Equal Temperament, history of, judgement of

[John Delacour]

At 1:01 AM -0600 24/2/02, Richard Moody wrote:

>The refinement of ET, I believe,  was not possible until the temperament
>could be checked against the beating of 3rds and 10ths. (However I would be
>willing to offer an example of ET composed of only 5ths and 4ths, to a
>panel for tuning evaluation.) The beast of 3rds were known but their
>nature, ie - how fast or how slow was not possible until Herman Helmholtz
>published his findings on the mathematical relation of partials.  But he
>did not elucidate on the beat rate of 3rds.   Even in the translation by
>Ellis ca 1880, the method of computing specific beat rates for specific
>intervals was given but I did not find a mention of checking ET by 3rds, or
>the 3rd--10th test, or that the beat rate of 3rds doubles each octave.

Here is the scale as laid at the Brinsmead factory at the latest in 
the 1860's.  Helmholtz came very late on the scene.  Already in 1834 
Scheibler had produced devices that enabled very exact tuning by 
means of forks that produced precisely four beats per second when 
sounded with each note in the scale.  Sets of forks could be produced 
for any pitch or temperament.

I know one tuner who was trained with the aid of a beat counter but 
learning to recognise the roughness of the thirds is something that 
has to come with long practice, I'm sure aural tuners will agree, and 
an experienced tuner no longer actually counts the beats.

I would emphasize again that I am not interested in running down 
"historic temperaments" but only in clarifying the nature of the 
history.  When I hear of "Victorian" temperaments I am rather 
sceptical when it is established fact that for the great part of 
Victoria's reign equal temperament was the commercial norm although 
the country she reigned over was one of the least musically 
progressive.

As Hipkins told Ellis, "it takes a quick man three years to learn how 
to tune a piano well in equal temperament by estimation of the ear. 
Tuners have not time for any other method".

_____

Tune C_52 to fork
Tune C_40 a perfect octave (without a wave)

C_40 + tune< G_35 a trifle flat
G_35 + tune< D_42 a shade flat
D_42 + tune< A_37 slightly flat
C_40 + tune> F_34 a shade sharp

TEST F_34 + A_37 + C_40

F_34 + tune> Bb_38 a wave sharp

TEST F_34 + Bb_38 + D_42

A_37 + tune< E_34 slightly flat

TEST G_35 + C_40 + E_34

E_34 + tune< B_39 slightly flat

TEST G_37 + B_39 + D_42

B_39 + tune< F#_45 slightly flat

TEST F#45 + A_37 + D_42

F#45 + tune< C#_41 slightly flat

TEST F#_34 + A#_38 + C#_41

Bb_38 + tune> Eb_43 a shade sharp

TEST G_35 + Bb_38 + Eb_43

Eb_43 + tune> Ab_48 a shade sharp

TEST Ab_36 + C_40 + Eb_43

TEST G#_36 + B_39 + Eb_44

TEST Db_41 + Ab_36

TEST  F_33 + Ab_36 + Db_41

"...the trial C# to G# will then prove if the 'fornt' and 'back 
scales' are properly tuned, as G# should be slightly flat to C#
The gradations of sharp and flat intervals have been mentioned as 
though well defined, but this is not the case in tuning, as they 
should be perceptible to the practised ear of the tuner alone."

JD