Physics 36 / Music 36 Duke University Spring 2008 Handout 9

Piano Lecture Notes

preface

• many pianos manufactured, intense competition among builders
• detailed record of evolution available through patent disclosures
• relatively few grands sold, but reputations of best builders based mainly on their concert instruments [examples of promotional schemes]

look at a grand piano, noting features likely to have acoustic effects:

• case: bottom always open, five possible positions for lid (incl off)
• soundboard: shape, thickness, ribs across grain on bottom
• bridges: shape, number, positions on soundboard
• strings: length, thickness, density, tension, number, some wrapped
• dampers: not on all strings, raised by pedals or by keys
• hammers: sizes, locations on strings, shapes, materials
• keys: how linked to hammers and dampers
• pedals: damper, una corda, sostenuto

now for more detailed look, beginning with the oscillators, the STRINGS:

• range of the piano = about 7 octaves
• consider a couple of naive design approaches:
  • (1) choose a single string material, diameter and tension [for a "nice uniform sound"] and vary only LENGTH: Then 7 octaves means changing the fundamental frequency by a factor of 2 to the 7th power or 128. Since the fundamentals of the strings will be inversely proportional to their lengths, the longest and shortest will differ by a factor of 128 also. If the longest is 6 feet then the shortest will be only about 1/2 inch! If we make the shortest at least a couple of inches to allow space for a hammer then the longest would have to be at least 24 feet!
  • (2) how about keeping the length the same and varying only the tension ["making the case symmetrical and easier to construct"]? The fundamental frequency would be proportional to the square root of string tension, so the tension would have to vary by a factor of 128 squared, or 16,384. If the minimum tension were 5 pounds, then, the maximum would exceed 40 tons!
• In practice, builders vary length, density, and diameter of the strings and keep the tension roughly constant at 150-160 pounds. The typical grand has about 240 strings and a TOTAL tension of about 20 tons.
• Both ends of the speaking length of each string are fixed (by bridge at one end, by agraffe or capo d'astro bar at the other) thus normal modes include all harmonics of the fundamental
  • [demo: sympathetic vibrations when we raise the damper from one string and then play a glissando above it, or play only notes whose fundamentals equal its harmonics]
• but upper notes grow increasingly sharp, with INHARMONIC partials [demo: contrasts over several octaves from Beethoven Sonata 23 ("Appassionata") 1st movement: arpeggios ok, multi-octave jumps show intonation differences)]
  • such inharmonicity results from stiffness of strings, which provides an additional restoring force. The aditional force is greater when the bending radius is smaller, i.e. when nodes are closer together -- for the shortest strings and highest modes of all strings.
• how to minimize effects of stiffness: (what matters is magnitude of extra restoring force RELATIVE TO THE TENSION)
  • (1) maximize tension
  • (2) decrease the string diameter
• A disadvantage of (2) is that it will increase impedance mismatch at the bridge and decrease the loudness of the instrument. A solution is to use SEVERAL STRINGS instead of just one for each note. (Such groups of strings are not independent, so the combined effect is greater than we'd calculate for combining independent sources.)
• the use of multiple strings also provides control over decay rates
  • Even a single string has two different exponential decay rates: vertical vibrations (initial amplitude greater by factor of 10), and horizontal vibrations (which become significant as vertical decays rapidly)
  • note different impedance matches for different directions of vibration, due to the design of the bridges
  • slight detuning allows control over relative phases and thus decay rates:
  • impedance match depends on how much bridge moves
    • strings in phase => more bridge movement => better impedance match => more rapid decay
    • two string example:
      • identical frequencies:
        • 180 degrees out of phase: virtually zero net force on bridge => bridge effectively rigid => very slow decay
        • perfectly in phase: net force on bridge doubled => bridge motion doubled => rapid decay
      • slightly mistuned frequencies:
        • slow phase shift (cf. beating) changes net bridge force and, thus, decay rate as function of time
        • hammer both: initially in phase w rapid decay, then out of phase with slower decay
        • [demo: damp one of pair and observe more rapid decay]
    • change in decay rate by hammering fewer strings of each set:
      • same slightly mistuned pair example:
        • hammer one: other immediately driven by bridge, 180 degrees out of phase, giving slower decay from the start
      • [demos: (1) una corda pedal use, with unhammered string free to vibrate and then damped (2) hammer pair; wait; dampen one, making sound louder and decay more rapid]
    • [for triplet of strings, more complicated but same idea]
• near the upper end of the piano's range, the best compromise turns out to involve reducing string length by a factor of 0.532 per octave and reducing the diameter by a factor of 0.946 per octave. The net result is that tension drops slightly and inharmonicity increases as the upper end is approached
• At the lower end the same scaling would make the longest string about 16 ft long. If the tension were kept constant and the diameter increased to compensate, the lowest string would be a 3mm steel rod. The solution here is to use slender steel strings wrapped with copper, increasing the density without much increase in stiffness. The tension is allowed to increase some (up to 50%) at the low end.
• The net effect is that a piano expertly tuned for maximum internal consonance (i.e. to minimize beating between fundamentals and the upper harmonics of lower notes) will be sharp at the upper end and flat at the lower (about 30 cents in each case). This is called STRETCH TUNING. Making the low strings as long as possible can reduce the problem there, but nothing is gained by lengthening the hghest pitched strings.
• tuning: tools (hammer, fork, wedges, felt strips)
  • protocols, use of beats among upper partials of fifths, etc.

