perfect fifth questions

[Kent Swafford]

(Sorry about the "mis-fire" of this post before.)

pianoman wrote:

>Dear list,
>With the process of doing the perfect fifths tunings on the SAT, questions.
>
>1. Are the perfect fifths used only as a way to stretch the octaves further
>or is it just as important that the fifths be perfect as well as the
>octaves stretched further.

Wide tunings are desirable for a number of reasons, including the fact 
that melody notes of octave 6 and thereabouts just sound so wonderfully 
"in-tune" for whatever reason when the tunings are wide.

>2.  In using the RCT, since there is a choice of 10 different octave
>stretch choices, is there on of them that comes closest to the perfect
>fifth results of the SAT?

No.  This will vary from piano to piano.  Pianos vary in both general 
level of inharmonicity and in the inharmonicity curves throughout the 
scales.  If piano tuners put a more-or-less uniform amount of stretch in 
the A3-A4 octave, say 1/3 beat per second at the 4:2, then the width of 
the fifths in pianos will vary.  This means that Jim Coleman's suggestion 
of adding an extra 1.5 cents to the reading for the A in FAC will have a 
different effect on different pianos.  Caution is in order.  That same 
extra 1.5 cents will be conservative on some pianos and extreme on others.

Kent Swafford

I posted the following to some of the RCT users on September 1.  The 
other RCT users seemed to agree.

I have experimented with some wider tunings during the time that these 
tunings have been under discussion on the lists.  My conclusion is that 
there are very good reasons for the standard practices that we employ.  A 
pure fifths temperament may be an "extreme" tuning unsuitable for 
everyday use.  On pianos that are in use by a given pianist day in and 
day out, I think those wide octaves and fast fourths are going to become 
excruciating.  And if, as happened on a B that I tuned wide, the tenor 
drops, the octaves will simply be over-the-line and unacceptable.  
12-tone equal temperament has a built-in safety factor that you will 
surely lose with wide tunings.

Now, having said that, I do think that there is a place for some wide 
tunings on concert instruments that will have little opportunity for 
drift between tuning and concert.  A tuning done up somewhat differently 
from an artist's everyday piano might be just the thing to inspire the 
pianist into a transcendent performance.

Now, wide tunings and calculating them:  As I have said, a pure fifths 
equal temperament is an extreme temperament.  Instead of being a tuning 
based on the 12th root of 2 as is 12-tone to the octave equal 
temperament, 7-tone to the pure perfect fifth equal temperament would be 
based on the 7th root of 1.5.  The difference between the two is not 
minor;  pure fifths equal temperament is a _different_ tuning and and we 
should not necessarily expect our VTD's to calculate them.

Based on looking at the 3:2 5ths that Ch2 calculates directly, and 
assuming that Ch2 is calculating high quality equal temperament, and 
seeing that the fifths are often still 2 cents contracted at the same 
time the octave is expanded 2/3 cents at the 4:2, I am extremely 
skeptical that an FAC tuning with an extra cent in the A number will 
result in pure fifths, although they may _sound_ aurally mostly pure.  
Theoretically, it would take well over 3 cents expansion in the octave to 
result in pure 5ths.

In other words I think Jim Coleman, Sr. isn't _really_ tuning pure 5th 
equal temperament, but more like 1 cent contracted 5ths equal 
temperament, which I have found to be plenty extreme in itself, at least 
on some pianos.

You know, since Jim Coleman, Sr. has been talking about all this, some 
things have been made clear:  There is a clear discrepancy between one 
thing tuners _believe_ and what appears to be the _reality_ of piano beat 
rates.  Dr. Sanderson always said that pianos are scaled so that their 
beat rates can resemble those of the mathematical model.  Tuners often 
repeat that piano 5ths are tuned at 1 cent contracted instead of 2 as a 
part of normal stretching.  Based on what I have seen in Ch2 and my 
experiences trying out some wide tunings that I found to be way too 
extreme, I would say Dr. Sanderson was absolutely right -- we tune pianos 
with 2 cent contracted 5ths, and the oft-repeated tale of 1 cent fifths 
is, mostly, a simple myth.

Kent