HT and ET

[Billbrpt@aol.com]

In a message dated 98-02-04 14:44:50 EST, you write:

<< I may have missed this point so I am asking Bill Bremmer to please explain
again his meaning of "cycle of 5ths based HT's ". How does a "cycle of 5ths
based HT" differ from one that is not "cycle of 5ths" based? Does "cycle of
5ths" mean *anything* else but a succession of 5ths that leads back after 12
steps to the initial tone thereby being capable of arranging in a circle? >>

     This is my own expression and not one that any published writer on the
subject may have used so you'll probably not see it anywhere per se.

    The idea comes from the rules that Andreas Werkmeister wrote for a Well-
Tempered tuning (WT) (also called Well-Temperament).  To paraphrase these
rules, a WT must have its slowest beating 3rd at C  (the "top" of the cycle).
As you add either sharps or flats, the speed of the 3rds increase in alignment
with the cycle, not chromatically.  There can be no "imbalances" in the
Temperament.  The 3rd of the key of A cannot beat faster than the 3rd for the
key of E, for example.  There can be two or more in a row that are the same
but they must ultimately progress to being fastest at F#, the "bottom" of the
cycle.  

    The Meantones and Modified Meantones have similar patterns but are a
little "lopsided" in that they favor the sharp side of the Cycle.  In these
temperaments, Ab or Db often has the fastest 3rds.

    There are also rules about 5ths in both kinds of temperaments but in my
view, it is the Rapidly Beating Intervals (RBI) which are the primary element
in what is known as "key color".  The Slowly Beating Intervals (SBI) are
secondary in importance.
    
    Since most of the classical literature we perform was composed on
keyboards that were tuned in one of these temperaments (all the way into the
20th Century, I believe) the key signature that a composer gave a piece was
given for a very deliberate reason, certainly not chosen at random or for
pitch reasons alone.

    Some other kinds of ancient tunings where only pure intervals were tuned
and no tempering was done are not really very useful today, even if they were
tuned in a pattern that went through the Cycle of 5ths like the Temperaments
do.

     There are also some theoretical tunings that base notes of the scale on
tuning pure unisons with the natural harmonics of the lowest C on the piano.
There are also temperaments that have 19 and 31 notes to the octave.  

    None of these last two groups are the kinds of HT's I practice or
advocate.   I only practice and advocate  Well-Temperaments, Meantones and
Modified Meantones.  Even among these there are a great variety.  Some may
consider only a very few of them to even be something to consider when tuning
a modern piano.  I tend to lean in that direction.  There are only a small
number of the many, many varieties that will work well for the modern piano
except in special circumstances.  To discover which ones they are has taken a
lot of trial and error on the part of many people.

    There is one more type of temperamment which might be considered which is
not truly Cycle of 5ths based.  It is the Quasi-Equal Temperament ("quasi"
meaning "almost").  When you hear someone speak of a "Marpurg" temperament,
that is what they are talking about.  There was one I mentioned in a recent
post where the temperament was essentially ET, only the note "C" was raised by
1¢.

    When you see the term "Reverse-Well" it generally refers to a temperament
which was erroneously constructed and ends up with the 3rds progressing
opposite (or more or less opposite) to the way I described above.
Unfortunately, it is very common today among aural tuners.

  This is the reason:  Most aural tuners who are attempting ET use a bearing
plan (or temperament sequence) that begins on A or C and tune the white keys
first.  They don't like the sound of a tempered 5th and tend to get it a
little too pure.  When they reach the end of the cycle of 5ths, it is
dicovered that the last 5th won't resolve with the octave and so they back up
through the cycle, over tempering the black key 5ths, to about midway where
things seem to have "evened out".  This is exactly opposite of the way you
might approach a true WT.  In most WT's the black key 5ths are pure or nearly
pure and the white key 5ths are tempered more than in ET.  This results in the
"alignment with the Cycle of 5ths" that I often write about. If the technician
has not been trained to listen carefully to 3rds & 6ths and doesn't know such
principals as the "Ratio of 4:5 of contiguous 3rds" and other very useful
checks among the RBI's, the resultant temperament can easily end up being this
unfortunate "reverse-well" pattern and syndrom. 

    The late, great, John Travis in his book, "Let's Tune Up" suggested that
if one used a C# or F#  fork,  a tendancy to "err towards the just 5th" could
be avoided.  He actually said the tuning would be "more equal".  This was only
a theory however that was written about during a time when not nearly as much
was known about tuning as there is today.  From a purely analytical and
technical point of view, it shouldn't really matter what note you begin with.
Everybody knows however that there are certain Temperament Sequences which
seem to work better than others.  I have always wondered if by starting on the
black keys, John Travis might actually have produced a very mild WT at times.
If he did, or his temperament ended up truly equal, his pianos would have
sounded superb either way.  It is the "Reverse-Well" mistake that truly makes
for a bad tuning. 
   Just as pianists learn to adapt to true HT's, and a true ET, they learn to
adapt to Reverse-Well too.  It's not a good thing.  It's analogous to a
pianist who always "pounds hard" because he/she has learned to play on a
poorly regulated action.  "Reverse-Well", in my opininion is a true scourge on
the tuning profession because it is so often offered as and believed firmly to
be an ET.  In my view, this is where the real concern ought to be regarding
people who are not tuning a true ET and "not telling the customer".  If it
weren't for the fact that the technician doing it really doesn't know any
better, it would be fraud.

Bill Bremmer RPT
Madison, Wisconsin