temperaments:clear as mud?

[Ron Koval]

OK everyone,

Here's a new angle I haven't heard before.  In my spreadsheed meanderings, 
I've come across a simple fact that might possibly add a whole new dimension 
to the ET/HT discussion.

It is said that ET has no key color, that all the thirds are the same, while 
optomizing temperaments have key color because there are differing widths to 
the thirds.  There are a few people out there that claim to hear key color 
in ET.  This can't be possible if all the thirds are the same, can it?

I've spent a fair amount of time graphing many different temperaments, and 
then setting them on pianos and comparing what the graphs predict, verses 
what I hear.  The graphs are a very seductive tool, with their nice bell 
shaped, inverted V's, all seeming to fit in a circle of fifths order.  
However, most temperaments don't 'play' quite as predicted.  Take the Thomas 
Moore temperament, which I use as an introduction quite often.  It has a few 
keys that are closer than pure and then most keys are predicted to be evenly 
less pure. When tuned, however, the third Ab-C SOUNDS less pure than the 
thirds that are supposed to match, according to the graphs.

Soooo..... What is it that we hear?  Oh yeah, those beat thingys!  Think for 
a moment about ET.  Does it sound equal?  Not likely.  Take our 'landmark' 
F-A third of around 7 beats/second.  What's the beat rate an octave higher?  
Yup, around 14 beats/second.  So the thirds in the keys are NOT the same, 
they progress in a chromatic fashon from really slow in the bass to really 
fast in the high treble, doubling every octave.  Go up, the energy 
increases, go down the energy decreases.

An optomizing temperament changes this equation.  Remember Bill Bremmers 
directions for EBVT?  In one octave the F, G, and C major thirds all beat 
THE SAME.  Bb and A thirds beat THE SAME. Gb and Ab beat THE SAME.

Only by using an optomizing temperament can you get equality, sameness, or 
chords that match.  ET only gives us varying thirds.  There is NO equality 
to beats (sound, tension, etc) in equal temperament, only equal differences 
between beat rates of equal steps.

(picture the end of Soylent Green: "THERE IS NO EQUALITY IN EQUAL 
TEMPERAMENT!")

Hmmmmmmmm......

more to chew on.

Ron Koval
Chicagoland
(Yes, I've been graphing predicted beat rates, yes, it's confusing and 
interesting at the same time!)

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