F

Five lectures on

THE ACOUSTICS

OF THE PIANO

Anders Askenfelt, editor

1990 Royal Swedish Academy of Music

Contents

Preface ............................................................................................................................................................................... 5

Introduction .................................................................................................................................................................... 7

Piano design factors .................................................................................................................................................. 17

From touch to string vibration ............................................................................................................................. 39

The hammer and the string.................................................................................................................................... 63

The coupled motion of piano strings .................................................................................................................. 85

The strings and the soundboard .......................................................................................................................... 97

Lectures:

Harold A. Conklin Jr.:

Piano design factors - their influence on tone and acoustical performance

Anders Askenfelt & Erik Janson

From touch to string vibrations

Donald E. Hall:

The hammer and the string

Gabriel Weinreich

The coupled motion of piano strings

Klaus Wogram

The strings and the soundboard

Preface

This volume contains five lectures given at a public seminar at the Royal Institute of

Technology in Stockholm, May 27, 1988. The lectures are based on accumulated experience

in piano design as well as recent experimental and theoretical studies -all presented in a

popular style.

The seminar day was preceded by two days of discussions between the lecturers and invited

representatives from piano manufacturers. Two representatives from Steinway & Sons,

Daniel T. Koenig, Vice President of Manufacturing, and William Y. Strong, Director of

Research and Development, joined the speakers in a closing panel session at the seminar,

answering questions from the audience and pondering future improvements in piano design.

The seminar day was closed by a remarkable concert -"From harpsichord to concert grand"

-in which the development of the piano was illustrated. The stage featured six instruments

representing piano design from 1813 to 1980, and a harpsichord as a reference to the

keyboard instruments before the piano epoch. Three pianists performed on the instruments

playing music contemporary to each instrument. Excerpts from this concert are included on

two gramophone records accompanying this book. The concert was recorded by The

Swedish Radio Company and later broadcasted.

The seminar was initially proposed by the Music Acoustics Committee of the Royal Swedish

Academy of Music. Later a Keyboard Committee of the same academy was founded, which

ran the seminar and additional events in cooperation with the Department of Speech

Communication and Music Acoustics at The Royal Institute of Technology and the Swedish

Radio Company.

The editing of this volume was considerably facilitated by the continuous and thoughtful

support of my colleague Erik Jansson. Due thanks are given to Si Felicetti, Gudrun Weiner-

Rispe and sa Wallner for patient assistance in the processing of the manuscripts and figures.

Stockholm in January, 1990

Anders Askenfelt, editor

Introduction

Background The scientific study of the acoustics of the piano goes back to Hermann von Helmholtz (1821

- 1894), a German physician and scientist, active in both neurology, optics, electricity and

acoustics. He compiled much of his thinking about sound, musical instruments and hearing

in a book "On the Sensations of Tone", which still is very much worth reading.(*) Helmholtz's

interest in musical instruments was strongly coupled to the perception of their sound. In

view of his limited measurement equipment - in which his ears played a central role - he

made remarkable contributions to the understanding of the tonal characteristics of several

musical instruments, among them the piano. In a series of Appendices, of which some have

become more famous than the text itself, he also presented theoretical analyses, including

the case of a string struck by a hammer.

Helmholtz was followed by occasional studies during the decades around the turn of the

century. These early investigators dealt in particular with the interaction between the

hammer and the string, a question which in fact still not has been completely settled. After

important pioneering works on almost every aspect of the piano in the 40's and 50's, by the

use of what we would call rather modern equipment, the study of the acoustics of the piano

has gained a renewed interest during the last decade. Although many, many questions

remain to be answered, a deeper understanding of the sound generation in the piano now

seems less remote than for several other instruments, in particular the bowed instruments.

The piano was invented in the 18th century, developed to its present design during the 19th

century - a period during which the bulk of classical piano music was written - and

produced on a large scale and frequently used in all kinds of music during the 20th century.

However, a complete understanding of the acoustics of the instrument will probably not be

reached until the next century. This may sound a little discouraging from a scientific point of

view, but the same statement holds true for almost all traditional instruments. The situation

is nothing but a result of man's incredible ingenuity in developing sound sources which not

only produce a pleasant sound, but which can also be intimately controlled by the player.

This evolution has resulted in musical instruments for which the acoustical function turns

out to be extremely complex, despite the fact that the instruments are based on seemingly

simple principles and made of common materials.

The piano is a representative example among the string instruments. The principle of its

function is indeed simple; a felt hammer strikes a metal string which is connected to a large

wooden plate. The string is set in vibration by the impact, and the vibrations are transferred

to the plate which radiates the sound. However, for none of the steps in this process - the

collision of the hammer with the string, the transmission of the string vibrations to the

wooden plate, and the radiation of sound from the plate into the air - the physics is well

enough understood to permit a detailed description of what actually happens in the real

instrument. In addition, "simple" materials like felt and wood turn out to have very complex

properties - different from sample to sample! -which further increases the difficulty of

describing the phenomena.

All this would have been enough, but the most cumbersome step is yet to come. The quality

of a traditional instrument is rated using our hearing as the ultimate test instrument. This

means that results of acoustical measurements should always be viewed in the light of how

they relate to the perceived sound. But this may not even be possible, because the

perception of sound, especially musical sounds, is a field which unfortunately is very poorly

explored. There are still many gaps in our knowledge of the relationship between physical

and perceptual properties of sounds. For this reason, many interpretations of experimental

results must remain on the level of advanced guesses.

With these difficulties in mind it is not surprising that it was possible to put a man on the

moon before the acoustics of a traditional instrument like the piano had been thoroughly

explained.

Landmarks in piano history In contrast to most other traditional instruments like the violin or the trumpet, whose

origins vanish in the haze of the past, a specific year and name can be attributed to birth of

the piano. In 1709 the Italian harpsichord maker Bartolomeo Cristofori replaced the

plucking pegs in a harpsichord by small leather hammers which he let strike the strings.

Since this new design allowed the notes to be played either soft or loud depending on how

the key was struck(**), he called his new instrument gravicembalo col piano e forte ("a large

harpsichord with soft and loud"). Soon the grandiose name was shortened to pianoforte or

fortepiano and eventually to piano.

Cristofori's piano was developed from the harpsichord and consequently rather small and

made entirely out of wood. As time passed, however, the development of larger instruments

with more and heavier strings at higher tensions - all in order to increase the volume of

sound - necessitated a more rigid construction. The wooden frame was successively

reinforced with more and more pieces of iron, and in 1825 the complete cast iron plate was

introduced by the American piano maker Babcock. The iron plate could withstand the

increased string tension, and prevented the instrument from gradually changing shape as

the wooden instruments did. Also, it now became possible to keep the tuning stable over

longer periods of time.

The hammers of the early pianos were tiny, light pieces made out of leather. However, the

introduction of coarser strings at higher tensions demanded larger and heavier hammers. In

1826, felt hammers were tried for the first time by an ingenious piano maker in Paris named

Pape. The success was immediate and lasting. An incredible amount of work was devoted to

the development and refinement of the actions. A prominent name in this connection is the

French piano manufacturer Erard who invented the so-called double repetition action in

1821, which is the type of action still used in the grand piano. The construction was refined

by another French manufacturer named Herz around 1840. Smaller improvements were

made during the following decades, but since then no essential changes have been made. A

simpler type of action, the Viennese action, lived a parallel life before it eventually vanished

during the first decades of this century.

The compass of the piano has increased successively during its history. Cristofori's piano

had only four octaves. Today a piano with a standard setup of 88 keys will cover more than

seven octaves (A0 = 27.5 Hz to C8 = 4186 Hz), no less than the pitch span of the modern

symphony orchestra. Furthermore, the acoustic output at fortissimo - small as it might seem

(of the order of 0.1 W) - surpasses all other string instruments. This power is enough to fight

even the largest ensemble (although brute force not always is the best way of making a solo

instrument heard above the orchestra).

The early pianos were of the type we now call a grand piano. During the 19th century the

manufacturers discovered a market for smaller and cheaper models, and squares and

uprights were constructed, both instruments being economy versions of the "real" piano

and filled with compromises. Both the grand and upright pianos as we know them today

developed during the 19th century, which saw a wealth of patent applications during its

latter half. The period of development declined shortly before the turn of the century,

indicating that the construction was perfected, at least for the time being.

