Posting an open question...

The normal diatonic scale is said to have octave equivalence.

Which means, you can start anywhere in the scale, move an

octave, and you won't know that anything happened.

The diatonic scale is semi-periodic at the 3:2. For moves of a

single 3:2, very little changes (say, from LLsLLLs to LLsLLsL).

How can we quantify this?

-Carl

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

> Posting an open question...

>

> The normal diatonic scale is said to have octave equivalence.

> Which means, you can start anywhere in the scale, move an

> octave, and you won't know that anything happened.

>

> The diatonic scale is semi-periodic at the 3:2. For moves of a

> single 3:2, very little changes (say, from LLsLLLs to LLsLLsL).

>

> How can we quantify this?

>

> -Carl

Hi Carl . . .

Only an MOS scale generated by approximate 3:2's will have this

property (only one note changes with transpositions by a 3:2). Bigger

examples include 19-out-of-31 and 29-out-of-41.

My omnitetrachordality property is closely related. If

omnitetrachordality holds, moves of a 3:2 will only change the notes

within a single 9:8 span.

>Only an MOS scale generated by approximate 3:2's will have this

>property (only one note changes with transpositions by a 3:2).

Yes but I'm interested in a general measure for any scale. I'm

thinking of a function which can take a scale and an interval as

an input, and spit out the "extent" of the scale's symmetry at

that interval.

>My omnitetrachordality property is closely related. If

>omnitetrachordality holds, moves of a 3:2 will only change the

>notes within a single 9:8 span.

Is definition in your paper still the most recent?

-Carl

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

> >Only an MOS scale generated by approximate 3:2's will have this

> >property (only one note changes with transpositions by a 3:2).

>

> Yes but I'm interested in a general measure for any scale. I'm

> thinking of a function which can take a scale and an interval as

> an input, and spit out the "extent" of the scale's symmetry at

> that interval.

Sounds good . . . for an MOS scale, it depends on how many generators

make up the 3:2 . . . for example, any MIRACLE MOS will have 6 notes

change when you transpose it by a 3:2.

>

> >My omnitetrachordality property is closely related. If

> >omnitetrachordality holds, moves of a 3:2 will only change the

> >notes within a single 9:8 span.

>

> Is definition in your paper still the most recent?

>

I don't use the word "omnitetrachordality" in my paper but the

concept is there.

>>Yes but I'm interested in a general measure for any scale. I'm

>>thinking of a function which can take a scale and an interval as

>>an input, and spit out the "extent" of the scale's symmetry at

>>that interval.

>

>Sounds good . . . for an MOS scale, it depends on how many

>generators make up the 3:2 . . . for example, any MIRACLE MOS

>will have 6 notes change when you transpose it by a 3:2.

I wonder if Gene, or somebody else with any knowledge of group

theory could help. Is there any notion like being 'partially

closed' under a certian transformations?

>>I don't use the word "omnitetrachordality" in my paper but the

>>concept is there.

Is there a place where you do use omnitetrachordality?

-Carl

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

> >>Yes but I'm interested in a general measure for any scale. I'm

> >>thinking of a function which can take a scale and an interval as

> >>an input, and spit out the "extent" of the scale's symmetry at

> >>that interval.

> >

> >Sounds good . . . for an MOS scale, it depends on how many

> >generators make up the 3:2 . . . for example, any MIRACLE MOS

> >will have 6 notes change when you transpose it by a 3:2.

>

> I wonder if Gene, or somebody else with any knowledge of group

> theory could help.

Help how? Aren't we done?

> >>I don't use the word "omnitetrachordality" in my paper but the

> >>concept is there.

>

> Is there a place where you do use omnitetrachordality?

On the tuning list. It means the same thing as the property mentioned

in my paper.

>>I wonder if Gene, or somebody else with any knowledge of group

>>theory could help.

>

>Help how? Aren't we done?

I won't speak for everybody else, but I'm not done before I have

a metric. Something that can return more than a boolean... as in,

"such-and-such scale in x % symmetrical at the 3:2", or something.

What makes sense here? I've got:

"The percentage of a scale's modes containing a 3/2, in which the

pattern of intervals generating scale degrees between 1/1 and 3/2

generates no pitches outside the scale when started at 3/2."

