>>This is exactly what we want. Except, instead of comparing

>>the scale and its rotation to a degree near a 3:2, we compare

>>it with its transposition by an exact 3:2, simply taking the

>>lowest value for all rotations of the transposed version,

>>using a simple ordered pairing of intervals in each case.

>

>Can _you_ illustrate with an example?

Maybe I shouldn't have said exactly. Since we're dealing with

transposition by a perfect 3:2 we can't compare 2nds because

they're the same. However, since I can losslessly convert a list

of 2nds into a list of "pitches (intervals measured from a 1/1)

and back again, I assume I can plug a list of "pitches" (1/1 9/8

5/4 etc.) into the formula and preserve the equivalence to the

list of intervals in the set of all subsets of the scale that we

agreed on...

Pentatonic scale in 12-tET:

scale- 0 200 500 700 900, sum of squares= 1,590,000

*(3:2), mod (2:1)- 702 902 2 202 402, corr= 0.430566

mode 2- 902 2 202 402 702, corr= 0.638113

mode 3- 2 202 402 702 902, corr= 0.971446

mode 4- 202 402 702 902 2, corr= 0.669559

mode 5- 402 702 902 2 202, corr= 0.487169

tetrachordality index= 0.971446

...Using pitches may goof up the formula, though, since lists of

2nds must sum to an octave while lists of "pitches" do not. I

don't know enough about how the correlation formula works

statistically to tell if this is a problem. I roughly remember

using a "correlation" in Physics class to measure the fit of a

linear function to a set of data... finding the best fit... but

I don't remember the math.

I am sure the first pitch = zero causes a problem...

6-tET:

scale- 0 200 400 600 800 1000, sum of squares= 2,200,000

*(3:2), mod (2:1)- 702 902 1102 102 302 502, corr= 0.648181

mode 2- 902 1102 102 302 502 702, corr=

mode 3- 1102 102 302 502 702 902, corr=

mode 4- 102 302 502 702 902 1102, corr= 1.139090

mode 5- 302 502 702 902 1102 102, corr= 0.866363 -- d'oh!

mode 6- 502 702 902 1102 102 302, corr=

...Here mode 5 is closer to 1 than mode 4. There's probably a

way around this, but maybe we don't need correlation at all --

just the mean difference between the pitches in the best-fit mode

of the transposed scale from the pitches in the original scale.

It might also be argued that we don't want to enforce order of

the pitches during comparison. Rather than testing each

transposed mode for the best fit, we could just line up the new

pitches with their nearest partners in the original scale, in any

order. I suspect this would break our agreed-upon equivalence

with the intervals of all subets. But do we care about an

interval pattern or pitches, as far as the perceptual basis for

this measure? You there, Paul?

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> But do we care about an

> interval pattern or pitches, as far as the perceptual basis for

> this measure? You there, Paul?

>

I'm afraid I've lost you on this. I defined omnitetrachordality on

the tuning list, and Gene said he was going to come up with some

theorems about it . . .

> I'm afraid I've lost you on this. I defined omnitetrachordality on

> the tuning list, and Gene said he was going to come up with some

> theorems about it . . .

Do we want to enforce scale order on the transposed pitches, or

just let them fall where they are closest to the original pitches?

Earlier in this thread, I claimed we don't care about scale order,

and I still claim it. But I was enforcing order in the example

in my last mail.

I don't like your definition of omnitetrachordality because:

() I can't tell if it's equivalent to What We Want^TM. That is,

I can't tell what sort of semi-periodicity it enforces. This

may be due to my utter lack of tools to eval. periodicity. A

request for which spun this thread.

() I doesn't punish approximate fifths -- only admits or rejects.

-Carl

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> > I'm afraid I've lost you on this. I defined omnitetrachordality

on

> > the tuning list, and Gene said he was going to come up with some

> > theorems about it . . .

>

> Do we want to enforce scale order on the transposed pitches, or

> just let them fall where they are closest to the original pitches?

Don't know what you mean by "enforce scale order".

> Earlier in this thread, I claimed we don't care about scale order,

> and I still claim it. But I was enforcing order in the example

> in my last mail.

>

> I don't like your definition of omnitetrachordality because:

>

> () I can't tell if it's equivalent to What We Want^TM. That is,

> I can't tell what sort of semi-periodicity it enforces. This

> may be due to my utter lack of tools to eval. periodicity. A

> request for which spun this thread.

Hmmm . . .

>

> () I doesn't punish approximate fifths -- only admits or rejects.

>

Same here.

>> Do we want to enforce scale order on the transposed pitches, or

>> just let them fall where they are closest to the original pitches?

>

> Don't know what you mean by "enforce scale order".

() Take the scale

() Transpose it by 3:2.

() Now compare the pitches of the two scales.

