Reply to Gerald Eskelin

Gerald Eskelin wrote,

>On the other hand, when I brought up the subject in regard to ear-training
>classes in our own department meeting, the chairman said, in effect,
"Jerry,
>why do you persist in this crusade; our music system is firmly based on
>twelve-tone equal temperament."

You should have your chairman study the history of Western tuning, and point
out that in England, meantone tuning remained standard for all keyboard
instruments through much of the 19th century (having been standard for well
over 300 years!). Let alone what non-fixed-pitch instruments could have been
doing! "Our music system" could mean the music system of Debussy and
Schoenberg; maybe Schubert; but certainly not the composers who brought
about common-practice tonality.

Gerald Eskelin wrote,

>Why is it important? Just because Paul E mentioned it maybe 86 times? Okay,
>I guess that's important enough. Did Forte actually say something
>significant? Perhaps I was too young to notice.

Look, you can drop the attitude -- John said he hadn't studied classical
tonal analysis and I thought I'd give him an example where he could learn
about diatonic chord functions, linear harmonic devices, etc., in the
context of the Western common-practice repertoire. I suppose you'd do away
with all classical tonal analysis and replace it with . . . ?

Gerald Eskelin wrote,

>This suggests that
>singers do NOT tend to sing a 4:5:6 major triad. (Forget the Pythagorean
>third. It is way too sharp.)

Gerald, this seems odd. In our off-list discussion, you told me that the
singers sing a major third _higher_ than the 12-tET one in a major triad,
and yet now you say the Pythagorean third is _way_ too sharp. There is only
8 cents difference between the two. 8 cents is about the smallest
distinguishable melodic interval.

>(It also seems unlikely that this "high third"
>is a product of culture, since it appears to be arrived at in terms of
>consonance, as opposed to learned preference.)

I don't know how you propose to back up that assertion. In fact, studies
have shown that untrained listeners tend to fall into two groups, one group
preferring a just major triad, and another group preferring a major triad
with the third either 15 cents sharp (as in 12-tET) or 15 cents flat, over
the just major triad. I would argue that these listeners have been
conditioned by a lifetime of exposure to the characteristic beating thirds
of 12-tET.

>Also, diminished triads, contrary to Mr. Wolf's description, in my
>experience seem NOT to conform to a tempered configuration.

Gerald, Mr. Wolf was speaking specifically of diminished _seventh_ chords,
not diminished triads. Besides augmented triads, another class of chords
that cry out for tempering are those consisting of 4-7 consecutive fifths,
such as a major chord with added 6th and 9th.

>Paul, read my paragraph again. There is a compliment to you hidden not very
>deeply under my facetious remarks.

Sorry, I thought I detected sarcasm.

>(To the new reader: The point of all of this is that I may be able to
supply
>a copy of the book in question. Paul omitted that part.)

I did not see that in the original e-mail.

>BTW, wouldn't that be THREE
>groups? 1) just, 2) 15 cents sharp, 3) 15 cents flat. ??? If the "another"
>group was responding to "either" the sharp or the flat tunings, we still
>don't know whether they might have preferred some other tuning. Perhaps the
>"another" group were lumped together as simply "wrong." (Many questions
>unanswered here.)

There were 5 options: (a) 30 cents flat, (b) 15 cents flat, (c) just, (d) 15
cents sharp, (e) 30 cents sharp. One group always preferred (c), gave
intermediate ratings to (b) and (d), and low ratings to (a) and (e). The
other group always preferred (b) and (d), gave intermediate ratings to (c),
and low ratings to (a) and (e).

Gerald Eskelin wrote,

>Perhaps the answer lies in the combination of partials emanating from two
>fundamentals (the root and the fifth, in this case), each sponsoring
certain
>frequencies and perhaps "agreeing" and "disagreeing" at certain points.
What
>do you think?

Gerald, this line of reasoning, when including the major third as a third
fundamental, _could_ provide a solution to your problem. If a C-E major
third is tuned 4:5 = 1/5:1/4 (0� 386�), the lowest common overtone is an E
two octaves higher, and if this overtone "stands out" and is heard in some
semi-conscious way, it melodically agrees with, or reinforces the chord. If
a G is now added to the chord, with tuning 4:5:6 = 1/15:1/12:1/10 (0� 386�
702�), the lowest common overtone is B three octaves higher, and if this B
is heard in some way, it may not be desirable since it is not one of the
notes in the chord. However, if the major third is raised, as you claim, and
specifically, the chord is tuned 1/24:1/19:1/16 (0� 404� 702�), the lowest
common overtone is a G four octaves higher, agreeing with the chord.

It could be that to reliably approximate the equal-tempered major triad
they're used to hearing, singers have learned to focus on the 16th harmonic
(four octaves above) the G, and learn to eliminate the beating among the
partials of the three notes at that pitch by adjusting the tuning of the
major third to 24:19. When the C-E third is unaccompanied, the sensation at
that high G is too weak and the singers have to resort to a simple 5:4.