Several of the recognized piano makers have had a long tradition including connections with

famous composers. Mozart played a Stein piano from Austria, Beeethoven preferred an

English Broadwood, and Chopin's piano was made by Pleyel in France - instruments from

eminent makers which today, however, are out of business or operating on a very low level.

Liszt and Wagner, on the other hand, used grands from Steinway & Sons (New York,

Hamburg) which were very close to the instruments we still are used to hearing 100 years

later. Other old, recognized piano manufacturers still in operation are Bsendorfer (Vienna),

Bechstein (Berlin), Baldwin (USA) and Yamaha (Japan).

The 20th century has been rather quiet as regards the development of the piano, but a

dramatically increased production has manifested itself in an undesirable way. The

beginnings of a lack of suitable wood and felt for piano purposes can be discerned. This will

successively put pressure on the manufactures to search for new materials which can

replace the traditional ones. This could, or probably will, demand changes in the design of

several major parts in the piano, and the possibility of an active period of development like

the one a century ago cannot be ruled out.

Thinking about the future Today, the piano is challenged by synthesizers, especially so the economy versions of

upright pianos. These pianos do not perform particularly favorably either in price or in tone

quality compared to dedicated piano synthesizers ("digital pianos, samplers"). Still, the

production of traditional pianos is large, estimated at 900 000 instruments a year

worldwide (1988). In particular, the grand piano seems to continue to attract professional

keyboard players of all genres, apparently for a number of reasons. Although the quality of

the sound probably is the main cause of its fascination, the mechanical response from the

instrument via the keys and the vibrating structure also seems to be very important.

In view of the rapid development of new instruments based on digital sound generation, it is

tempting to speculate about the future for the piano and the other traditional instruments. It

seems reasonable to suppose that the singing voice will be recognized as a musical

instrument as long as we use speech in communication. The vowels in speech and singing

will familiarize us with harmonic sounds, i. e. sounds which are associated with a distinctive

pitch. As long as pitch is used as a mean of communication in music, string and wind

instruments will take an exclusive position, because strings and pipes are the only tools

available for generating such sounds mechano-acoustically. A piano-like instrument with

struck strings could thus be assumed to be a natural member also of a future instrument

inventory, should the traditional way of generating sounds survive.

However, it is also possible that in the future most music will be performed on electronic

devices. This technique gives a much wider freedom in designing the sounds, including

imitation of the traditional instruments. Such imitations could also include extrapolations to

new pitches and dynamic levels, not accessible by the original instruments. It is hard to

deduce a priori if the piano sounds belong to the group of traditional musical sounds which

will survive in the long run, when transferred to a family of new instruments. However, in

view of the present popularity of the piano and recognizing the slow change in taste of

musical sounds hitherto, it is an advanced guess that pianolike sounds will be used and

enjoyed for at least another century.

Basics of piano acoustics In this section, a survey of basic piano acoustics is given for those of the readers who want

an introduction to the lectures. The fundamental principles which govern the acoustics of

the piano are presented in a somewhat simplified form. A detailed and more realistic story

of the sound generation in real pianos follows in the lectures.

Construction

A schematic view of the piano is shown in Fig. 1.

A steel string is suspended under high tension between two supports (the agraffe or capo

d'astro bar and the hitch pin) fastened in the metal frame (the plate). Close to the hitch pin

end, the string runs across a wooden bar, the bridge, which is glued to a large and thin

wooden plate, the soundboard. The level of the bridge is slightly higher than the string

terminations, thus causing a downbearing force on the bridge and the soundboard. The

soundboard is reinforced by a number of ribs glued to the underside, one reason being to

make the soundboard withstand the downbearing force. The string is struck by a felt

hammer, which gains its motion from the key via a complicated system of levers, the action.

Fig. 1. Principal sketch of the piano, designating the main components.

String motion

Physically, the string motion can be described in the following way. As the hammer strikes

the string, the string is deformed at the point of collision (see Fig. 2). The result is two waves

on the string, travelling out in both directions from the striking point. The wavefronts

enclose a pulse, or hump, which gradually gets broader.

Fig. 2. The evolution of the propagating pulse on the string after

hammer impact.

However, as the string is struck close to its termination at the agraffe, one of the wavefronts

(the one travelling to the left in the figure) soon reaches this end and is reflected. The

reflection at a rigid support makes the wave turn upside down. This inverted wave starts

out to the right and restores the string displacement to its equilibrium level.

The surprising situation has now developed that the wavefront initially travelling to the left

in the figure, has turned into the trailing end of a pulse of fixed width, propagating to the

right towards the bridge. At the bridge, the entire pulse is reflected, the effect being that the

pulse starts out in the opposite direction upside down. A new reflection at the agraffe turns

it right side up again, and soon the pulse has completed one round trip and continues out on

the next lap. If the key struck happens to be A4 = 440 Hz ("concert A"), the pulse completes

440 such round trips per second.

Pitch, partials and inharmonicity

The propagation velocity of the pulse on the string is determined by the tension and mass

per unit length of the string, a higher velocity the tauter and lighter the string. The number

of round trips per second, the fundamental frequency (closely related to the perceived

pitch), also depends on the distance to be covered - the longer the string the longer the

round trip time (fundamental period), and hence, the lower the pitch. The pitch of a string is

thus determined by a combination of its length, tension, and mass per unit length. In

particular, string length can be traded off against mass per unit length in order to reduce the

size of the instrument. This can be seen in the bass section, where the strings are wrapped

with one or two layers of copper in order to make them heavy and thus relatively short. The

advantage of a wrapped string over a plain string is that the mass can be increased without

reducing the flexibility drastically. A piano string need not be perfectly flexible, but a too

stiff a string would have a detrimental influence on the tone quality as will explained below

A piano string, like all other strings, has a set of preferred states of vibration, the resonances,

or modes of vibration (see Fig. 3). When a string is vibrating at one of its resonances, a

condition which usually only can be reached in the laboratory, the motion of the string is of

a type called sinusoidal. The corresponding sound is a musically uninteresting sine wave. In

normal use, however, where the string is either struck, plucked or bowed, all resonances are

excited, and the result is a set of simultaneously sounding sine waves, partials, forming a

complex tone.

Fig 3. The four lowest modes (resonances) of a rigidly

supported string. Sometimes these elementary states of

vibration are referred to as standing waves, because the

amplitude contour does not change with time.

Such a tone is conveniently described by its spectrum, which shows the frequencies and

strengths (amplitudes) of the partials (see Fig. 4, bottom). As mentioned, the pitch of the

tone is related to the frequency of the lowest member in the spectrum, the fundamental. To

be more specific, it is the frequency spacing between the partials - which for a piano tone is

closely the same as the fundamental frequency - which is the closest physical correlate to

the perceived pitch. The relations between the amplitudes of the partials and their evolution

in time contribute to our perception of tone quality.

The pulse running back and forth on the piano string has a most surprising connection to

the string modes (resonances). It can be shown mathematically that the travelling pulse is

made up of a sum of all the string modes! The shuttling pulse and an (infinite) sum of string

modes of appropriate amplitudes are equivalent; they are just two ways of representing the

same phenomenon (cf. Fig. 4). So while our eyes will detect the pulse motion (if slowed

down enough by the use of a stroboscope) our ears prefer to analyse the string motion in

terms of its partials or Fourier components, so named after the French mathematician who

first described this equivalence.

Fourier also stated that if the motion is periodic, that is, the same events will repeat

indefinitely with regular intervals, the frequencies of the corresponding partials will be

harmonic. This means that the frequency ratios between the partials will be exactly 1 : 2 : 3 :

4 . . . , which will be perceived as a sound with a clearly defined pitch and steady tone quality.

The statement can also be turned the other way around; if the resonance frequencies of a

string are strictly harmonic, the resulting motion of the string will always be periodic.

Fig. 4. Schematic illustration of the equivalence of the pulse motion on

the string (top) and a sum of the string modes (resonances) (middle).

The properties of the tone are conveniently summarized by its

spectrum (bottom), showing the frequencies and amplitudes of the

components (partials).

