geometrical mean (Digest 1513 Topic 4)

Hi Jan (et al),

At 04:01 PM 8/24/98 -0400, you wrote (in Digest 1513 Topic 4):
>From: Jan Haluska
>Subject: geometrical mean
>
>if X , Y are two tone frequencies,
>how to tune the value Z=sqrt(X*Y) acousticaly
>( sqrt(X*Y) denotes the geometrical mean of the tones X and Y )?
>Demonstrate the method on X = 1, Y = 5/4 .

I'll give it a try (while I have insomnia).

It's easy if you're in an Equal Temperament. Take 12tET on a conventionally
tuned piano. If you number the piano keys left to right from 1 to 88, you
can just take the average (AKA the "arithmetic mean") of the two key numbers
to get the tone in question.

For example, take the C and E at the bottom (left) of a piano. The C is the
4th key, and the E is the 8th key. The "answer" becomes (4+8)/2, = 6. The
6th note is D.

The geometric mean of the frequencies is the arithmetic mean of the note
numbers. It works this way because as you go up from note to note, the
frequencies go up exponentially. The math is kind of indirectly relying on
this property:
S^(M+N) = S^M * S^N
where "*" means "times", and "^" means "to the (power of)".

Z is the "midpoint" between X and Y, in the sense that you want the interval
from X to Z to equal the interval from Z to Y. As long as there is an even
number of semitones between X and Y, the Z will also be an integer. But if
they're an odd number apart, Z will lie on a "quarter tone". Examples:

X Y Z
- -------------------- --------------------------------------
C C (an octave higher) F# (between the C's)
C G (a "fifth" higher) E half-flat (halfway between E and Eb)

With Just Intonation it's the same math, but... If you're using "justly
intoned" notes, it's less likely that the resultant "average" note will also
be justly intoned. Examples:

X (1/1) Y Z
------- --------------------- ---------------------------------
C G# (25/16, two "major sqrt(25/16), = 5/4, = E
thirds" higher) (one "major third" up from the C)

C E (5/4, as in sqrt(5/4), approx 1.118, approx D
your example)

As long as the interval from X to Y amounts to two of the same just interval
side by side, the result is just one of that interval. But if the result of
the square root is irrational, then you're departing from Just Intonation,
since it's based on ratios (AKA "rational" numbers).

I hope this info leads you to what you're looking for (and I hope all this
makes sense in the daytime).

--Mark
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| Mark Nowitzky |
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