Which adds a modal aspect to the metric. But I wonder if there's

a raw measure of symmetry?

>>>I don't use the word "omnitetrachordality" in my paper but the

>>>concept is there.

>>

>>Is there a place where you do use omnitetrachordality?

>

>On the tuning list. It means the same thing as the property

>mentioned in my paper.

Cool. I've seen it on the lists. Just wondering if a Monzo

dictionary entry, or something, had been made.

-Carl

I wrote...

> "The percentage of a scale's modes containing a 3/2, in which the

> pattern of intervals generating scale degrees between 1/1 and 3/2

> generates no pitches outside the scale when started at 3/2."

>

> Which adds a modal aspect to the metric... But I wonder if there's

> a raw measure of symmetry?

One try is to root each mode of the scale on a random pitch, then

mulitply all those pitches by 3:2, then count the number of

new pitches and divide by the number of notes in the scale.

-Carl

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

> >>I wonder if Gene, or somebody else with any knowledge of group

> >>theory could help.

> >

> >Help how? Aren't we done?

>

> I won't speak for everybody else, but I'm not done before I have

> a metric. Something that can return more than a boolean... as in,

> "such-and-such scale in x % symmetrical at the 3:2", or something.

>

> What makes sense here?

How about the number of notes that stay the same, divided by the

total number? So for the diatonic scale, it's 6/7 . . . for

blackjack, 15/21 . . . for canasta, 25/31 . . . my decatonics would

be 4/5 . . . one can easily calculate this as

1-(number of generators in 3:2)/(number of notes).

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

> One try is to root each mode of the scale on a random pitch, then

> mulitply all those pitches by 3:2, then count the number of

> new pitches and divide by the number of notes in the scale.

I don't see what the mode has to do with it. What changes when you

label one note as the root, as opposed to not doing so?

>>One try is to root each mode of the scale on a random pitch, then

>>mulitply all those pitches by 3:2, then count the number of

>>new pitches and divide by the number of notes in the scale.

>

>I don't see what the mode has to do with it. What changes when you

>label one note as the root, as opposed to not doing so?

Quite possibly nothing... I'm cautious because I'm already thinking

through a layer of abstraction -- specifically, we don't actually

care about pitches moving (as we do for transpositional coherence),

we care about how much of an interval pattern is destroyed when we

_aren't_ allowed any new pitches. I'm trying to think if this is

really equivalent to counting the new pitches we'd need. I think

it is, but a lot of what I've thought has historically failed when

tested against "unusual" (improper, non-just) scales...

Assuming counting the changing pitches works, then we need to ask

if looking at a single mode, or even all the modes, is equivalent

to what we really care about. Namely, every possible interval

pattern in the scale is generated, transposed by a fifth, the number

of changing notes is divided by the length of the pattern in each

case, and then the whole lot of results is averaged.

-Carl

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

> Which adds a modal aspect to the metric. But I wonder if there's

> a raw measure of symmetry?

Do you care about the relative proportions of the scale steps?

Carl, how about the autocorrelation of the scale with its

transposition to the degree representing 3/2?

In Scala, look up the interval class for the nearest interval

to 3/2, do SHOW TRANSPOSE and look at the value for this

interval class.

It wouldn't be defined if the scale is not CS though, in any

case not if there's more than one i.c. for the fifth.

Or we can think further for a way to define it in that case

too, which may not be useful.

Manuel

>> Which adds a modal aspect to the metric. But I wonder if there's

>> a raw measure of symmetry?

>

> Do you care about the relative proportions of the scale steps?

Not sure what you mean... sounds like the answer would be yes...

maybe an example would help make it clear.

-Carl

> Carl, how about the autocorrelation of the scale with its

> transposition to the degree representing 3/2?

> In Scala, look up the interval class for the nearest interval

> to 3/2, do SHOW TRANSPOSE and look at the value for this

> interval class.

Don't really know what how autocorrelation is calculated. How

many lines of code does it take? You can take your pick between

posting sample code and attempting a verbal explanation.

> It wouldn't be defined if the scale is not CS though, in any

> case not if there's more than one i.c. for the fifth.

