Phi, Fractions & Musical Boundaries

I'm going a little bit crazy here, I think, and I need you guys to tell me if I'm on a wild goose chase.

Margo's paper about interval classes got me thinking about phi and noble mediants, and how the noble mediants between two simple-integer ratios (i.e. where n*d</=70) seemed to correlate to maxima of harmonic entropy. Then I remembered that Erv Wilson seemed to have thing for phi as well, and when I pulled out his diagram of the scale tree, I realized that he had labelled the noble mediants as well! Then it hit me that the scale tree uses fractions just like JI, only instead of representing frequency ratios, they represent fractions of the octave.

So, I've been wracking my brains trying to figure out how the noble mediant between two branches on the scale tree has an analogous meaning to the noble mediant between two simple JI ratios, and I can't quite figure it out. I've realized that when you do a series of classical mediants, they oscillate around the noble mediant, approaching it but never reaching it; for instance:
CM of 2\5 and 3\7 is 5\12; 5\12 and 3\7 is 8\19; 5\12 and 8\19 is 13\31; 8\19 and 13\31 is 21\50. These oscillate around (2+3(phi))\(5+7(phi)).

The thing that I can't figure out is what these noble scales have to do with MOS families. If anything, these noble scales seem to suggest minima of "melodic entropy" (if such a concept makes any sense); what I mean is, at each node of the scale tree, you have an EDO, and the lower the EDO, the more enharmonic equivalents there are. For instance, in a 19-EDO diatonic scale, augmented seconds are distinct from minor thirds, but in 12-EDO, they are the same. In 12-EDO, major and minor thirds are distinct, but in 7-EDO, they are the same. At the noble scale, though, there will never be any enharmonic equivalents, no matter how far out you extend the circle of fifths. So I use the concept of "melodic entropy" to refer to the possible identities of a given note. 7-EDO would have high melodic entropy, because a given note represents the intersection of a whole lot of interval classes, whereas in 19-EDO, a given note represents the intersection of substantially fewer. At the noble scale, a given note does not represent the intersection of ANY interval classes.

In harmonic entropy, the noble mediants seem to be equally far from the "fields of attraction" of various intervals, making them high in harmonic entropy. So just what the heck is going on here? I feel like there is something very obvious and very relevant staring me in the face, but I just can't see it.

Am I nuts? Is there some connection here, or have I just been spending too much time thinking about numbers and patterns?

-Igs

cityoftheasleep wrote:

> Then it hit me that the scale tree uses fractions just like JI,
> only instead of representing frequency ratios, they represent
> fractions of the octave.

A coincidence indeed. The scale tree, when viewed instead as a
dyad tree, looks very nearly like an upside-down harmonic entropy
curve.

http://yhoo.it/chvSLU

Note the minima are mostly or totally related by 'freshman sums'.

-Carl

The scale tree was meant to be used in both ways harmonically and as log -,ET,
The latter becomes more important when using it to map keyboards.

He added on to the Stern Brocot ( well we didn't know about this till the late 90's at the earliest) the noble numbers of the series up till the level he worked out.
Of course it is implied there are more.
The thing is when you get to these places another harmonic coincidence often kick in and makes it all hold together which is troublesome for HE.

He objects to Brocot name being included though as he stated he could not find in the work sited an actual description of this tree.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> cityoftheasleep wrote:
>
> > Then it hit me that the scale tree uses fractions just like JI,
> > only instead of representing frequency ratios, they represent
> > fractions of the octave.
>
> A coincidence indeed. The scale tree, when viewed instead as a
> dyad tree, looks very nearly like an upside-down harmonic entropy
> curve.
>
> http://yhoo.it/chvSLU
>
> Note the minima are mostly or totally related by 'freshman sums'.
>
> -Carl
>

Erv has no theory of consonance and dissonance. He leaves that choice up to the user.
And who knows what tomorrow will bring?
He finds that both can hold together either simple ratios or summation triplets

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I'm going a little bit crazy here, I think, and I need you guys to tell me if I'm on a wild goose chase.
>
> Margo's paper about interval classes got me thinking about phi and noble mediants, and how the noble mediants between two simple-integer ratios (i.e. where n*d</=70) seemed to correlate to maxima of harmonic entropy. Then I remembered that Erv Wilson seemed to have thing for phi as well, and when I pulled out his diagram of the scale tree, I realized that he had labelled the noble mediants as well! Then it hit me that the scale tree uses fractions just like JI, only instead of representing frequency ratios, they represent fractions of the octave.
>
> So, I've been wracking my brains trying to figure out how the noble mediant between two branches on the scale tree has an analogous meaning to the noble mediant between two simple JI ratios, and I can't quite figure it out. I've realized that when you do a series of classical mediants, they oscillate around the noble mediant, approaching it but never reaching it; for instance:
> CM of 2\5 and 3\7 is 5\12; 5\12 and 3\7 is 8\19; 5\12 and 8\19 is 13\31; 8\19 and 13\31 is 21\50. These oscillate around (2+3(phi))\(5+7(phi)).
>
> The thing that I can't figure out is what these noble scales have to do with MOS families. If anything, these noble scales seem to suggest minima of "melodic entropy" (if such a concept makes any sense); what I mean is, at each node of the scale tree, you have an EDO, and the lower the EDO, the more enharmonic equivalents there are. For instance, in a 19-EDO diatonic scale, augmented seconds are distinct from minor thirds, but in 12-EDO, they are the same. In 12-EDO, major and minor thirds are distinct, but in 7-EDO, they are the same. At the noble scale, though, there will never be any enharmonic equivalents, no matter how far out you extend the circle of fifths. So I use the concept of "melodic entropy" to refer to the possible identities of a given note. 7-EDO would have high melodic entropy, because a given note represents the intersection of a whole lot of interval classes, whereas in 19-EDO, a given note represents the intersection of substantially fewer. At the noble scale, a given note does not represent the intersection of ANY interval classes.
>
> In harmonic entropy, the noble mediants seem to be equally far from the "fields of attraction" of various intervals, making them high in harmonic entropy. So just what the heck is going on here? I feel like there is something very obvious and very relevant staring me in the face, but I just can't see it.
>
> Am I nuts? Is there some connection here, or have I just been spending too much time thinking about numbers and patterns?
>
> -Igs
>