temperaments - Ellis

[Ron Koval]

After questioning how Alexander Ellis measured temperaments a few weeks 
back, and getting no response, I decided to just dig in and read Helmholtz 
myself.  Well, I found that everything we've been discussing comes not from 
Helmholtz, but the appendix written by Ellis.  Very interesting, indeed, the 
cleverness used to get reliable pitch references.  Simply put, what he used 
were a series of tuning forks, carefully calibrated to be graduated by 4 
beats between forks. (about 15 cents at A440)  Then by comparing the beats 
between the note to be sampled, and two separate forks, the distance between 
could be found mathematically.  To achieve any amount of precision, beat 
counting was spread over intervals of 10, 20 or 30 seconds.

Hmmmmmm... Do you see a problem there?  Sure, this will work for an organ, 
maybe a harmonium, but a piano?  Any modern ETD user can tell of the 
non-stable nature of the pitch of a struck piano string.  Additionally, this 
process would lead to errors, considering the non linear comparison between 
the fundamental tone measured, and the upper partials used when tuning.  I 
think what we see is a false sense of precision based on a mathematical 
rendering of beat comparisons.  In fact what Mr. Ellis seems most interested 
in does relate more to the tuning of organs, where his main ideas might be 
summarized by:

The only good interval is a Just interval.

What's the big difference between the organ and the piano?  The ability to 
control the volume by touch alone.  The meantone temperaments, while having 
just intervals, do have those wolves, and there simply isn't any way to 
moderate the strength of those wolves on an organ.  But a sensitive pianist 
will be able to either hide, or bring out dissonence by the way they play.  
(This speaks to the recent idea of using a disclavier to test 
temperaments.... what's lost is the artistic interplay between performer and 
tuning.)

How does Mr. Ellis come to grips with the reality of tuning a fixed pitch 
instrument?  Convinced by the beauty of the mathematics of just intervals, 
he suggests that the only answer is to venture into microtonality, the 
splitting of the octave into smaller and smaller intervals, until any 
interval can be played to the mathematical justness he seeks.

He goes on to put all tunings in two boxes, which simply leaves no room for 
the Well temperaments.  He uses the terms linear(MT) and cyclic(ET) to 
separate the pythagorean/meantone from the myriad of equal temperaments 
using differing octave divisions.

Some quotes below from the Helmholtz/Ellis book - the Ellis appendix

     The advantage of the Cyclic over the Linear temperaments consists 
chiefly in a power of endless modulation - a very questionable advantage 
when harmoniousness is sacrificed to it. (p 433)

On the tuning of ET 36 note scale:

     The accumulation of almost insensible into intolerable errors besets 
all attmpts to tune by a long series of similar intervals.  Even octaves are 
rarely tuned accurately through the compass of a grand pianoforte.  But for 
major thirds and minor sixths ther is no chance at all (except by a real 
piece of haphazard luck) to get even one interval tuned with absolute 
correctness by mere appreciation of ear.

    It follows that all attempts to tune by ear must have grievously failed, 
wherever they depended upon considerable alterations of just intervals, and 
that even the laborious and careful training of modern tuners for obtaining 
the very slightly altered fifths and fourths of equal temperament can only 
lead them to absolute correctness 'by accident.' (p 484)


     In their endeavours to avoid the 'wolves' of meantone temperament 
musicians invented numerous really unequal temperaments, which it would be 
uncharitable to resuscitate. (p 435)

Here is the only mention of another type of temperament.... Proposed by 
musicians, yet dismissed by Ellis.  What's more, the temperaments that he 
measured, clearly unequal by his own measurements are lumped under the 
heading of ET to nicely fit into his theories.  So what we seem to have here 
is a mathematician's analysis of pitch and combined pitches.  No mention of 
artistry, or music, or performers.  What is missing is at the crux of 
musical expression; contrast, tension /release, loud /soft, fast/slow.  This 
is the chief component of the Well temperaments that have been described, 
and are currently being tested by list members.

Ron Koval








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