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

[Bernhard Stopper]

See Comments below, regards, Bernhard Stopper

Jason Kanter wrote:
> 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.
agree
> 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.
correct for standard ET temperament
> Inharmonicity stretches them.
not very correct.
inharmonicity stretches *everything, still keeping the problem of the 
pythagorean comma.*
tuning twelfths pure an ET, is the result of a *completely different* ET 
temperament, dividing the pythagorean comma to the octaves.
See more about my work on this at http://www.stopper-scale.com

> 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 
> <mailto:formsma at gmail.com>> wrote:
>
>     Comments interspersed.
>
>     On 8/21/07, Matthew Todd <toddpianoworks at yahoo.com
>     <mailto: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
>
>
>
>
> -- 
> |  ||  |||  ||  |||  ||  |||  ||  |||  ||  |||  ||  |||  ||  |||
>          jason's cell 425 830 1561 
-------------- next part --------------
An HTML attachment was scrubbed...
URL: https://www.moypiano.com/ptg/pianotech.php/attachments/20070822/82fa46ae/attachment-0001.html