history of tuning Pythagorean

[Fred Sturm]

	A couple relatively brief sources are articles in the New Groves (big, 
multi-volume reference book on music and musicians) with articles on 
Pythagorus and on Pythagorean tuning; and more briefly in the Harvard 
Dictionary of Music. Any college or university music library should have 
both.
	Pythagorus is mostly known to us through Aristotle, who commented on many 
of his predecessors and contemporaries. The general lore that came down to 
us about Pythagorus, in reference to music, has to do with whole number 
ratios as they pertain to musical intervals. Particularly the ratios 2 : 1 
(octave), 3 : 2 (5th), 4 : 3 (4th) and the derivative 9 : 8 (second), which 
is the difference between where the 4th and the 5th lie within the octave. 
These ratios were seen in terms of length of string, or possibly weight of 
object (there is an apocryphal story about Pythagorus hearing two metal 
smiths hammering against a couple anvils, which produced a tonal difference 
of a 5th. And that he weighed them and found the ratio of their weights to 
be 3 : 2).
	This general information was very much a part of medieval thinking, and 
music was one of the seven Liberal Arts of the time, closely associated 
with astronomy. Both were studied in numerical terms, dealing with ratios. 
The problem of dividing the octave was puzzled over for centuries by many 
minds, and the obsession has continued to modern times. Pythagorus noted 
that a "circle of 5ths" doesn't quite equal an octave, but comes 
tantalizingly close. 3 : 2 simply won't resolve to 2 : 1. He also is 
credited with discovering the "irrational" ratios - ratios between the 
diagonal and side of a square or between diagonal and circumference of a 
circle. He proved that these ratios couldn't be resolved to whole numbers, 
no matter how large (an idea we take for granted, and see as endlessly 
receding numbers past a decimal point). Quite an amazing breakthrough in 
thought, and one that he found very difficult to accept.
	Anyway, bottom line, besides perfect 5ths and 4ths, what is known as the 
"Pythagorean tuning" (not temperament, as it isn't tempered) is mostly 
known for its very wide major thirds (around 16 bps in the temperament 
area). It is the difference between these "Pythagorean thirds" and "pure" 5 
: 4 thirds that is termed the "syntonic comma."
	That's plenty of that. Back to bulk tuning in preparation for our 
ridiculously early fall semester (1 1/2 weeks to go).
Regards,
Fred Sturm
University of New Mexico


--On Tuesday, August 12, 2003 12:56 AM -0500 Richard Moody 
<remoody@midstatesd.net> wrote:

> 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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