1. October 25, 2004 Expanding our Ideas About Musical Instruments
Lecture 15
October 25, 2004
Expanding our Ideas About
Musical Instruments

2. Examination #2

3. From before the test
We used “simple ratios of string lengths to create a musical scale.
If a tone found itself outside of the original octave, we divided the result by 2.
The ratios that we used were 2/3, ¾, … all “small number ratios”.
This was clearly consistent with religious views about music being “perfect”.
And it sounded good!

4. The result of this particular method (there are others)
We have a scale called the “JUST” scale.

5. And LooK!

6. Definition
Semitone
Tone

7. Houston … we have a problem
The scale starting on C is pretty good.
If you start the scale on A, you get a set of frequencies that don’t quite match those of the C starting scale.
This led to real problems with instruments that are of fixed tunings … like strings.

8. Music tends to transition to other scales …

9. You might want to look at Escher’s drawings!

10. This led to things like this:
And worse!

11. A Compromise The Well-Tempered
The Answer –
A Compromise The Well-Tempered
Scale

12. We did all of this because
The tones from a stringed instrument sound pretty good with these ratios.
God would never create a musical scale that wasn’t “perfect” and the Just scale was somewhat perfect because of the nice ratios.
Only certain frequencies or wavelengths were “stable” on a string fixed at both ends.
What about “other” instruments??

13. Like these …..

14. Let’s just do the flute

15. Times Change
40,000 years

16. A very simplified flute
AIR
Atm. Pressure
SOUND
Atm. Pressure

17. A very simplified flute
open
open
SOUND
open
open
closed

18. Nodes, antinodes and other stuff
open
closed
The molecules can “pile up”
at the closed end so that
the pressure is a maximum –
an ANTI-NODE
Pressure at this end
is always open to the
atmosphere which makes
it a pressure NODE
Motion of the molecules
at the close end is a minimum
because it is stopped by the wall
Motion of the molecules
is a maximum here.

19. The Rule OPEN END CLOSE END IN BETWEEN THE ENDS
Pressure has its NODE at the open end because it is pinned to the atmosphere.
CLOSE END
Pressure has its ANTI-NODE at the closed end because the pressure can build up there.
IN BETWEEN THE ENDS
There can be anti-nodes and nodes as long as the waves match the requirements above for the ends.

20. Standing Waves The waves going back and forth will combine
Puff of air starts
down along the tube
as a sound wave
Wave hits the end
and bounces back
The waves going back and forth will combine
to form a STANDING WAVE if the wavelength
is just right, otherwise they will add up to not
much musical.

21. Why in the world would I bounce BACK?????
WAIT A MINUTE !!!
I’m Moving
I get to the end
of the tube
Why in the world would I bounce BACK?????

22. The Bounce From Nothingness..
Lower pressure
than atmosphere

23. A HARD GRAPH

24. The top Graph L l
l=2L

25. A HARD GRAPH
l=2L
l=L
l=(2/3)L
l=(1/2)L

26. What tones can you get from a flute?
ALL (INTEGER) MULTIPLES OF THE FUNDEMENTAL ARE POSSIBLE

27. The Edge Tone
How do you get the tone
going down the flute?

28. Cover more of the opening leads
to a gradual transition from an open end to a closed end.
Blowing faster (harder) increases
the frequency of the edge tone
The “returning waves” resonate with the incoming wave to create
a resonance. This leads to the particular frequency being selected.

29. A Permanent Closed End
The clarinet f0
f0,f3,f5, … etc

30. Over-blowing the flute

31. Controlling the Flute’s Pitch

32. Site