an interesting generalization

for anyone who might be interested, several years ago i came up with a simple method to create harmonic series triadic identities and subsequent 2-D lattices given any, initiating and random 1-D generator.

Lately i was looking at it again, and noticed that it always gave results that were "proper" (according to Rothenberg, Balzano, et al).
Though i understand how it works very well, I'm not sure how to explain all this without confusion as i'm not mathematically versed.

However, should anybody be interested just ask for a random n-note scale, and i'll give you the generalizations resulting triad and scale starting at its simplest generator, as well as its possible temperaments. And as far as i can tell, the results always seem to be pretty interesting

Daniel,

I'm intrigued but I'm not sure I understand you correctly.

Are you saying you can produce all triad for say 17deo or... say all just
triad for say melodic minor 12 edo?

I'd be very interested in the 17 edo case.

Thanks,

Chris

On Mon, Dec 28, 2009 at 11:01 PM, daniel_anthony_stearns <
daniel_anthony_stearns@...> wrote:

>
>
> for anyone who might be interested, several years ago i came up with a
> simple method to create harmonic series triadic identities and subsequent
> 2-D lattices given any, initiating and random 1-D generator.
>
> Lately i was looking at it again, and noticed that it always gave results
> that were "proper" (according to Rothenberg, Balzano, et al).
> Though i understand how it works very well, I'm not sure how to explain all
> this without confusion as i'm not mathematically versed.
>
> However, should anybody be interested just ask for a random n-note scale,
> and i'll give you the generalizations resulting triad and scale starting at
> its simplest generator, as well as its possible temperaments. And as far as
> i can tell, the results always seem to be pretty interesting
>
>
>

no, what i'm getting at is more along the lines of something like this, "what's the simplest harmonic series segment triad that can make a 5-note scale?"

in that case it would be a 6:7:8 "triad" and using the "syntonic diatonic" template, a 5-note scale of

0 267 498 702 969 1200
0 231 435 702 933 1200
0 204 471 702 969 1200
0 267 498 765 996 1200
0 231 498 729 933 1200
this in turn could be seen as a scale consisting of 1 small step, 2 medium steps, and 2 large steps, thereby giving its simplest equal temperament rendition as a 5-out of-9 out of-16 5-note scale in 16-tone ET:

0 300 525 675 975 1200
0 225 375 675 900 1200
0 150 450 675 975 1200
0 300 525 825 1050 1200
0 225 525 750 900 1200

using the single generator, "Pythagorean" model would give:
0 294 498 702 996 1200
0 204 408 702 906 1200
0 204 498 702 996 1200
0 294 498 792 996 1200
0 204 498 702 906 1200

and 5-note scale of 3 small steps and 2 large steps which would find its simplest EDO, tempered expression in 7-tone ET:

0 343 514 686 1029 1200
0 171 343 686 857 1200
0 171 514 686 1029 1200
0 343 514 857 1029 1200
0 171 514 686 857 1200

using a "meantone" model, and dispersing the comma to make the "third" of the "triad" pure would give:

0 267 489 711 978 1200
0 222 444 711 933 1200
0 222 489 711 978 1200
0 267 489 756 978 1200
0 222 489 711 933 1200

In any event, this works for any given n-note scale, and seems to always give proper scales as well.

You can also use the generalization algorithm to give something like a nominal triad for any given 1-d generator. As an example, if you have 3/2 as your generator you then have a 4:5:6 triad, but if you generalize this and suppose the generator is a 11/6 in place of the 3/2, you'd get a 12:17:22 "triad" (and a nice 7-note scale, as well).

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Daniel,
>
> I'm intrigued but I'm not sure I understand you correctly.
>
> Are you saying you can produce all triad for say 17deo or... say all just
> triad for say melodic minor 12 edo?
>
> I'd be very interested in the 17 edo case.
>
> Thanks,
>
> Chris
>
> On Mon, Dec 28, 2009 at 11:01 PM, daniel_anthony_stearns <
> daniel_anthony_stearns@...> wrote:
>
> >
> >
> > for anyone who might be interested, several years ago i came up with a
> > simple method to create harmonic series triadic identities and subsequent
> > 2-D lattices given any, initiating and random 1-D generator.
> >
> > Lately i was looking at it again, and noticed that it always gave results
> > that were "proper" (according to Rothenberg, Balzano, et al).
> > Though i understand how it works very well, I'm not sure how to explain all
> > this without confusion as i'm not mathematically versed.
> >
> > However, should anybody be interested just ask for a random n-note scale,
> > and i'll give you the generalizations resulting triad and scale starting at
> > its simplest generator, as well as its possible temperaments. And as far as
> > i can tell, the results always seem to be pretty interesting
> >
> >
> >
>