well-temperament comparator 2

Some of you may remember my original "temperament
comparator" spreadsheet. Here's an updated version:

http://lumma.org/stuff/WellTemperamentComparator.zip

Now included are 24 well temperaments, plus 12-tET.

The RMS error, and the sum of the Tenney-weighted absolute
errors, of certain 5-limit consonaces (2:1, 3:2, 5:3, 5:4)
are given for each 'key' of each scale. There's also a
7-limit Tenney-weighted sheet which adds 7:4 to the above
intervals. The rationale behind this choice of intervals:

. Don't assume pure octaves, therefore the error of 2:1
must be considered.

. Nevertheless, most WTs have pure octaves, or very nearly
pure octaves. And so inversions like 5:4 and 8:5 will
tend to have equal but opposite errors -- improving one
will tend to make the other worse. The thing is, 5:4 is
simpler than 8:5 (n*d = 20 vs. 40) and therefore the error
of 5:4 should be more important. Therefore simply omitting
the error of 8:5 in the sum can be justified. Tenney
weighting takes care of this for us, and so all intervals
should be included, but to save effort in Excel I think a
good approximation is to again omit the contribution of
these more complex intervals -- which will be comparatively
small in the weighted scheme -- from the sum. Maybe that's
not right but it seems like it might be. :)

. There's the question of which 5:3 should be measured for
a 'key' -- in C, should we measure Eb-C, or C-A? I opted
for the latter. The error of the Eb-C sixth will count
against the key of Eb in this scheme. It's debatable which
choice is more natural.

Anyway, this gives you the .scl files, let's you see the
key contrast (hopefully), and if the temperament has more
total error than 12-tET.

Comments appreciated.

-Carl

> The RMS error, and the sum of the Tenney-weighted absolute
> errors, of certain 5-limit consonaces (2:1, 3:2, 5:3, 5:4)
> are given for each 'key' of each scale. There's also a
> 7-limit Tenney-weighted sheet which adds 7:4 to the above
> intervals. The rationale behind this choice of intervals:

One could also say I considered all 5- and 7-limit
intervals with Tenney Height (num * den) less than 30
in the error calculations.

-Carl

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> Some of you may remember my original "temperament
> comparator" spreadsheet. Here's an updated version:
>
> http://lumma.org/stuff/WellTemperamentComparator.zip
>
> Now included are 24 well temperaments, plus 12-tET.
>
> The RMS error, and the sum of the Tenney-weighted absolute
> errors, of certain 5-limit consonaces (2:1, 3:2, 5:3, 5:4)
> are given for each 'key' of each scale. There's also a
> 7-limit Tenney-weighted sheet which adds 7:4 to the above
> intervals. The rationale behind this choice of intervals:
>
> . Don't assume pure octaves, therefore the error of 2:1
> must be considered.

OK, good point.

> . Nevertheless, most WTs have pure octaves, or very nearly
> pure octaves. And so inversions like 5:4 and 8:5 will
> tend to have equal but opposite errors -- improving one
> will tend to make the other worse. The thing is, 5:4 is
> simpler than 8:5 (n*d = 20 vs. 40) and therefore the error
> of 5:4 should be more important. Therefore simply omitting
> the error of 8:5 in the sum can be justified. Tenney
> weighting takes care of this for us, and so all intervals
> should be included, but to save effort in Excel I think a
> good approximation is to again omit the contribution of
> these more complex intervals -- which will be comparatively
> small in the weighted scheme -- from the sum. Maybe that's
> not right but it seems like it might be. :)

The effort you speak of is only the effort required to set up a
formula for cloning, isn't it? I can see that you can't just take
the formulae you have presently and copy them across a range;
however, perhaps a clever way could be found to do this, perhaps
using xLOOKUP functions. But this is just mechanics, and if the
reward were great enough, perhaps someone would be motivated to do
the tedious editing needed to add in further intervals, eg m3 and
m6, and possibly M2 and M7.

More important, I think, are these two questions:
(1) Whether including other intervals would materially affect the
results, ie the sensitivity of the analysis. Given that octaves may
be *quite* impure, perhaps it sometimes would.

(2) Which other intervals need to be included. I suspect the
answer to this question is: that depends on your objectives.

Two other questions occur to me:
(3) Whether "Tenney weighting" is the most appropriate weighting to
use. (AFAICT, that seems to be weighting JI intervals inversely to
the product of numerator and denominator, BICBW.) It would be nice
for particular users to be able to try out other weighting schemes
that may seem more natural to them. Other candidate weighting
schemes include:
a) 1/Square-root(numerator * denominator), or inverse geometric
mean, weighting.
b) 1/Max(numerator, denominator), or inverse max, weighting.
c) 1/Mean(numerator, denominator), or inverse mean, weighting.

