well tempered temperaments

[A440A@aol.com]

John writes:

<<  I was indeed asking how to tune a well 

temperament for an aural tuner, though all that didn't seem to come out 

:)


A few questions if you don't mind:


>>What is 1/12 of a comma?
Just a shade under 2 cents,  real near an ET fifth.  



>>When doing step 3 in the P.B. '93 well temperament (and step 3 of 

Kirnberger) where tuning the Eb, Ab, etc. via just fifths I presume you 

are to cover over 2 octaves? <<

I assume Paul would have you extrapolated via the octave to keep the bearing 
plan with one.  
 
>>And when it says "Spread this around the 

keyboard" I presume that is tuning those notes in different octaves 

around the keyboard?<<

That is the way I assumed it to mean.  Paul can give you more details, (I 
expect him to be jumping in here for updates on his temperament, soon enough).


>>I am particularly interested in the Young 1799 temperament, which 

seems to be highly regarded. Does anyone have a details that could help 

an aural tuner?<<

  Yes,  The Young temperament was an ideal of perfect balance, however, on 
the piano, it may be that the 21 cent F#-A# is too much for some.  I used to use 
this as an introduction to temperaments for the uninitiated, but found that 
too high a percentage of customers were put off by it.  If you want to try it, 
I would suggest that you follow Thomas Young's abreviated version of 
instruction as published by Owen Jorgensen.  

 "In practice, nearly the same effect (temperament),  may be very simply 
produced, by tuning from C to F, Bb, Eb, Ab, Db, Gb,  six perfect fouths; and 
again, from C, tune six equally imperfect fifths, (ie, C-G-D-A-E-B-F#) .   This 
results in a Valotti that is transposed up a fifth, which leaves C as the most 
consonant key rather than Valotti's F.  
Notice that you end up with a Gb when tuning by perfect fourths from C, and 
that same note is an F# when you approach it from C by imperfect fifths.  This 
will leave a rather active third between F# and A#, but that will be the 
widest third in the temperament.  

Now,  if you would like to excercise control over how strong the temperament 
is, the following is a magical procedure from Jorgensen that allows the tuner 
to make decisions in regard to allotting the dissoannce.  

Tuning To Personal Taste (Jorgensen)

All fourths are wide or Just,  all fifths are narrow or Just

1.   tune C to fork
2.   tune C4-E4 according to taste  (Just or wide)
3.   tune E4 - E3    Just
4.   tune G#3 between E3 & C4 so   Ab-C is 1.25 times as fast as  E-G#

Tune the following intervals Just for Well temperament, (or very slightly 
tempered for Victorian if C-E is beating at least 4 bps).
5.  G#3-C#4
6.  C#4 - F#4
7.  F#4 - F#3  
8.  Ab3 - Eb4
9.  Eb4 - Bb3

10.  tune D4 between Bb3 & F#4 so D -F#  is1.25 times as fast as Bb-D
11.  tune G3 from C4 & D4 so G-C beats 1.3 as fast as G-D
12.  tune A3 from E3 & D4 so A-D beats 1.3 times as fast as E-A
13.  tune B3 between F#3 and E4 so B-E beats 1.3 times as fast as F#-B

    Check 
G3-D4 is slower than A3-E4
G3-C4 is faster than A3-E4
A3-D4 is faster tahn G3-C4

14.  tune F4 from Bb3 & C4 so C-F beats 1.5 times as fast as Bb-F
Beat rates should increase in order from 
E3-B3
Bb3-F4
B3-F#4
B3-E4
E4-A4

    If steps 5 - 9 are tuned in Just intonation, the C-E third controls the 
harmonic balance.  If it is Just, a temperament like Kirnberger(Prinz) results. 
 If C-E is tuned faster than 4 bps and less than7 bps, only F#-A# is 21.5 
cents wide and the results are more "Victorian".  
 
Enjoy! 

 
Ed Foote RPT 
http://www.uk-piano.org/edfoote/index.html
www.uk-piano.org/edfoote/well_tempered_piano.html