perfect fifth questions

[Jim Coleman, Sr.]

Hi Kent:

Since my name has been used, may I humbly disagree with a few of your
comments?

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).

When I do Pure 5ths equal temperaments, I can and often do check the
5ths with either machine. Due to the individual inharmonicities of the
notes involved, all the 5ths may not be perfectly pure (some may be 
wide). Until version 2.0 of the RCT comes out, it will not be possible
on many pianos to tune pure 5ths temperament with the custom feature, 
but at least an approach can be made.

If we didn't have inharmoncity in pianos, my calculations show that it
would be necessary to stretch the octave by 3.26 cents in order to 
result in pure 5ths. Here is a case where normal inharmonicity in pianos
helps us have quieter 5ths with just our normal tunings.

Kent, you are right in suggesting caution. The pure 5ths temperament
is not for everybody. For low inharmonicity pianos the octave stretching
will be more radical in producing pure 5ths. For pianos with greater
inharmonicity, the octave additional stretch requirement will be less.

On the question of myths. if you will tune a pure 5ths temperament
with your SAT and then check it with your RCT set for tuning by the
3rd partials, you will see how close the 5ths come out to pure when you
set your RCT on the bottom of the interval, stop the display with the
pitch adjustment on the right of the screen, and then watch how still
the display is when the upper note of the 5th is played. Of course one
could do a similar thing with just the SAT alone if he puts the SAT in
TUNE mode and adjusts it to the 3rd partial of the bottom note of the 
5ths and then plays the upper note of the 5ths. It's quicker to stop
the dots on the SAT than to stop the pattern on the RCT for the 
comparisons.

Jim Coleman, Sr.


On Sat, 4 Oct 1997, Kent Swafford wrote:

> (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
>