THIS THURSDAY APRIL 14 We will meet at the Cathedral of St. John in Downtown ABQ

1. THIS THURSDAY APRIL 14 We will meet at the Cathedral of St. John in Downtown ABQ For a tour & demonstration of the largest pipe organ in NM http://www.stjohnsabq.org/organ.php 318 Silver Ave. SW

2. A standing waveform is shown for a pipe with one closed end. What is shown? A] Pressure B] Displacement Ans. B. Displacement node at the wall.

3. What OVERTONE is this? (NOT which harmonic… I’ll ask that next.) A] This is the fundamental B] This is the first overtone (2nd longest wave that resonates) C] This is the 2nd overtone D] This is the 3rd overtone E] This is the 5th overtone Ans. C. Draw the fundamental, 1st OT etc.

4. This is the 2nd overtone. What harmonic is this? A] It’s the 2nd harmonic, with twice the frequency of the fundamental B] It’s the 4th harmonic, with four times the freq of the fundamental C] It’s the 5th harmonic … D] It’s the 8th harmonic… E] It’s not a harmonic overtone Ans. C. The fundamental has one quarter wave, this has five quarter waves.

5. If this pipe is 5 meters long, what frequency is this mode? Speed of sound = 344 m/s. A] 344 ÷ 5 Hz B] 344 ÷ (4/5) Hz C] 344 Hz D] 344 x 5 Hz Ans. B. frequency x wavelength = speed, So frequency = speed / wavelength

6. At one instant in time, the displacement of air in the pipe is shown, curve A. (+ is a displacement to the right, - to the left) Which curve shows the pressure in the pipe at this instant? One-quarter cycle later, which function shows the displacement in the pipe?

7. One-quarter cycle later, there is no displacement. Where is the sound (wave) energy? A] There is sound energy in the pressure differences along the length of the pipe B] There is sound energy in the velocities of the molecules C] Both A&B D] There is no sound energy at this instant Ans B. The molecules, even though they have no displacement, are moving. Their inertia will carry them past the “rest position”, just as for a mass & spring.

8. Let’s look at some sound spectra from half-closed & open pipes. Very important: The edge in a flue pipe or flute acts like an open end. That means that the pressure there is always very nearly atmospheric, but there are large displacements of air.

9. Reed Instruments. Rather than having vibration driven by and edge tone, the vibration is driven by a flexible strip of material. This is the definition of “reed instrument” that we, and other musical acousticians, use. With this definition, a trumpet is: A] a blown pipe B] a reed instrument Ans. B. The reed is the trumpeter’s lips.

10. In addition to blown organ pipes (“flue pipes”), there are also organ reed pipes. Reed oscillation is a fluid-flow instability, driven in part by the pressure difference between the boot & the shallot. The Bernoulli force also helps (demo.) The reed also has mass&spring properties… inertia and a restoring force if bent.

11. When the reed is closed, the air flow is zero. When it is open, it is small. On the other hand, the pressure changes in the shallot are large, as the reed opens and closes. Therefore: for acoustic purposes, the reed end of a pipe is better treated: A] as a closed end B] as an open end Closed. If this doesn’t make sense to you, consider trying to blow a horn when the EXIT end is closed… you can’t do it. But you can blow organ flue pipes with closed exits.

12. Which curve shows the displacement in the lowest mode (the fundamental) in a reed-driven pipe? Which curve shows the first overtone? Demo. Need a horn player volunteer.

13. The flute and the clarinet (a reed instrument with a cylindrical bore, like the flute) are about the same length. How do their ranges compare? A] They are about the same B] The flute is about an octave lower C] The clarinet is about an octave lower *Although those of you familiar with these instruments will know the answer from experience, everyone can get the right answer by considering the lowest mode in an open pipe vs. a half-closed pipe.

14. Reeds are called “soft” if they can vibrate at a frequency determined by feedback from the resonator. They are colled “hard” if their frequency is determined by their own stiffness and mass. The same soft reed needs to vibrate at many different frequencies. (For example, a clarinet reed is soft.) Considered as a mass & spring with damping, does a soft reed have A] a lot of damping B] very little damping A] a lot of damping gives a broad response

15. So: in an instrument where each note has a separate reed, the reeds are usually hard (like in an organ.) Where each note has to be played by the same reed, and the note is thus determined by the resonator attached, the reed must be soft. The human voice, as you probably know, comes from the vibration of the vocal chords. These are: A] soft reeds B] hard reeds B !!! You can sing different notes, but NOT by changing the volume or length of your vocal tract! (You use your muscles to change the tension in your vocal chords, so technically you use different reeds for different notes.

