Yahoo Tuning Groups Ultimate Backup tuning-math re: Tetrachordality and Scala

>>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

🔗Paul Erlich <paul@stretch-music.com>

🔗Carl Lumma <carl@lumma.org>

> 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

🔗Paul Erlich <paul@stretch-music.com>

--- 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.

🔗Carl Lumma <carl@lumma.org>

>> 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

🔗Paul Erlich <paul@stretch-music.com>

--- 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.

🔗Carl Lumma <carl@lumma.org>

>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

🔗genewardsmith@juno.com

🔗Carl Lumma <carl@lumma.org>

>> 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