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.
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