ramblin' (tuning by pure 5ths)

[Jim]

Hi Frank:

You asked the question: What does tuning by 5ths give us? In simple terms
only the addition of tempered octaves. The additional beat rate of the 
4ths is hardly noticeable. The 3rds and 6ths remain basically the same.
As the octaves are widened in the center, there is better match overall 
throughout the piano. This is what makes the piano sound bigger and
better. The increase is somewhat akin to the effect of a well temperament
which is quite mild. I must admit that it doesn't give the same effect
as in the Vallotti, for everything is still in equal temperament. The
5ths have practically no progression, The 4ths have very little 
progression, the 3rds progress the same as in normal ET. The 6ths progress
the same as in ET. As most of us know, there has been very little 
progression of the 4ths and 5ths in standard ET due primarily to the
inharmonicity curve of most pianos. This is a bit of a departure in
comparison to tuning harpsichords and organs which do have a definite
progression of beats in the 4ths and 5ths due to their lack of inharmoni-
city..

There are many variations in approaching pure 5ths equal temperament. For
quite some time aural tuners have had a tendency to tune the 5ths as clear
as possible in standard ET. The more they stretch the octaves, the more
pure become the 5ths, all other things being equal. For example, it was
only 20 years ago when most fine aural tuners were insisting that the 
3rd-10th test for an octave be kept absolutely equal. This makes the
octave pure at the 4-2 relationship. For the last several years, the
majority of good tuners have been stretching the octave so that the 10th
would beat up to half a beat faster than the 3rd in proving the octave
relationship. This was 4-2 type tuning plus a little bit. For sometime
there have been good aural tuners who used the 6-3 octave test for the
temperament area as well as for Bass octaves. As you well know, this
involves keeping the lower minor 3rd of an octave exactly equal in beat
rate to the complementary Major 6th above. In most pianos, this stretches
the octave a little more. The result is that the 5ths slow down a little
more and the 4ths speed up a little more. The other intervals remain
relatively the same, so it is still called equal temperament. All 
intervals still progress normally as in standard ET.

The distinct advantage of tuning pure 5ths, or greater stretched octaves
is that the triple, and quadruple octave matching sounds better. The high
treble sounds definitely better. There is a solidity about the overall
tuning of the piano. What little added beat in the center single octaves
is not noticeable at all during the playing of music. This was reinforced
by the voting of a group of 25 technicians (many of whom played well) who
favored the pure 5ths tuning by a vote of 179 to 97 at the University of
Texas on the last day of May '97.

In the math department, the question is not whether n*log2(2/3) will ever
equal n, but what octave ratio is needed so that the 7th half step will
theoretically be in a ratio of 3/2 in respect to the bottom note of the
octave. If you start with a frequency of A=220 and enlarge the octave
ratio sufficiently, the frequency of E=330 can be realized. In this 
theoretical case the octave ratio will be 2.004 approximately. Of course
all of this changes when we consider that inharmonicity is prominent in
pianos and the more complicated formulae used in the plotting of tuning
scales in the Accutuner takes all of this into consideration. I'm not
knowledgeable, able or permitted to explain Dr Sanderson's later formula 
for tuning with varied amounts of inharmonicity, but it works.

In the Music department, I'm glad you have the freedom to experiment with
various historical tunings. Back in Jan 97 I posted the results of a blind
test between equal Temp. tuning, Stretched octave tuning (almost pure 
5ths), and a quasi Well temp. which I had devised about a year ago or so.
Two different classes voted on which was their preference. The Well
temperament came out second for both classes. I really like some of the
Well temp tunings. I did a Vallotti for a private recital on an SD10
Baldwin 2 years ago. The graduate student who played it didn't know what
was different, but he sure liked it. I kept my own piano tuned like that
for about a month and played thru my entire hymnbook. It was interesting
to hear different kinds of sounds. I haven't heard Jazz on a Vallotti,
but I have heard just about every kind of music in Well and I like it.

Jim Coleman, Sr.

On Mon, 23 Jun 1997, Frank Weston wrote:

> Jim wrote:
> > 
> > Hi Richard:
> > 
> > One little thing caught my eye in your last reply where you said that
> > a pure 4th and a pure 5th could be tuned within a pure octave. Well, for
> > one pair that is true, but you can't have both continuing in a
> > temperament.
> > As you spread the octave in order to fit in the pure 5ths, the 4ths
> > actually get faster. Now I'm learning that they will always be faster than
> > the stretch octaves. Sometimes more than twice the speed of the octave.
> > 
> > I hope this clue helps those who are actually trying to set a scale with
> > pure 5ths. Doing it with the SAT was so easy, that almost all the pianos
> > I tune now are tuned this way. But doing it aurally is quite another
> > matter.  I am working out a foolproof method now for doing it aurally.
> > It should be on the list in just the next few days.  I would like to
> > actually do it a few more times before I publish.
> > 
> > Jim Coleman, Sr.
> 
> Of course you can't have pure 5ths and pure 4ths in the same
> temperament!  There are two immutable laws of the Universe which should
> concern persons attempting to tune a keyboard instrument.
> 
> 1.  You never get something for nothing.
> 2.  n*log2(3/2)  will never be equal to n for any value of n except
> zero.
> 
> The history of temperament is the history of attempts to reconcile these
> two laws.  The tradeoff has always been more pure intervals vs. more
> freedom to modulate.  When a new system of tuning is proposed, the
> question must be asked, "What does it give us?"  Does the new system
> provide intervals which are more pure?  Does the new system result in
> more freedom to modulate?  Does the new system have some historical
> significance to the music that will be played?  If the answer to these
> questions is no or mostly no, then what is the benefit of the new
> system?  We must bear in mind that the purpose of a tuning system or
> temperament is ultimately to make music, NOT to be convenient to the
> person doing the tuning or to the device used to do the tuning.
> 
> So, the question:  What does tuning by pure 5ths give us except for
> convenience?  More musical intervals?  No.  Perfect fifths are traded
> for less than perfect octaves, thirds and sixths.  More freedom to
> modulate?  No.  Modern equal temperament is the ultimate in this
> regard.  Historical significance to music of a particular era? 
> Possibly.  According to Jorgensen, some English  harpsicord tuners in
> the early 18th century claimed to tune 5ths perfect with good results.  
> Now ask yourself, what English harpsicord music from the 1720’s would
> you like to reproduce?
> 
> For me, the ultimate temperament will always be a well temperament, and
> Vallotti is my favorite.  Why?  Because the musical intervals are
> relatively more pure (or at least more interesting) than equal or
> attempts at equal, and they are well suited to the characteristics of
> the piano.  Because there is key-coloration.  And finally because I am a
> big fan of Baroque and Classical Music.   Actually a good
> well-temperament is usually suitable for most music except for modern
> art-music and competently played jazz.
> 
> My two cents worth.
> 
> Frank Weston
>