> Or we can think further for a way to define it in that case

> too, which may not be useful.

What I'm after here should definitely be independent of CS.

-Carl

Carl, you and I independently came up with the same measure here, so

aren't we done? That is, once you realize that mode is irrelevant.

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

> > Do you care about the relative proportions of the scale steps?

> Not sure what you mean... sounds like the answer would be yes...

> maybe an example would help make it clear.

Suppose we look at 12-note scales of the form abababaababa, formed by

iterating fifths within octaves. We have 7 a and 5 b, and if we take

scale steps such that a=2, b=1, we get the 19 et, a=3, b=2 gives the

31 et, a=4 b=5 the 53 et, and a=5 b=4 the 55 et. One way of measuring

self-similarity would make all of these the same, however we could

also look at it in proportion to the average number of et intervals

making up a scale step. In all cases a shift of a fifth interchanges

an a and a b; for the 19-et that means an exchange of 2 and 1, in a

sitation where the average step size is 19/12, whereas for the 55-et

it is and exchange of 5 and 4 compared to an average size of 55/12;

both are an exchange of one et interval so we might measure the

goodness of the first by 12/19 and the second by 12/55.

This is related to the approximation of the fifth of the et by 7/12;

a standard measure here would measure the relative goodness of the

approximation to x by the reduced fraction p/q by q^2 |x - p/q|. If

we do this, we find 12^2 |7/12 - 11/19| = 12/19 and

12^2 |7/12 - 32/55| = 12/55, etc. If a scale of n steps is generated

from a single cycle of a generator in an m-et, this suggests

n^2 |a/n - b/m|, where b/m is the generator and a/n is what we might

call the scale-step generator, as a measure of self-similarity.

We can also adapt it to more than one cycle of the generator; for

instance a 12-cycle in the 22 et gives us 12^2 |7/12 - 13/22| = 12/11

as a measure of self-similarity, whereas if we have two cycles of six

tones a half-ocatave apart, we have two cyles each of which has a

self-similarity measure of 6^2 | 1/6 - 2/11| = 6/11, suggesting the

total self-similarity should be the same at 12/11.

>Carl, you and I independently came up with the same measure here,

>so aren't we done? That is, once you realize that mode is

>irrelevant.

Yeah, we may be done. Just so long as nobody comes along and says

that the effect of a transposition on the entire scale doesn't

nec. represent the effects of transpositions on all of its subsets.

Also, I'm interested to find out how the autocorr. thing works in

Scala. And maybe gene has a group-theoretical way of looking at

it.

-Carl

>Suppose we look at 12-note scales of the form abababaababa, formed

>by iterating fifths within octaves. We have 7 a and 5 b, and if we

>take scale steps such that a=2, b=1, we get the 19 et, a=3, b=2

>gives the 31 et, a=4 b=5 the 53 et, and a=5 b=4 the 55 et. One way

>of measuring self-similarity would make all of these the same,

All of these are the same under the measure Paul and I are pushing.

>however we could also look at it in proportion to the average number

>of et intervals making up a scale step. In all cases a shift of a

>fifth interchanges an a and a b; for the 19-et that means an

>exchange of 2 and 1, in a sitation where the average step size is

>19/12, whereas for the 55-et it is and exchange of 5 and 4 compared

>to an average size of 55/12; both are an exchange of one et interval

>so we might measure the goodness of the first by 12/19 and the

>second by 12/55.

Hmm... what would that do... this may be useful for comparing one

scale to another when they otherwise share the same value on the

measure Paul and I suggest (as in the example). In general, though,

I was after a way to compare a scale only with itself. So the

answer is, I don't care about the proportions.

>This is related to the approximation of the fifth of the et by

>7/12; a standard measure here would measure the relative goodness

>of the approximation to x by the reduced fraction p/q by

>q^2 |x - p/q|. If we do this, we find 12^2 |7/12 - 11/19| = 12/19

>and 12^2 |7/12 - 32/55| = 12/55, etc. If a scale of n steps is

>generated from a single cycle of a generator in an m-et, this

>suggests n^2 |a/n - b/m|, where b/m is the generator and a/n is

>what we might call the scale-step generator, as a measure of self-

>similarity.

