Equal Temperament, history of, judgement of

[Richard Moody]

----- Original Message -----
From: <Billbrpt@AOL.COM>
To: <pianotech@ptg.org>
Sent: Wednesday, February 27, 2002 8:22 AM
Subject: Re: Equal Temperament, history of, judgement of


| >     Braid White, I suppose, did not explain the 4:5 ratio of contiguous
ET
| > 3rds because it does not exist. The ratio is actually 1.25992105, Which
is
| > sharp of the ratio of 5/4 by 13.686 cents.  (1.25992105/1.25)
| >
|
|{this} statement must be some kind of misunderstanding.
| Contiguous 3rds in ET will beat at a ratio of 4:5.

This might seem so because multiplying the fifth and fourth partials by 4
and 5
produces the beats of the 3rds, and looks like they should also have a
ratio of
4/5.   However if you actually compute the ratio of the beat rates you find
they come to 1.2599.  The ratio of 5/4 is 1.25.   If you think this
difference is close enough think what would happen if the fundamental
frequencies of the 3rds were computed at 5/4.  You would have quarter comma
meantone, a huge difference from ET

Take the rates of F3--A3   and A3--C#4 or

                8.73/6.93     the calculator shows  1.25974025974

        Taking the rates to 3 decimals it is    1.25992105 very close to

                            the ratio of 5/4  or 1.25

If this seems insignificant the difference between in cents
is.
               actually 13.685 when 5 decimal places

This also happens to be the cents difference of 3rds sharp from pure.

It is interesting to note that when the fundamental frequencies are
compared

                F = 174.614    A=220     220 divided by 174.614  equals

                                1.259921

Therefore the rates of the 3rds follow the ratio of the the fundamentals
 of  3rds in ET.   That rate is close to 1.26   not 1.25 or 5/4

The cents difference between 1.26 (to 5 places) and 1.25 is 13.686

The difference between pure and ET 3rds is 13.686.

|Not | understanding this concept will inevitably lead to uneven
progressions
|of 3rds & 6ths.

Using the ratio of 4/5 of beat rates will lead to 3rds being off and will
not produce the most important concept of all, that 3rds double in rate
every octave, that a 3rd from tonic will beat the same as a 10th from
tonic. because the octave formed by four 5/4 3rds will be flat.  Here is
the math.

Take    F3--A3--C#4(Db)--F4--A4 .   Here F3-A3 beats at 6.93.  Therefore
F4-A4 beats at double that or 13.86.

Now if 6.93 bps of F3--A4  increases by 5/4 then
6.93*5/4 = 8.6625 or rate of A3--C#4
 then...   8.6625*5/4 =  10.83 or rate of C#--F4
 then    10.828125*5/4 =  13.535  of F4--A4

This may seem close to the theoritical 13.86 bps but actually differs by
53.525 cents from 13.535 derived by 5/4 increase.


|Show me a single | piano that was ever tuned so that all of the beat rates
exactly |matched what  Braide White wrote.  If you could (and you can't),
I'll tune a piano right
|next to it that sounds a hell of a lot better, in ET, not to mention EBVT.

It is of course important to realize the beat rates are theoritical, they
are just a guide. But the most precise guide we have. As far as exactness,
one can argue there will never be a piano exactly on 440 so that alone will
alter the rates from the tables.

The concept of exactness in tuning expressed by Wm Braid White......

    "If we can estimate the number of beats that should occur between the
sounds of any given equally tempered interval, we can always tune such an
interval in the Equal Temperament by noting the number of beats and
adjusting this to the theoritical numbers in the  calculations.    It is
not possible to accurately to follow the number of beats that are supposed
to be be between any given intervals in Equal Temperament even when the
pitch of the tonic is precisely the same as in the calculations.    It is
not possible, therefore, in practice, to tune with such accuracy as theory
would demand, but an approximation may be obtained. "  (p 126)  _Theory and
Practice of  Piano Construction_  (1906)

---ric









Now concerning the misunderstanding that "Contiguous 3rds in ET will beat
at a ratio of 4:5."    How can the beat rates of ET contiguous 3rds be 4/5
when the frequency ratio of ET 3rds is NOT 4/5?   Or do you mean that the
3rds beat according to the ratio between the 5th of the tonic and the 4th
partial of the 3rd above?

Contiguous 3rds would be for example C4--E4, E4--G#4.
Beat rate of C3--E4     10.38
Beat rate of E4--G#4      13.08

Ratio of 10.30/13.08 =  0.7935779816514
ratio of    4/5              =  .8

The difference between .8 and .7935 might not look significant but it is
actually a difference of -13.954 cents.   If the beat rate doubles each
octave then a beat rate of 5/4  increase would leave a series of 4
contiguous 3rds flat or beating slower than theoritical.
 Take    F3--A3--C#4(Db)--F4--A4 .   Here F3-A3 beats at 6.93.  Therefore
F4-A4
beats at double that or 13.96.

Now if 6.93 bps of F3--A4  increases by 5/4 then
6.93*5/4 = 8.6625 or rate of A3--C#4
 then...   8.6625*5/4 =  10.83 or rate of C#--F4
 then    10.828125*5/4 =  13.535  of F4--A4

Now in cents the difference between 13.535 bps and 13.96 bps is  53.525
cents.  WOW  this seems astounding because of only half a beat per second
difference.  Yet in cents that is what my spread sheet shows.   But since
it is an accumulation of 4 congitougous 3rds in error of 13.686 cents in
the beat rate, then 13.686* 4 = 57.44 cents.  Remember it is the 4th
contiguous 3rd that actually has the double rate of the starting 3rd.  Here
a  3 cent difference is probably due to rounding off.  (See that wonderful
article in the Journal called "Making Sense of Cents"  ; )   Or the
computer's accuracy in calculating logs.

Anyhow  the  4:5 rate of contig. 3rds does give an indication that they
decrease proportionally.   However it is easy to find more meaning in the
5/4 ratio.   It is the fifth partial of the lower note (or tonic) that is
coincendent to the fourth partial of the upper note. This calculated for
all the 3rds reveals the beats double each octave.  From this we realize
that the beat rate of the "3rd from the tonic equals the 10th from the
tonic. " This can be expressed as a 5/4/2 ratio.  Or that a  5/4 ratio
beats the same as a 5/2 ratio in beats in ET.   Now if the interval was a
pure 5/4/2 ratio where the fifth partial of the tonic equals the fourth
partial of the 3rd above and that partial equals the second  partial of the
octave from the 3rd AND the10th above the tonic, all these are equal,
there would be no beats.  However it is the difference of the partials from
a 5/4 ratio that makes the 3rds beat, so if the 3rds beat then it is not
"exactly" a 5/4 3rd.     Whether congruous or successive or contiguious, it
is the (tempered) 3rds not exactly equal to 5/4 in frequency that beat.
For ET to be "exactly" right,  it is the ratio of nearly 1.26 instead of
1.25 (or 5/4) that accounts for the actual rate of the 3rds in ET.


And it is the numbers that "Braid White wrote" that enable the closest
approximation.


 ---ric