In real pianos, the resonance frequencies of the strings are not exactly harmonic. The

frequency ratios are slightly larger than 1 : 2 : 3 : 4 . . . , more like 1 : 2.001 : 3.005 : 4.012 . . . ,

which is referred to as inharmonicity. The inharmonicity in piano strings, which is caused by

the bending stiffness of the steel wire, is a desirable property as long as it is kept within

limits. According to Fourier, the string motion will now not repeat exactly periodically as the

note decays, but change slowly which gives a "live" quality to the note.

Returning to the excitation of the string by the hammer impact, not only the amplitude of

the initial pulse on the string changes with the strength of the blow, but also its shape. This

is due to a remarkable property of the felt hammer, more specifically the characteristics of

its stiffness. The stiffness increases (the hammer becomes progressively harder to compress)

the more the hammer already has been compressed, a phenomenon referred to as nonlinear

stiffness. This means that a harder blow not only will give a larger amplitude but also

sharper corners of the pulse on the string. Again, according to Fourier, sharper wiggles in

the waveform correspond to more prominent high frequency partials in the spectrum.

Consequently, the piano tone will attain a different ("more brilliant") tone quality at forte

(loud) compared to piano (soft).

Sound radiation and impedance mismatch

The vibrating string contains all the partials we would like to hear, but unfortunately the

string is in effect unable to radiate sound. The difficulty is well described by the English

saying: "You can't fan a fire with a knitting needle!" The reader may easily verify this

statement by making the experiment, but can also notice that by means of a large object like

a tray instead of the needle, it is quite possible to fan a fire even from a distance. The point is

that a certain flow of air must be pumped back and forth per second in order to radiate a

"fan wave."

This can be achieved with a limited motion of the tray having a large cross section, while the

needle would have to make unreasonably large movements to reach the same effect. The

acoustic engineer would "explain" the situation by saying that the radiation resistance of the

tray is much higher than that of the needle. In other words, because of its larger area, the

tray is much better than the needle as a transmission link between the motion of the arms

and the motion of the air.

Returning to the piano, we now realize that as the thin string cannot radiate a sound wave

itself, its motion has to be transferred to a much larger object which can serve as a more

efficient radiator of sound. This is readily done by incorporating a soundboard in the design,

including a bridge as a connecting element to the string(s). But now the piano designer

meets with a new difficulty. The soundboard is much heavier than the string, which means

that the string will not be able to vibrate the soundboard efficiently and the vibrational

energy will still be trapped in the string. Only slowly the energy will leak into the

soundboard during repeated reflections of the string pulse at the bridge.

In engineering terms, there is a mismatch between the mechanical impedance of the string

and that of the soundboard. The mechanical impedance is a property that tells us to what

degree an object resists (impedes) motion. From the point of view of the string, the

soundboard has a very high (input) impedance; it can be thought of as a very heavy stone, or

a very stiff spring, which must be vibrated vigorously. The experienced reader will certainly

agree that this is a most uncomfortable task with little chance of success.

Loudness versus "sustain"

However, conditions can be improved, or in other words, the impedance mismatch can be

diminished, by increasing the (characteristic) impedance of the string. This is easily done by

making it heavier and by increasing its tension. But a heavier string usually means a thicker

string, which automatically gives a higher stiffness and hence more inharmonicity, which

soon spoils the desired piano timbre. Piano designers circumvent this problem in two ways,

either by wrapping a rather thin steel core with copper (which also influences the pitch as

mentioned), or by "splitting" a thick plain string into two or three strings, tuned to (almost)

the same frequency, a technique called multiple stringing. Now the vibration energy is

transmitted more efficiently from the string(s) into the soundboard and the note sounds

louder, perhaps "too" loud. Because here the next difficulty appears; the gain in loudness

does not come for free.

It stands to reason that the pianist cannot feed energy continuously to the string like the

violinist via the bow. Consequently the piano tone is condemned to decay and die. The

question is then how to spend the energy quantum delivered at the key stroke in the best

way. If a loud and thus necessarily shorter note is desired, the impedance mismatch

between string and soundboard should be decreased by making the strings heavier and

tightening them even harder.

On the other hand, the note can be made longer by using lighter and less tense strings, but at

the expense of loudness. The trade-off between loudness and duration, or "sustain," of the

tone is a difficult problem in piano design, especially as the impedance of the soundboard

can vary wildly from note to note, due to its inherent resonances. It is easy to get a piano in

which some notes are loud and short while adjacent notes are much softer and longer, a

musically most unsatisfying situation. Fortunately, such fluctuations between notes as well

as the basic conflict between loudness and sustain can be alleviated in an almost miraculous

way by multiple stringing, a phenomenon which is covered in detail in one of the lectures.

The imperfect soundboard

The soundboard radiates sound much better than the strings do, as mentioned, but

nevertheless it has several severe shortcomings. One occurs at very low frequencies and is

due to the fact that both sides of the soundboard are directly exposed to the surrounding air.

The reason is the following.

Let the soundboard be moving upwards, pushing the air above its upper surface together.

This causes a temporary excess of air molecules in a region above the soundboard, a

compression, corresponding to an increased pressure. The underside of the soundboard is

also moving upwards, so there is at the same moment a temporary loss of air molecules

beneath the soundboard, a rarefaction, corresponding to a reduced pressure. As nothing

prevents the compressed air on the upper side from flowing into the lower region, this

pressure difference will soon be neutralized. Half a period later, when the soundboard is

moving downwards, the process repeats but now the air flows from the lower to the upper

side. So, at low enough frequencies - as long as the motion of the soundboard is slow enough

to allow the exchange of air to take place before the direction of its motion has reversed -the

soundboard will uselessly pump air from its upper side to its lower side and back again

instead of radiating sound. The phenomenon is called acoustic short-circuiting, and can be

avoided by separating the two radiating sides of the soundboard by an (almost) closed

sound box, as in the guitar or in most harpsichords.

A similar phenomenon can be observed also at higher frequencies. Now the soundboard no

longer vibrates as a unit but spontaneously divides into smaller vibrating areas separated

by thin regions of no motion (nodal lines). Depending on frequency, the vibrating areas form

different patterns; the higher the frequency, the smaller and so the more numerous are the

areas. These preferred states of vibration are called the eigenmodes (modes), or often, the

resonances, of the soundboard. Adjacent vibrating areas vibrate in what is called opposite

phase, which means that while one area is moving upwards its neighbour is moving

downwards and vice versa. Also in this case, it is easy to imagine that a useless exchange of

air between adjacent areas can occur instead of the desired sound radiation.

That's all!

This closes the short survey of basic piano acoustics. Once again, it is to be understood that

the explanations are simplified, dealing only with the basic aspects of the phenomena.

Against this background, the lectures that follow will illustrate the wealth of complications

which arise in real instruments.

A note on units In this volume, the use of metric (SI) units is encouraged. While the use of meters and

kilograms probably will cause English and American readers only minor problems, the force

unit Newton (N) might be less familiar. As a rule of thumb, 1 N corresponds to the weight of

an apple (mass 100 g)!(***) Likewise, 10 N corresponds approximately to the weight of a

mass of 1 kg, for example 1 litre (1 US quart) of milk.

The naming of octaves and pitches follows the straightforward nomenclature given by

American standards. In this notation the "middle octave" is indicated by number four

(middle C = C4). The lowest note on full size piano is A0 and the highest C8.

Departure After these introductory passages, it is time for a detailed voyage into the world of the

acoustics of the piano, guided by experts in the different areas. The lectures follow in the

same (logical) order as they were given on the seminar day, but as the contributions are

essentially independent the readers may feel free to follow their own paths.

In the first lecture, Harold Conklin, an experienced piano design engineer, outlines the

design principles of the parts of the piano, and makes comparisons between the early and

the modern instruments.

Secondly, Anders Askenfelt and Erik Jansson, researchers in music acoustics with a focus on

string instruments, present measurements from the initial steps in the tone production,

from the moment when the pianist touches the key up to and including the string vibrations.

Then follows a theoretical study by Donald Hall, a physics professor with a strong personal

interest in keyboard instruments, who describes a computer model of what actually

happens during the collision between the hammer and the string, and the implications for

the string vibrations.

The decay of the piano tone, and in particular the influence of multiple stringing is covered

next by Gabriel Weinreich, also a physics professor with a strong interest in music acoustics.

Finally, the sound radiation and its connection to the properties of the soundboard are

described by Klaus Wogram, a researcher with many years of experience in investigating

musical instruments, in particular brass instruments and the piano.

