further comparisons of well temperaments

What does anybody think of...

http://www.lumma.org/tuning/WellTemperamentComparator.xls

?

(There are RMS and TOP worksheets in it.)

-Carl

> What does anybody think of...
>
> http://www.lumma.org/tuning/WellTemperamentComparator.xls
>
> ?
>
> (There are RMS and TOP worksheets in it.)
>
> -Carl

I wanted to implement Absolute TOP error, but the Excel
formulas got to be unwieldy, as I'd seemingly have to use
if statements to avoid catastrophe wherever the error
from JI was zero.

-Carl

Carl Lumma wrote:
> What does anybody think of...
> > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> > ?
> > (There are RMS and TOP worksheets in it.)

I guess that's useful if you want to see the error of the entire temperament, but I'd think that with well temperaments you'd want to weight the error so that the errors in the keys with fewer sharps and flats is more significant than the errors in distant keys. You'd expect the overall error to be worse than 12-ET (after all, that's the point of 12-ET, to be able to play in any key with equally bad errors in all keys), but it would be interesting to see how a well temperament compares with 12-ET when shorter chains in the cycle of fifths are compared.

When I was playing around with well temperaments a couple of years ago, I put together a Microsoft Works spreadsheet that determined the best and worst major thirds of each temperament. I used the size of the fifths to represent the temperaments instead of the tuning of each note, but one can be derived easily enough from the other.

http://www.io.com/~hmiller/music/well-temperaments.xlr

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> What does anybody think of...
>
> http://www.lumma.org/tuning/WellTemperamentComparator.xls
>
> ?
>
> (There are RMS and TOP worksheets in it.)
>
> -Carl

The TOP formulas don't appear to be correct. For one thing, it looks
like you're using a weighted L1 (sum of absolute weighted errors)
measure, while TOP is a weighted L-infinity (maximum absolute error)
measure. For another thing, you have a list of intervals in this sum
which doesn't appear to relate to the TOP paradigm at all.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > What does anybody think of...
> >
> > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> >
> > ?
> >
> > (There are RMS and TOP worksheets in it.)
> >
> > -Carl
>
> I wanted to implement Absolute TOP error, but the Excel
> formulas got to be unwieldy, as I'd seemingly have to use
> if statements to avoid catastrophe wherever the error
> from JI was zero.
>
> -Carl

Why is that? Just use max(abs(...),abs(...),abs(...),...) and there
should be no catastrophe. Of course, there's still a question as to
what abs terms should be in this expression, but I would think you'd
probably be evaluating a particular triad so there should be three
terms(?) . . .

> > What does anybody think of...
> >
> > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> >
> > ?
> >
> > (There are RMS and TOP worksheets in it.)
> >
> > -Carl
>
> The TOP formulas don't appear to be correct. For one thing, it
> looks like you're using a weighted L1 (sum of absolute weighted
> errors) measure, while TOP is a weighted L-infinity (maximum
> absolute error) measure. For another thing, you have a list of
> intervals in this sum which doesn't appear to relate to the TOP
> paradigm at all.

Hiya Paul -- thanks for taking the time to check out the
spreadsheet.

Once again, I'm abusing terminology. I call the worksheet "TOP",
but if you read it, it says, "pairwise Tenney-weighted errors of
a 4:5:6:8 chord". Whether this is any good is another story.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> Once again, I'm abusing terminology. I call the worksheet "TOP",
> but if you read it, it says, "pairwise Tenney-weighted errors of
> a 4:5:6:8 chord".

OK . . . If you add the words "sum of" before that, I think the
description will agree with the calculations.

> > > What does anybody think of...
> > >
> > > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> > >
> > > ?
> > >
> > > (There are RMS and TOP worksheets in it.)
> >
> > I wanted to implement Absolute TOP error, but the Excel
> > formulas got to be unwieldy, as I'd seemingly have to use
> > if statements to avoid catastrophe wherever the error
> > from JI was zero.
>
> Why is that? Just use max(abs(...),abs(...),abs(...),...) and
> there should be no catastrophe. Of course, there's still a
> question as to what abs terms should be in this expression,
> but I would think you'd probably be evaluating a particular
> triad so there should be three terms(?) . . .

