RE: Meaner Tones (Paul E)

From: PAULE

>He suggested a tuning with Major Thirds (TM) of 372 cents,
>Fifths (F) of 720 and minor thirds (tm) of 348 and asks that I
>compare this to 15-tet. I can't tune up the triads easily today, but
>I did calculate the following:

> Fifth Major T minor t (in cents)
>PE's tuning 720 372 348
>15-tet 720 400 320

>720-F 18.05 (720-F)^2 325.62
>T-372 14.3137 (T-372)^2 204.88
>400-T 13.683 (400-T)^2 187.31
>348-tm 32.36 (348-tm)^2 1047.09
>320-tm 4.36 (320-tm)^2 19.0

>Now let us sum the squared errors of the fifth and the Major
>Third for both tunings:

>15-tet 325.62 + 187.31 512.93
>PE's 325.62 + 204.88 530.5

>15-tet is the more consonant tuning by this test, though not by much.

>Now let us add the squared error of the minor third to each.
>15-tet 512.93 +19531.93
>PE's 530.5 + 1047.09 77.59

>I will stipulate that this is a very large difference, but we already
>knew that 15 tet was the more consonant.

Grr! OK, PE's is now redefined (for the third time) to have a perfect fifth
of 720 cents and a major third whose error is exactly equal in magnitude and
opposite in sign to that of 15-equal. There. Now tell me that this isn't a
glaring counterexample to judging triadic consonance by looking at perfect
fifths and major thirds only!

By the way, if we want to take these sum squared errors and interpret them
as measures of consonance, we should divide by the number of terms in the
sum to obtain a mean squared error, then divide by twice the squared
tolerance of the ear, and exponentiate the negative of that result (in case
you didn't get it, that's a normal distribution model, normalized to give a
value of one for just intonation). The tolerance of the ear is about 1% so
the squared tolerance is about 300 square cents.

>Another question is whether the sum of the squared or even absolute
>errors
>is the proper measure of out-of-tuneness. Errors in the flat direction
>may be more serious than sharp ones, but both our techniques conflate
>the two.

For the case I'm trying to make with the "PE's" tuning, it doesn't matter
whether you use squared or absolute error. As for preferring sharp
intervals, presumably this would apply to all intervals as well as their
inversions, and since we are considering all intervals along with their
inversions, all such asymmetry will cancel out.

What does it mean to prefer a sharp unison over a flat unison? (that's just
a joke, but there is a serious (hehe) debate on this topic over on
rec.music.compose)

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I have always been under the impression that "European As" were 442/43-
but I suppose it could have been a dream!

Allen Strange

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I spoke with one of the better tuners in Frankfurt today. It seems that
chamber music players can range from 435 to 442, western German orchestras
and new music groups are at 442/443 and eastern orchestras (include the
Berlin Phil) are up around 444/445. Early music groups are all over the
place (although 415 and 440 have the largest following), but most pop and
commercial recording is done at 440 (if they wish to push the pitch they go
ahead and transpose).

I hesitate to subscribe to Ray Tomes' extraction of pitches from naturally
occuring frequencies (which do vary - sometimes in regular patterns, but
often eccentrically - from the values he gives(most reference books give
rounded-off values, anyways)), but La Monte Young's pragmatic use of house
current frequencies does have something to recommend itself (even if he
does have to modulate when he crosses the ocean).

It is curious that Tomes claims a fixed pitch for Ni. Besides the fact that
singers and instrumentalists vary widely in their choice of Sa (I have
heard everything from (in terms of 440A) G to Eb with C# the most frequent
), Ni can have several levels and would seem, from a musical-functional
viewpoint, to be an odd choice for a pitch standard.

