Holy smokes. . .

>Some of my latest posts have been geared toward disproving _exactly_ this
>belief. For example, the natural resonances of just intervals lead one most
>naturally to what kind of pentatonic scale? A just one? No.

I would say this grossly depends on how the pentatonic scale is going to be
used.

-Carl

> OK -- with the most informationless prior about how it's going
> to be used, one finds that tempering it is best. Think of it as
> the problem of tuning a set of harmonic-series-timbre windchimes,
> or a keyboard that a monkey or serialist composer is going to
> play . . .

I don't see how the minimum-entropy scales are better than, say,
harmonics 5-10 under these circumstances.

-Carl

>>I don't see how the minimum-entropy scales are better than, say,
>>harmonics 5-10 under these circumstances.
>
>You may not have noticed the line
>
>0 268 498 703 885 40.473
>
>in my results, but that corresponds to the scale you're talking about. Would
>you like me to show, interval by interval, how this number is arrived at, as
>opposed to how the 39.506 of
>
>0 195 390 699 891 39.506
>
>is arrived at; or should we step outside this framework and talk about its
>validity?

Show first, please.

-Carl

>I'd like to develop a true odd-limit harmonic entropy measure, but with that
>the general pattern would be the same. Both lists start with two ratios of
>three. But then, my scale has one ratio of 3 for every ratio of 5 in your
>scale and one ratio of 5 for every ratio of 7 in your scale. Finally, both
>scales have 3 intervals that are dissonant in the 7-limit. So my scale
>clearly wins, and the small about of tempering required doesn't change that
>result.

Thanks- that's as I suspected. I have no problem with averaging the
inversions for _this_ result, but unless you specify that you intend the
scale to convey different fundamentals with mode changes (per my original
complaint), then my scale must have a lower overall harmonic entropy than
yours, since it's simply a saturated 9-limit otonal chord. I suggest this
would be reflected by a true higher-adic harmonic entropy measure.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

> Thanks- that's as I suspected. I have no problem with averaging the
> inversions for _this_ result, but unless you specify that you
intend the
> scale to convey different fundamentals with mode changes

How about just simply chord changes? I sure prefer chord changes that
imply different fundamentals rather than the same fundamental over
and
over again.

>(per my original
> complaint), then my scale must have a lower overall harmonic
entropy than
> yours, since it's simply a saturated 9-limit otonal chord. I
suggest this
> would be reflected by a true higher-adic harmonic entropy measure.

Undoubtedly. However, if you're playing the scale two notes at a
time,
the diadic measure I've been using is quite appropriate.

>>Thanks- that's as I suspected. I have no problem with averaging the
>>inversions for _this_ result, but unless you specify that you
>>intend the scale to convey different fundamentals with mode changes
>
>How about just simply chord changes?

That's (more or less) equivalent to what I said.

>I sure prefer chord changes that imply different fundamentals rather
>than the same fundamental over and over again.

Maybe I prefer it too, but the issue was saying so. I'm big on saying
so. Surely, lots of interesting stuff can happen by subsetting a series
over a fundamental. I've enjoyed this at various points in Carter
Scholz's work. Denny used to play the resonant filter on the ARP 2500.
Ozric uses the effect in many synth gurgles. If music is imitation-speech,
then perhaps this is the most basic type of music, but hardly the least
desirable.

>>(per my original complaint), then my scale must have a lower overall
>>harmonic entropy than yours, since it's simply a saturated 9-limit otonal
>>chord. I suggest this would be reflected by a true higher-adic harmonic
>>entropy measure.
>
>Undoubtedly. However, if you're playing the scale two notes at a
>time, the diadic measure I've been using is quite appropriate.

I guess that depends on how much stock we decide to place in "echo".
I'm open to suggestions, but your own examples have taught me, over
the years, a great respect for echo.

-Carl