Yahoo Tuning Groups Ultimate Backup mills-tuning-list Origin of Harmonic Tonality

"Asmussen, Robert" erote:
>I have recently posted a paper that explores tuning by ratios. Included
>in this paper are illustrations, programs, WAV files, Csound examples
>and a retuned two-part invention by J.S. Bach.
>http://www.terraworld.net/users/r/robert/default.htm

Robert, on arriving at your pages I had a feeling of deju vu.
Your three coloured waves logo with periods in the ratio 2:3:5 is quite
similar to one on my pages. The problem with which you are dealing,
that of working out the exact correct frequency of every note in any
composition, is the same one that I address in my AJI pages at
http://www.kcbbs.gen.nz/users/rtomes/aji-main.htm

Robert, congratulations on a nice clear presentation.

In your pages you say:
>Another point to note is that most pieces of traditional tonal music,
>even short and seemingly simple pieces by Mozart and Haydn, cannot be
>translated without modification from traditional notation into integer
>ratios. Chromatic passages are especially difficult to recast into this
>new framework.

Right! I have considered a number of options for this and believe that
in general the best solution is one based on a sort of information
theory idea. The following may be a bit heavy, but once grasped can be
seen as a powerful tool.

To explain this idea, consider that each chord played has notes related
by ratios to the tonic. Suppose that these ratios for a particular
chord are 5/4, 15/8, 3/1. Then it is necessary to find the LCM (lowest
common multiple) of the numerators and divisors which are 15 and 8. My
system then consists of counting demerit points for the factors of these
two numbers. We have 2^3 and 3^1*5^1 and so must score demerits by
weighting the value for the indices here. Generally ratios of 2 are
most acceptable and then 3 and then 5 etc. My preferred formula for
demerits is in proportion to p*ln(p) so that if 2 is set as unity each 3
and 5 will score 2.4 and 5.8 respectively. In that case 8 and 15 will
score 3 and 2.4+5.8 for a total of 11.2 demerits. If there is a better
way to "read" the chord then it will score less.

Some intervals such as a minor sixth may be interpreted as 7/4, 16/9 or
9/8 depending on circumstances. The one that minimises the demerits is
to be considered better because it is more simply related to the other
notes present or even previously played.

BTW, there is a missing graphic on one of your pages. Unfortunately I
just pressed the wrong key and lost everything including the reference.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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🔗"Asmussen, Robert" <robert@...>

Dear Mr. Tomes,

Thank you for your comments regarding my paper. I have in turn been
reading your paper on just intonation, which is located at:

http://www.kcbbs.gen.nz/users/rtomes/aji-main.htm

I think you have many fine ideas, such as translating passages of
traditionally notated music with variable pitch sets. I will take more
time later to delve into the details.

I have devised a mechanism similar to the keyboard system in your paper.
In my translator, a Csound orchestra stores variable pitch sets (12
members each) in wave tables, then looks them up as they are needed
according to values stored in the score.

For the purposes of pedagogy, especially at the introductory level, I
try to avoid references to traditional tuning systems. Doing so
eliminates the unnecessary step of translating from a limited, fixed
system into an unlimited one. For this reason, I left comparisons and
translations between equal temperament and just intonation out of the
most recent version of my paper.

As to some specific details, I believe you are incorporating intervals
in your Mozart example that are not in keeping with the classical style.
For example, you use octave equivalents of 7/4 as some of your frequency
ratios. I challenge you or anyone to create a dominant seventh chord,
using the number seven as a multiple in the denominator of the chord�s
seventh, that would not sound out of place in a piece of chamber music
by Mozart.

I would suggest giving it the acid test, which would be to make a WAV
sound file of your Mozart score using Csound. With the tempo nice and
slow, try 7/4, then 16/9 as the relative frequency for the seventh of
your dominant chord.

Under such conditions, nearly everyone would agree the ratio 16/9 is a
much better selection for music of this period. I cannot provide an
airtight proof of why this is the case, but my ear and 30+ years
experience as a musician tell me it is so. I can point out that from the
major scale provided by Helmholtz in his book, �Sensations of a Tone�,
the relationship of 16/9 for the seventh of a dominant seventh chord may
be inferred.

You state, �Some intervals such as a minor sixth may be interpreted as
7/4, 16/9 or 9/8 depending on circumstances.�

Without going into the potential merits of your statistical approach to
pitch selection, I must point out that 7/4 is approximately a minor
seventh; 16/9 is an in-tune minor seventh; and 9/8 is a major second.
Certainly none of these ratios is a minor sixth. I assume this is just a
grammatical error.

When I stated in my paper that pieces of traditional tonal music could
not be translated without modification from traditional notation into
integer ratios, I could have elaborated further with the following two
points:

1. Often, such as when employing chromatic scales, the composer is
simply filling in the space from point A to point B with notes. Such
passages do not need to be translated, nor often can they be, into
ratios. One might even argue that such passages are not really music,
but rather more like dust on a mirror.

2. Composers of the past wrote their music using limited instruments and
tuning systems that were available at the time. We should not expect to
translate their imperfect creations into a purely mathematical
framework; in fact, we should be surprised when it is possible to do so.

