Meantone, my personal adventure reply to R. Moody

[Billbrpt@AOL.COM]

In a message dated 3/16/99 12:19:51 AM Central Standard Time,
remoody@easnet.net writes:

<< Your description of you tuning pure thirds and fifths and fourths up to A
 and E is quite interesting.  You must have been precocious, or it is
 intuitive that the note between the third C..E must lie halfway between,
 or at the mean.  And that is how meantone (the sources say) got its
 name. But that has to do with pure thirds is beyond me. Braid White
 explains the mean between C and E is the square root of the ratio 5/4
 (1.25). After this he looses me. How this D or that mean relates to the
 calculation of the rest of the scale, I didn't get.  >>


You are right about the way Meantone got its name.  The only Meantone with
pure 3rds is 1/4 Comma.  As in all the Meantones, 1/4 Comma is a series of
5ths, each tempered exactly the same amount.  That amount is 1/4 of the
Syntonic Comma (21.5 cents).  Theoretically, a 1/4 Comma 5th is 5.4 cents
narrow.  If you tune a string of 5.4 cent narrow 5ths around the Cycle of 5ths
but never going past Eb or Ab, you will get 9 pure 3rds.

The data I published for the 1/7 Comma Meantone is based on the same idea.
1/7 of 21.5 is 3.1.  Because the modern piano has inharmonicity, the actual
width of each 5th will be somewhat less than the theoretical.  In fact, even
on a low inharmonicity piano, you must tune a series of -2.9 cent 5ths.  A
medium will get -2.8  5ths and a high will get -2.7.

Owen Jorgensen asked me to tune a piano in 1/4 Meantone with conservative
octaves for the Convention in Dearborne.  I rarely do this temperament and I
was concerned that my aural skills could not guaranty either the size and
consistency of my 5ths or my 3rds.  A 3rd could be slightly on the narrow or
wide side and still sound "pure".

I got the idea I had simply from reading what the description of a Meantone
Temperament is in Owen's first publication, "Tuning the Historical
Temperaments by Ear".  It said that all of the 5ths are tempered by exactly
the same amount.  A light went on when I read that because I knew the SAT
could provide that.

When I first tried the idea to see if it would create a 1/7 Comma Meantone, I
reasoned that you could drop the .1 cents from the 3.1 cents because the 5th
would be at least that much wider than theoretical.  I saw that it would be
very easy to program an SAT to make a series of -3 cent narrow 5ths.  The SAT
already has -2 cents built in between each 5th.  All you would need to do is
keep adding the number 1 around one side of the Cycle from your starting point
and subtracting it going the other way.

I tried it and it almost worked.  I finally decided that maybe I should try
another figure, 2.9.  It worked better but it was still not quite right.  I
saw that I could go all the way to 2.7 perhaps before I would begin to be in
the realm of another Comma, namely 1/8 of which there are no MT's only WT's.
Since I was tuning a Steinway, which has relatively high inharmonicity, that
is what I needed to do, tune -2.7 cent 5ths.  I programmed in multiples of .7
cents.

This time it worked like a charm.  All of the aural checks worked perfectly.
I was pretty thrilled about this idea and went to Owen's room at the
Convention to work out the same plan for 1/4 Comma MT.  On paper it really
seemed to check out.  The piano I was to tune was a Kawai 7 foot in the low
end of medium inharmonicity.

I plugged in multiples of the number 3.2 around the cycle of 5ths.  It turns
out that my 3rds were not technically pure, they were .8 cents wide.  But this
is a pure 3rd *adjusted* for inharmonicity.  They sounded pure to me and did
to Owen too.  Then he went through all of the aural verification tests that he
knew (many more than I know) and was delighted to tell me that everything
checked out perfectly.

To tune the most conservative octaves possible could not have been easier.
You program your SAT so that the 3rd, 4th and 5th octaves all read on octave 5
(the way the PTG RPT exam does).  Then you program in the same numbers you
have in your F3-F4 temperament octave over these three octaves.  When you get
to the 6th octave, you play the note from the octave below it, stop the lights
and tune whatever it says.  Same for the 7th octave.  This gives the piano the
least expanded octaves possible.

I know there are FAC deviations for 1/4 Comma MT but I think doing it this way
insures the accuracy of the relationship that all of the notes have with each
other.  Since the FAC program is one kind of calculation and the cents
deviation firgures are another, I wonder if the FAC method could really
produce the same kind of consistency.   Perhaps someone might want to compare
the results of a 1/4 Comma MT tuned by the FAC method and the Direct Interval
approach that I discovered.

Sincerely,
Bill Bremmer RPT
Madison, Wisconsin