Thomas Jefferson Tuning Scheme I

[Richard Moody]

A portion of  the Thomas Jefferson tuning scheme....

Note      Freq      TEST    5ths      3rds

A 4	441.000		0.000	27.563
G# 4	418.605	 20.672	-16.90	-2.361
G 4	392.000 -24.50	0.000	24.500
F# 4	372.094		0.000	-2.098
F 4	348.444		0.000	21.778
E 4	330.750		0.000	20.672
Eb 4	309.728	-19.358	0.000	19.358
D 4	294.000		0.000	18.375
C# 4	279.070		0.000	-1.574


The "Test thirds" beat rates.  
	E4--G#4.. 20.6    Eb4--G4...19.3  G4--B4...24.50.  

Now these are based on frequencies arrived at by tuning pure fifths
according to the scheme Jefferson wrote down.  This is a sample
of the temperament.  I will post the whole scheme if it will appear
with the columns intact. 
We DO NOT know that Thomas Jefferson indeed tuned pure fifths.
However to research what this tuning might have been, I decided 
to start with pure fifths.  The figures were arrived at by using a 
spread sheet which I will send by email if any one desires. It is in 
MS Works, and can be sent in flavors of Quatro Pro, and perhaps
EXcel. Let me know what your software is and I will see if MSWorks
saves it to that.  
	The beat rates in the   3rds    column are based on the root. 
Thus the C#4 rate (-1.5) is that of C#4--F4.   D4--F#4 third beat
rate 
is 18.375.   The beat rates in the TEST column are from the root ex-
cept the G#4  whose root is E4.  It is that way bacause that is the 
way Jefferson noted it. 
	The rates may appear very fast, BUT as noted in the first 
post, they are actually "sub-harmonics" of the root frequency.  In 
Wm Braid White, "Piano Tuning and Allied Arts" under Coincident
Harmonics, (p. 55) he notes that the difference of the two
frequencies of a 
perfect fourth (G35 and C40) is 65.41 which he states "at this speed
they
will coalesce into a continuous sound sensation of ... C16 two
Octaves 
below C40." 
	I am wondering if the rapid beat rates of the thirds which are 
ratios of the root will "coalesce" the same way, thus giving the
sensation
of pure thirds rather than beating thirds.   Very soon I will have a
piano 
to try this out.  If anyone has investigated this phenomenon it would
be
interesting to hear about.  I tuned some Pythagorean (pure fifths)
temps
last summer, but did not know from calculations the predicted beat
rates.
I remember some very slow and also very fast beat rates.  But I can 
see that perhaps a very slight error in the fifths might create a
very fast third, 
instead of a "coalesced" third. 

Richard Moody 



----------
> From: Billbrpt
> 
.  The curious ">>" marking may have been Jefferson's "fudge"
interval.
....he may have been sensitive to how
dissonant he would
> allow his "wolf" interval to be. 
---------

Actually the curious marking is   <<  as it appears in C#<<G#.  This
is the interval before the wolf which is G#--Eb.  Now if this
interval is flat because of tuning pure fifths, one could imagine
that << means tune G#  flat, or "towards" C# in an attempt to widen
the narrow fifth.   But the wolf is so flat(19 bps), he would have
created two wolves instead of one.  
	The other possibility is that Jefferson knew to tune the fifths as
flat as the ear would bear. (the universal tuning instruction) This
would come under "ear tuning" Jorgensen mentions. (tuning intervals
harmonically  rather than by beats).  But that is for another spread
sheet, contracting 5ths until something dramatic happens to the
thirds, but still we need the  perfect fifths sheet for comparison.
rm 


Jeffersons tuning scheme. 
G3--G4,  G4--D4,  D4--A4,  A4--A3,  A3--E4,  E4--E5,  E4--B4,

		Test G4 - B4

B4-- 3,  B3--F#4,  F#4--F#3,  F#4--C#5,  C#5--C#4,  C#4<<G#4
G#4 -- G#3. 
		Test: G#4 -- E4. 

G4--C4,  C4--C5,  F4--G5[*], F4--F5,  F5--Bb4,  Bb4--Bb3,  
Bb4--Eb4,  Eb4--Eb5, 
		Test: Eb4 -- G4.
			rm