Self-similarity? I'm not sure I'm making that connection. I

should warn you that I'm pretty slow when it comes to the

language of math. I can do simple algebra, but it takes me a

long time. I'm not afraid of it, but maybe I should be. That

sort of thing.

>We can also adapt it to more than one cycle of the generator; for

>instance a 12-cycle in the 22 et gives us 12^2 |7/12 - 13/22| =

>12/11 as a measure of self-similarity, whereas if we have two

>cycles of six tones a half-ocatave apart, we have two cyles each

>of which has a self-similarity measure of 6^2 | 1/6 - 2/11| = 6/11,

>suggesting the total self-similarity should be the same at 12/11.

Hrm. This sounds interesting, but I'm not sure how it relates

to symmetry at the 3:2. If I had to brute-force the problem, here's

how I'd do it:

() Tune up the scale.

() Find all the subset of it.

() Transpose each subset by a perfect 3:2.

() Measure how far outside of original complete scale

each trasposed subset is.

() Take the mean of these.

Keep in mind that I need to be able to look at scales which are

not MOS, not even a periodicity block at all. Is there a standard

way to measure partial symmetry? Come to think of it, how do we

even write normal octave equivalence for a non-PB? Write a seed,

and then a transformation?

-Carl

This autocorrelation value measures the scale step similarity.

From the help file:

[...] the normalised autocorrelation values for the logarithmic

intervals between consecutive pitches. It is a measure how similar the

interval sequence is when the scale is transposed to the given key. A

value of 1.0 means identical, a value of 0.0 means no similarity.

So if P(i) is scale pitch i and S(i) is the step P(i)-P(i-1) and n the

number of notes and k the interval class representing the fifth,

then the autocorrelation for the scale step pattern is

n

sum S(i)*S((i+k) mod n)

i=1

------------------------

n

sum S(i)*S(i)

i=1

But apparently you want a more general measure.

>() Tune up the scale.

>() Find all the subset of it.

>() Transpose each subset by a perfect 3:2.

>() Measure how far outside of original complete scale

> each transposed subset is.

>() Take the mean of these.

The fourth step is unclear to me. What are you measuring if

the scale doesn't contain just fifths? What's the criterion

for deciding if the tones of the subsets are inside the

original scale or not?

Manuel

>So if P(i) is scale pitch i and S(i) is the step P(i)-P(i-1) and

>n the number of notes and k the interval class representing the

>fifth, then the autocorrelation for the scale step pattern is

>

> n

> sum S(i)*S((i+k) mod n)

> i=1

> ------------------------

> n

> sum S(i)*S(i)

> i=1

Is S(i) a log-freq. size?

> But apparently you want a more general measure.

I'm not sure autocorr. is too special, but it breaks if there

isn't a well-behaved k for 3:2. Unlike propriety, and in

particular transpositional coherence, we don't care about the

mapping between scale intervals and acoustics here.

> >() Tune up the scale.

> >() Find all the subset of it.

> >() Transpose each subset by a perfect 3:2.

> >() Measure how far outside of original complete scale

> > each transposed subset is.

> >() Take the mean of these.

>

> The fourth step is unclear to me. What are you measuring if

> the scale doesn't contain just fifths? What's the criterion

> for deciding if the tones of the subsets are inside the

> original scale or not?

I left that intentionally vague, because there are many ways

to do it, none of which have any bearing on the point that

whatever procedure we use, it should be equivalent to looking

at all the subsets of the scale.

The two main families of #4 choices are:

4a; We allow appoximations of a 3:2 to be as good as a true

3:2 early on, and then only measure the relative change

in the pattern of scale steps after the transposition.

That's what Paul and I are working with so far.

4b; We consider things to get smoothly worse as distance from

the 3:2 increases, making comparisons in log-freq. space

at the end.

-Carl

Carl wrote:

>Is S(i) a log-freq. size?

Yes.

And earlier:

>Namely, every possible interval

>pattern in the scale is generated, transposed by a fifth, the number

>of changing notes is divided by the length of the pattern in each

>case, and then the whole lot of results is averaged.

