linear approximation

Paul wrote:
>> Ok, it's not so difficult. Imagine the scale in the X-Y plane.
>> Horizontally the degree numbers and vertically pitch in cents. Then
>> imagine the straight line going through the origin and coming as
>>close
>> as possible to the points that represent the tones. Some points
>>will be
>> above the line and some below. If it would be equal to the average
>> interval, the line would go through the octave point, which is not
>> required.

>Can you write me off-list, or on the still-extant tuning-math list,
>with a simple, perhaps pentatonic, example? I need to see the details
>of what you're doing.

Let's take
0: 1/1 0.000 unison, perfect prime
1: 4/3 498.045 perfect fourth
2: 3/2 701.955 perfect fifth
3: 2/1 1200.000 octave

Then SHOW DATA or the first line shown by FIT:
> linear LS approximation: 392.9968 cents, 3.053460/oct. RMS diff: 78.6109 cents
will show this value. If we create this ET:

0: 1/1 0.000 unison, perfect prime
1: 392.997 cents 392.997
2: 785.994 cents 785.994
3: 1178.990 cents 1178.990

Show the difference with the original scale:

1: -105.048 cents -105.048
2: 84.039 cents 84.039
3: -21.010 cents -21.010
Total absolute difference : 210.0964 cents
Average absolute difference: 70.0321 cents
Root mean square difference: 78.6109 cents
Highest absolute difference: 105.0482 cents

So this RMS difference of 78.61 cents is the lowest.
The RMS difference with 3-tET would be 80.0534 cents.

It's exactly the same as the best LS generator for
a Pythagorean scale, with the generator being one step here.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> It's exactly the same as the best LS generator for
> a Pythagorean scale, with the generator being one step here.
>
The best LS generator . . . Manuel, there are several _strange_
features in this calculation. First of all, you're including _both_
the 1/1 _and_ the 2/1, while every other pitch class appears only
once. Second, you're forcing the fit line to pass through the 1/1.
These are very odd features and I'm not sure how you'd justify them.
As far as I can tell, if you do it correctly, the best LS step has to
equal the mean step.

>The best LS generator . . . Manuel, there are several _strange_
>features in this calculation. First of all, you're including _both_
>the 1/1 _and_ the 2/1, while every other pitch class appears only
>once.

It's only an example. The user can decide which intervals he wants
to approximate.
Whether or not calculations are strange is also something I want to
leave the user to decide. If some calculation may be useful to
someone I consider adding it. I'm not trying to impose a set of
values whether features are good or bad onto users.

>Second, you're forcing the fit line to pass through the 1/1.

That's because the points are equidistant horizontally. If that's
not done, the line would pass through the x-axis at a non-integer
coordinate in most cases.

>As far as I can tell, if you do it correctly, the best LS step has to
>equal the mean step.

That would imply a different definition of LS difference then.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> >The best LS generator . . . Manuel, there are several _strange_
> >features in this calculation. First of all, you're including _both_
> >the 1/1 _and_ the 2/1, while every other pitch class appears only
> >once.
>
> It's only an example. The user can decide which intervals he wants
> to approximate.
> Whether or not calculations are strange is also something I want to
> leave the user to decide. If some calculation may be useful to
> someone I consider adding it. I'm not trying to impose a set of
> values whether features are good or bad onto users.

Can you explain to me a situation in which this calculation, as it
appears in the tutorial, is in a form that tells something useful to
someone?
>
> >Second, you're forcing the fit line to pass through the 1/1.
>
> That's because the points are equidistant horizontally. If that's
> not done, the line would pass through the x-axis at a non-integer
> coordinate in most cases.

What's wrong with that?

>Can you explain to me a situation in which this calculation, as it
>appears in the tutorial, is in a form that tells something useful to
>someone?

If you have a set of roughly equidistant tones then it tells which
ET is the best LS approximation to those without octave repetition.
For a more elaborate approximation, not with step sizes 1, 2, 3, etc.
but with higher integers, the FIT command is available.

>What's wrong with that?

It's assumed that 1/1 is on degree 0. If you'd create an ET with
the step size given by the slope of the line it wouldn't be the
best approximation anymore because the whole line would shift.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> >What's wrong with that?
>
> It's assumed that 1/1 is on degree 0. If you'd create an ET with
> the step size given by the slope of the line it wouldn't be the
> best approximation anymore because the whole line would shift.

It seems to me the opposite would be true. Allowing the whole line to
shift would allow, in general, a better approximation to the step
sizes, wouldn't it?

>It seems to me the opposite would be true. Allowing the whole line to
>shift would allow, in general, a better approximation to the step
>sizes, wouldn't it?

But then 1/1 is not included in the approximation. Of course,
with one tone less, the approximation can be better.
To do it that way, do "delete 0" and then check the value with
"fit" or "show data" again.

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> >It seems to me the opposite would be true. Allowing the whole line to
> >shift would allow, in general, a better approximation to the step
> >sizes, wouldn't it?
>
> But then 1/1 is not included in the approximation.

I was thinking that it was included. Just because the line doesn't have to pass exactly through
1/1 doesn't mean that the line is not _affected_ by the presence of the point at 1/1.

Paul wrote:
>I was thinking that it was included. Just because the line doesn't have to
pass exactly through
>1/1 doesn't mean that the line is not _affected_ by the presence of the
point at 1/1.

You're right, jee I wasn't having my day.
Well I think if you're trying to approximate a set of pitches with an
ET, then a regular linear regression is a possibility.
What I'm trying to do is approximating a set of intervals and then if
the line doesn't pass through the origin and 1/1 shifts, then _all_ the
intervals are affected. So I chose to keep it fixed in order to avoid that.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:

> What I'm trying to do is approximating a set of intervals and then if
> the line doesn't pass through the origin and 1/1 shifts, then _all_ the
> intervals are affected. So I chose to keep it fixed in order to avoid that.

But if the line passes through the origin, and 1/1 is at the origin, 1/1 shifting by +x would be
tantamount to 1/1 remaining fixed and all each of the other pitches shifting by -x. If you think of it
that way, shouldn't _that_ affect all the intervals?

Yes, this is exactly what I meant.

Manuel

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
>
> Yes, this is exactly what I meant.
>
> Manuel

So we agree . . . and yet we disagree. I'm confused.

Do we disagree that doing a linear regression to the pitches
is not a least squares approximation to the intervals?
If we agree to that, I hope we can agree too that my method is
not strange.