In my paper, I have included one such surprise, complete with a WAV
sound file generated in Csound. This entire two-part invention, a lovely
gem by Bach, can be tuned according to the principles given 130 years
ago by Hermann Helmholtz. The Csound orchestra, score and WAV file are
located at:

http://www.terraworld.net/users/r/robert/htmlpap7.htm

In the Csound score for this two-part invention, every interval is
provided in the last two parameter fields, columns 5 and 6. The bars are
numbered clearly to facilitate comparison with an actual score.

Regarding the problems you are having with the graphics in my paper,
would you please contact me if you experience additional difficulties? I
have reloaded all ten HTML files off my server quite easily using only 8
megs of RAM while running a minimum of applications and windows.

I look forward to studying your paper, as well as examining your many
other papers, in greater detail.

Best regards,

Robert Asmussen

P.S.

For those seeking information about Csound, a good starting point is:

http://ccrma-www.stanford.edu/~tkunze/Csound/title.html

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🔗rtomes@kcbbs.gen.nz (Ray Tomes)

Robert Asmussen wrote:

>As to some specific details, I believe you are incorporating intervals
>in your Mozart example that are not in keeping with the classical style.
>For example, you use octave equivalents of 7/4 as some of your frequency
>ratios. I challenge you or anyone to create a dominant seventh chord,
>using the number seven as a multiple in the denominator of the chord�s
>seventh, that would not sound out of place in a piece of chamber music
>by Mozart.

And yet if you look at the derived fundamental frequency (defined below)
for the three bars I think that the "correct" interpretation is
undoubtedly F for the 1st and 3rd bars and C for the second. This fits
quite naturally with the 7 ratios but not with your interpretation.
See http://www.kcbbs.gen.nz/users/rtomes/aji-xmpl.htm for this Mozart
example and the Beethoven example mentioned below.

>I would suggest giving it the acid test, which would be to make a WAV
>sound file of your Mozart score using Csound. With the tempo nice and
>slow, try 7/4, then 16/9 as the relative frequency for the seventh of
>your dominant chord.

I certainly don't know enough about music history to argue on that score
and anyway accept that 16/9 is sometimes the correct value for that
note. However I would argue on the basis of logic. If you have a
dominant 7th chord and the other notes have frequency ratios of 4:5:6:8
why should the extra note be 64/9 in that ratio scheme when 63/9 would
cancel down nicely to 7 and make an elegant 4:5:6:7:8? Whatever was
actually played historically I still feel that the intention or meaning
of such a chord is 4:5:6:7:8.

>You state, �Some intervals such as a minor sixth may be interpreted as
>7/4, 16/9 or 9/8 depending on circumstances.�

>Without going into the potential merits of your statistical approach to
>pitch selection, I must point out that 7/4 is approximately a minor
>seventh; 16/9 is an in-tune minor seventh; and 9/8 is a major second.
>Certainly none of these ratios is a minor sixth. I assume this is just a
>grammatical error.

Oops, I really went to sleep on that one didn't I? Two typos.
What I meant to say was that a minor 7th could be 7/4, 16/9 or 9/5.

>When I stated in my paper that pieces of traditional tonal music could
>not be translated without modification from traditional notation into
>integer ratios, I could have elaborated further with the following two
>points:

>1. Often, such as when employing chromatic scales, the composer is
>simply filling in the space from point A to point B with notes. Such
>passages do not need to be translated, nor often can they be, into
>ratios. One might even argue that such passages are not really music,
>but rather more like dust on a mirror.

>2. Composers of the past wrote their music using limited instruments and
>tuning systems that were available at the time. We should not expect to
>translate their imperfect creations into a purely mathematical
>framework; in fact, we should be surprised when it is possible to do so.

I have to agree at the surprise, but believe that the results often tell
us more than we expected to find.

Have a look at my second example, Beethoven's "Romance".

The melody goes b c d g g g b g a b b g b c d but the fundamental
frequency b c d g g g b c d g g g b c d is calculated.

(I define fundamental frequency as the frequency which has ratio 1 when
all the ratios are reduced to integers, so it divides all the played
frequencies). To me it seems certain that the part with melody b g a
has been correctly interpreted because the fundamental frequency echoes
the other parts that go b c d. Notice that the ratio 7 occurs in this
short passage several times and is necessary to give this beautiful
structure.

Actually I cheated a wee bit, because one of those b's is actually an
incidental b/a.

Thanks for the information about Csound. I will access this.

>Regarding the problems you are having with the graphics in my paper,
>would you please contact me if you experience additional difficulties? I
>have reloaded all ten HTML files off my server quite easily using only 8
>megs of RAM while running a minimum of applications and windows.

I accessed all your pages and there was only one missing graphic.
Unfortunately when I went to gets its name I hit the wrong key and lost
it all. You already saw above how butterfingered I was that night from
my typing :-) If I find this again I will let you know.

Robert, thanks for the interesting comments, Ray.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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Origin of Harmonic Tonality

🔗Mark Bradlyn & Claire Sherard <mkbird@...>

🔗rtomes@kcbbs.gen.nz (Ray Tomes)

🔗"Asmussen, Robert" <robert@...>

🔗rtomes@kcbbs.gen.nz (Ray Tomes)