Is the fact that you want to give each subset equal weight the reason

for evaluating each subset?

And if you'd give each tone equal weight instead, wouldn't it be

necessary anymore to evaluate subsets?

Manuel

> Is the fact that you want to give each subset equal weight the

> reason for evaluating each subset?

Yes.

> And if you'd give each tone equal weight instead, wouldn't it be

> necessary anymore to evaluate subsets?

As far as I can tell, giving each tone equal weight is equivalent

to giving each subset equal weight, sinc the set of all subsets

should have the same proportion of intervals as a single instance

of the complete scale.

-Carl

Paul,

Looks like the measure we both came up with can be done in

Scala, like this:

() LOAD a scale, or make one

() use KEY to rotate the scale until a 3/2 shows up

() SHOW TRANSPOSE

() observe the row corresponding to the key with the 3/2

() observe the first number in the third column of that row

() divide that number by the number of tones in the scale

There's a problem, though. If there are different approx.

3:2's in the scale, you may get diff. results depending on

which row you observe from SHOW TRANSPOSE. It seems this

is a reason to go over to something like autocorrelation...

unfortunately, it enforces degree order (but it wouldn't

have to, right Manuel?).

Maybe SHOW DIFFERENCE can help:

>If the scales do not have the same number of notes, the amount

>compared is the maximum of the two scale sizes.

?

> /NEAREST

>

>Show the differences not between corresponding scale degrees but

>between the pitch of the current scale and its nearest counterpart

>in the given scale.

That sounds like what we want.

>At the end the number of pitches which are different is given

Ehhhxcellent. Manuel, there are a couple of precision variables

in Scala (TOLERANCE with SHOW LOCATIONS, for example). Does this

final number go through any of them?

-Carl

>As far as I can tell, giving each tone equal weight is equivalent

>to giving each subset equal weight, sinc the set of all subsets

>should have the same proportion of intervals as a single instance

>of the complete scale.

Ah yes, that should be right.

Manuel

>() use KEY to rotate the scale until a 3/2 shows up

Or use SHOW LOC 3/2 to see where one is right away.

>There's a problem, though. If there are different approx.

>3:2's in the scale, you may get diff. results depending on

>which row you observe from SHOW TRANSPOSE.

Yes it only looks for exactly corresponding intervals.

>It seems this

>is a reason to go over to something like autocorrelation...

>unfortunately, it enforces degree order (but it wouldn't

>have to, right Manuel?).

Then it probably needs a more complex definition. Perhaps an

average value for transpositions by a fifth starting on all

keys of the scale.

>>If the scales do not have the same number of notes, the amount

>>compared is the maximum of the two scale sizes.

Ah, that's unclear. What's meant is that the smaller scale is

implicitly octave-extended to the size of the larger scale.

So that's not what you want.

>Ehhhxcellent. Manuel, there are a couple of precision variables

>in Scala (TOLERANCE with SHOW LOCATIONS, for example). Does this

>final number go through any of them?

No.

Manuel

>>It seems this is a reason to go over to something like

>>autocorrelation... unfortunately, it enforces degree

>>order (but it wouldn't have to, right Manuel?).

//

>Perhaps an average value for transpositions by a fifth

>starting on all keys of the scale.

Can you illustrate with an example?

>Then it probably needs a more complex definition.

//

> n

> sum S(i)*S((i+k) mod n)

> i=1

> ------------------------

> n

> sum S(i)*S(i)

> i=1

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.

n

sum S(i) * S(i')

i=1

-------------------

n

sum S(i)^2

i=1

...where i' is an interval from the transposed scale.

-Carl

I wrote...

>> n

>> sum S(i)*S((i+k) mod n)

>> i=1

>> ------------------------

>> n

>> sum S(i)*S(i)

>> i=1

>

>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,

Did I say lowest value? I meant highest value, or 'best'

value. =1 means perfect correlation. Actually, isn't it

possible to get >1 ??

-Carl

[me]

Perhaps an average value for transpositions by a fifth

starting on all keys of the scale.

[Carl]

Can you illustrate with an example?

Let's forget it, on second thought it was a bad idea.

>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?

Manuel