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

[Jason Kanter]

Double octave, yes. But within this, check all the twelfths -- they should
be as close to pure, beatless as possible and this will guarantee the right
amount of stretch. The test for a perfect 12th is a sixth below the lower
note. That is: to test C4-G5, use Eb3 against the C4 (a sixth that beats at
the frequency of G5) and Eb against G5 - should beat the same. This will
almost always give you an octave stretch that is the sweet spot between 4:2
and 6:3.
Note - mathematically perfect ET twelfths in a world without inharmonicity
would be narrow. Inharmonicity stretches them. The spot of the perfect 12th
turns out to be a great choice for the stretch because the 3rd partial is
usually very strong.

Perfect twelfths are also an excellent test up into the high treble.


On 8/22/07, John Formsma <formsma at gmail.com> wrote:
>
> 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
>



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