That's where we left off. We were counting the number of changing

pitches. But what if they all change slightly. I suggest comparing

them statistically in log-freq. space. Maybe just mean-deviation is

okay here.

The question is, when comparing the scales, do we insist on lining

up the pitches in order, for some rotation of the transposed scale.

Or do allow a re-ordering of pitches to get the closest matches.

For CS scales with a 3:2 in them, in order will always be the best

order, I think.

>> () I doesn't punish approximate fifths -- only admits or rejects.

>>

> Same here.

That was supposed to be "it doesn't", referring to your measure.

As you can see above, I want to punish approximate fifths, or at

least formalize what we will allow as an approximate fifth.

-Carl

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> >> Do we want to enforce scale order on the transposed pitches, or

> >> just let them fall where they are closest to the original

pitches?

> >

> > Don't know what you mean by "enforce scale order".

>

> () Take the scale

> () Transpose it by 3:2.

> () Now compare the pitches of the two scales.

>

> That's where we left off. We were counting the number of changing

> pitches. But what if they all change slightly. I suggest comparing

> them statistically in log-freq. space. Maybe just mean-deviation is

> okay here.

>

> The question is, when comparing the scales, do we insist on lining

> up the pitches in order, for some rotation of the transposed scale.

> Or do allow a re-ordering of pitches to get the closest matches.

> For CS scales with a 3:2 in them, in order will always be the best

> order, I think.

>

> >> () I doesn't punish approximate fifths -- only admits or rejects.

> >>

> > Same here.

>

> That was supposed to be "it doesn't", referring to your measure.

> As you can see above, I want to punish approximate fifths, or at

> least formalize what we will allow as an approximate fifth.

>

> -Carl

How about a rule of thumb. 1.2% is what I'm currently using for both

harmonic entropy and for these kinds of judgments. That's 20.775¢, or

about 1 step in 58-tET.

>How about a rule of thumb. 1.2% is what I'm currently using for

>both harmonic entropy and for these kinds of judgments. That's

>20.775¢, or about 1 step in 58-tET.

I'm perfectly happy to use a binary function like this, because

badness doesn't continue to increase smoothly as you deviate

from the 3:2 anyway -- you hit a wall before going towards 7:5,

or whatever.

How does it fit into the rest of the procedure, though? For

cleanness I still like multiplying by a perfect 3:2 and seeing

how much stuff changes.

-Carl

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I'm afraid I've lost you on this. I defined omnitetrachordality on

> the tuning list, and Gene said he was going to come up with some

> theorems about it . . .

I asked if it would be a good idea, since I think it should be

possible. I need to take projects a few at a time, though.

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> That's where we left off. We were counting the number of changing

> pitches. But what if they all change slightly. I suggest comparing

> them statistically in log-freq. space. Maybe just mean-deviation is

> okay here.

I thought you'd rejected the idea of caring about the sizes of

intervals?

>> That's where we left off. We were counting the number of changing

>> pitches. But what if they all change slightly. I suggest

>> comparing them statistically in log-freq. space. Maybe just mean-

>> deviation is okay here.

>

> I thought you'd rejected the idea of caring about the sizes of

> intervals?

While I'm not sure which intervals you're referring to here, it's

safe to say I haven't rejected anything yet. :)

What I care about is modeling an effect that Paul cooked up way

long ago -- listeners use the 3:2 to chunk together pitches they

here, much the same way they use the 2:1 to do this.

So what I proposed in the mail you're replying to, was to transpose

the scale by a 3:2, and call that the thing the listener has in

his/her brain when hearing the original scale. Then ask how well

the original scale fits this mental space.

The only things I'm not sure of are:

1. We don't want to acknowledge the existence of the "scale"

as such. We want the measure to represent the average sequence

of music made with the scale. But for ease of calculation, this

won't due. I'm reasonably convinced that the entire scale

represents the set of all subsets of itself fairly well for most

purposes, but I really have no idea.

2. What does it mean to allow or forbid a re-arrangement of the

transposed scale to best fit the original scale? For example,

transposing the diatonic scale by a tritone, the 4th and 7th

degrees of the original scale will appear in the transposed

scale, _but in no mode of the transposed scale will these

pitches be a 4th and a 7th_. Should we just do a 'picture

pages' match-up? My feeling is yes, we should, since our

perceptual model (equivalence at the 3:2) doesn't assume our

listener will be keeping track of scale degrees. But I'm open

to suggestions here.

Paul would like to forget these details, and transpose by

an interval that appears in the scale. But Paul, how does

this work if the scale contains more than one size of

acceptable 3:2? And this measure isn't defined on scales

without an acceptable 3:2 (granted such scales will all be

"bad" on my measure, anyway).

Does this help clear anything up for you, Gene?

-Carl