thirds and sixths

[Tom Cole]

martin store wrote:
> 
> what is the relationship between thirds and sixths?
> 
> for inctance:
> 
> f3 - a3  if this one beats 6,9 bps what is the
> f3 - d4 sixth suppose to beat?

The sixth in this case would be 1bps faster. I hasten to add that the
6.9 figure is only a theoretical number and your mileage will definitely
vary with real pianos. Whatever tuning method you use should have a way
of verifying these early settings before you go too far out on a limb.
> 
> and what should these intervalls beat if they are correctly tuned?
> 
> f3 - d4 sixth
> a#3- d4 third
> 
> how do this really work?

F - D, 8bps
A# - D, 9.5bps

If you next tune F#3 - A#3 (7.5bps) and D3 - F#3 (5.5bps + good octave
with D4 and 5 to 4 ratio beat speeds as you progress up the contiguous
thirds), you can check the middle third with F3 - A3 for a 1/2 beat
difference. If there is too much difference, speed up F3/A3 by lowering
F3 slightly and recheck. If they're too similar, you'll need to slow
down F3/A3.

When you ask, "how do this really work?", I'm assuming you mean what
produces the beats. The wavering of an interval is caused by the
coincident harmonics of the two tones beating against each other. In the
case of major thirds, it's the 5th partial of the lower beating against
the 4th partial of the upper. To test this, silently depress and hold
down the keys of a major third, say F3 - A3, and briefly play A5, this
interval's coincident harmonic. Listen for the 6.9 or whatever beats per second.

For a major 6th, you hear the 5th partial of the lower beating against
the 3rd partial of the upper. Using the above example, hold open the
dampers of F3 - D4 and again play A5 to hear the approx. 8bps. 

I know I'm glossing over some of details. If this isn't clear, feel free
to email me privately.

Tom Cole
mailto:tcole@cruzio.com