Interval analysis of Bach's WTC

I suddenly became intensely curious about what tuning might best fit
the entire set of Bach's "Well Tempered Clavier", Book I.

So, I went up to prs.net and pulled down John Sankey's complete set,
concatenated all 24 of them together, and fed the result to my program
for a 5-limit Calculated Optimum Fixed Tuning analysis. I am VERY
surprised at the result! The ideal tuning that I calculate is extremely
close to 12-tET.

Perhaps, given the fact that the pieces ARE in all 24 keys, this seems
only self-evident, or, at the very least, expected. Yet I thought I had
gathered from posts on this list (which I can't remember in detail at
the moment), that Bach had tended to arpeggiate chords in the more
distant keys, while stressing nice major thirds in the keys most favored
by Well Temperaments in use at that time.

Instead, I find the following:

12-tET Total spring pain: 9599911.701
Werckmeister III spring pain: 11581027.133
Kirnberger III Total spring pain: 11879895.829
Thomas Young Total spring pain: 10845223.537
COFT Total spring pain: 9576719.049
After relaxing, Total spring pain: 4037621.105
Final vertical spring pain: 1958782.615
Final horizontal spring pain: 325804.112
Final grounding spring pain: 1753034.378

All of the three Well Temperaments I threw at it are significantly
worse than 12-tET, and my own COFT number is less than 1% better than
12-tET. The intervals are very evenly distributed in this set of
pieces. The COFT tuning values themselves that I calculate range from
a high of +0.8 cents more than 12-tET for A, down to a low of 0.5 cents
below 12-tET for F. We're talking minuscule deviations from 12-tET
here!

I do not claim, of course, that these results imply that Bach used
12-tET. Still, it seems surprising to me, if he had a Well Temperament
under his hands as he wrote these pieces, that he didn't skew his chord
selection at all (as far as I can tell) in a way that would make
favorable use of its qualities to tend toward JI.

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
> I suddenly became intensely curious about what tuning might best fit
> the entire set of Bach's "Well Tempered Clavier", Book I.
>
> So, I went up to prs.net and pulled down John Sankey's complete set,
> concatenated all 24 of them together, and fed the result to my
program
> for a 5-limit Calculated Optimum Fixed Tuning analysis. I am VERY
> surprised at the result! The ideal tuning that I calculate is
extremely
> close to 12-tET.
>

Hello John!

This is a very interesting study, but isn't it predicated on the
assumption that Bach really WANTED just intonation?? So the "spring
pain" is calculated upon that assumption??

And, doesn't 12-tET spread the comma around more equally than the
others so there would NATURALLY be a lower spring value if Bach used
all the keys about equally?? Or am I getting something wrong??

It rather means, then, that Bach uses all the 12 notes and KEYS about
equally in modulation, particularly in Werckmeister, as Johnny
Reinhard attests. According to Johnny, this is what makes a "well
temperament" NOT an EQUAL temperament...

HOWEVER, use of all the keys and 12 pitches doesn't mean that the
CHARACTER of the intervals and keys aren't different depending on the
key as in Werckmeister even if all the notes are used equally.

Is this making any sense??

_______ _____ __ __
Joseph Pehrson

[Joseph Pehrson:]
>This is a very interesting study, but isn't it predicated on the
>assumption that Bach really WANTED just intonation?? So the "spring
>pain" is calculated upon that assumption??

Yes.

>And, doesn't 12-tET spread the comma around more equally than the
>others so there would NATURALLY be a lower spring value if Bach used
>all the keys about equally?? Or am I getting something wrong??

You are correct. Any sequence, for example, that was 12 identical
fragments played in 12 keys, would give a COFT of exactly 12-tET.

>It rather means, then, that Bach uses all the 12 notes and KEYS about
>equally in modulation, particularly in Werckmeister, as Johnny
>Reinhard attests. According to Johnny, this is what makes a "well
>temperament" NOT an EQUAL temperament...

Of course, the analysis says nothing at all about the tuning Bach
actually used. It only suggests that he didn't let an unequal tuning
influence his choice of intervals and chords to any significant degree.
This I do find surprising, but perhaps others don't.

>HOWEVER, use of all the keys and 12 pitches doesn't mean that the
>CHARACTER of the intervals and keys aren't different depending on the
>key as in Werckmeister even if all the notes are used equally.

True.

>Is this making any sense??

Sure. It would be perfectly possible, for example, to tune a keyboard
to 1/4 comma meantone, then proceed to write music that uses all keys
equally. The keys that crossed the "wolf" would be very sour to most
people's ears, but that doesn't make the act impossible.