16. Pipe Organs use a source of air pressure, about 10 cm of H2O (used to be human powered bellows) A set of pipes with the same timbre is called a RANK A STOP is a control that allows the organist to admit air to one or more ranks, when a key is pressed on the keyboard. The basic stop is called 8’, because largest pipe C2 is 8’ long. Some sets of pipes are designed to add overtones.

17. Overtone & other stops: 4’ (up an octave) sends air to a C3 pipe when C2 is pressed. 2’ (up two octaves) 16’ (down an octave) 2 2/3’ (= 8/3, one third as long as the standard pipe.) This is called a mutation stop. Which harmonics does the 2 2/3’ mutation stop sound (with open pipes) ? A] 2,4,6,8… B] 3,6,9,12… C] 4,8,12,16…

18. The 2 2/3’ stop opens pipes sounding at 3f. The overtones of these pipes will be at 6f, 9f, 12f, etc. So, when the organist hits the C3 key, if this stop is open, the note of G4 will sound, in addition to its overtones. (Up an octave and a fifth.) There are also “mixture” stops, that control several mutation and overtone stops together.

19. Lastly: how do the valves get opened? “Action” Mechanical (or tracker) action - best sense of control for the organist Pneumatic action - valves are opened by air pressure, sent through tubing from the console. 19th century. Electropneumatic action - valves are still opened by air pressure, but via a small electromagnetic valve Direct electric action - Big electromagnetic valve - can be loud!

20. Woodwinds

21. Note: All of these instruments have overtones that are approximately, but not exactly harmonic! Doesn’t a conical bore change the overtones to make them badly anharmonic? No (rather remarkably).

22. Even more remarkable: ALL Harmonics are present, not just odd harmonics! Clarinet tends to miss even harmonics, since it is cylindrical.

23. The clarinet has a cylindrical bore and therefore no even harmonics. When overblowing a clarinet, the first harmonic that can be produced is the third. If you want to design a clarinet that can be played over several octaves, how many different fingered notes do you need? A] 12 (a separate arrangement of open holes for every semitone in 1 octave) B] 19 C] 24 D] 12 per octave

24. Saxophones vs. Oboes. Cone angle 4° vs. 1.4°. The saxophone has a much larger bore. Which instrument will be richer in HIGH HARMONICS? (Recall our discussion of organ pipes… what happens if the wavelength is small compared with the pipe diameter?) A] Sax B] Oboe

25. Clickers 0E02DCD0 13EB639B PLEASE identify the mammal that is pressing your buttons.

26. Summary of Woodwinds • Woodwinds (and all reed-driven pipes) are effectively closed at the reed end. • Regardless of conical or cylindrical bore, overtones are nearly harmonic • Cylindrical bores have weak or absent even harmonics (Clarinet needs a lot of keys, because first overblown note is an octave + a fifth, 3° harmonic) • Narrower pipes give more high harmonics (just as with organ pipes)

27. During the organ tour, I was asked about pipes closed at both ends. These are not musically useful, because the sound can’t get out. But they are scientifically useful. How can you measure the speed of sound at 32,000 feet? If you know the air pressure & temperature, you can make a closed cell and measure the standing wave modes. An open pipe gives all harmonics A half-closed pipe gives odd harmonics A fully closed pipe gives: A] all harmonics B] odd harmonics only C] even harmonics only D] prime harmonics only Hint (?) •Boundary conditions •Pressure & Displacement are the Yin & Yang of sound waves

28. More fun with the closed-end organ pipe. Why is the overblown note “flat”?

29. Brass instruments all use “double reeds” (lips) But there are the (quasi) cylindrical bores: trumpet, trombone, french horn & (quasi) conical bores: cornet, tuba

31. Even the highest modes cannot extend beyond the bell. So it is best to think of the lower modes as being shortened (compared to what they would be in an instrument of the same total length but no bell). What does this do to the low mode frequencies? A] lowers them B] raises them

32. The back bore & cup of the mouthpiece can lower the frequencies of the high harmonics. NOW, the frequencies are a harmonic series of a DIFFERENT FUNDAMENTAL!