Notes

(*) Hermann von Helmholtz: Die Lehre von Tonempfindungen als physiologische Grundlage fr

die Theorie der Musik, first edition 1862, English translation of the fourth edition in 1885 by A.

J. Ellis: On the Sensations of Tone as a Physiological Basis for the Theory of Music, reprinted

(paperback) by Dover Publications Inc., New York 1954.

(**) It is true that also the harpsichord can be played at somewhat different dynamics

depending on how the key is depressed. Compared to the piano, however, the dynamic range is

narrow, and dynamics are usually not prescribed in harpsichord music.

(***) This useful remark was given by one of the lecturers (G. Weinreich) on an earlier occasion.

Piano design factors

- their influence on tone and acoustical performance

Harold A. Conklin Jr

Introduction My presentation will be an overview of some of the ways in which the design of a piano

affects its tone and acoustical performance. It is not possible in a short lecture to mention all

the important factors, because so many things in a piano affect its sound. Fig. 1 shows the

oldest existing piano, the Cristofori instrument of 1720, which is in the Metropolitan

Museum at New York City. In a recorded excerpt we can hear this historic instrument

followed by the familiar sound of a contemporary concert grand (sound example 1).

Fig. 1. Cristofori piano of 1720.* (By

permission of the Metropolitan Museum of

Art, New York: The Crosby Brown

Collection of Musical Instruments, 1889.

Piano Forte (89.4.1219): compass 4

octaves and one quarter (C - F), Italian,

Florence, 18th C., 1720. Maker:

Bartolomeo Cristofori.).

It is obvious that the extreme differences in design between these two instruments produce

extreme differences in tone quality. From time to time we will refer again to the oldest piano,

but in order to concentrate on acoustical factors we will ignore or mention only briefly some

of the important mechanical differences. For example, it is evident that the action of the

1720 piano (see Fig. 2, Pollins 1984) is much simpler and less controllable than that of a

modern grand. After listening to the recording of the old piano one can say almost with

certainty that the music that was played on the modern instrument could not be played

properly with the action of Cristofori.

Fig. 2. Action of 1720 Cristofori piano. (By

permission of the Journal of the American

Musical Instrument Society).

The hammers The hammers of a piano not only define the instrument: they also are among the most

important factors in determining its tone quality. The hammers in the 1720 piano have

wooden heads which are covered with leather (Fig. 3). Modern piano hammers are covered

with wool felt that is compressed and stretched over a wooden molding (Fig. 4). Often two

layers of felt are used. In Fig. 5 a modern hammer with the outer felt loosened is shown.

Fig. 3. Hammers of 1720 Cristofori piano (By permission of

the Metropolitan Museum of Art).

Fig. 4 and 5. Modern grand piano hammers as normal and

with outer felt loosened.

A view of the action from the 1720 piano is seen in Fig. 6. Cristofori glued his hammers onto

wooden shanks as we are still doing today. In a modern grand piano the mechanical system

of the hammer head on its somewhat flexible shank exhibits a major vibrational mode

around 260 Hz as installed in the piano. This mode is not normally audible in the lower half

of the piano's compass, but it can be heard in the treble register, as part of the "knock"

component of the tone, beginning somewhere around A4 (key 49), and can be shown to

affect noticeably the tone of the instrument. An improvement in the tone can sometimes be

obtained by shifting the frequency of this resonance. The motion of the hammer as it

contacts the strings is very complex, and is only recently becoming clear (Hall 1986, Hall

1987a, 1987b, Hall & Clark 1987, Boutillon 1988, Hall & Askenfelt 1988).

Fig. 6. Cristofori action being played (By

permission of the Metropolitan Museum of

Art)

The hammers of the Cristofori piano are all about the same size. In a modern piano, the size

and weight of the hammers increase from treble to bass in order to achieve the best

compromise between tone quality, loudness, and playability. Fig. 7 shows typical hammer

head weights for a modern grand piano.

Fig. 7. Typical weight curve for modern grand

hammer heads.

The largest bass hammers may weigh around 11 grams. The smallest treble hammers may

weigh as little as 3.5 grams each. Somewhat more output could be obtained at the extreme

treble end of the scale if the hammers were somewhat lighter, but this would increase

manufacturing problems. In the bass, tones having somewhat more fundamental energy

could be obtained by using heavier hammers but then the piano would become harder to

play. An increase in hammer weight can be counterbalanced statically by installing

additional lead weights near the fronts of the key levers so that the force required to

depress a key very slowly will remain at its nominal value (usually around 50 grams).

However, this cannot compensate dynamically for increased hammer mass. Key velocities

corresponding to higher musical dynamic levels can require a finger force corresponding to

several kilograms, and the value of this force increases noticeably with an increase in

hammer weight.

The heavier a piano hammer is, the longer it will stay in contact with the string(s). There is a

critical region of the piano's compass, between about G4 and G6 on the keyboard. Within

this range the contact time of the hammer against the strings becomes equal to the

roundtrip travel time for the initial pulse on the strings (Benade 1976). Below this range the

hammer leaves the strings before the arrival of the first major reflection from the far end;

above this range the hammer normally is still in contact. If the hammer is still in contact at

the time of the first reflection, losses occur that decrease the output of the piano and may

cause an undesirable quality of tone. In order to produce a graceful tone within this critical

range, it is important to have an optimum hammer striking position along the strings, to

have the hammer strike all of the strings of a note equally, and to keep the hammers from

being too heavy. These factors are interdependent. If the hammers in the treble are too

heavy, the tone will not be as loud. If the hammers of the 1720 piano were to be used in a

modern instrument, the tone of the bass and middle registers would sound too thin and

bright, and the treble tone probably would be harsh.

The hardness of a piano hammer directly affects the loudness, the brightness, and the

overall tone quality of the instrument. In order to produce the best tone, each hammer must

have its hardness within a certain range. Also, the hardness should have a gradient such that

the string-contacting surface is softer than the inner material. If there is no gradient, the

result can be poor tone or undesirable noise components. In Fig. 8 a special tool called a

durometer is shown in use to measure the hardness and indicate the gradient of a hammer.

This measurement can indicate whether the hammers have the right hardness to make a

good piano tone. You could also find this out just by listening to the piano, if the hammers

were already in place. But by measuring the hardness first it can be determined in advance

whether the hammers can sound good, and it will be indicated how much work will be

required to voice them.

Fig. 8. Durometer in use to measure

hammer hardness.

Fig. 9 shows the measured hardness for three different hammers of similar size and weight.

To demonstrate the relation between hardness and tone quality I have made a recording of

the tone produced by each of these hammers when installed at G5 (key 59) in the same

piano (sound example 2). First you will hear the softest hammer played six times, then the

harder hammer, and finally the hardest hammer. (The amount of difference you hear in the

tone may depend on where you are sitting in relation to the loudspeakers.) As I hope you

can hear, the softest hammer produces a pleasant tone that is perhaps a bit too soft (dark);

the middle hammer produces a significantly brighter and louder tone, and the hardest

hammer produces a still brighter but somewhat harsh tone that contains excessive noise

components.

Fig. 9. Shore A hardness for three hammers.

The optimum hardness for a hammer varies widely with its keyboard position. In order to

produce tones of uniform loudness all across the scale, the extreme treble hammers must be

much harder than the middle or bass hammers. The need to make the hammers harder in

the treble usually begins, probably not by coincidence, in the critical region where the

roundtrip time becomes equal to the hammer-string contact time.

Fig. 10 indicates approximately how the relative hardness of hammers should vary across

the scale in order to produce tones of equal loudness for an equal key effort. Of course the

optimum value for hardness also depends on how bright a tone the listener prefers, so this

graph gives only a general indication.

Fig. 10. Approximate relative hardness of

piano hammers for equal loudness.

Hammers can be "voiced" by a skilled piano technician to make them harder or softer, in

order to produce the best tone and smooth response from note to note. In voicing, the felt

may either be softened by piercing it with needles at certain carefully chosen locations, or it

may be hardened, either by filing away the soft outer felt with sandpaper, or by applying a

chemical hardening agent. Voicing has little measurable effect on the lower partials of bass

tones. In the treble, all of the partials are affected.

Good hammers, properly voiced, are necessary to make a fine piano, but they are not

sufficient. The other parts of the instrument are at least equally important. The hammers

merely provide the exciting force for the strings. A bad piano equipped with even the best

hammers will still be judged a bad piano.