Here is the distribution of major thirds that I found in the WTC:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 -0.29 4 -0.26 3403.570 386.643 400.030 45563.433 304977.804
1 -0.01 5 -0.49 3436.930 386.593 399.517 44418.579 287030.859
2 0.44 6 -0.15 3308.893 386.635 399.409 42266.708 269950.490
3 0.02 7 0.33 3436.466 386.587 400.306 47144.081 323379.405
4 -0.26 8 0.27 3141.660 386.681 400.528 43505.127 301225.502
5 -0.49 9 0.80 2793.153 386.642 401.287 40905.269 299525.492
6 -0.15 10 -0.43 3261.698 386.599 399.717 42785.109 280615.404
7 0.33 11 -0.23 3346.193 386.664 399.440 42752.191 273108.864
8 0.27 0 -0.29 3723.893 386.588 399.442 47864.695 307612.062
9 0.80 1 -0.01 3370.952 386.691 399.196 42153.511 263563.330
10 -0.43 2 0.44 2977.159 386.625 400.874 42420.919 302223.444
11 -0.23 3 0.02 3087.812 386.536 400.254 42358.463 290535.736

Ignore most of the fields and focus on the Strength. The first record
shows notes 0 and 4: C and E, to have a strength of 3403.570. Notes
1 and 5 (Db and F) weigh in at 3436.930. Close, or what? And so on
down the line. Oddly, F-A seems to be the LEAST represented.

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/17562

> >It rather means, then, that Bach uses all the 12 notes and KEYS
about equally in modulation, particularly in Werckmeister, as Johnny
> >Reinhard attests. According to Johnny, this is what makes a "well
> >temperament" NOT an EQUAL temperament...
>
> Of course, the analysis says nothing at all about the tuning Bach
> actually used. It only suggests that he didn't let an unequal
tuning influence his choice of intervals and chords to any
significant degree. This I do find surprising, but perhaps others
don't.
>

Hello John!

But, basically what you are talking about is "frequency" of usage of
intervals, in a statistical sense. If I understand it correctly,
your study has "concatenated" all the pieces in all the keys -- so
certain primary "nearer to just" intervals could still be stressed in
EACH key and the total result would still turn out as entirely
"democratic"... (???) The "democratic" usage DOES seem to
substantiate the contention that Well Temperament practice used all
the keys equally, but that has nothing to do with the particularly
"flavor" of Well Temperment used...or the frequency of usage of
"nearer to just" intervals within any ONE GIVEN key... (??) Is this
right (??)

_____ ____ ___ __ _
Joseph Pehrson

John deLaubenfels wrote,

>I do not claim, of course, that these results imply that Bach used
>12-tET. Still, it seems surprising to me, if he had a Well Temperament
>under his hands as he wrote these pieces, that he didn't skew his chord
>selection at all (as far as I can tell) in a way that would make
>favorable use of its qualities to tend toward JI.

That seems to be a defect of your method of calculation, if the results of
Barnes, Kellner, and van Eck are to be believed. Their statistical analyses
found that certain thirds were preferred quite significantly over others. I
refer you to their original papers.

[I wrote (before obtaining results):]
>>for something like, say, Bach's Well-Tempered Clavier,[...]
>> . Concatenate the entire set of sections together and get COFT
>> numbers to be used for grounding the entire work.

[Paul E wrote:]
>John, something like this was attempted by several researchers,
>including Barnes, Kellner, and van Eck. See the Microtonal
>bibliography.

Sorry, what exactly is the "Microtonal bibliography"?

[JdL:]
>>The results of the first option would be very interesting to compare
>>against what was present in Bach's time!

[Paul:]
>The above researchers generally came up with temperaments roughly along
>the lines of Werckmeister/Kirnberger philosophy, differing only in the
>details.

Hmmm, interesting! Wonder how they arrived at this?

[JdL (after obtaining results):]
>>I do not claim, of course, that these results imply that Bach used
>>12-tET. Still, it seems surprising to me, if he had a Well Temperament
>>under his hands as he wrote these pieces, that he didn't skew his
>>chord selection at all (as far as I can tell) in a way that would make
>>favorable use of its qualities to tend toward JI.

[Paul:]
>That seems to be a defect of your method of calculation, if the results
>of Barnes, Kellner, and van Eck are to be believed. Their statistical
>analyses found that certain thirds were preferred quite significantly
>over others. I refer you to their original papers.

Well, it is of course always possible that there are one or more
"defects" in my methods of analysis! The same would apply to anyone
else's, potentially. Where can I find these other results?

JdL

John deLaubenfels wrote,

>Sorry, what exactly is the "Microtonal bibliography"?

ftp://ella.mills.edu/ccm/tuning/papers/bib.html

The references in question are:

Barnes, John. "Bach's Keyboard Temperament: Internal Evidence from the
Well-tempered clavier", Early Music vol. 7 no. 2, April 1979, pp. 236-249.
Followup correspondence in Early Music vol. 8, 1980, pp. 511-513 (Philip P.
Jones); vol. 9, Jan. 1981, p. 141 (H.A. Kellner); April 1981, pp. 219-221
(Cl. di Véroli); Oct. 1981, p. 579 (J. Barnes)

Kellner, Herbert Anton. Lots of references -- see biblio

Van Eck -- seems to be missing -- perhaps there is a more recent version of
the bibliography that I should be pointing John to?

Paul, you are talking about Barnes and his content analysis of interval
proximity and sustain. It found that there was a linking between consonant
and just fifths with greater musical attention, and consequently with major
thirds.

However, JdL is correct that JS Bach would not be blatant about these
distinction. He could have done much more if he wanted to bring attention to
these distinctions. They are subtle, relatively.

