Klarinet Archive - Posting 000352.txt from 2004/11

From: "Keith" <100012.1302@-----.com>
Subj: [kl] RE: pitch standard
Date: Tue, 9 Nov 2004 02:18:02 -0500


Orm1ondtoby,

The Pythagorean system seems to be used for intonation (minimising beats)
but is not used to derive the diatonic scale. Apart from the "comma" (the
difference between five octaves and eight fifths) that you point out, there
is no Pythagorean harmonic anywhere near a fourth, until you get so way up
high that amplitudes are very low. And the minor thirds are dodgy.

The construction of the scale is most sensibly seen (I think!) by the
approach first given by Helmholz (in Tonempfindung, available in Dover "On
the sensations of tone"). We can tune octaves, fifths and triads easily by
absence of beats. So we get the scale by the major triads based on the tonic
(I), the dominant (V) and the subdominant (IV). The subdominant is tuned a
fifth below the tonic.

Note that in this method, which is probably used instinctively in orchestral
tuning, the major thirds in the scale (eg C to E, F to A and G to B) do not
have the same ratios. So how they are tuned will depend on the harmonic
context.

Joe Fasel's point about choirs being urged to tune thirds wide, does have a
little mystery. I believe it happens in string quartets too. The actual
ratio will vary a little, depending on whether the prevailing harmony is
tonic, dominant or subdominant, and I don't know whether this is the
explanation. Another possibility is that the third is urged high, not so
much to preserve pitch as to make very clear the "major" character of the
chord; the tension between major and minor modes being a pivotal part of
music since the baroque. However, you can hear a good choir lock in to
freedom from beats at exact whole number ratios at both the major and minor
thirds.

Piano tuning is an art all to itself (as you say, string masses and
perceptions come into it) and has nothing to do with orchestral tuning
except the choice of A. Orchestras do not play in equal temperament.

The statement "G# is not the same as Ab" is true but only a bit of the
story, and I think it is misleadingly inadequate. Because G# is different
from G#, depending on the harmonic context (e.g. whether it is the seventh
of the tonic or the third of the dominant in the key of A).

Keith Bowen

-------------------------------------------------
<ormond1toby wrote>

"Pythagorean" temperament:

if you begin at (say) Ab and move upwards by repeated multiplications of
3/2 --- that is, if you move upwards by fifths --- you will hit all twelve
semitones (but not in alphabetical order) and eventually you will
reach G# in a higher octave. Since we 'feel' a strong similarity
between two frequencies that are exactly one or more octaves apart (2/1,
4/1, 8/1, etc), we can collapse this series of frequencies into a single
octave without damaging our 'sense' of fifths. Finally, if we arrange
the frequencies in numerical order, we have the Pythagorean temperament....
ooops! except that....

Notice I said that you will reach G# --- not Ab. It turns out that
there is no way express the octave relationship *exactly* as multiples
of fifths. That is, there is no 2^n which exactly equals (3/2)^n. In
the Pythagorean scheme of fifths that I outlined above, the final G# is
not quite the same as Ab. In a similar way, your "intuition" that A#
and Bb aren't identical is correct. An adjustment of some sort is
required in order to maintain octave relationships. In the case of a
piano, other adjustments are required because piano strings have varying
masses and don't behave in the 'ideal' manner.

(If you do all the arithmetic, you'll find that the final G# in my example
is actually 531441/4096 = 129.746, not the clean-cut 128 for which you would
hope.)

"Just" temperament:

If you still 'feel' that each interval should be a relationship between
small integers, but you don't like the Pythagorean approach, a different
list of frequencies can be obtained by appropriate multiplications and
divisions with 2/1, 3/2, 4/3, etc. (I think you need to skip over many
of the integers and go up into the 20's in order to obtain all twelve
semitones.)

Once you have these twelve "Just" ratios, you can arbitrarily choose A=440
and multiply it by the ratios in order to obtain frequencies for all twelve
semitones.

-------------------------------------------------

---------------------------------------------------------------------
Klarinet is a service of Woodwind.Org, Inc. http://www.woodwind.org