33. BRASS INSTRUMENTS are a lot like TIMPANI! • Harmonic series reshaped (by bell & mouthpiece, or kettle) to give 2nd, 3rd, 4th etc. harmonics. • Fundamental still off-pitch. 2nd mode is lowest used. With Timpani, the fundamental dies out quickly. With Brass, we just don’t play in the fundamental mode. Brass players make sure their lips vibrate at a high enough frequency to excite only higher modes. If your lowest mode is the 2nd (but you can also use the 3rd, 4th, etc), how many different fingerings do you need to play an octave? A] 2 B] 5 C] 7 D] 12

34. You need to be able to play all notes in an interval of a fifth. That’s seven semitones. But how does a trumpet do it with only three valves?

35. Trumpet valves add length to the trumpet airway. The problem is that, for each semitone down, we need to extend the airway by a factor of 1.0595. When we push two valves down, the additional lengths will ADD.

36. Trumpet valves add length to the trumpet airway. Valve 2 makes the trumpet 1.0595 times longer. (down a semitone). If Valve 1 is to lower the pitch two semitones, how long should the trumpet be when this valve is pressed? A] 1.05952 L = 1.1225L B] 1.119 L Ans A. So valve 2 adds 0.0595L and valve 1 adds 0.1225L. With both pressed, the added length is 0.0595 + 0.1225 = 0.182L BUT 1.05953 = 1.189L, an added length of 0.189L. Pressing both valves does not quite get us down 3 semitones. The trumpet player must compensate!

37. Nonlinear Physics Nonlinear means that the response to twice the driving force is not twice as large. Simple example: Imagine driving your lips sinusoidally. (If your horn is sounding mode 2 alone, that wave will drive your lips sinusoidally.) You can imagine a sinusoidal low amplitude variation of your mouth opening, but what happens when the amplitude increases? Can it remain sinusoidal? At large amplitudes, the mouth area cannot change sinusoidally. It reaches limits: it closes, or it opens maximally.

38. Any reed is intrinsically nonlinear, since it spends part of a cycle closed. But larger amplitudes are more nonlinear (less like a sine wave) than smaller amplitudes At large amplitudes, the mouth area cannot change sinusoidally. It reaches limits: it closes, or it opens maximally.

39. The most important consequence of nonlinearity in reeds Each mode affects the others. “Nonlinear Coupling.” If your horn is sounding mode 2, it will make your lips move in a non-sinusoidal motion (because of the limits of motion) This means your lips will have motion at other frequencies, which are harmonics of the frequency of mode 2! This mode mixing is unavoidable, but it is more extensive at higher amplitude (louder notes)

40. For loud notes, nonlinear mode coupling causes energy originally present in one mode to leak or spread to other modes. Trumpet pp mp fff

41. Input Impedance Curves “Impedance” is generally a driving force ÷ response. For sound, we tend to think of pressure as a driving force and displacement as a response. Note, however, that we want a big pressure change at the reed, in order to provide the strong positive feedback from the resonant pipe. The participation of a mode tends to be proportional to the height of the input impedance peak (especially at fff.)

42. Suppose I blow the 3rd harmonic of this pipe. What overtones (of the 3rd pipe harmonic) sound? Answer: only the odd ones. (Another way to see this: only the odd multiples of 3 are odd.)

43. Oboe has more high harmonics than sax. Note also that “harmonics” are not perfectly harmonic! What does this mean?

44. The note B flat MUST consist ONLY of 233.1 Hz and integer multiples! A periodic waveform may always be written as the sum of sine waves of the harmonic series! The dots show what overtone frequencies CAN contribute to this note. How well each mode can resonate is given by the input impedance. Thus, higher dots indicate modes that resonate more easily and thus loudly.

45. The note B flat MUST consist ONLY of 233.1 Hz and integer multiples! A periodic waveform may always be written as the sum of sine waves of the harmonic series! The dots show what overtone frequencies CAN contribute to this note. How well each mode can resonate is given by the input impedance. Thus, higher dots indicate modes that resonate more easily and thus loudly.

46. Regime of Oscillation - several modes resonate to make for a playable note.

47. Aside from an insufficient number of valves, is there any reason for a trumpet play NOT to use mode 1? Why begin with mode 2? The regime of oscillation gets little support from the trumpet’s higher modes. This leads to lack of stability and “regime change”.

48. When a trumpet player plays B3 flat at 234 Hz (circled), which solid dots show the harmonics? Choose any correct dot (there are more than one!)

50. Radiation Efficiency - Higher frequencies (shorter wavelengths) escape better.