BRIDGES AND SOUNDBOARD

• board is gently arched piece of spruce, roughly 10 mm thick
• impedance matching important:
  • bridge-soundboard and soundboard-air:
    • in both cases one wants maximum transmission, minimal reflection
    • bridge glued to soundboard (must transmit both vertical and horiz vibes)
  • string-bridge:
    • more complicated; substantial reflection required to obtain standing wave modes, substantial transmission needed for instrument to be heard; too good a match causes too rapid decay, too poor a match causes insufficient loudness
      • notice how many things are varying with pitch here:
        • sensitivity of listeners' ears, number of strings per key, density and thickness of strings, tensions on strings, etc.
        • Impedance match between string and bridge must also be varied as function of pitch.
          • (1) vary the downbearing at the bridge
          • (2) use different bridges (wrapped vs unwrapped, pair vs triple)
          • (3) vary size and shape of bridges along length
      • bridges positioned on board to allow effective coupling into modes but without noticeable resonances. The positions of the string ends are thus determined with respect to the soundboard.
• Ribs give the board more nearly the same elasticity across the grain as along it.
• Sound radiates into the air from the board, even at the lowest frequencies.
  • Note that the bottom of the case is always open, so reflective properties of the floor will be important.
  • The change in tone whenthe lid is closed involves interaction between the board and reflected sound within the case.

ACTION

• experiment: (1) hit key and note sounds, (2) sound not muffled if key held down, so hammer must not remain on string, (3) sounds longer if key held down [most keys], (4) push key faster and sound is louder, (5) if push VERY slowly there is no sound, (6) can repeat without fully releasing key
• mechanism: key, wippen, jack, hammer, jack regulator, back check, repetition lever, jack reset, damper
• "TOUCH": only the speed of the hammer and the timing of its hitting the string can be controlled by the performer. the hammer is disconnected from the key before striking the string. LOUDNESS AND TONE QUALITY ARE NOT INDEPENDENT IN THE PIANO

HAMMERS

• The duration of the contact between string and hammer (i.e. the impulse) is important
  • varying the size and shape of the hammer and the softness of the felt
    • (softer at upper end; adjusting felt softness with picks and sandpaper is a large component of TONE REGULATION, which is discussed in detail in one of our assigned handouts)
• Details of the interaction are important:
  • The mass of the hammer controls the brief motion of the combined hammer-string system.
  • The position of the hammer along the string (between 1/7 and 1/9 of its length) determines how much of each mode is excited.
  • On some strings the [inverted] reflected pulse from the far end actually knocks the hammer off the string.
  • For the shortest strings, multiple early reflections from the far end -- before the hammer leaves the string -- serve to fill in high frequency modes that would not be excited otherwise because of the hammer's relatively large size.

Sequence in which relative positions of components are determined:

• hammers in line
• striking position on each string and length of each string
• capo d'astro bar, agraffe, and bridge positions
• soundboard position
• downbearing requirements
• hitch pin positions
• frame design
• case design

certain noises are characteristic parts of the instrument's sound:

• thump of initial hammer impact
• damped string ringing
• ringing of dead string segments
• damper release and seating sounds
• sympathetic vibrations on strings when damper pedal activated
  • [Bösendorfer adds extra sympathetic-only strings to low end of some of its largest concert grands, so even the lowest playable strings will have the advantage of still lower sympathetic resonators.]

Comparisons with related instruments:

• clavichord: tangent (metal wedge) strikes string and remains in contact, tangent attached directly to key, fixes speaking length of string from bridge (other end damped with felt) as well as providing impulse, no action, key pressure can vary string tension somewhat during note
• harpsichord: lower tensions, lighter strings, closed bottom of case, no significant inharmonicity cf. piano, plectrum plucks string, loudness not variable except by coupling more strings to each key, action connected to key right up to release of string, damper for each key, plectrum hinged to slip past string as key is released but before damper reseats