Klarinet Archive - Posting 000319.txt from 2004/11

From: orm1ondtoby@-----.net (Ormondtoby Montoya)
Subj: Re: [kl] Re: pitch
Date: Mon, 8 Nov 2004 09:27:56 -0500

Rien wrote:

> You give three different frequencies, and I
> know that an octave means a doubling of
> frequency [snip] On a piano A-sharp and
> B-flat are the same key, and thus the same
> frequency. But to my intuition B-flat is slightly
> lower then A-sharp.

How to divide the octave? My tuner has settings for nine different
temperaments. I presume that musicians have devised more than nine
methods over the centuries.

"Equal" temperament looks for a number which, when multiplied by itself
twelve times (since there are twelve semitones in the Western system)
will yield exactly 2.000 (one octave). This number turns out to be
approximately 1.05946. If A is arbitrarily chosen to be 440, then A#
will be approximately 440 x 1.05946 = 466.16 (depending on how you round
the numbers).

All the other temperaments are 'unequal' by definition. Most (all?) of
them are based on the idea of harmonic ratios --- that is, ratios of
small *integers*. Thus an octave is 2/1 rather than (say) 2.01/0.99.
A fifth is 3/2 rather than (say) 2.99/2.01.

"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.

And last but not least, you can say: "These particular pitches don't
sound quite right in this particular situation, so I'll move one of them
a little bit...."

....and then your embouchure wobbles and you miss the pitch anyway....
):-0

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