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?

take care,

--Dan Stearns

--- 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.

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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--- 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).

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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>

>

> Your use of Yahoo! Groups is subject to

http://docs.yahoo.com/info/terms/

>

>

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> 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?

how about the rotation where the scale's center of gravity is

closest to 600 cents?

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 :-)

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 :-)

>

>

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>

>

In-Reply-To: <002601c1af92$f08c5820$a061d63f@stearns>

D.Stearns wrote:

> 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.

In fact, I have little script that tries to follow Terhardt's formula for

finding the root of a chord. That might do the trick. I'll dig it out

sometime.

Graham