuning/topicId_41347.html#41383
>>
>> /tuning-math/message/5132
>> /tuning-math/message/5136
>>
>> -Carl
>
>yes, i am very interested!!

My interest regards a systematic search for scales with
between 5 and 10 notes, in which a single pattern of scale
degrees yields more than one consonant m-ad in a majority
of modes, where the m-ads are approximated at least as
well as the 5-limit in 12-tET (unweighted RMS).

How would one begin such a search? At the outermost level,
would we step through m? If so, I suggest 2 <= m <= 6.

Within a given m, we need to identify promising commas.
Probably cents/(n*d) is a good metric for that...

Beyond that, how do we set things up? Should we limit
ourselves to linear temperaments? What's the method for
finding which chords/scale pattern will be involved given
a set of commas and the commatic/chromatic designations?
Does this method work when three or four different m-ads
appear on the scale pattern (rather than just two)?

I'm willing to turn English into Java or Scheme...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Probably cents/(n*d) is a good metric for that...

why do you want cents to be high?

>> Probably cents/(n*d) is a good metric for that...
>
>why do you want cents to be high?

Whoops, that should be 1/(n*d)cents.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Probably cents/(n*d) is a good metric for that...
> >
> >why do you want cents to be high?
>
> Whoops, that should be 1/(n*d)cents.
>
> -C.

why do you want cents to be low?

>> Whoops, that should be 1/(n*d)cents.
>>
>> -C.
>
>why do you want cents to be low?

I want the simpler ratios in a given size range to
fare best.

I guess I should figure out if we want the scores
to be periodic as the interval size changes. The
mean complexity of smaller ratios is higher than
that of larger ratios (for ratios in lowest terms).
We'll have to adjust for that if we do want scores
to be comparable in different size ranges.

I count the number of ratios n/d in lowest terms,
1 < n/d < 2, n > d, for...

(n*d <= 100) = 21
(n*d <= 1000) = 210
(n*d <= 10000) = 2111
(n*d <= 100000) = 21069

For n*d <= 3000, there are 633 such ratios. They
are in this spreadsheet...

http://lumma.org/stuff/thinger.xls

...Excel's chart Wizard again completely confounds
me. I just want to see how (n*d)cents changes with
respect to cents. Will go to bed instead.

-Carl