This is the best hypothesis I can think of to support your observation,
Gerald. Whether it's valid or not, I'm sure you'll find, once you measure
some examples, that your "high third" of stable, held major triads is far
closer to 404� (24:19) than to 435� (9:7). And again, I think the acoustical
pull of 4:5:6 is much greater than that of 1/24:1/19:1/16 and the former is
used in contexts, such as barbershop and Renaissance works, where a sense of
complete blending is the desideratum.

I wrote,

>>Gerald, you're being misled by these ratios. 45/32 and 27/16 form a 4:5:6
>>triad with 9/8, so with 9/8 there, they would be very easy to "lock into".
>>To be fair, I agree that these ratios alone would be too difficult to lock
>>into on their own, but that was not the context here -- it was a full chord.

Jerry wrote,

>Thanks, Paul. I'm still wrestling with the number jargon here on the List.
>Every little bit helps.

>Jerry

>>That is the usual take on Indian music, where the just (5/4) and Pythagorean
>>(81/64) major thirds are sruti #s 7 and 8 out of a 22-sruti octave. Now it
>>is true that the Pythagorean major third wouldn't lock harmonically as part
>>of a triad, so for the third time I suggest the possibility of
>>1/24:1/19:1/16 for your locked major triad with high third.

>For now the second time in a post to you (and the third if you count another
>to someone else) I am requesting a clarification of what appear to be
>fractions separated by colons. Please unlock the code. For example, how does
>the expression 4/5 relate to the expression 4:5?

They often mean the same thing, but 4/5 (usually 5/4) often has the addition
significance of representing a pitch 5:4 above a root or tonic, which is denoted
as 1/1.

>Also, what does 1/x
>represent?

Just what you would think. For example, the usual JI minor triad has a 5:6
between root and third, a 4:5 between third and fifth, and a 2:3 between root and
fifth. These ratios can be combined in (at least) two ways to give a three-term
"ratio" for the chord. They are:

10:12:15
1/6:1/5:1/4

If you calculate the interval between root and third in the first chord, you get
10/12=5/6. Similarly, if you calculate the interval between root and third in the
second chord, you get (1/6)/(1/5) = 5/6. And so on. These two representations of
the chord are called "otonal" and "utonal" respectively, since they relate to the
overtone series and the undertone series. The undertone series is best understood
as the series of fundamentals under a common overtone. Every JI chord can be
expressed both ways, though many chords look simpler one way or the other. For
example, the minor triad above uses smaller numbers in the utonal representation,
so it is usually considered a utonal chord. The major triad's two representations
are

4:5:6
1/15:1/12:1/10

and since the first is simpler, it is usually considered an otonal chord.

The earlier example was of a chord with pitches 1/1 7/4 9/8 45/32 27/16. I made
the assertion that the 45/32 and 27/16 would be easy to tune with the 9/8 since
they form a 4:5:6 major triad with it. Check: (9/8)/(45/32) = 4/5, (9/8)/(27/16)
= 2/3, and of course, (45/32)/(27/16) = 5/6.

>I really would like to know what you are saying here. Can someone
>whose math training ended at college algebra have a fighting chance of
>understanding? I hope this is simply a matter of form rather than of depth.

So far, no algebra, just multiplying and dividing fractions.

Jerry wrote,

>Perhaps you missed it, Paul. The whole idea here was to use
inexperienced
>students for this experiment in order to minimize cultural
conditioning, a
>factor that has appeared important to you (and to me as well).

But these students are from the Western Hemisphere (right?), so
heard 12-tET their whole lives. Unlike a group of experienced singers
who may have had a chance to develop their own tuning paradigm, these
students may rely primarily on their memory of how these intervals
"should" sound. So this may maximize, rather than minimizing, the
influence of cultural conditioning.

>When I got it home, I noticed the flaws that I described on my post.
>Nevertheless, it clearly demonstrated the phenomenon, so I went with
>it--mess and all.

This is how I heard your example: The basses come in quite sharp of
the tonic, and with very uncertain pitch. The sopranos then sing a
third that might be a 2:5 over the basses, but due to the sharpness
of the basses they are only a few cents flat of the 12-tET third. As
soon as the piano third is played, the sopranos adjust up to match
it, and then, as the fifth is introduced (with a bit of a struggle),
the interval between the bass and soprano does not appear to change
in any essential way -- perhaps they both drift up a few more cents,
so that the piano third sounds flat in comparison. I certainly don't
hear an
acoustical "locking" occuring at two distinct values for the major
third in this example.

P.S.
When I said the example was a "mess", I meant that purely from the
point of view of the mathematical problem of determining what
ratios were being represented by the intervals (it would be a messy
problem). I did not mean that it was ugly or unmusical.

I wrote,

>> Have you read Daniel Wolf's reply and Margo's preview of a reply
to the
>> issue of whether tonal music arose from tempered instruments or
from
>> free-pitch instruments? I really think you are operating under an
inaccurate
>> picture of how this music developed.

