Yahoo Tuning Groups Ultimate Backup tuning-math any ideas?

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> This is a question that keeps coming back and interesting me from
time
> to time, and it's one that I've never come up with a nice answer for
> either--maybe someone here can?
>
> If you take any random scale, what type of a method could be applied
> to the rotations so as to result in some pleasing "best" rotation?
>
> By "pleasing" I mean elegant in the math or aesthetic sense, and by
> "best" I could mean just about anything depending upon
how "pleasing"
> is defined!
>
> Anyway, any ideas?

I like to leave considerations of "rotation" and "choosing a tonic"
free to be informed by the particular style in which a scale is to be
used. For Indian-type styles, I tend to center around whatever is the
closest thing I can find to a central 1/1-3/2 dyad. But it varies.
Also, the question of "rotation" is somewhat ambiguous, as
distinct "octave species", "finalis", "dominant", and other aspects
of this question could be asked independently.

🔗D.Stearns <STEARNS@CAPECOD.NET>

Paul,

Well of course you're absolutely right about context, but I'm more
interested in a broad generalization here. For some reason this one
keeps coming back to haunt me, and uninvited visitors always get my
attention!

Anyway, I think you've got to look at this sort of a thing as an
interesting conundrum rather than a particularly utilitarian musical
question if you want to generalize to the degree that I'd hope to.
Because it nags at me, I have reason to believe it means some
something or the other that I'll find useful, but it doesn't
necessarily have to either... just think of it as a brainteaser.

Anybody have any ideas?

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Thursday, January 31, 2002 7:42 PM
Subject: [tuning-math] Re: any ideas?

> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> > This is a question that keeps coming back and interesting me from
> time
> > to time, and it's one that I've never come up with a nice answer
for
> > either--maybe someone here can?
> >
> > If you take any random scale, what type of a method could be
applied
> > to the rotations so as to result in some pleasing "best" rotation?
> >
> > By "pleasing" I mean elegant in the math or aesthetic sense, and
by
> > "best" I could mean just about anything depending upon
> how "pleasing"
> > is defined!
> >
> > Anyway, any ideas?
>
> I like to leave considerations of "rotation" and "choosing a tonic"
> free to be informed by the particular style in which a scale is to
be
> used. For Indian-type styles, I tend to center around whatever is
the
> closest thing I can find to a central 1/1-3/2 dyad. But it varies.
> Also, the question of "rotation" is somewhat ambiguous, as
> distinct "octave species", "finalis", "dominant", and other aspects
> of this question could be asked independently.
>
>
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>

🔗paulerlich <paul@stretch-music.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

Here's something along the lines of what I'm looking for:

All maximally even subsets favor palindromic symmetry. In fact, these
subsets are always the most palindromic or least skewed rotations or
modes--and if they're actual palindromes, then they're also unique, as
they have zero skew and therefore no inversion amongst their
rotations.

This is the sort of a special condition that I had in mind... though
I'd want it to work with any given scale, not just scales with
Myhill's property. Any ideas?

take care,

--Dan Stearns

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Thursday, January 31, 2002 9:42 PM
Subject: [tuning-math] Re: any ideas?

> --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Paul,
> >
> > Well of course you're absolutely right about context, but I'm more
> > interested in a broad generalization here.
>
> I like avoiding generalizations in this context. But in general, I'd
> get the 1/1-3/2 happening, and then if there were more than one
> possibility, I'd look for rare intervals to resplve to, and rare
step-
> sizes to act as signposts or leading-tone figurations toward, notes
> in the tonic triad (or its equivalent).
>
>
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>
>

🔗paulerlich <paul@stretch-music.com>

🔗jon wild <wild@fas.harvard.edu>

Hey Dan - I think that's an interesting question too. Why don't you list a
bunch of scales that each have one rotation you definitely prefer over
others? Then it might be possible for people to reverse-engineer some sort
of algorithm that produces those rotations, and only those rotations, from
the scales listed. Then if it also works on other scales, we've found a
winner.

I'd suggest listing each scale in a so-called "normal form", i.e. as
compact, then as left-packed as possible, and then give your favourite
rotation. For 12-tet diatonic, if you prefer Lydian, that'd be like this:

normal: 1221222
preferred: 2221221

Ok, this obviously only works for subsets of equal temperaments.

I don't know about the palindrome thing--do you prefer Dorian over the
other 12-tet diatonic modes? (re Gene's suggestion, Dorian is also the
only mode that has its centre of gravity at 600c)

In case a precise definition of "compact and left-packed" is needed, you
can use this: the most compact left-packed rotation of a scale is the one
whose representation as a binary number (lowest note as least-significant
digit) is smallest.

