Rules for Well Temperament

[Paul N. Bailey]

Richard Moody wrote:
>`Thanks Paul for posting the "Rules". It is one more incentive (esp
>to me) to buy his book.  However I am wondering if 
>"Tuning The Historical Temperaments By Ear" by Owen Jorgensen is a
>different work from  "Tuning" (by Jorgensen) listed as #572629 in the
>American Piano Supply Co  (APSCO) catalog. 
>What I am really wondering is,  are there "Instructions" as to how
>tune WT? :

        To my knowledge, there are three books by Owen Jorgensen:

1. TUNING THE HISTORICAL TEMPERAMENTS BY EAR
        Northern Michigan University Press, 1977
        
        Unfortunately, this is out of print. Ch.I is HOW THE MUSICAL SCALE
DEVELOPED THROUGH THE JUST TUNINGS.     This book is a great resource for
education and 
also for practical use.

2.      THE EQUAL-BEATING TEMPERAMENTS-A Handbook for tuning harpsichords
and forte-pianos,with tuning techniques and tables of fifteen historical
temperaments.
                        Raleigh: the Sunbury press, 1981.
        I think you might find this at:
                                        HARPSICHORD CLEARING HOUSE
                                        Glenn Giuttari
                                        9 Chestnut St.
                                        Rehoboth MA, 02769
                                        800-252-4304
        It's a slim paperback,only about 35 pages. In a sense it's the most

concentrated wealth of practical knowledge of the three. The equal beating
temperaments don't depend on accurate estimation of absolute beat rates, or
pitch levels. The results are often more musical than "theoretically
correct"
temperaments. Equal Beating Temperaments are probably more similar to what
was
done in the past.

        Fred Tremper has worked out the cents dev. from ET for the 15
temperaments
of THE EQUAL BEATING TEMPERAMENTS, perhaps he would still send copies to
those who
send him a stamped, self-addressed envelope.

3.      TUNING  Containing The Perfection of 18th Century Temperaments,
                The Lost Art of 19th Century Temperament        and
                The Science of Equal Temperament        with
        Instructions for Aural and Electronic Tuning.
                
                        Michigan State University Press
                                East Lansing
                                    1991

 
>My novice, (some might argue tyro) understanding of temps other than
>ET, the size of the intervals is dictated (for lack of a better word)
>by where one starts. Usually  from  middle C  (older) or A above
>middle C (modern).   Would WT also be thus affected? 

        The sizes of the thirds is determined by the amount of tempering
of the four conjunct fifths that intervene. 'for instance, C E is 21.5
cents
wider than pure if      CG , GD , DA and AE are all just fifths. If each
fifth
is contracted from pure by  1.95 cents, CE  will be 21.5 -( 4 x
1.95)=13.7cents;
and if each fifth is reduced by (21.5 / 4 ) cents, CE will be pure.
        One reason most 'older' temperaments started from C is; many 
well temperaments have just fifths from C to F to Bb to Eb to Ab to
Db to Gb (F#) ; and then the remaining 'long' keys are tempered to 
create equal beating pairs of intervals. For instance, since Bb and F# were

tuned in the partial cycle of just fifths , D could be tempered to create 
two equal beating thirds between Bb D and D F#. Several well temperaments
have 
this characteristic. 
        If the conjunct thirds Bb D  and D F#  have the same beat rate,
they 
are equal beating thirds. If Bb D  and D F# have  a beat rate ratio of 4:5
they will be the same width in cents, and "theoretically correct", in 
contrast to equal beating.
        
>Now Jim mentioned  figuring it out by spread sheets.   They (mine)
>can only give beats per second of intervals, or cents deviation  from
>each note in ET.  I am willing to give it a try.

>Richard Moody 

        You can learn a lot by sketching circles of fifths, thinking of the
fifths as just  until you deduct so many cents from them, then the thirds
will
be the size of 21.5 cents minus the total cents tempering in the four
in-between
fifths. Play around with this on paper, and patterns will emerge....keep in
mind you 
have a total of 23.4 cents of tempering of the fifths to play with before
you
will induce a wide fifth and the resultant 'harmonic waste'.    

                                Paul Bailey RPT