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