Jerry wrote,

>I'm sure that exchange would be interesting, but time restraints
being what
>they are, I can't afford the luxury. The reason I assume that it may
not be
>relevant is that tonal music appears to have been developed vocally
in most
>world cultures. Sure, that is an assumption on my part, but I am
choosing to
>believe that instruments--whether tempered or free--had little to do
with
>the development of tonal music. My experience with "new ears" (as
opposed to
>"tone deaf) shows that they frequently sing a fourth or fifth below
a given
>pitch to be matched. I find that very significant and relevant to
the point.
>I have to believe that the ancients "discovered" tonal music in a
similar
>way.

Fifths and fourths arose vocally -- I believe you're absolutely right.
Tonal music, with functional triads, major and minor keys, and
modulation -- those only arose in Europe in a specific environment --
one laden with composers at meantone-tuned keyboards. Even in the
century before Monteverdi, the triadic language was to some extent
predicated on the compromises of meantone -- see Blackwood's book.
Jerry, if this issue interests you (and since you are an author), I
suggest you at least read Daniel Wolf's recent, and Margo Schulter's
forthcoming, comments on this issue.

Particularly interesting for you, though on a slightly different
topic, would be for you to read the comments of musicians in various
decades of the 18th century on the issue of temperament. Many of them
were clearly as attuned to the tuning practices of their time as you
are to those of today, and reading their reactions, and the way their
reactions change between the 1720s and 1750s may force you to
reconsider your apparent assumption about the level of ignorance with
respect to tuning that existed among musicians at that time.
Jorgenson's big book _Tuning_ contains a few enlightening quotes along
these lines.

I wrote,
>
>>Perhaps you missed it, Paul. The whole idea here was to use
> inexperienced
>>students for this experiment in order to minimize cultural
> conditioning, a
>>factor that has appeared important to you (and to me as well).

And Paul Erlich responded:
>
> But these students are from the Western Hemisphere (right?), so
> heard 12-tET their whole lives. Unlike a group of experienced singers
> who may have had a chance to develop their own tuning paradigm, these
> students may rely primarily on their memory of how these intervals
> "should" sound. So this may maximize, rather than minimizing, the
> influence of cultural conditioning.

Paul, I think your reasoning here is a bit suspect. If my hundreds of
students were relying on a memory of 12-tET, why do they tune the third
below piano pitch before the fifth is sounded, and why do they consistently
move it above piano pitch while the fifth is sounding? Granted my "demo" had
problems, but it did show a narrower third (albeit nearly at the piano pitch
because of the basses sharp root) near the beginning of the example and a
larger third near the end. To my ear, neither of these thirds were 12-tET.

There is no doubt that many if not most student singers tend to imitate
piano pitch, but after they gain the concept of locked tuning, they
consistently tune the major third as I have described. I repeat, I'm not
here to prove anything--merely to understand it. If you simply dismiss my
description in these illogical ways, neither one of us will learn much.

I said:
>
>>When I got it home, I noticed the flaws that I described on my post.
>>Nevertheless, it clearly demonstrated the phenomenon, so I went with
>>it--mess and all.

Paul said:
>
> This is how I heard your example: The basses come in quite sharp of
> the tonic, and with very uncertain pitch. The sopranos then sing a
> third that might be a 2:5 over the basses, but due to the sharpness
> of the basses they are only a few cents flat of the 12-tET third.

I agree.

> As soon as the piano third is played, the sopranos adjust up to match
> it,

Actually, it is _one of the three sopranos that initially moves at that
point. It was the new one, for which tuning to something other than the
piano is still somewhat uncertain. Therefore, when she hears the piano's
third she likely went for it out of habit. This does not influence the final
placement of the third, however, as the demo shows. If the singers were
being influenced by memory of 12-tET, they would have stayed with the
piano's third. They clearly did not.

> and then, as the fifth is introduced (with a bit of a struggle),
> the interval between the bass and soprano does not appear to change
> in any essential way -- perhaps they both drift up a few more cents,
> so that the piano third sounds flat in comparison.

Upon listening to the demo again, I admit I've heard better locks on the
high third. However, I don't think the basses moved again after they settled
into their pitch before the fifth sounded. And, most importantly, the new
position of the sopranos results in a "brighter" third than the initial one,
a "flavor" characteristic of the high third.

> I certainly don't
> hear an acoustical "locking" occuring at two distinct values for the major
> third in this example.

Do you mean you don't hear _any locking? Or do you mean the "two" thirds are
not different? As you would know, your answer to these questions is
important. If the two thirds sound the same to you, it means we are hearing
differently.

> P.S.
> When I said the example was a "mess", I meant that purely from the
> point of view of the mathematical problem of determining what
> ratios were being represented by the intervals (it would be a messy
> problem). I did not mean that it was ugly or unmusical.

Excuse my tender skin. Although, I'm not sure I would call this demo
"musical." ;-)

Thanks for your feedback.

Jerry