re : Optimal mixolydian and Dorian

> Subject: Re: Optimal Tuning of the Mixolydian and Dorian Modes
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > > Hello!
> > >
> > > I'd like to thank everyone who replied! Let me explain more
> clearly
> > > what I have in mind:
> > >
> > > The tuning would be a chain of seven fifths i.e. some form of
> > > meantone. The dominant seventh chord on the mixolydian tonic
> would
> > > have to approximate 4:5:6:7 and the minor sixth chord on the
> dorian
> > > tonic would have to approximate its utonal counterpart in root
> > > position, that is 1/1:6/5:3/2:12/7 (although 16:19:24:27 might be
> > > interesting too). I'd think maximizing consonance for these with
> > > something like minimized unweighted RMS-errors or minimax would
> > make
> > > the fifth much wider than meantone fifth, something close to
> > > pythagorean (around 702 or 703 cents). But that makes major
> thirds
> > > wide and I think these modes still sound better in 12-equal. Why
> > > won't the optimal tuning sound better than 12-equal?
> >
> > I calculated an rms-optimum at 701.797 cents for 4:5:6:7 and
> > 16:19:24:27. I'm not sure if this is correct but the tetrads sound
> a
> > bit better than in 12-equal. But the thirds sound worse in
> isolation.
>
> Then I got 702.226 cents for 7-limit consonances only. The minor
> sixth chord sounds pretty much similar to the above.

Well, you realize that the basic problem is that two of the things you
are trying to optimize are at complete cross purposes. A fifth of 715c
satisfies the seventh, and 696c satisfies the third.

Do your solutions to you really sound like a 4:5:6:7 chord? In my opinion
12et C:E:G:Bb desn't sound like a 4:5:6:7 chord, and your solutions are
very close to 12et.

With some enharmonic magic you can get closer in a single chain of fifths.
Re-spell mixolydian as C D E E# G A A# C (from the contiguous chain
C G D A E B F# C# G# D# A# E#) and a typical meantone tuning (696.7c
looked good) will make a much better 4:5:6:7 chord. For what its worth,
you also get a 4:5:6:7 on the V chord.

But the VI chord is hosed.

And the Dorian is 1/1 9/8 7/6 4/3 3/2 27/16 7/4, awesome, but maybe not
what you were looking for.

Bob

--- In tuning@yahoogroups.com, Bob Valentine <BVAL@I...> wrote:

> Well, you realize that the basic problem is that two of the things
you
> are trying to optimize are at complete cross purposes. A fifth of
715c
> satisfies the seventh, and 696c satisfies the third.

True but that's what is so interesting in this case! And isn't
temperament mostly about making a compromise between opposing forces?

> Do your solutions to you really sound like a 4:5:6:7 chord? In my
opinion
> 12et C:E:G:Bb desn't sound like a 4:5:6:7 chord, and your solutions
are
> very close to 12et.

They sounds like non-JI 4:5:6:7s and a bit more "locked" than 12-
equal. I compared a just 4:5:6:7 chord with its best approximation in
12-equal and those two chords definitely have a perceptual
similarity. The only difference is the phasing, beating and whooshing
of the 12-equal. I used static sawtooth waveforms.

Does this mean that you don't hear 12-equal mixolydian scale the way
Paul Erlich argues it is heard? What about the dorian?

Kalle

Kalle,

IMHO, there's much a difference between 4:5:6:7 and 12-eq Dominant
7th chord.

The latter is quite unstable (really UNSTABLE), and useless as a
resting sound, although appropiate for a drive into the tonic.

The JI-tetrad, well... I could hear it for days without needing a
resolution onto I. :D

Max.

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

...
> They sounds like non-JI 4:5:6:7s and a bit more "locked" than 12-
> equal. I compared a just 4:5:6:7 chord with its best approximation
in
> 12-equal and those two chords definitely have a perceptual
> similarity. The only difference is the phasing, beating and
whooshing
> of the 12-equal. I used static sawtooth waveforms.
>
...
> Kalle

--- In tuning@yahoogroups.com, giordanobruno76@y... wrote:
> Kalle,
>
> IMHO, there's much a difference between 4:5:6:7 and 12-eq Dominant
> 7th chord.

Hi, Max.

I recognize their difference but I also recognize their similarity.

> The latter is quite unstable (really UNSTABLE), and useless as a
> resting sound, although appropiate for a drive into the tonic.

Yes, that is how it works in the major scale but I was talking about
the mixolydian mode. I don't advocate the use of the rough 12-equal 7-
limit tetrad approximation as a tonic chord. The mixolydian tonic
chord would be the tonic triad but the seventh might be heard as
moderately consonant when played against it. Please read my reply to
Bob.

> The JI-tetrad, well... I could hear it for days without needing a
> resolution onto I. :D

It doesn't need a resolution. So there is no I. <-- Wow, that sounded
deep! :)

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> I compared a just 4:5:6:7 chord with its best approximation in
> 12-equal and those two chords definitely have a perceptual
> similarity. The only difference is the phasing, beating and
whooshing
> of the 12-equal. I used static sawtooth waveforms.

Different waveforms, and especially different musical contexts, may
elicit different evaluations than this.

> Does this mean that you don't hear 12-equal mixolydian scale the
way
> Paul Erlich argues it is heard? What about the dorian?

I use the word "possibly" in my paper to purposely leave the question
open, rather than to "argue it is heard" one way or the other.
However, I must say your later points about "jamming" were very well
taken, being a big fan and practitioner of "jam" music myself.