For ATE you need to do logs of errors. When the errors are
zero, Excel's log function blows up. Where do the logs fit in
the above expression? I suppose max() could come first?

This reminds me that I never liked the minimax nature of
TOP. It's because...

3 3 3
mean = 3
max = 3

0 0 9
mean = 3
max = 9

...I think mean is closer to the truth here than max. Though
I'm not aware of this ever having been established with a
listening test.

-Carl

> > Once again, I'm abusing terminology. I call the worksheet "TOP",
> > but if you read it, it says, "pairwise Tenney-weighted errors of
> > a 4:5:6:8 chord".
>
> OK . . . If you add the words "sum of" before that, I think the
> description will agree with the calculations.

Actually "sum of" is before that! :)

But I see I also incorrectly called that sheet "RMS". Updated
version is now up in the same place...

http://lumma.org/tuning/WellTemperamentComparator.xls

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > What does anybody think of...
> > > >
> > > > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> > > >
> > > > ?
> > > >
> > > > (There are RMS and TOP worksheets in it.)
> > >
> > > I wanted to implement Absolute TOP error, but the Excel
> > > formulas got to be unwieldy, as I'd seemingly have to use
> > > if statements to avoid catastrophe wherever the error
> > > from JI was zero.
> >
> > Why is that? Just use max(abs(...),abs(...),abs(...),...) and
> > there should be no catastrophe. Of course, there's still a
> > question as to what abs terms should be in this expression,
> > but I would think you'd probably be evaluating a particular
> > triad so there should be three terms(?) . . .
>
> For ATE you need to do logs of errors. When the errors are
> zero, Excel's log function blows up.

My guess is that you're doing something wrong. What is the formula
you're trying to use exactly? You shouldn't be taking the log of
anything with units of cents, and on your current spreadsheet it
doesn't seem you are.

> Where do the logs fit in
> the above expression? I suppose max() could come first?

I don't get what you're asking. It seems you just need to replace
the "outer" sum in your formula with a max statement, and then
everything will be allright, and nothing will blow up.

> This reminds me that I never liked the minimax nature of
> TOP. It's because...
>
> 3 3 3
> mean = 3
> max = 3
>
> 0 0 9
> mean = 3
> max = 9
>
> ...I think mean is closer to the truth here than max. Though
> I'm not aware of this ever having been established with a
> listening test.

If you advocate the mean (or the sum, which amounts to the same thing
as regards tuning comparisons), then you're advocating P=1.
Meanwhile, Gene's "poptimal" is based on a range 2<=P<=infinity.
Another reason for Gene to reconsider?

The nice thing about minimax, though, is that once you've established
it over the intervals in a tuning system, you know that the minimax
for any particular chords in the tuning system will be less than or
equal to the tuning's minimax. Not so for mean or for any
intermediate value of P.

Want to move this to tuning-math?

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > Once again, I'm abusing terminology. I call the
worksheet "TOP",
> > > but if you read it, it says, "pairwise Tenney-weighted errors of
> > > a 4:5:6:8 chord".
> >
> > OK . . . If you add the words "sum of" before that, I think the
> > description will agree with the calculations.
>
> Actually "sum of" is before that! :)
>
> But I see I also incorrectly called that sheet "RMS".

Wait a minute -- do you mean the *other* sheet? Otherwise, I'm
confused.

> Updated
> version is now up in the same place...
>
> http://lumma.org/tuning/WellTemperamentComparator.xls
>
> -Carl

Sum-squared and RMS amount to exactly the same thing (always give the
same answer) when assessing whether one tuning or chord is "better"
than another, and thus they yield the exact same "optimal"
tunings . . .

