Musical scales

Bill Hibbert bill at
Sun Dec 3 22:15:49 GMT 2006

Andrew Wilby:

> I find this discussion rather surprising and depressing.

This is necessarily a rather involved reply to Andrew's note, 
correcting some possible errors of fact and also advancing some 
theories of my own. Non-musicians move on now!

> It's all JS Bach's fault! By developing equal temperament ... by 
writing the 48 Preludes and Fugues for "the well tempered clavier"

I believe it is now generally accepted that Bach was not composing 
for equal temperament but for an unequal temperament of his own 
devising. The pieces in the '48' are written to take advantage of, 
and avoid the disadvantages of, the different effects of the various 
keys, i.e. it is clear from the details of the compositions that each 
key would have sounded different. The reason there are 48 is simply 
that there are two pieces for each of the 24 major and minor keys. 
Instruments did exist around Bach's time with split 'black notes' 
providing both versions of e.g. C# and Db but I don't think Bach was 
composing for these. 

> the "sweet" (ie in tune key) was say C, a more distant key such as 
say E would sound rather coarse and distorted.

The classic example of this is to play in Ab major on a keyboard 
tuned in meantone, a sound so vile as to drive one in pain from the 
keyboard. What Bach was demonstrating in the 48 was not that all keys 
sounded the same in his 'well' temperament, but that the worst 
horrors of the remote keys in meantone had been removed while still 
keeping a distinct difference between keys.

> Equal temperament has done quite a lot of damage. I was listening 
to a demonstration of a digital virtual organ ...

Just as it is not possible to demonstrate bell partials by striking 
chords on a piano (cf the Coventry case in the 20s) it is also of 
limited value to demonstrate different temperaments on an electronic 
instrument. In traditional instruments (stringed keyboard 
instruments, organs etc.) there is quite a degree of coupling between 
the various vibrating elements. In the case of pianos and 
harpsichords, the strings are coupled via the soundboard. To hear 
this, play a single note on a piano with the sustain pedal up and 
down. In the latter case, multiple strings will vibrate in sympathy 
with the one hit. In organs, the coupling between pipes is both 
through the air around the pipes and also through the shared air 
supply. This is a problem for organ tuners trying to tune unison 
pipes - they can sound in unison when played together even if they 
are tuned slightly apart.

The effect of this coupling in non-electronic instruments, I believe, 
is to pull slightly mistuned strings or pipes into tune when they are 
played together. This means that some of the dullness and beating 
heard in equal temperament on an electronic instrument is mitigated 
in a 'physical' instrument and equal temperament sounds sweeter than 
one might expect from theoretical or electronic considerations.

I discovered to my surprise a while ago when analysing recordings by 
very good unaccompanied choirs, that choirs do not sing in Just 
tuning as theory might suggest, but in something nearer equal 
temperament. The reason I believe is that we have become accustomed 
over the last 100 years or so to the sharper thirds of equal 
temperament - the flatter 'correct' thirds of Just sound, well, flat! 

Finally, I believe that none of this applies to the tuning of the 
nominals of change-ringing bells, though it does apply to carillons. 
The differences between the various temperaments and tunings are in 
general rather less than the changes in bell pitch due to disposition 
of the higher partials, which can be up to 1/4 of a semitone or even 
more in very thin or thick bells.

I do agree that, whenever the subject of enharmonic notes comes up on 
this list, we seem to repeat the same debate, when the underlying 
musical theory and terminology is well established (e.g. and many other references).


Bill H


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