Where should the hammer hit the string

The hammer striking ratio (d/L) for the 1720 Cristofori piano and for two representative

modern pianos is shown in Fig. 11. Here L stands for the speaking length of the string and d

is the distance from the closest string support (the agraffe) to the point where the hammer

strikes. The values for the 1720 piano seem to wander over a wide range to no apparent

purpose. Early makers did not fully appreciate the effect of varying d/L but by the late 18th

century, piano makers began to know what values work best (Harding 1933). Many books

about pianos state that the best place for the hammer to strike the strings is between 1/7

and 1/9 of their speaking length. (Good 1982, Marcuse 1975, Mc Ferrin 1972, Briggs 1951,

White 1946, Wood 1944, Vant 1927, Ortman 1925, Wolfenden 1916, White 1906, Hansing

1888, Brinsmead 1879, Helmholtz 1863).

Fig. 11. Striking ratio (d/L) for two

contemporary pianos and for 1720

Cristofori.

This is certainly not true for all the notes of modern pianos. In the best modern grand pianos

the smallest treble hammer (C8) is always positioned at the factory for each piano

individually and is set to produce the loudest tone. This normally occurs for a d/L-value

much smaller than 1/9, usually in the range between 1/12 and 1/17. As you can see from

the curves labeled "contemporary" in Fig. 11, d/L in the bass is a little less than 1/8, and it

decreases gradually up to around A4 (note 49), and then decreases rapidly. How d/L should

vary across the compass depends on a number of factors and is decided by the designer of

the piano.

In the mid-treble, the best striking ratio often is a compromise between maximum first

partial energy and the most graceful tone. Reducing the striking distance in this region

generally makes the tone sound thinner because less fundamental energy is present.

Increasing the striking distance makes the tone fatter, but may produce an unclear, muddy

quality. Here, hammer weight is also an important factor.

In the lower part of the scale, hammer contact time is small in comparison with the

roundtrip time for the pulse on the string - from the striking point to the bridge and back

again. Consequently, damping due to the hammers is small. Moving the striking points of the

hammers changes the tone quality primarily by rearranging the relative amplitude of the

partials. If the hammer should strike the string at a nodal point, or near, where the string

motion is small, then the amplitude of the corresponding partial will also be small.

Fig. 12 shows how the measured output of one particular string varied as the hammer

striking ratio was changed. The graph shows partials 5 through 9. The amplitude of each

partial passes through a distinct minimum point as the striking ratio is increased. If you

were listening to the tone of the string you would hear obvious differences in timbre as the

hammer striking distance was changed, and I am sure you would like the tone at certain

d/L-values better than at others.

Fig. 12 Output vs. striking ratio (d/L) for

partials 5, 6, 7, 8, and 9.

Fig. 13 shows the instantaneous peak output spectrum for two different values of the

hammer striking ratio. For d/L = 0.019 (1/53), the lower partials all have very small

amplitude. This is because the hammer is striking almost at the very end of the string. For

such a small d/L the tone sounds thin and weak. For a longer striking distance, d/L = 0.143

(1/7), the lower partials have gained in amplitude and the 7th partial is almost completely

missing. At one time it was believed that the 7th and 9th partials were dissonant and ought

to be eliminated by a proper choice of the striking distance. Personally, I do not believe that

any string partial should be deliberately minimized.

Fig. 13. String spectra for short and long striking ratio (d/L

= 0.019 and 0.143).

Soundboards

Fig. 14 shows the top of the soundboard of the 1720 Cristofori piano. The original

soundboard was made of cypress wood, 3.5 mm thick, which may have come from Crete

(Pollins 1984). The original soundboard was replaced in 1938 with what is said to be an

accurate copy. The bottom of the soundboard can be seen in Fig. 15. In contrast, Fig. 16

shows a contemporary concert grand. Note that much of the contemporary soundboard is

covered by the cast iron string plate. The soundboards of modern pianos usually range in

thickness between 6.5 and 9.5 mm approximately. In the U.S.A., spruce, and particularly

Sitka spruce, has been the favored soundboard material for high quality pianos.

Fig. 14 (left) Plan view of 1720 Cristofori

(By permission of the Metropolitan

Museum of Art).

Fig. 15 (right) Bottom of 1720 Cristofori

(By permission of the Metropolitan

Museum of Art).

Fig. 16. Plan view of contemporary concert grand.

How does a soundboard vibrate?

Soundboards vibrate more readily at their modal or resonance frequencies than at other

frequencies. The photos in Figs. 17 - 20 show how a piano soundboard vibrates at some of

its modes (resonances).

The lowest frequency at which a soundboard can vibrate strongly is called the first mode. In

Fig. 17 we see an experiment in which a concert grand piano soundboard has been vibrated

at its first mode. The vibration generator, the circular object that can be seen to the left in

the photos, has been connected mechanically to the soundboard at a point near its edge. For

such a test the procedure is the following: Before being vibrated the soundboard is covered

uniformly all over its surface with a mixture of fine particles (in this case sand and "glitter").

Then the vibration generator is turned on and tuned slowly until its frequency coincides

approximately with that of a soundboard mode, as will be indicated by a noticeable increase

in sound level from the soundboard. Then the generator level is increased until the

acceleration of the particles exceeds "1 g" (the acceleration of gravity, 9.8 m/s2) and the

particles begin to dance on the soundboard. As they dance, the particles gradually collect in

those areas that are not moving at all or are moving with minimum velocity. This produces a

pattern called a Chladni figure, so named after the famous German physicist.

The first mode of this soundboard occurred at 49 Hz. In this mode, it is the center of the

soundboard that moves most violently; the edges, where you see most of the particles,

nearly stand still. A piano soundboard rapidly loses its effectiveness as a sound radiator at

frequencies below that of the first mode, so notes below the first modal frequency usually do

not have very much energy in the first partial.

In Figs. 18, 19, and 20, you can see how the soundboard moves at some of its other modes.

Remember that the particles collect where the soundboard is moving least.

Fig. 17. First (lowest) soundboard mode at 49 Hz. Fig. 18. Second mode at 67 Hz.

Fig. 19. Third mode at 89 Hz. Fig. 20. Eighth mode at 184 Hz.

The modal frequencies are determined by many factors, the primary ones being the material,

size and shape of the soundboard, its thickness and grain direction, and also the material,

dimensions, and placement of its ribs. Secondary factors include the characteristics of the

rim or case to which the soundboard is attached. In general, the thicker the soundboard, the

louder the piano but the less the duration of its tone. Soundboard design is often a

compromise.

Today there is a better way, called modal analysis, to study the vibration of piano

soundboards. Using this method, the soundboard is tapped with a special hammer that is

fitted with a force transducer. An accelerometer attached to the soundboard responds to

vibrations caused by the hammer and the force and acceleration signals are stored digitally.

The tapping is repeated at a number of different preselected points on the soundboard, and

after all the data has been taken, a computer analyzes it and identifies the modes (Suzuki

1986). With modern equipment it is possible to see an animated display of the modal

motion of the soundboard on a TV-screen, a technique which will be described in more

detail in the lecture by Klaus Wogram.

Fig. 21 presents modal information in another way: it is a graph giving the velocity of

motion of the soundboard at one particular point (for a constant driving force), as a function

of frequency. This plotted quantity is called mobility, and is the reciprocal of mechanical

impedance. Each of the large peaks you see in Fig. 21 corresponds to a particular

soundboard mode like those earlier shown in the Chladni patterns.

Fig. 21. Driving point velocity vs. frequency for concert

grand soundboard with no strings or plate (top), and with

the piano fully assembled and tuned (bottom).

The frequency and shape of soundboard modes are affected by the strings and the cast-iron

plate. For the graph at the top in Fig. 21, the plate and strings were removed from the piano.

The lower graph in the same Fig. shows the mobility at the same point on the same

soundboard with the strings and plate in place and the piano fully tuned. Notice how much

the picture has changed: the first mode has shifted upward in frequency from 48 Hz to

around 60 Hz, and the modal peaks are broader than before and not so high. In pianos of

this size (concert grand) you can often identify the first mode by playing single notes up and

down the scale. You may feel a slight increase in the vibration level of the case, usually

around C2 - D2 (keys 16 - 18), and you may hear an increase in the sound level of the first

partial.

