May the 4ths and 3rds be with you

[Richard Moody]

Robin wrote....
    > the actual ratio of  beating of equally tempered
> major thirds necessarily  being the same as the ratio of the
tempered
> frequencies of the major thirds, that is 2^1/3 as you point out.


Thanks,

    While it might seem to most to be a picky-unish point to make,
because it really only has relevance in tuning theory it does seem
that a lack of understanding of the nature of beats held up the
accurate tuning of ET since fixed tone keyboards started to be
tuned.  (1300's?)
    Had someone realized the beats of the thirds in ET should
increase gradually and double each octave, the instructions of
Braid White could have been written in 1615 instead of 1915.  And
this could have been proven at any time in history with a tuning
hammer, no monochord needed.  The cause of beats was finally
discovered by Helmholtz in the 1860's, but it took 40 more years
to construct a tuning scheme for ET using 3rds as proofs.   At
least that is the present state of knowledge about the history of
tuning ET.   I am actively researching this history and have not
come up with using 3rds in ET before 1900.
     The James Broadwood "article" suggests using twelve tuning
forks, that was in 1811.  He also offered a crude way to guess how
much the 5ths should be flattened, but it seems highly
impractical, the twelve forks probably the most practical way for
ET until the beats of 3rds were used.   His suggestions were to
the home or musician tuner with no indication how it was done in
his factory. In the article are references to another publication
(Stanhope) and an encyclopedia article (Ree's).
    When one attempts to tune Meantone by ear you soon realize how
to be content with an approximation rather than getting each 5th
contracted the same to produce the pure 3rd.  Their understanding
of  meantone (1500-1700) shows knowledge of  the two commas,
schizmas, dieses resulting from various schemes for scales and
temperaments but their actual tuning results must have varied
widely from the theoretical.  Today with taking the theoretical
knowledge of 3rds and applying to meantone, I am sure we can get
much closer and consistent results than the tuners before.

----ric  (ramblinagain)

"But, being still of opinion, in spite of all I have read and
heard, that mathematical speculations cannot be of any practical
use in directing the tuning-hammer (a mere mechanical operation,
guided by the ear, as the brush of the painter is by the eye)"
James Broadwood 1811



----- Original Message -----
From: Robin Hufford <hufford1@airmail.net>
To: <pianotech@ptg.org>
Sent: Tuesday, August 20, 2002 4:05 AM
Subject: Re: May the 4ths be with you


> Ric,
>      You are right on, something which I have for quite some
time been
> myself aware of,  as to the actual ratio of  beating of equally
tempered
> major thirds necessarily  being the same as the ratio of the
tempered
> frequencies of the major thirds, that is 2^1/3 as you point out.
For
> that matter, this applies,  similarly,  to all other  tempered
> intervals.   I think it does behoove us professional "Piano
Tuners" to,
> at least, understand this correctly, something which, at the
moment
> appears to be, in general, somewhat misconstrued  particularly
as
> represented by postings to this list.