history of tuning Pythagorean

[Richard Moody]

Nothing written by Pythagoras himself has come down to us.  He is
famous for the Pythagorean comma which is the ratio of the
difference of twelve notes tuned to pure 5ths from the pure
octave.  This was a mathematical exercise in early Greek history.
He didn't advocate a temperament because there were no instruments
in his time that benefited from tempering.
    When keyboard instruments came about in the 1300-1400's some
historians believe that some people tuned them by perfect 5ths and
endured a wolf, or the difference between the 12th 5th and the top
note of the starting octave. Because no actual historical records
have been found of this procedure, this seems to be the
"provincial tuning" alluded to by some writers.
    The most comprehensive explanation and exploration of
"Pythagorean tuning" I have come across is in a pamphlet with the
unlikely title, "Theory and Tuning: Aron's Meantone Temperament
and Marpurg's Temperament I"  (I, as the capital of i) by John W.
Link, Jr.  This was published in 1972 by Tuners Supply but is most
likely out of print now. I have a copy machine copy of it and
would be glad to lend it to you.
It covers such things as how and why seemingly pure 3rds occur in
Pythagorean tuning, (they are actually augmented 2nds or
diminished 4ths I forget which).    He also presents a theory
based on resultant beats that attempts to explain why this
temperament sounds so good for certain intervals.
    This booklet runs 130 pages  (65 8x11 1/2 typewritten sheets
folded).  I think it would be worth while to scan and OCR if
indeed it is out of print.  I'd volunteer but with a 40 dollar
scanner and its "free software"  I don't think so.  Much easier to
mail it to you and you send it back.  Unless there are 10 or more
interested in spending 10 bucks on a digital version.   Hmm I
wonder if Kinkos does scanning and OCR ing.....?

    Richard Moody

ps From an educated guess to tune  a "Pythagorean", start from
middle C (C4) and tune pure 5ths up and pure 4ths down until C# 5
then C#4 down.  Next from C4  tune pure 5th down (F3)  pure 4th up
(Bb3)  pure 4th up (Eb4) pure 5th down (Ab3), now a  pure fourth
up would be Db4 but this has already been tuned as C# 4.   This
interval Ab3 and C#4  (augmented 3rd?) (yet looks like a 4th on
the keyboard) sounds awful, but this is the Pythagorean comma, the
discrepancy predicted 1000 years before the keyboard was invented.
    Another way might be to tune up to G# and only down to Eb
leaving the wolf between G# and Eb which is where the meantone
wolf is usually found.   This is also covered by John
nk.   ----rm


----- Original Message -----
From: Elwood Doss, Jr. <edoss@utm.edu>
To: College and University Technicians <caut@ptg.org>
Sent: Monday, August 11, 2003 9:25 AM
Subject: history of tuning


Anyone have a good source regarding Pythagoras and his
contribution to tuning--especially the Pythagorean temperament.
I've studied Jorgenson's big red book (I happened to purchase one
in 1994--even though I had little money at that time and less
sense) and have gotten some information.  Any class notes, books,
etc. would help.  I'm teaching a tuning class this fall and want
to begin with a brief history of tuning.  Any help would be
appreciated.

Joy!
Elwood

Elwood Doss, Jr., RPT
Technical Director/Piano Technician
Department of Music
225 Fine Arts Building
University of Tennessee at Martin
731/587-7482
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