In order to do the analyses just mentioned we have to test an actual soundboard. So we need

first to build a piano before we can measure it. But now, by still another new technique

called finite element analysis (FEA), we can construct a model of the proposed soundboard

with computer software. Then, using a computer, we can find out how the soundboard will

move before we build the piano!

The varnish

The varnish on a piano soundboard does not have a significant effect on the tone, as far as I

have been able to discover. However, the varnish has a very significant effect on the tuning

stability of pianos. Without varnish on its soundboard a piano can go rather quickly out of

tune if the humidity should change. This is because the size and weight of any piece of wood

depends on the relative humidity and temperature of the air around it. Wood absorbs

moisture until the amount in the wood is in equilibrium with the surrounding environment.

Fig. 22. Equilibrium moisture content of wood vs. relative

humidity at 24o C.

Fig. 22 shows how much moisture will be in a piece of wood at equilibrium for different

values of relative humidity at a temperature of 24o C (75o F) (see Wood Handbook 1987).

The amount will be about the same for all species of wood. As you can see, at 50% relative

humidity about 9% of the weight of the wood will be moisture. The varnish on a soundboard

slows down the rate at which moisture can enter and leave the soundboard, and so lets the

piano stay in tune longer. I once gave a copy of this graph to a friend who had just bought a

new grand piano. A couple of days later he telephoned, sounding somewhat upset: it seems

he had calculated that his new $30,000 piano contained 7.2 gallons (27 litres) of water and

that each gallon had cost $229!

New materials

Why can't we use a material for soundboards that is not affected by humidity? Some early

research on soundboard materials other than wood apparently was done here in Sweden by

a man named Fridolf Frankel. Fig. 23 shows the cover of a booklet, written in English and

dated 1923, describing a soundboard that is said to have been made of steel, 0.65 mm thick.

I found the booklet in some old files of an American piano company. Despite the favoured

testimonials for the performance of his pianos (Fig. 24), few instruments seem to have

survived.

Many years later, in 1961, the American harpsichord builder, John Challis, constructed a

piano having a metal soundboard and bridge (Challis 1963). Such an instrument was

demonstrated by the pianist Arthur Loesser at a concert in New York City in 1967 (Henahan

1967). An excerpt from a recording of this concert can be heard in sound example 3.

Apparently, the Challis piano was not suitable for playing a wide range of standard piano

literature, for even at this concert it was used by the performer only for a few 18th century

pieces.

Fig. 23. Cover of Frankel booklet describing steel

soundboard (1923).

Fig. 24. Testimonials for Frankel's pianos.

In 1969, a U.S. patent was issued to P.A. Bert describing a soundboard for pianos employing

sandwich construction with a cellular core and plastic facings. I have been told that at least

one such instrument was built, but so far as I know, they were not marketed. I personally

believe strongly that researchers today have available better materials for piano

soundboards than ever before, and that only diligent applied research is needed in order to

produce the next significant improvement in the piano.

When we try to "improve" the piano we must remember that we dare not change its

essential character. If we do, it is almost certain to be rejected. Pianists who have spent

years in learning to deal artfully with existing instruments quite naturally do not want to

have to relearn their skills. They are incredibly sensitive to changes! I was present at one

occasion on which an instrument of rather exotic construction was being tested. It looked

like an ordinary piano and its sound was extremely pleasant, though slightly unusual. An

excellent pianist was called in to give an opinion. He played at some length. To us in the

audience the instrument sounded quite beautiful. Finally our pianist said it had a "nice

sunny sound" and would be very good for Spanish piano music. After this encouraging initial

response some minor changes were made and a rather more famous pianist was called in.

This pianist reported that the instrument was good only for French piano music! Still further

changes were made but this particular instrument never moved across the border and into

Germany, musically speaking.

The piano case

As you can see from Figs. 25 and 26, a modern grand case is very substantially made. The

rims of the best modern grand pianos are usually made from heavy hardwoods such as

maple or beech, and may have a total thickness between 80 - 90 mm. The case for a piano of

this size may weigh 150 - 200 kg. The acoustical benefit of this is that it provides a massive

termination for the edges of the soundboard. This means that the vibrational energy will

stay as much as possible in the soundboard instead of dissipating uselessly in the case parts,

which are inefficient radiators of sound.

Fig. 25. Grand rim with keybed attached.

Fig. 26. Grand rim nearly completed.

Cristofori had a totally opposite idea about the soundboard as the sketch in Fig. 27 shows

(Pollins 1984). His soundboard, 3.5 mm thick, was glued to an extra inner vertical case wall,

only about 4 mm thick. This was mechanically decoupled from the main outer walls of the

case. Cristofori must have felt that connecting the soundboard directly to the outer case

would impede its vibration. Fig. 28 is a view of the underside of the 1720 piano with its

bottom board removed. The large cross members are not ribs but rather stiffening members

that are connected to the sides of the outer case.

Fig. 27. Cross section of Cristofori

case. (By permission of the Journal

of the American Musical

Instrument Society).

Fig. 28. Close-up of underside of

1720 Cristofori piano. (By

permission of the Journal of the

American Musical Instrument

Society).

The cast-iron plate Piano makers gradually learned that pianos could be made louder by increasing the weight

and tension of the strings with the result that wooden frames soon became inadequate to

support the increased stresses. Until around the beginning of the 19th century the load-

bearing structure of pianos was made entirely of wood. By the end of that same century

almost all pianos had cast-iron string plates. The metal plate (Fig. 29) brought improved

tuning stability and, at least to most modern ears, better tone.

Fig. 29. View of contemporary

concert grand plate.

The need for a stronger supporting structure for the strings is clearly indicated by Fig. 30,

which shows the pulling force per string (tension) and the total string pull, calculated for the

1720 Cristofori and for a contemporary concert grand. The average pull of the strings of the

Cristofori is only about 70 N (16 lbf, "pound force"), versus about 830 N (190 lbf) for the

contemporary piano. The total string load is roughly 7500 N (1700 lbf) for the Cristofori,

compared with about 210 000 N (47 000 lbf) for the modern concert grand.

Fig. 30. String pull for 9-ft (274 cm)

contemporary grand and for 1720

Cristofori.

The plate must be strong enough not to break under the load of the strings, and it should

also be stiff enough to provide good tuning stability. Beyond this, the design of the plate

affects the tone of the instrument in many less obvious ways, of which only their general

direction will be indicated.

In any stringed instrument the speaking length of each string has two ends. In a piano, one

end is connected, via the bridge, to the soundboard, which is expected to radiate sound

efficiently. The other end is always connected in some way to the frame of the instrument,

which is invariably an inefficient radiator of sound. In modern pianos the forward string

termination (the agraffe) is located on the iron string plate, as shown in Fig. 31 This part of

the plate should be designed so as not to steal energy away from the strings and the

soundboard. The plate should not vibrate appreciably at string frequencies as the piano is

being played. Generally, this requires that the plate be rather massive. Plates of concert

grand size may weigh 160 - 180 kg.

Fig. 31. Close-up of forward termination

(agraffe panel) of modern grand.

Strings

The fundamental frequency of a stretched string is given by the expression below, known

since the beginning of the 17th century, in which L is the length of the string, T is its tension

or pull, and M is its mass per unit length.

The fundamental frequencies of a modern piano are known in advance because A4 has a

standard frequency (440 Hz) and because the frequencies of adjacent notes all across the

equally tempered scale have a ratio equal to the twelfth root of 2, about 1.05946. The most

common calculation in designing a piano scale is the tension, not the frequency of the string.

(The scale is the distribution of the string lengths and gauges over the compass of the

instrument). The above expression can easily be rearranged to give the tension. Also, a

factor (F) can be added to the formula above to allow for the use of wrapped strings, and a

changeable constant (k) may be employed in order to permit calculation for any string

material in any system of units. Then the tension of a string may be written as:

In this formula, dc is the string diameter (or the core wire diameter, in the case of a wrapped

string). F is unity (1) for a plain string but has some larger positive value for a wrapped

string. For a string of steel music wire for which the length and diameter are given in

centimeters, the tension will be given in Newtons (N) for k = 4096. I show this formula to

make a certain point: there's a lot of multiplying here! In the old days it could take a long

time to calculate just one string. What if you had to do this for 88 different strings with only

pencil and paper?

