HTs using cents offset from ET

[Robert Scott]

What is involved in creating a "cents offset from ET" program for
tuning an historical temperament?  This is what people want to know
if they are going to tune an historical temperament by means of a
visual tuning aid.  But it may not always be possible to come up
with such a program.

Let us take, as the definition of an historical temperament, an
aural method.  These aural methods rely on beat rates of specified 
intervals.  And the beat rates are not always exactly
specified.  For example, the method may call for adjusting two
intervals for "equal beating" without specifying exactly what the
beat rate will be.  For a particular piano with a particular
set of inharmonicities, this beat rate is well-defined.  But for
a different piano with different inharmonicities, the "equal-beating"
beat rate might turn out to be something different.

This brings me to Jorgensen.  In his book on tuning, he gives about
60 samples of "cents offset from ET" tables for historical temperaments
that are otherwise defined in terms of an aural method.  So what
inharmonicity did Jorgensen assume in coming up with his cents offsets?
My inclination is to say "zero inharmonicity".  The reason that I
cannot say this for sure is that when I assume zero inharmonicity and
follow his aural methods, I get cents offsets that deviate slightly
from what Jorgensen gives in his tables.  I suspect the difference is
due to different degrees of round-off and approximations.

For example, [if you don't like math, skip this paragraph] in
chapter 61, Jorgensen describes a particular well-temperament of 
1785.  After tuning middle C, he says to tune the F below so that
the fifth F3-C4 is narrow by 0.8 beat per second.  According to
his offsets table, that C4 is sharp from ET by 2.55384 cents,
which would give it a fundamental frequency of 262.0117886 Hz.
>From here derive the frequency for F3 by multiplying by 2, adding
0.8 (the beat rate), and dividing by 3, which gives F3=174.9411924 Hz.
Compared to ET, this is sharp by 3.23981 cents, while his table gives
3.40512 cents for the offset for F from ET.  I tried various
approximations and still could not get 3.40512, but it is very 
close to my "zero inharmonicity" figure of 3.23981 cents.  Any
real piano would likely have more than .16 cents difference between
the 2nd and the 3rd partials, so it still looks like Jorgensen
assumed essentially zero inharmonicity.

If Jorgensen did assume zero inharmonicity, then that fact should
cause us to take those "offsets from ET" tables with a grain of
salt.  Furthermore, if we are to apply these offsets to an ET
tuning, the question still remains, "which ET?".  There are as
many different equal temperament tunings as there are degrees of
octave stretching, so some assumptions must be made here.

When variations in inharmonicity are taken into account in an
ET tuning, they are generally reflected in the only parameter
open to adjustment - the degree of octave stretching.  This is
essentially what the FAC program does.  So what is the equivalent
phenomenon with an HT defined by an aural method?  That depends
on the method.

If you have a piano with really extreme inharmonicity and you
try to follow one of Jorgensen's aural methods, you will find
that it is impossible to get all the checks to work out.
That's because these checks can only co-exist with each
other under a given set of inharmonicities - the inharmonicities
under which the aural method was first developed.  If the
inharmonicities are changed, it is not at all obvious how to
modify the aural HT method to best retain the original "essense"
of the HT.  The essense is not contained in the method.  Unless
you are very clever, you cannot even predict the character of an
HT just by looking casually at the method used to construct it.
So, faced with drastically different inharmonicities, the tuner of
an HT would be forced to go "back to the drawing board" to
essentially re-invent the method to achieve the desired result.

This brings me to my proposal.  Bill Bremmer recently commented
on the difficulty in duplicating an aural HT tuning using cents
offsets from ET.  Here's how I think it can be done.

First verify a method aurally.  That means tune a modern instrument
using a proposed aural method and verify that all the required
checks work out on that piano.  I suspect that some of the methods
in Jorgensen, for example, will not pass this test.  Second, measure
the inharmonicities involved in the temperament area.  These must
include measurements for all the partials that are used in the
aural method, including the checks.  For example, if the major third
F3-A4 is to be tuned for a specific beat rate, the inharmonicity
measurements must include the 5th partial of F3 and the 4th partial
of A4 (assuming it was a 5/4 beat and not a 10/8 beat).  Finally,
feed all these inharmonicity measurements and the beat rates
for the aural method into a computer program.  From these data it
should be possible to calculate the cents offset from ET.  It might
be interesting to see if tables developed this way differ markedly
from the tables in Jorgensen.  I would be glad to develop such a
computer program if sufficient data were generated on which to use it.

Any takers?

Robert Scott
Ann Arbor, Michigan