March 2, 2007
Understanding the Western Tempered Scale (Part 10 of 15):
Soprano and Alto Recorders Show Us A New Number

Another great and simple science fair project (like last week's!):

Look at two recorders, a Soprano and an Alto. Measure the distance from the sound hole near the mouth piece (not where you blow, but further down from there, where the sound comes out) to the end of the recorder (past where your right hand's little finger goes).

Do that for both the Soprano and the Alto - and divide the big length by the small length. How many half steps lower is the Alto than the Soprano?

The lowest note on the Alto recorder is an F, while the lowest note on the Soprano recorder is C - a 5th above F. How many half steps between F and C?

	1	2	3	4	5	6	7
F	-	G	-	A	Bb	-	C
Do	-	Re	-	Mi	Fa	-	Sol

( "-" represents a half step that is not played - i.e., whenever you see a "-", that means a whole step separates the two notes.)

So, from Do to Sol, from F to C, we need to go up 7 half steps. Or 7 factors of 1.059463 - which is.... 1.059463^7 = 1.498306

In other words, the Alto should be 1.498 times longer than the Soprano. From whistle hole to far end, my Yamaha recorders measure 16.0 and 11.0 inches. 16/11 = 1.4545... which is NOT EXACTLY 1.498306, but is close. It turns out that vibrating air columns have some other effects going on right at the ends that change the frequency of vibration, so they are further off from our mathematical model than the vibrating string.

But what about that number, 1.498306 --- it is pretty close to 1.500000 - is something special going on here? Stay tuned!

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