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

Examination #2

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!

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

And LooK!

Definition Semitone Tone

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.

Music tends to transition to other scales …

You might want to look at Escher’s drawings!

This led to things like this: And worse!

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

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??

Like these …..

Let’s just do the flute

Times Change 40,000 years

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

A very simplified flute open open SOUND open open closed

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.

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.

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.

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?????

The Bounce From Nothingness.. Lower pressure than atmosphere

A HARD GRAPH

The top Graph L l l=2L

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

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

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

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.

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

Over-blowing the flute

Controlling the Flute’s Pitch

Site http://hyperphysics.phy-astr.gsu.edu/hbase/music/flute5.html