Illustration--as binary numbers:

Dorian = 11010101101 = 1709
Phrygian = 10110101011 = 1451
Lydian = 101011010101 = 2773
Mixolyd. = 11010110101 = 1717
Aeolian = 10110101101 = 1453
Locrian = 10101101011 = 1387
Ionian = 101010110101 = 2741

so we get the ordering:

Locrian, Phrygian, Aeolian, Dorian, Mixolydian, Ionian, Lydian.

(Note that adjacent rotations in the final ordering differ by a power of
two in each case, since they result from one bit-swap.)

And I'll just say again that this is not meant as a way of generating a
preference ordering, but instead a "normal" ordering that one could use as
a starting point for quantifying which rotations are preferred.

Finally, I personally think it unlikely that we will be able to come up
with a generally-applicable algorithm, even if we restrict it to the
preferences of just one person, Dan Stearns :-)

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Jon,

Thanks for the ideas, here's a couple more:

1) Most rooted--as in the rotation most closely resembling a power of
2 harmonic series. I guess this would only apply to RI scales. One way
to do this would be to simply take the LCM of the rotations'
denominators, and the one that best matches a power of 2 wins. In the
classic just and Pythagorean scales that would be the Lydian mode.

2) Another RI method would be to take the scale with the simplest
average--in other words the rotation with the smallest mean of all its
terms wins. In the classic just and Pythagorean scales that would be
the Ionian and Dorian modes respectively.

What I'd prefer is a built in condition like the ME, palindromic
uniqueness measure. Unfortunately I don't see a sensible way to
generalize that condition to any given scale rendered as cents.

Any ideas?

thanks,

--Dan Stearns

----- Original Message -----
From: "jon wild" <wild@fas.harvard.edu>
To: <tuning-math@yahoogroups.com>
Sent: Sunday, February 03, 2002 5:46 PM
Subject: [tuning-math] Re: any ideas?

>
> Hey Dan - I think that's an interesting question too. Why don't you
list a
> bunch of scales that each have one rotation you definitely prefer
over
> others? Then it might be possible for people to reverse-engineer
some sort
> of algorithm that produces those rotations, and only those
rotations, from
> the scales listed. Then if it also works on other scales, we've
found a
> winner.
>
> I'd suggest listing each scale in a so-called "normal form", i.e. as
> compact, then as left-packed as possible, and then give your
favourite
> rotation. For 12-tet diatonic, if you prefer Lydian, that'd be like
this:
>
> normal: 1221222
> preferred: 2221221
>
> Ok, this obviously only works for subsets of equal temperaments.
>
> I don't know about the palindrome thing--do you prefer Dorian over
the
> other 12-tet diatonic modes? (re Gene's suggestion, Dorian is also
the
> only mode that has its centre of gravity at 600c)
>
> In case a precise definition of "compact and left-packed" is needed,
you
> can use this: the most compact left-packed rotation of a scale is
the one
> whose representation as a binary number (lowest note as
least-significant
> digit) is smallest.
>
> Illustration--as binary numbers:
>
> Dorian = 11010101101 = 1709
> Phrygian = 10110101011 = 1451
> Lydian = 101011010101 = 2773
> Mixolyd. = 11010110101 = 1717
> Aeolian = 10110101101 = 1453
> Locrian = 10101101011 = 1387
> Ionian = 101010110101 = 2741
>
> so we get the ordering:
>
> Locrian, Phrygian, Aeolian, Dorian, Mixolydian, Ionian, Lydian.
>
> (Note that adjacent rotations in the final ordering differ by a
power of
> two in each case, since they result from one bit-swap.)
>
> And I'll just say again that this is not meant as a way of
generating a
> preference ordering, but instead a "normal" ordering that one could
use as
> a starting point for quantifying which rotations are preferred.
>
> Finally, I personally think it unlikely that we will be able to come
up
> with a generally-applicable algorithm, even if we restrict it to the
> preferences of just one person, Dan Stearns :-)
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
> Get your FREE credit report with a FREE CreditCheck
> Monitoring Service trial
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> --------------------------------------------------------------------
-~->
>
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>
>
>
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>
>

any ideas?

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗paulerlich <paul@stretch-music.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗paulerlich <paul@stretch-music.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗paulerlich <paul@stretch-music.com>

🔗jon wild <wild@fas.harvard.edu>

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗graham@microtonal.co.uk