more re: perfect fifth questions

[Jim Coleman, Sr.]

Hi again Kent:

The last time Steve gave his class on this subject, was before he and
I had long discussions. Since then, he has modified his program to
allow additional stretch, not only in his two octave temperament, but
variation is now possible in all octaves. That man is a genius. I have
tracked all of his formulae. He has a great understanding of all that
is involved. With his system it is now possible to tune by pure 5ths or
any size octave and have all of the resultant intervals displayed in
either cents widths or in beats. His program also has graphing 
capability so that it is plain to see what is going on with any type of
tuning.

I know Dean has a copy of the program and probably utilizes a good bit
of that information in the RCT.

In an article which I am now preparing for the January issue of PTJ,
I will have a chart of the actual inharmonicity of the temperament area
of a particular piano. The pertinent partials are layed out in spread-
sheet form so that one can with just simple addition and subtraction
create an ideal tuning on paper just as one would do it aurally. This
is even more exact than Steve's presentation because it is dealing with
actual partials instead of calculated partials. When this comes out, I
hope many take up the challenge to work out a good tuning. It should be
relatively easy since no hammer technique will be involved and stability
will not be an issue. The main difficulty with this challenge is that
inharmonicity is not as smooth in real life as we would like it to be.

Jim Coleman, Sr.

On Sun, 5 Oct 1997, Kent Swafford wrote:

> Jim Coleman, Sr. wrote:
> 
> >Equal temperament on pianos is not the 12th root of 2 due to the effect
> >of inharmonicity which in and of itself stretches the octaves some. 
> >This is why the 5ths in normal equal temp. tunings are less than 2 cents
> >and the 4ths are more than 2 cents. You can easily check this with your
> >RCT or the SAT (I have done both).
> 
> I thought of a way to check this from existing data that many would 
> consider to be very credible.  In April of 1989, Steve Fairchild gave a 
> class in which he presented his Super Three Octave Temperament in which 
> he measured most _every_ string of a piano and calculated a tuning for 
> each note individually, rather than just calculating a smooth curve.  His 
> idea was to find a tuning that would take into consideration a number of 
> tuning intervals and let each interval progress as smoothly as possible, 
> compromising one interval for the overall smoothness of the others where 
> necessary.  The idea, I believe, was to use enough data to take into 
> consideration the "fits and starts" of inharmonicity, and fully emulate 
> an aural tuning.
> 
> As a part of the class Steve calculated a tuning for a Steinway B, a 
> piano model that some have described as being perfectly scaled.  I still 
> have the data from that tuning.
> 
> Looking at the width in cents of contiguous intervals centered on A4:
> 
>  A3-A4, 0.9 cents at 4:2
>  A4-A5, -0.2 cents at 4:2
> 
> We may say we we expand the octave, but as can be seen here, it does not 
> always work that way.
> 
>  D4-A4, -1.5 at 3:2
>  A4-E5, -1.9 at 3:2
> 
> Just as Jim Coleman predicted, fifths are less than 2 cents contracted.
> 
>  E4-A4, 1.7 at 4:3
>  A4-D5, 1.5 at 4:3
> 
> However, instead of the fourths being more than 2 cents expanded, these 
> 4ths are _narrower_ than 2 cents.
> 
> Others can draw their own conclusions, but I would say that the effects 
> of inharmonicity are too unpredictable to be reduced to neat 
> generalizations.  Inharmonicity has always and will always bedevil our 
> tunings.  It is clear to me though that tuning any _one_ interval, be it 
> octaves, fifths, or whatever, to a uniform width, whatever that width 
> might be, will surely introduce undesirable unevenness in other intervals.
> 
> Be that as it may, in looking over the whole body of data for Fairchild's 
> tuning of this B, it appears that the generalization (that fifths will be 
> less than 2 cents and fourths will be more than 2 cents) actually holds 
> quite true, but only for those intervals below A4.  And remember this is 
> a piano that is "perfectly scaled."  Anyone know how the fifths and 
> fourths compare on a spinet?
> 
> Kent Swafford
>