(2) Whether to use sums (as you have) or averages for comparison.
For example, to conduct a sensitivity analysis on whether to include
m3 and m6, using averages would produce directly comparable values,
which using sums would not.

> . There's the question of which 5:3 should be measured for
> a 'key' -- in C, should we measure Eb-C, or C-A? I opted
> for the latter. The error of the Eb-C sixth will count
> against the key of Eb in this scheme. It's debatable which
> choice is more natural.

Indeed. One could even argue that including all intervals of the
diatonic scale was most natural.

> Anyway, this gives you the .scl files, let's you see the
> key contrast (hopefully), and if the temperament has more
> total error than 12-tET.

Is there a reasonably natural measure of key contrast? Perhaps the
standard deviation of the mean weighted errors for all intervals for
each key would serve.

> Comments appreciated.

I've tried. ;-) Let me finish by saying I appreciate the work
you've put into the spreadsheet.

Regards,
Yahya

> > . Nevertheless, most WTs have pure octaves, or very nearly
> > pure octaves. And so inversions like 5:4 and 8:5 will
> > tend to have equal but opposite errors -- improving one
> > will tend to make the other worse. The thing is, 5:4 is
> > simpler than 8:5 (n*d = 20 vs. 40) and therefore the error
> > of 5:4 should be more important. Therefore simply omitting
> > the error of 8:5 in the sum can be justified. Tenney
> > weighting takes care of this for us, and so all intervals
> > should be included, but to save effort in Excel I think a
> > good approximation is to again omit the contribution of
> > these more complex intervals -- which will be comparatively
> > small in the weighted scheme -- from the sum. Maybe that's
> > not right but it seems like it might be. :)
>
> The effort you speak of is only the effort required to set up a
> formula for cloning, isn't it?

Partly -- and the formula does get rather long and unweildy --
but also the way I set up this spreadsheet was a bit stupid (but
haven't thought of a better way yet) and the cloning breaks
in both directions now, due to a mixture of relative and
absolute pointers, among other things.

> Two other questions occur to me:
> (3) Whether "Tenney weighting" is the most appropriate weighting
> to use. (AFAICT, that seems to be weighting JI intervals
> inversely to the product of numerator and denominator, BICBW.)

The log of the product, yes. It's open to debate whether it's
best. (I'm currently working on a totally new version of this
spreadsheet that will test that a bit!)

> It would be nice
> for particular users to be able to try out other weighting schemes
> that may seem more natural to them. Other candidate weighting
> schemes include:
> a) 1/Square-root(numerator * denominator), or inverse geometric
> mean, weighting.

That's the harmonic entropy approximation. log(n*d)
gives the same ranking.

> b) 1/Max(numerator, denominator), or inverse max, weighting.
> c) 1/Mean(numerator, denominator), or inverse mean, weighting.

Yes, if I were Dave Keenan I'd be able to have a user interface
for all this in my spreadsheet.

> > . There's the question of which 5:3 should be measured for
> > a 'key' -- in C, should we measure Eb-C, or C-A? I opted
> > for the latter. The error of the Eb-C sixth will count
> > against the key of Eb in this scheme. It's debatable which
> > choice is more natural.
>
> Indeed. One could even argue that including all intervals of the
> diatonic scale was most natural.

Here each 'key' is typically used for both major and minor
scales. But the question is which consonances matter most.
If you consider *all* the intervals, all the keys would come
out the same. My new spreadsheet has answered this with some
insight offerred by a paper in which the authors perform
spectral analysis of a huge corpus of human speech sounds in
several languages.

> I've tried. ;-) Let me finish by saying I appreciate the work
> you've put into the spreadsheet.

Thanks Yahya!

-Carl

> Here each 'key' is typically used for both major and minor
> scales. But the question is which consonances matter most.
> If you consider *all* the intervals, all the keys would come
> out the same. My new spreadsheet has answered this with some
> insight offerred by a paper in which the authors perform
> spectral analysis of a huge corpus of human speech sounds in
> several languages.

I should have said, *will* answer this (I'm still working on
it). -C.

Hi Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> > Here each 'key' is typically used for both major and minor
> > scales. But the question is which consonances matter most.
> > If you consider *all* the intervals, all the keys would come
> > out the same. My new spreadsheet has answered this with some
> > insight offerred by a paper in which the authors perform
> > spectral analysis of a huge corpus of human speech sounds in
> > several languages.

Good to know that the corpus linguistics approach (actually collecting
a large body of linguistic data to improve its chance of being truly
representative of its domain) now extends to spectral analysis for
phonetics. That means that ad hoc and a priori theories will no
longer be the best we have to work with; science approaches, albeit
incrementally, closer to understanding reality.

I do like the fact that you're taking insights for speech analysis
into musical theory.

> I should have said, *will* answer this (I'm still working on
> it). -C.

I look forward to seeing it.

Regards,
Yahya