Stretching

[mitchkiel@olywa.net]

>     I recall my instructor at the Chatauqua Institution stating that if you
>tune beatless octaves above the temperament octave you will automatically
>stretch these octaves.
>From one who still uses the "human computer" I'm curious.  I'm ashamed to
>admit I haven't spent much time studying these matters but do you electronic
>folks impart a beat to these octaves to stretch them?
>
>Bob Sadowski
>Erie, PA

Bob,
    You're confusing two differerent meanings of "stretch." I like to 
separate
them into "natural stretch" and "artificial stretch."
  Natural stretch is what happens naturally (!) when tuning beatless 
octaves.
Because piano strings have inharmonicity, the frequency of an upper 
partial
is (usually!) a tiny bit greater than an exact multiple of the 
fundamental's frequency..
Therefore you "naturally stretch" a coincident partial of another note to 
meet it.
   For example, if we tune A4 at the first partial to 440 Hz, its second 
partial
is higher than a simple doubling (880 Hz). Due to inharmonicity, it's 
maybe
at 881. If you tune A3 to A4 using the 3rd:10th test, you're tuning a 4:2 
octave 
(the 4th partial of A3 to the 2nd partial of A4). But the 
4th partial of A3 also has inharmonicity, and because it's a higher-order
partial it's probably more "off-line" than A4's second partial. Therefore
A3 is pushed downwards, even though A4's second partial is pushed 
slightly upwards.
    But when we tune the piano we don't have to worry about any of this!
If we just tune the octave beatless, it all takes care of itself. 
This is what your instructor meant.
  (By far the best demonstrations I've ever seen of this are Daniel
Levitan's tuning classes, and the best written explanation is Daniel's 
series
of articles in the Journal a few years back. He writes
very clearly and the prop he uses in his class is simply brilliant.
As you can tell, I think he's one damn smart guy.)

  Then there's a second kind of stretch I like to call "artificial" 
stretch.
If you make any interval wider or narrower than its "natural" stretch, 
you're creating "artificial" stretch. This is what we do when we tune 
A3-A4 a half beat wide by using the 3rd:10th test and making the 10 a 
half beat faster than the 3rd.

    All modern ETDs incorporate both kinds of stretch, albeit in 
different ways. 

   Reyburn CyberTuner takes care of natural stretch the same way aural
tuners do: by matching partials. When RCT takes samples, it records
many partials of many notes. For example, it records partials 1, 2, 3, 
and 4
of A3 and of A4. So when it computes the A3-A4 octave, it simply matches
the 4th partial of A3 to the 2nd partial of A4. Bingo au naturelle!
    To add artificial stretch, RCT gives you several choices. You can
click an on-screen button to choose any of ten pre-set overall stretches. 
For example,
if you choose Octave Tuning Style #1 then the A3-A4 4:2 octave will be
.20 bps wide, and if you choose Octave Tuning Style #9 then A3-A4 will
be .72 bps wide (and the overall stretch will increase proportiionately 
automatically).
   If you want even more control of the artificial stretch, you can use 
RCT's Custom Equalizer,
which lets you micro-adjust the stretch of every octave of any Octave 
Tuning Style
by raising or lowering a slider located above A0, A1, A2, A3, A5, A6, or 
A7 in 
increments as little as 0.02 bps.
    For example, in Custom EQ you might first select Octave Tuning Style 
#5
but want an eensy bit more stretch at A7 and a itty bit less stretch at 
A3. 
So you click on the OTS #5 button, then click and raise the slider over 
A7 till
the graph or numbers make you happy, and then click and lower the slider
over A3 likewise. Custom EQ's graph can show you the changes you made
to OTS #5 and predict interval widths in beats and cents! for various 
octaves,
double octaves, fifths, twelfths, etc. When you're happy, you click the 
Calculate button and, in less than a half second, RCT calculates an 
88 note tuning with the exact stretch you want.

Mitch

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Mitch Kiel, RPT
authorized Reyburn CyberTuner sales and support
1-888-I-LUV-RCT (1-888-458-8728)

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email: mitchkiel@olywa.net
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