scala stability logic

Manuel,

Any reason you don't display Rothenberg stability for improper
scales?

-Carl

Carl wrote:

>Any reason you don't display Rothenberg stability for improper
>scales?

None that I remember, I must have assumed it's not defined for
improper scales, but it might well be. I don't have the paper
at hand, do you think he didn't make that requirement?

Manuel

>None that I remember, I must have assumed it's not defined for
>improper scales, but it might well be. I don't have the paper
>at hand, do you think he didn't make that requirement?

I seem to think he didn't, though I don't have the paper handy
either.

And how are you deciding when to say "ambiguous key"? As soon
as all possible keys are not distinct? For example, the
wholetone scale has 1 group of keys, the octatonic scale has 2,
and the diatonic 7. How are you calculating efficiency in
these cases?

-Carl

>And how are you deciding when to say "ambiguous key"? As soon
>as all possible keys are not distinct?

Yes.

>For example, the
>wholetone scale has 1 group of keys, the octatonic scale has 2,
>and the diatonic 7.

I determine that they have 6, 4 and 1 repeating blocks. So if
that's more than one I add the text "ambiguous key".

> How are you calculating efficiency in these cases?

Strictly by R.'s definition.

Manuel

In-Reply-To: <OFF5004A07.0C575491-ONC1256B9E.002AB5D9@office.novaterra.nl>
Manuel wrote:

> Carl wrote:
>
> >Any reason you don't display Rothenberg stability for improper
> >scales?
>
> None that I remember, I must have assumed it's not defined for
> improper scales, but it might well be. I don't have the paper
> at hand, do you think he didn't make that requirement?

D. Rothenberg, "A Model for Pattern Perception with Musical Applications
Part II: The Information Content of Pitch Structures" Math. Systems
Theory, 1978, p.356 "Note that stability only applies to proper scales."

Carl agreed with this a few days ago.

Graham

Almost 11 hours without appearing... -C.

>>And how are you deciding when to say "ambiguous key"? As soon
>>as all possible keys are not distinct?
>
>Yes.
>
>>For example, the wholetone scale has 1 group of keys, the octatonic
>>scale has 2, and the diatonic 7.
>
>I determine that they have 6, 4 and 1 repeating blocks. So if
>that's more than one I add the text "ambiguous key".
>
>>How are you calculating efficiency in these cases?
>
>Strictly by R.'s definition.

Excellent.

>D. Rothenberg, "A Model for Pattern Perception with Musical Applications
>Part II: The Information Content of Pitch Structures" Math. Systems
>Theory, 1978, p.356 "Note that stability only applies to proper scales."

Rats! Looks like Lumma stability is all we have for things like the
Pythagorean diatonic, then.

-Carl

In-Reply-To: <4.2.2.20020417193556.01ed2d20@lumma.org>
Carl Lumma wrote:

> Rats! Looks like Lumma stability is all we have for things like the
> Pythagorean diatonic, then.

No, the Pythagorean diatonic works fine as a proper subset of a 12 note
chromatic. See 13. Distinct Scales and Mistunings on p.365, "That there
is only a finite number of equivalence classes of any cardinality is
musically significant in that only finitely many significantly differing
musical scales may be constructed. Also note that, if a scale is
conceived of as an ordering, the use of finer tunings and smaller
intervals does not necessarily produce new scales." He does also say the
tuning should initially preserve propriety, but the 12 note scale should
be familiar enough to most listeners.

For Rothenberg stability and efficiency to work properly, you need to
define each diatonic on a chromatic. I suggest the chromatic should
always be strictly proper, and have high Lumma stability. There should
also be a maximum number of notes allowed in a chromatic, somewhere below
53.

Graham

>> Rats! Looks like Lumma stability is all we have for things like the
>> Pythagorean diatonic, then.
>
>No, the Pythagorean diatonic works fine as a proper subset

That's funny, since it is improper.

>Distinct Scales and Mistunings on p.365, "That there is only a finite
>number of equivalence classes of any cardinality is musically
>significant in that only finitely many significantly differing musical
>scales may be constructed.

Right.

>Also note that, if a scale is conceived of as an ordering, the use of
>finer tunings and smaller intervals does not necessarily produce new
>scales."

Right.

>For Rothenberg stability and efficiency to work properly, you need to
>define each diatonic on a chromatic.

? What terminology is this?

-Carl

--- In tuning-math@y..., graham@m... wrote:

> For Rothenberg stability and efficiency to work properly, you need
to
> define each diatonic on a chromatic. I suggest the chromatic
should
> always be strictly proper, and have high Lumma stability. There
should
> also be a maximum number of notes allowed in a chromatic, somewhere
below
> 53.

i, for one, am opposed to defining an "extra-sensory" chromatic, as i
complained before in reference to balzano and clough. i'm glad to
hear mark gould is with me on this -- sorry if i implied otherwise,
mark.

In-Reply-To: <a9n8mm+abjt@eGroups.com>
emotionaljourney22 wrote:

> i, for one, am opposed to defining an "extra-sensory" chromatic, as i
> complained before in reference to balzano and clough. i'm glad to
> hear mark gould is with me on this -- sorry if i implied otherwise,
> mark.

I'm not suggesting anything "extra-sensory" either. If your target
listener senses the 612 cent interval as belonging to a distinct interval
class, rather than being a tuning of a "tritone" of around 600 cents,
we'll consider the Pythagorean diatonic independent of the 12 note
chromatic. If they can hear a 612 cent interval as being wider than a 588
cent interval, when they're at unrelated pitches, we can even call the
Pythagorean diatonic improper.

Graham

In-Reply-To: <4.2.2.20020418114756.01ece7b0@lumma.org>
Carl Lumma wrote:

> >For Rothenberg stability and efficiency to work properly, you need to
> >define each diatonic on a chromatic.
>
> ? What terminology is this?

Mark Gould defines diatonics, pentatonics and chromatics. I'm removing
the distinction between diatonics and pentatonics. Treating the chromatic
as an equal temperament, even if it's tuned differently, is needed for
Rothenberg's conclusions about the ambiguity of the tritone to be valid in
meantone.

Graham