> > > > I wanted to implement Absolute TOP error, but the Excel
> > > > formulas got to be unwieldy, as I'd seemingly have to use
> > > > if statements to avoid catastrophe wherever the error
> > > > from JI was zero.
> > >
> > > Why is that? Just use max(abs(...),abs(...),abs(...),...) and
> > > there should be no catastrophe. Of course, there's still a
> > > question as to what abs terms should be in this expression,
> > > but I would think you'd probably be evaluating a particular
> > > triad so there should be three terms(?) . . .
> >
> > For ATE you need to do logs of errors. When the errors are
> > zero, Excel's log function blows up.
>
> My guess is that you're doing something wrong. What is the formula
> you're trying to use exactly? You shouldn't be taking the log of
> anything with units of cents, and on your current spreadsheet it
> doesn't seem you are.

I'm not doing ATE, but I wanted to. For that, you take the
base n*d log of the error, right?

> > This reminds me that I never liked the minimax nature of
> > TOP. It's because...
> >
> > 3 3 3
> > mean = 3
> > max = 3
> >
> > 0 0 9
> > mean = 3
> > max = 9
> >
> > ...I think mean is closer to the truth here than max. Though
> > I'm not aware of this ever having been established with a
> > listening test.
>
> If you advocate the mean (or the sum, which amounts to the
> same thing as regards tuning comparisons), then you're
> advocating P=1. Meanwhile, Gene's "poptimal" is based on
> a range 2<=P<=infinity. Another reason for Gene to reconsider?

I've always agreed with John deLaubenfels that errors seem to
get worse with the square of their size. So I think I'm a P=2
guy. But it'd be great to get some tetrads and triads to
try to compare infinity with a low P. It's not immediately
obvious to me how to calculate examples like 009 vs. 333 since
the pairwise errors in a chord are not independent.

> The nice thing about minimax, though, is that once you've
> established it over the intervals in a tuning system, you
> know that the minimax for any particular chords in the
> tuning system will be less than or equal to the tuning's
> minimax. Not so for mean or for any intermediate value
> of P.

Ok, that's a key point. However, isn't the "tuning's minimax"
here just another "particular chord", ultimately? It's nice
to think of JI as a lattice, where none of the weighted errors
exceeds a certain bound. I agree that's nice. But really it
seems that only the primary intervals matter... yes, I suppose
mean says that a subset of your basic chord could be worse
than the entire basic chord. That's a pain in the butt, but
what if it corresponds to reality? I dunno, I hate to dig up
the error thing, but it never sat quite right with me. I
know Gene and Graham have delivered TOP-like results based on
RMS... I never took the time to see what those were like.

> Want to move this to tuning-math?

I'm copying this there.

-Carl

> > > >Once again, I'm abusing terminology. I call the
> > > >worksheet "TOP", but if you read it, it says,
> > > >"pairwise Tenney-weighted errors of
> > > > a 4:5:6:8 chord".
> > >
> > > OK . . . If you add the words "sum of" before that, I think
> > > the description will agree with the calculations.
> >
> > Actually "sum of" is before that! :)
> >
> > But I see I also incorrectly called that sheet "RMS".
>
> Wait a minute -- do you mean the *other* sheet? Otherwise, I'm
> confused.

There are two "worksheets" in this Excel file (check out the
tabs at the bottom). I've fixed this terminology snafu now,
I think...

> > Updated version is now up in the same place...
> >
> > http://lumma.org/tuning/WellTemperamentComparator.xls
>
> Sum-squared and RMS amount to exactly the same thing (always
> give the same answer) when assessing whether one tuning or
> chord is "better" than another, and thus they yield the exact
> same "optimal" tunings . . .

Yes, but they differ on how much "better", which is important
when you're asking things like "how much worse than 12-tET
triads are 15-tET triads?".