Much in early piano design obviously was empirical. Empiricism seems to have persisted for

longer than one might expect. Would you believe that even in the 20th century, piano

designers still didn't know how to calculate the tension of a wrapped string, and had to find

it by actual measurement. (Wrapping the core of a string helically with turns of another wire

is a very old method, still in use, to make a bass string heavy without having its core too

thick and thereby too stiff to produce a good tone.) The following sentence appears in a

supplement, dated 1927, to Wolfenden's well-known book about piano design:

"It is remarkable that, at this date, after spun strings have been in use for, say, a matter of

two centuries, neither in this country nor any other, as far as many enquiries have shown, is

there in trade use, a method by which the tensional stress upon a spun string, tuned to a

given pitch, can be approximately ascertained" (Wolfenden 1927).

Equations for calculating the pull of wrapped piano strings are now well known to at least

some piano manufacturers and also to many piano technicians. Also, with computers we can

calculate piano strings and scales very accurately and much more quickly than ever before.

Longitudinal string modes

Of course there is more to designing good piano scales than merely calculating the tension of

the strings. In 1967, I applied for a patent under the heading, "Longitudinal Mode Tuning of

Stringed Instruments" (Conklin 1970). I found the technique outlined in this patent to be

such a powerful tool in scale design work, especially in the design of wrapped strings, that

today I would not consider designing a piano without it.

In longitudinal modes of vibration, energy propagates lengthwise along the string (as

periodic compressions of the string material) without sidewise (transverse) motion of the

string. Longitudinal and transverse vibrations of a piano string can occur simultaneously.

However, the lowest-frequency longitudinal mode of a piano string is always more than ten

times the frequency of the lowest-frequency transverse mode.

It has long been known that the strings of pianos and other musical instruments can have

longitudinal modes of vibration (Rayleigh 1877, Knoblaugh 1944, Leipp 1969). My patent

simply teaches what the designer should do about it in order to make the best sounding

instrument. I learned the importance of the longitudinal mode by accident; one day, while I

was installing a new string, I noticed that the string sounded better when it was tuned to the

wrong frequency! After some study it became apparent that the reason had to do with the

longitudinal mode. The first longitudinal mode of a piano string normally occurs at a

frequency somewhere in the range between 3 octaves plus a fifth and 4 octaves plus a third

above the "normal" fundamental transverse frequency of a string. This range is determined

by certain design constraints related to the properties of piano wire that are common to all

present-day pianos.

A piano tuner tunes the transverse, or flexural, modes of the strings by changing the tension

of the strings as he turns the tuning pins. A piano tuner can do nothing to affect the

frequency of the longitudinal mode because turning the tuning pins doesn't change it. The

longitudinal frequency of a plain steel string in a piano can be changed only by altering its

speaking length. In the case of wrapped piano strings, the longitudinal mode can be tuned

only in two ways: either by changing the speaking length or by changing the weight of the

wrapping wire in relation to the weight of the core wire. So, the tuning of the longitudinal

mode is established, either deliberately or accidentally, by the designer of the piano; and, as

a practical matter, it cannot be changed after the piano has been built.

In designing a piano nowadays, it is possible to tune the longitudinal modes of its strings to

those frequencies that will make the piano sound best. In sound example 4, you can hear

what kinds of changes in the timbre of piano tones are produced by changing the tuning of

the longitudinal mode.

I think you will agree that each string sounds different from the others. However, all those

six strings were tuned to the same transverse frequency by a piano tuner: all the notes were

low G1 (key 11) on the piano! They sounded different because each had a different tuning of

the longitudinal mode. In sound example 5, you can hear a little tune which is known as

"Yankee Doodle". The tune is played in two different keys.

I am sure you will recognize that a tune is being played. However, the tune was played on

strings that were all tuned to the same transverse frequency! The tunes could be heard

because each string was designed so that its longitudinal mode differed in frequency by a

semi-tone (100 cents) from that of the preceding string. (The common transverse frequency

was not the same for the two versions in different keys.)

Next, listen to some chords, each chord followed by a bass note having a different tuning of

the longitudinal mode but the same tuning of the transverse mode (sound example 6).

As I hope you can hear, the longitudinal mode is important in determining the tone color of

the bass and tenor regions of the piano. The longitudinal mode creates a formant-like

emphasis in the tone at its own frequency, with the result that some tunings of the

longitudinal mode sound much better than others. In particular, it is desirable to have the

longitudinal mode tuned so that it blends harmoniously with the tone from the transverse

modes. This can be achieved by careful and deliberate choices in the design of the strings

and scale of the instrument.

In the examples you have heard so far the longitudinal mode was deliberately tuned at

intervals of a certain number of semi-tones with reference to the fundamental transverse

mode. Strange and undesirable things can happen to the tone if the longitudinal mode is

ignored or left to chance by the designer. Next you will hear some scales played on two

different pianos. The first piano has the longitudinal mode tuned by design, the second one

does not. As you can hear, the piano having deliberately tuned longitudinal modes has a

much more uniform and pleasing voice through the scale (sound example 7).

Physicists may want to know if it is possible to measure what we are hearing. Fig. 32 is an

acoustical spectrogram of piano note E1, with a fundamental frequency of about 41 Hz. The

"normal" transverse partials are identified by small dots near each peak. The longitudinal

mode can be seen between the 14th and 15th partials and is about 20 decibels lower in level

than the neighboring partials (Podlesack & Lee 1988).

Fig. 32. Spectrum of piano note E1 (41 Hz)

showing longitudinal mode (indicated by

the vertical line at about 600 Hz).

Machines similar to those shown in Fig. 33 have been used for a long time to wind the

copper covering wire onto wrapped piano strings. With these machines the characteristics

of the finished strings are strongly dependent upon the technique of the operator. Operator

technique varies, not only from person to person, but also from string to string. It was found

to be impossible with such machines to control the tuning of the longitudinal mode precisely

enough so that successive strings could be accurately tuned and alike in tone. For this

reason, it was necessary to design a new type of machine, also patented, with which the

characteristics of the finished string would be independent of the operator. With this

machine, optimum settings are predetermined for each type of string.

Fig. 33. Old string

machines.

The tuning pins One thing about pianos has hardly changed at all in the 268 years since the 1720 Cristofori

was built: the tuning pins! They are still small metal cylinders that are driven into holes

bored in a slab of wood. As all piano technicians know, tuning pins can have various

problems that interfere with accurate tuning of the instrument. I am very pleased to

announce - and this is the first public announcement - that I have devised a new type of

tuning pin that seems to eliminate problems encountered with conventional tuning pins. My

tests indicate that the new pin will make tuning easier, faster, and more accurate. Because of

patent considerations, I cannot yet describe it to you but before long I hope to be able to

convince piano manufacturers to use it!

References

Benade, A.H. (1976): Fundamentals of Music Acoustics (Oxford University Press, London) pp.

339-343.

Boutillon, X. (1988): "Model for piano hammers: Experimental determination and digital

simulation," J. Acoust. Soc. Am. 83, pp. 746-754.

Briggs, G.A. (1951): Pianos, Pianists and Sonics (Wharfdale Wireless Works, Bradford Rd.,

Bradford, Yorks) p. 37.

Brinsmead, Edgar (1879): The History of the Pianoforte (reissued by Singing Tree Press, Detroit,

orig. pub. by Novello, Ewer and Co., London, 1879) p. 47.

Challis, John (1963): "New: A 20th Century Piano," American Music Teacher, Jul-Aug 1963, p. 20.

Conklin, Harold A. Jr. (1970): U.S. Pat. 3,523,480, "Longitudinal Mode Tuning of Stringed

Instruments," Aug. 11, 1970.

Good, Edwin M. (1982): Giraffes, Black Dragons, and other Pianos (Stanford University Press) p. 9.

Hall, D.E. (1986): "Piano string excitation in the case of small hammer mass," J. Acoust. Soc. Am.

79, pp. 141-147.

Hall, D.E. (1987a): "Piano string excitation II: General solution for a hard narrow hammer," J.

Acoust. Soc. Am. 81, pp. 535-546.

Hall, D.E. (1987b): "Piano string excitation III: General solution for a soft narrow hammer," J.

Acoust. Soc. Am. 81, pp. 547-555.

Hall, D.E. & Clark, P.J. (1987): "Piano string excitation IV: The question of missing modes," J.