Johnny Reinhard

[Joseph Pehrson wrote:]
>>>It rather means, then, that Bach uses all the 12 notes and KEYS
>>>about equally in modulation, particularly in Werckmeister, as Johnny
>>>Reinhard attests. According to Johnny, this is what makes a "well
>>>temperament" NOT an EQUAL temperament...

[I wrote:]
>>Of course, the analysis says nothing at all about the tuning Bach
>>actually used. It only suggests that he didn't let an unequal
>>tuning influence his choice of intervals and chords to any
>>significant degree. This I do find surprising, but perhaps others
>>don't.

[Joe:]
>But, basically what you are talking about is "frequency" of usage of
>intervals, in a statistical sense. If I understand it correctly,
>your study has "concatenated" all the pieces in all the keys -- so
>certain primary "nearer to just" intervals could still be stressed in
>EACH key and the total result would still turn out as entirely
>"democratic"... (???) The "democratic" usage DOES seem to
>substantiate the contention that Well Temperament practice used all
>the keys equally, but that has nothing to do with the particularly
>"flavor" of Well Temperment used...or the frequency of usage of
>"nearer to just" intervals within any ONE GIVEN key... (??) Is this
>right (??)

If I'm understanding you correctly, this is right. In my previous
response, I used the example of writing a piece in 1/4 comma meantone
with equally distributed intervals, a more extreme example along the
same lines. My own ear would tend to shy away, to some extent at least,
from the intervals farther from JI, and would tend to embrace intervals
closer to JI.

JdL

Johnny Reinhard wrote,

>Paul, you are talking about Barnes and his content analysis of interval
>proximity and sustain. It found that there was a linking between consonant

>and just fifths with greater musical attention, and consequently with major

>thirds.

>However, JdL is correct that JS Bach would not be blatant about these
>distinction. He could have done much more if he wanted to bring attention
to
>these distinctions. They are subtle, relatively.

Not disagreeing -- simply pointing JdL to the fact that others performed the
same sort of analysis he did, but with differing results, and wondering what
might be causing the differences. It might even be that the other authors
had a goal in mind and fudged their methods to acheive the results they
wanted, while John is being more intellectually honest.

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/17577

> [Paul E wrote:]
> >John, something like this was attempted by several researchers,
> >including Barnes, Kellner, and van Eck. See the Microtonal
> >bibliography.
>
> Sorry, what exactly is the "Microtonal bibliography"?
>

Hello John!

This is the excellent and exhaustive (literally) bibliography
studiously assembled by Manuel op de Coul. He has placed it in the
file area of this very group!

http://www.egroups.com/files/tuning/docs/bib.html

>Van Eck -- seems to be missing -- perhaps there is a more recent version
of
>the bibliography that I should be pointing John to?

He's under the P, he has a double name: van Panthaleon van Eck.

But indeed, don't use the mills ref anymore. The file is still there,
but the file permissions disallow me to remove it. The recent version is
http://www.xs4all.nl/~huygensf/doc/bib.html

Manuel

Thanks Paul, Joe, and Manuel, for the refs. I'll see what I can dig up
regarding past analyses of Bach's WTC in the next couple of days (or
this weekend at the latest). I do have a fair degree of confidence in
my routines' COFT analysis, but it'd be dumb to claim I'm certain of my
results.

If anybody has in front of them one of the twelve major thirds which
past analysis has shown to be underrepresented in the WTC, I could pull
out an analysis of the presence of this third in each of the 24 parts.
Etc. Sometimes verifying methods (or proving them wrong) is not easy,
but choosing some specific focus can help.

JdL

[I wrote:]
>Thanks Paul, Joe, and Manuel, for the refs. I'll see what I can dig up
>regarding past analyses of Bach's WTC in the next couple of days (or
>this weekend at the latest).

Ha ha ha. I had sillyily imagined that there'd be actual info on the
web, but these are, for the most part, references to books and
periodical articles that exist somewhere in hardcopy. Somewhere I don't
have easy access to.

For now, we'll have to have an impasss on this question. I'll gladly
supply more details about my analysis if desired, and would welcome more
info on analyses that give different results.

(Thanks, though, Manuel, for maintaining this list!)

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

http://www.egroups.com/message/tuning/17602

>
> Ha ha ha. I had sillyily imagined that there'd be actual info on
the web, but these are, for the most part, references to books and
> periodical articles that exist somewhere in hardcopy. Somewhere I
don't have easy access to.

You know, it really is a shame there is not more tuning information
in .pdf format out on the Web... kind of like what Kraig Grady is
doing with his Wilson archive....

Living in NYC, I'm luckier than some, providing I take the time to
physically research such materials, but I have found that even BASIC
tuning references like Easily Blackwood's _The Structure of
Recognizable Diatonic Tunings__ are out of print. I've had a search
over at Amazon.com for that for MONTHS, and nothing is turning up.

(I could probably find his father's bridge books much easier!)

Of course, it would probably take some kind of grant to set up and
maintain such a tuning reference site.... but wouldn't that be
something to dream about???

(I guess Starrett and Monzo come the closest right now!)
________ _____ ____ __
Joseph Pehrson