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > > I wanted to implement Absolute TOP error, but the Excel
> > > > > formulas got to be unwieldy, as I'd seemingly have to use
> > > > > if statements to avoid catastrophe wherever the error
> > > > > from JI was zero.
> > > >
> > > > Why is that? Just use max(abs(...),abs(...),abs(...),...) and
> > > > there should be no catastrophe. Of course, there's still a
> > > > question as to what abs terms should be in this expression,
> > > > but I would think you'd probably be evaluating a particular
> > > > triad so there should be three terms(?) . . .
> > >
> > > For ATE you need to do logs of errors. When the errors are
> > > zero, Excel's log function blows up.
> >
> > My guess is that you're doing something wrong. What is the
formula
> > you're trying to use exactly? You shouldn't be taking the log of
> > anything with units of cents, and on your current spreadsheet it
> > doesn't seem you are.
>
> I'm not doing ATE, but I wanted to. For that, you take the
> base n*d log of the error, right?
>
>
> > > This reminds me that I never liked the minimax nature of
> > > TOP. It's because...
> > >
> > > 3 3 3
> > > mean = 3
> > > max = 3
> > >
> > > 0 0 9
> > > mean = 3
> > > max = 9
> > >
> > > ...I think mean is closer to the truth here than max. Though
> > > I'm not aware of this ever having been established with a
> > > listening test.
> >
> > If you advocate the mean (or the sum, which amounts to the
> > same thing as regards tuning comparisons), then you're
> > advocating P=1. Meanwhile, Gene's "poptimal" is based on
> > a range 2<=P<=infinity. Another reason for Gene to reconsider?
>
> I've always agreed with John deLaubenfels that errors seem to
> get worse with the square of their size. So I think I'm a P=2
> guy. But it'd be great to get some tetrads and triads to
> try to compare infinity with a low P. It's not immediately
> obvious to me how to calculate examples like 009 vs. 333 since
> the pairwise errors in a chord are not independent.
>
> > The nice thing about minimax, though, is that once you've
> > established it over the intervals in a tuning system, you
> > know that the minimax for any particular chords in the
> > tuning system will be less than or equal to the tuning's
> > minimax. Not so for mean or for any intermediate value
> > of P.
>
> Ok, that's a key point. However, isn't the "tuning's minimax"
> here just another "particular chord", ultimately? It's nice
> to think of JI as a lattice, where none of the weighted errors
> exceeds a certain bound. I agree that's nice. But really it
> seems that only the primary intervals matter... yes, I suppose
> mean says that a subset of your basic chord could be worse
> than the entire basic chord. That's a pain in the butt, but
> what if it corresponds to reality? I dunno, I hate to dig up
> the error thing, but it never sat quite right with me. I
> know Gene and Graham have delivered TOP-like results based on
> RMS... I never took the time to see what those were like.
>
> > Want to move this to tuning-math?
>
> I'm copying this there.
>
> -Carl

And that's where I replied.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:

> > > > >Once again, I'm abusing terminology. I call the
> > > > >worksheet "TOP", but if you read it, it says,
> > > > >"pairwise Tenney-weighted errors of
> > > > > a 4:5:6:8 chord".
> > > >
> > > > OK . . . If you add the words "sum of" before that, I think
> > > > the description will agree with the calculations.
> > >
> > > Actually "sum of" is before that! :)
> > >
> > > But I see I also incorrectly called that sheet "RMS".
> >
> > Wait a minute -- do you mean the *other* sheet? Otherwise, I'm
> > confused.
>
> There are two "worksheets" in this Excel file (check out the
> tabs at the bottom).

Yes, I knew that. You were talking about the "Tenney" one above, and
then you seemed to say that you had called *that* one RMS, but I was
pretty sure you had called the *other* one RMS . . .

> I've fixed this terminology snafu now,
> I think...
>
> > > Updated version is now up in the same place...
> > >
> > > http://lumma.org/tuning/WellTemperamentComparator.xls
> >
> > Sum-squared and RMS amount to exactly the same thing (always
> > give the same answer) when assessing whether one tuning or
> > chord is "better" than another, and thus they yield the exact
> > same "optimal" tunings . . .
>
> Yes, but they differ on how much "better", which is important
> when you're asking things like "how much worse than 12-tET
> triads are 15-tET triads?".