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I agree with Daniel Wolf that Indian tuning varies. Using the
solfeggio system, Sa Ri Ga Ma Pa Da Ni Sa, which is used similarly
to Do, Re Me ... , the starting point for fsingers is very flexible.
I've seen Sa shown most often in western notation starting on
C. So I would expect A to correspond to Da rather than Ni.
According to Danielou (Northern Indian Music) which is controversioal,
but which is the one that I've seen which actually gives a
precise tuning, the tuning for Da could be 5/3 or 27/16 if it
is major, depending on the raga, and 8/5 or 128/81 if it is minor
(da komal). The major seventh (ni tibravra)can be either 15/8 or 243/128 (which
is fairly sharp). Basically one of the principal tunings for
each pitch is just iand one is Pythagorean. Danielou's hypothesis
is that each raga uses a specific pitch, depending on which way
it's tuned. Some ethnomusicaologists don't agree with him, but
I've trined the tunings of some of his ragas, and they sound
realistic, so I think it's likely that his informants did tune
this way, at least at the time he made his study (would have
been early in th e20e w20th Century, I think). Now some fixed pitch
instruments and even equal temperament are also affecting Indian
tuning, to some degree, though on an instrument such as the
mandolin(?) S!) Srinivas does bend the pitch sometimes. Even so,
and even though he's a famous virtuoso, I'd rather hear a sitar
any day.

Best,

Lydia Ayers

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-------------------- Begin Original Message --------------------

Message text written by INTERNET:tuning@ella.mills.edu

"I have always been under the impression that "European As" were 442/43-
but I suppose it could have been a dream!
"


-------------------- End Original Message --------------------

The Berlin Philharmonic, at least under ol' Herbie's direction (I'm not
sure about when they play under Daniel Barenboim), tuned at AD8Hz.

In the mid 1800s, while Boehm developed the modern flute, A was
standardized at 435Hz, and the current-day flute does not fully account for
the upward drift (they merely shorten the head joint, and have not adjusted
the toneholes relative to each other.

In Bach's time absolute pitch was even flatter still. Rumor has it that
it was close to C%6Hz, although I don't know that for sure.

The only explanation I've heard for this upward drift was from my
bassoon teacher from back in high school: "Nobody wants to be flat".

OK...

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Daniel Wolf wrote:

>I spoke with one of the better tuners in Frankfurt today. It seems that
>chamber music players can range from 435 to 442, western German orchestras
>and new music groups are at 442/443 and eastern orchestras (include the
>Berlin Phil) are up around 444/445. Early music groups are all over the
>place (although 415 and 440 have the largest following), but most pop and
>commercial recording is done at 440 (if they wish to push the pitch theygo
>ahead and transpose).

>I hesitate to subscribe to Ray Tomes' extraction of pitches from naturally
>occuring frequencies (which do vary - sometimes in regular patterns, but
>often eccentrically - from the values he gives(most reference books give
>rounded-off values, anyways)),

This is true. The exact theoretical Schumann frequency is 7.49 Hz but
it certainly is not a steady hum.

> but La Monte Young's pragmatic use of house
>current frequencies does have something to recommend itself (even if he
>does have to modulate when he crosses the ocean).

Use 300 Hz, that fits in with both and the earth resonance :-)

>It is curious that Tomes claims a fixed pitch for Ni. Besides the fact that
>singers and instrumentalists vary widely in their choice of Sa (I have
>heard everything from (in terms of 440A) G to Eb with C# the most frequent
>), Ni can have several levels and would seem, from a musical-functional
>viewpoint, to be an odd choice for a pitch standard.

My information comes from the book "An Introduction to the Study of
Indian Music" by E Clements 1912. Note that there are 7 different
frequencies used for ni depending on the harmonic meaning. Clements
uses a very nice notation (I don't know if this is standard) with flats
that have a little 7 in them for the 7/4 ratio ones (Hi Robert :-)
See the diagram at http://www.kcbbs.gen.nz/users/rtomes/in_music.gif
which shows all the notes and the harmonic relationships including
ratios of 2, 3, 5 and 7. The ni I refer to is the ni natural.
This may explain the points made by Lydia Ayers .

It may be that this is just a typical scheme and that others also exist.
I don't know about that, but the detail and consistency of Clements
scheme is very sound to me. Note that the diagram is not his structure
just the note names and frequencies.

However in my bid for A=450 I have yet to play my master card... Billy
Joel! I will write a post about Harmony and Rhythm which will explain
this.

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