2. Chapter 8 Real Hammers and Plectra

3. Simulating Broad Plectra

4. Simulating Complex Shapes Use a collection of narrow plectra to simulate the broad one in any pattern. Initial vibrational pattern can be deduced by taking the individual patterns of the narrow plectra and combining.

5. Example Case ½ ¼

6. Fundamental Mode Excitations For the fundamental mode the central plectrum is at the antinode (maximum excitation). o Amplitude a 1 if it acts alone The plectrum at the ¼ point excites the fundamental 0.707 as much. o Amplitude b 1 = 0.707 a 1 if it acts alone When both act together, the displacement is constant between the plucking points and equal to a 1 + (⅔)b 1 Note: I’m not quite sure where the ⅔ comes from, but clearly the second position should get less weight in predicting the excitation of this mode

7. Net Mode 1 Excitation Mode 1 is excited to an amplitude of a 1 + (⅔)[(0.707)a 1 ] = 1.47a 1

8. Changing the Method of Plucking a 1 - (⅔)(0.707)a 1 = 0.53a 1 Notice the minus sign – the second plectra gives a negative contribution for this mode

9. Another Example Mode 7 Oscillation (3½ wavelengths) p’s mark positive maxima q’s mark negative maxima Both the magenta and red plectra grab the string at p positions

10. Constructive Use of Plectra Two plectra at p’s should give twice the initial amplitude as one at a p position. The following situation should also produce twice the initial amplitude as one plectrum. To see this ask yourself which way the string would move when the plectra are removed. Notice that both plectra reinforce the mode

11. Destructive Use of Plectra If only the left plectrum were there, then after release all p’s move down and all q’s move up. If only the right plectrum were there, then after release all q’s move down and all p’s move up. With both plectra in place, mode 7 oscillations are destroyed.

12. Generalize (Four Points) 1.The result of using several plectra is given by combining the results of each plectrum acting alone (call this the narrow plectra substitution model). 2.We must take proper account of the sign of the individual results. Some plectrum positions may cancel others.

13. Continued 3.For two plectra close together (less than ⅓ rd the spacing of mode maxima) and pulling in the same direction, we get nearly the same effect as pulling one back twice as far. 4.Two plectra pulling in the same direction separated by exactly ½ wavelength will cancel out that mode (the destructive example two slides ago). If the separation is approximately ½ wavelength, that mode is only weakly excited.

14. Application to Wide Plectra Recall that a wide plectrum (width W) acts the same as two narrow plectra (separted by W). W

15. Finger Picking a Guitar Use a thumb and forefinger on a guitar string (separation W about 2 cm) o modes are weakly excited whose wavelength is around 4 cm (W then would be ½ and point 4 applies). Since the length of the string is 60 cm (L), modes that fit this condition are n = L/W = 60 cm/4 cm = 15 (that is the 15 th harmonic would have a wavelength of 4 cm)

16. Results of model Mode 15 is absent and near 15 are weak. Point three tells us that modes of wavelength greater than three times W (>6 cm) are excited by a large amount and are accurately predicted by this narrow plectra substitution model n = L/(3W) = 60 cm/6 cm = 10 Modes up to 10 are predicted well Modes between 10 and 15 are not predicted

17. Using a Pick Pick width 0.2 cm Harmonics around mode 150 are missing and simple theory carries us to mode 100 o Since the amplitudes fall off with n (use of a pick is like striking the string), these are all the usable modes

18. Hammer Width for Struck Strings Force is not uniformly distributed Imagine the hammer hitting a wax block The indentation gives an accurate view of the forces at work

19. Restoring Force The simple model uses a linear restoring force F = -kx (Hooke’s Law) When a steady force is applied to the felt of a piano hammer, the felt becomes stiffer with use. o Larger force must be applied to produce the same compression. F = Kx p o p is the non-linearity coefficient

20. Comparing forces

21. Effective Nonlinearity Exponent Hammers taken from pianos o p = 2.2 – 3.5 Unused hammers o p = 1.5 – 2.8 Preferred range of values is 2 - 3 o p < 1 hammer is too linear and loud notes are simply amplified soft notes o p > 3 there is too much contrast - fortissimos are too harsh and pianissimos are too bland

22. Hammer Forces Spatial Variation of the Force F max WhWh Force ½ F max Distance Along String

23. Observations For W h < ¼ wavelength ( ) vibrational modes are excited just like a narrow plectrum. For W h  ½ excitations are about half as strong as a narrow plectrum For W h > that mode receives very little excitation.

24. Impact Duration Temporal Variation of the Force F max ThTh Force ½ F max Time

25. Observations When T h < P/4 (¼ Period), vibrational modes are excited that are the same as an instantaneous hammer strike (impulse). For T h  P/2 are excited at about half the strength as an instantaneous hammer strike. For T h > P that mode received little excitation.

26. Hard and Soft Hammers (expected linear response) Soft HammerHard Hammer Soft hammers apply the force over a longer period of time. Soft and hard blows last the same time Soft Blows Hard Blows

27. Real Piano Hammers Solid curves show results Dotted curves show attempts to fit data to various models

28. Soft and Hard Blows (real piano) Hard blow results from higher impact speed

29. String Stiffness So far we have considered flexible strings Stiffness makes the string shape rounded o Higher frequency modes are blocked

30. Summary Major assumptions of the method: o Excitation of a mode depends on the relative amplitude of the mode at the point of striking or plucking. o Extended plectra or hammers may be treated by adding the result of several narrow plectra or hammers arranged to give the same initial shape.

31. Continued More assumptions: o Modes whose wavelength are comparable to the width of the plectrum or hammer are not excited. The same holds if the time interval of the hammer strike is comparable to the period of the mode. o Plucked string modes decrease in initial amplitude by 1/n 2. Struck string modes decrease in initial amplitude by 1/n.

32. Examples Pluck guitar string 6 as normal and then damp at the mid-point. o The odd-numbered modes have antinodes at the mid- point and are damped. o Only the even-numbered modes remain. o The normal 329.6 Hz tone (E 4 ­ ) is replaced by 2*329.6 Hz = 659.2 Hz or E 5. If the damper is applied at the 1/3 rd point, then modes 3, 6, 9, 12 survive, giving a tone close to B 5.