The "how much better" question is completely separate from the
question of which P to use (sum-squared and RMS are both P=2) . . .
more on tuning-math. But you'd probably agree that using mean vs.
using sum doesn't, itself, amount to a substantive difference, since
it's just an overall multiplicative constant . . . (?)

> But you'd probably agree that using mean vs. using sum doesn't,
> itself, amount to a substantive difference, since it's just an
> overall multiplicative constant . . . (?)

Yes.

-Carl

On Sat, 27 Aug 2005, Carl Lumma wrote:
> > What does anybody think of...
> >
> > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> >
> > ?
> >
> > (There are RMS and TOP worksheets in it.)
> >
> > -Carl
>
> I wanted to implement Absolute TOP error, but the Excel
> formulas got to be unwieldy, as I'd seemingly have to use
> if statements to avoid catastrophe wherever the error
> from JI was zero.
>
> -Carl

Errrr ... How do you use it?

Yahya

--
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> > > What does anybody think of...
> > >
> > > http://www.lumma.org/tuning/WellTemperamentComparator.xls
> > >
> > > ?
> > >
> > > (There are RMS and TOP worksheets in it.)
> > >
> > > -Carl
> >
> > I wanted to implement Absolute TOP error, but the Excel
> > formulas got to be unwieldy, as I'd seemingly have to use
> > if statements to avoid catastrophe wherever the error
> > from JI was zero.
> >
> > -Carl
>
> Errrr ... How do you use it?

Hi Yahya,
Not sure what you're asking, but I'm going to be issuing
a new version of that spreadsheet 'any day now'... so maybe
you would like to wait until then. Or, could you rephrase
your question?
-Carl

Hi Carl,

On Fri, 07 Oct 2005, you wrote:
>
> > > > What does anybody think of...
> > > >
> > > > http://www.lumma.org/tuning/WellTemperamentComparator.xls
...
> > > > -Carl
> >
> > Errrr ... How do you use it?
>
> Hi Yahya,
> Not sure what you're asking, but I'm going to be issuing
> a new version of that spreadsheet 'any day now'... so maybe
> you would like to wait until then. Or, could you rephrase
> your question?

You may have already issued that - I'll keep a look out as
I trawl thru the messsages in the digests.

What I meant was: It seemed likely to me that a spreadsheet
called WellTemperamentComparator would offer a fairly
direct way to choose and compare two Well Temperaments.
If that was the intention, I couldn't see how to do this.

If OTOH your intention was to tabulate the interesting
characteristics of the chief historical WTs, then I think
you've provided a useful reference.

Best,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Anti-Virus.
Version: 7.0.344 / Virus Database: 267.11.13/126 - Release Date: 9/10/05

> > Hi Yahya,
> > Not sure what you're asking, but I'm going to be issuing
> > a new version of that spreadsheet 'any day now'... so maybe
> > you would like to wait until then. Or, could you rephrase
> > your question?
>
> You may have already issued that - I'll keep a look out as
> I trawl thru the messsages in the digests.

Nope, not yet. Hopefully soon.

> What I meant was: It seemed likely to me that a spreadsheet
> called WellTemperamentComparator would offer a fairly
> direct way to choose and compare two Well Temperaments.
> If that was the intention, I couldn't see how to do this.

Good point. Maybe I'll change the title.

> If OTOH your intention was to tabulate the interesting
> characteristics of the chief historical WTs, then I think
> you've provided a useful reference.

Thanks!

-Carl

> > > Hi Yahya,
> > > Not sure what you're asking, but I'm going to be issuing
> > > a new version of that spreadsheet 'any day now'... so maybe
> > > you would like to wait until then. Or, could you rephrase
> > > your question?
> >
> > You may have already issued that - I'll keep a look out as
> > I trawl thru the messsages in the digests.
>
> Nope, not yet. Hopefully soon.

Note: the current version contains errors!

-Carl