Acoust. Soc. Am. 82, pp. 1913-1918.

Hall, D.E. and Askenfelt, A. (1988): "Piano string excitation V: Spectra for real hammers and

strings," J. Acoust. Soc. Am. 83, pp. 1627-1637.

Hansing, Siegfried (1888): Das Pianoforte in seinen akustischen Anlagen (New York), p. 92.

Harding, Rosamond E.M. (1933): The Piano-forte - Its History Traced to the Great Exhibition of

1851 (Da Capo Press, New York 1973, reprint of 1933 edition), pp. 64-66.

Helmholtz, Hermann L.F. (1863): On the Sensations of Tone as a Physiological Basis for the

Theory of Music (Dover Publications Inc., New York 1954, first ed. Friedr. Vieweg & Sohn,

Braunschweig 1863) p. 77.

Henahan, Donal (1967): "Loesser, Pianist, Exhumes 'Ghosts' to Mark Halloween," (a review of the

concert), New York Times, Oct. 3, 1967.

Knoblaugh, Armond F. (1944): "The clang tone of the pianoforte," J. Acoust. Soc. Am., 19, p. 102.

Leipp, Emile (1969): The Violin (University of Toronto Press), pp. 97-99.

Marcuse, Sibyl (1975): Musical Instruments, A Comprehensive Dictionary (W.W. Norton & Co.,

New York), p. 405.

McFerrin, W.V. (1972): The Piano - Its Acoustics (Tuners Supply Co., Boston), p. 84.

Ortman, Otto (1925): The Physical Basis of Piano Touch and Tone (E.P. Dutton & Co., New York),

p. 96.

Podlesak, M. & Lee, A.R. (1988): "Dispersion of waves in piano strings," J. Acoust. Soc. Am. 83, pp.

305-317.

Pollins, Stewart (1984): "The Pianos of Bartolomeo Cristofori," J. Amer. Musical Instrument Soc.

10, pp. 32-68.

Rayleigh Lord (Strutt, John William, 1877): The Theory of Sound (MacMillan & Co., Ltd., London,

reprinted by Dover Publications, Inc., New York) Vol. I, p. 252 (a quotation referring to

longitudinal mode of piano and violin strings taken from p. 154 of Donkin's Acoustics).

Suzuki, H. (1986): "Vibration and sound radiation of a piano soundboard," J. Acoust. Soc. Am. 80,

pp. 1573-1588.

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30.

White, William Braid (1906): Theory and Practice of Pianoforte Building (Edward Lyman Bill,

New York, reprinted by Dover Publications, Inc., New York 1975) p. 34.

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45.

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reprinted by Unwin Bros., Ltd, Old Working, Surrey, U.K., 1975), pp. 50-51.

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From touch to string vibration

Anders Askenfelt & Erik Jansson

Introduction

This lecture will present a series of experiments exploring the initial stages of the sound

production in the piano - beginning with the motion of the key and ending with the string

vibrations. This chain of events is closely connected with the performance of the pianist,

who, by depressing the key, sets the parts of the action in motion, which eventually causes

the hammer to strike the strings.

In contrast to performers on other string instruments, like the violinist or the guitar player,

the pianist could be said to only have an indirect control of the string excitation. Using the

computer biased terminology of today, it is tempting to call the action an "interface"

between the pianist and the string. This interface is an interconnecting device, which at the

input end (the keys) is particularly adapted to the soft and sensitive fingers of the pianist,

while the output end is equipped with hard felt hammers, capable of exciting even the

thickest of the tense piano strings vigorously. The function of this "interface" in playing is by

no means simple.

We will illustrate some important properties of the action by presenting measurements of

the timing in the action under different conditions, and also show how the motions of the

key and hammer change, depending on how the key is depressed. Furthermore, the

resulting string vibrations will be closely examined and the manufacturer's, the piano

technician's and the pianist's influence on the spectrum of the piano tone will be compared.

During the presentation it will successively become clear that the successful piano

performer is accompanied by two mostly anonymous artists, the piano technician and the

tuner, sometimes but far from always combined in one and the same person. In contrast to a

widespread belief, the fact is that it is not sufficient to have the piano tuned prior to the

performance (in such a way that it stays in tune during the entire concert); the piano must

also be properly regulated in order to play well.

Despite the remote control of the actual string excitation by the hammer impact, pianists

pay great attention to the way the key is depressed. Often the term "touch" is used to denote

this process. Physicists and piano players have had contrasting views regarding the

importance of this point for a long period of time, and later on we will try to add some

material regarding this question. However, we must hasten to add that at the moment we

will not be able to resolve this conflict, but perhaps we can indicate in which directions the

answers can be sought.

Timing in the action First of all, let us present a rather detailed description of the function of the grand piano

action. The actions of all grand pianos of today are in principle identical, and the small

differences which do remain are limited to the design of the individual parts.

Fig. 1. View of the action of a modern grand piano (Steinway & Sons). The shaded areas

indicate felt and the broad lines indicate leather.

Principally, the action consists of four major parts: the key, the lever body with appurtenant

parts, the hammer and the damper (see Fig. 1). The successive steps in the operation of the

action during a blow is illustrated in Fig. 2.

2(a) Rest position. The hammer rests via the hammer roller on the spring-supported

repetition lever, a part of the lever body. The lever body stands on the key, supported by the

capstan screw. The weight of the hammer and the lever body holds the playing end of the key

in its upper position. The damper rests on the string, pulled down by lead weights.

2(b) Acceleration. When the

pianist depresses the key, the

lever body is rotated upwards.

The jack, mounted on the

lever body, pushes the roller

and accelerates the hammer.

The damper is lifted off the

string by the inner end of the

key.

2(c) Let-off. The tail end of

the jack is stopped by the

escapement dolly, and the top

of the jack is rotated away

from the hammer roller. The

hammer, which now is free,

continues towards the string.

The repetition lever is

stopped in waiting position

by the drop screw.

2(d) Check. The rebounding

hammer falls with the

hammer roller on the

repetition lever, in front of

the tripped jack, before it is

captured at the tail of the

hammer head by the check.

The stroke may now be

repeated, either by releasing

the key as usual, or by using

the double-repetition

feature (see text).

The action of the grand piano features a special construction for fast repetitions, the double-

repetition mechanism, not incorporated in the action of the upright piano. In order to use

the double-repetition feature, the key is let up only about a third of its travel after a stroke.

At this stage, the hammer has been released from the check and lifted slightly by the spring-

supported repetition lever (cf. Fig. 2 d). This allows the spring-loaded jack to slip back into

its initial position under the roller, and the action is set for a second blow. The double-

repetition mechanism enables very fast repetitions on the same key, without the damper

touching the string between notes.

A correct function of the action requires a careful regulation. Of crucial importance is the

distance between the top of the hammer at rest and the string, in the following hammer-

string distance (piano technicians term: "blow level"). This distance is adjusted with the

capstan screw (typical value 45 - 47 mm). Of equal importance is the setting of the release of

the jack ("let-off"). This is adjusted with the escapement dolly. The adjustment is made by

observing the distance between the string and the top of the hammer at the highest point of

its travel (let-off distance), when the key is depressed slowly. The let-off distance is typically

set between 1 and 3 mm, the actual value depending on such factors as the diameter of the

string, interval between regulations, and sometimes, the personal taste of the pianist.

In all contact points between moving parts, one of the surfaces is covered with felt or leather

in order to ensure a smooth and silent motion, free from backlash. In particular, thin shafts

with close tolerances, for example the shaft for the hammer shank in the flange, are

mounted in bushings of high-quality felt. The combination of wood and felt parts means that

the action will change condition not only because of wear, but also due to changes in

temperature and humidity. Periodic regulation is thus necessary in order to keep the

instrument in optimum condition.

Measuring the timing We measured the timing in the action electrically by a network of "micro-switches,"

integrated with the action. These switches consisted of copper foils and thin copper wires

glued and sewn to the contact surfaces. During a blow, the different switches turned off and

on as the parts moved, and a stepwise signal was generated at the output of the network.

In order to obtain blows which could be repeated with a high degree of reproducibility in

the timing experiments, we also had to develop "a mechanical pianist." It turned out that for

this particular purpose a long pendulum was ideal as a substitute for the player.

Fig. 3. Overview timing diagram of the grand action for a