2:1, 4:2, 6:3 octaves

[John Formsma]

Comments interspersed.

On 8/21/07, Matthew Todd <toddpianoworks at yahoo.com> wrote:
> I have been really, really studying tonight.

Good!  You will eventually get it if you keep studying the right stuff
and apply yourself.  I promise.

> Can someone please explain the system they use to tune 2:1, 4:2 and 6:3
> octaves.  I am so close to grasping this concept, but I think I need another
> tech to explain it to me besides Reblitz.

Get someone else besides Reblitz.  As was mentioned, Baldassin's On
Pitch, the newly revised version, and the PTG Tuning Exam Source Book
were great for helping me understand all this.

> In the octave interval, if I were to tune a 4:2, the fourth partial of the
> lower note theoretically has the same frequency as the 2nd partial of the
> upper note.  Do those partials normally dominate each octave?  How can I
> tell whether to tune a 4:2 or a 6:3?

Yes, you have a dominant partial pair.  Which is why you "normally"
tune certain octave sizes in certain places in the piano.  The
Baldassin book has a chart that tells which octave size generally fits
best.

However, you must get the best fit for each piano.  You can't just
begin by tuning a 4:2 plus a little bit in the A3-A4 octave in every
piano, and expect it to be the best.  Some pianos will require in
between a 2:1 and 4:2, and some pianos might allow for 6:3.

To help you know what octaves will work with the piano, I find it
immensely helpful to start by working within a double octave. You tune
A3-A4 first, then tune A2 from A3.  This will help you establish the
correct octave width b/c you are using two octaves rather than one.
(Otherwise, if you begin with a A3-A4 octave that is too wide, you
will end up with bass and treble octaves that beat too much. Working
with a double octave prevents this.)

What I do is this:  Tune A4, then A3.  Make it sound the best (you can
change it later).  Check to make sure it's close to a 4:2 octave with
the M3-M10 test (because usually that fits most pianos well).  Then
tune A2 from A3, and make it a 6:3 octave using the m3-M6 test.  Then
you want to use F2 with A2, A3, and A4 to see if those octaves will
work for that piano.

You want to first make sure that A2-A4 is not more than 1 bps.  This
is *very* important. Play F2-A2, then compare it with F2-A4.  (Listen
at A4.)

Now use F2 to check A4 and A3. F2-A4 will probably be a tiny bit
faster than F2-A3.  And F2-A3 will probably be a tiny bit faster than
F2-A2.  If you have correctly set these octaves, the tuning will fit
that piano very well.  There may be some strings in lesser pianos that
don't fit well, but they will be minimized if you get octaves right
from the first.  If you do this on a well-scaled piano, you will be
astonished at how good it can sound when you're done. Octaves
complement each other, and it is just delightful.

Clear as mud? Probably. <Grin>  Get the books, do the requisite
head-scratching and pulling out.  You'll get it sooner or later if you
don't give up.

You will also find there are multiple checks for octaves.  The M3-M10
is a check for a 4:2 octave.  But also another good one is the
"shared" P4/P5 test. Say you're checking A3-A4 to see if it's 4:2.  If
A3-D4 beats the same as D4-A4, it's a 4:2.  HOWEVER, the 4th must be
expanded, and the 5th must be contracted for this test to be valid.
So, it would go like this: Expand D4 to get a 4th beating to whatever
you can hear well.  Then check D4-A4 to see if it's the same. If it's
the same, it's a 4:2.  If it's faster, you have an octave smaller than
4:2.  If it's slower, you have an octave larger than 4:2.

I like to use this test because it's relatively easy to hear.  I find
it tricky to know if the M3-M10 beats are the same because of
competing higher partials that can fool the ear.

Wish we were able to sit down at a piano while explaining this. It
would be much more understandable.

JF