Quality of Piano Tones

THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA

Volume 34 Number 6

JUNE. 1962

HARVEY FLETCIIER,E. DONNEL• BLACKHAM,AND RICIIARD STRATTON Brigham Young University, Provo, Utah (ReceivedNovember 27, 1961)

A synthesizerwas constructedto producesimultaneously 100 pure toneswith meansfor controllingthe intensity and frequencyof each one of them. The piano toneswere analyzedby conventionalapparatus and methodsand the analysisset into the synthesizer.The analysiswas consideredcorrect only when a jury of eight listenerscould not tell which were real and which were synthetictones. Various kinds of synthetictones were presented to the jury for comparisonwith real tones.A numberof thesewere judged to have better quality than the real tones.According to thesetests synthesized piano-like tones were produced when the attack time was lessthan 0.01 sec.The decaycan be as long as 20 secfor the lower notesand be lessthan 1 secfor the very high ones.The best quality is producedwhen the partials decreasein level at the rate of 2 db per 100-cpsincrease in the frequencyof the partial. The partialsbelow middle C must be inharmonicin frequencyto be piano-like.

INTRODUCTION synthesizer,and (4) the frequencychanger. To these HISpaper isa reportof our efforts tofind an ob- facilitieshave been added, a sonograph,an analyzer, a jectivedescription of the qualityof pianotones as single-tracktape recorder,a 5-track tape recorder,and understoodby musicians,and also to try to find syn- other apparatususually available in electronicresearch thetic toneswhich are consideredby them to be better laboratories.A block diagram of the arrangementis than real-piano tones. shownin Fig. 1. The usual statement found in text books is that the pitch of a tone is determinedby the frequencyof EQUIPMENT vibration,the loudnessby the intensityof the vibration, 1. Anechoic Chamber and the quality by the waveform.This picture is far too simplefor any of thesethree subjective aspects of a An anechoic chamber was constructed for use as a tone. Pitch and loudnesshave received very extensive listeningroom. It wasbuilt accordingto the architect's study. In this paper an attempt has been made to drawingsloaned to us by the Bell TelephoneLabora- throw some additional light upon the quality of a tories. Therefore, it is a copy of the one at those piano tone. laboratories. It is true that the quality dependsupon the wave- It consistsof a rectangularblock of cementwith in- form. But it alsodepends upon the pitch, the loudness, side dimensions of 40X30X30 ft. The block rests on the decayand attack time, the variationwith time of sand and gravel, and is completelyseparate from the the intensity of the partials, the impact noiseof the rest of the Eyring ScienceCenter building. The room hammer,the noiseof the dampingpedal, and alsothe was treated with 6-ft acousticalwedges on each side, characteristicending of the tone by the dampingfelt, thus reducingthe size 12 ft in each direction.A wire- etc. mesh floor was constructedby stretching steel wires In order to study the relative importance of these across the steel I-beams on the sides. These wires were variousfactors, the followinglaboratory equipment and separatedby 2 in., and there were two sets at right room facilities have been developed, namely, (1) anglesto each other. This resulted in meshes2 in. anechoicchamber, (2) loudspeakersystem, (3) tone square.The floor is 10 ft belowthe ceilingedge of the 749

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FREQUENCYSHIFTER TONE SYNTHESIZER PowerSupply The requirementsplaced upon the power supply i•0I [00ank;oo, - CPS I consistedof (a) supplyingadequate power at the ap- [ R . propriatevoltages and (b) maintenanceof a low level of noiseand hum. The secondof the two requirementswas Ampl tier themost difficult to •atisfy,and was especially difficult 'I in this casebecause of the surgesof power involvedin the operation of the attack and decay amplifier. To obtain this low noiselevel, a 6-v dc battery wasused for TAPE I RECORDER#1 the attack and decay amplifier vacuum-tubegrid and filament supplies, and for the transistor oscillators' power requirements.The use of this simple battery

M icrcphone eliminatesthe inherent problemsposed by alternating Input • • TAPE current supplies.To obtain the requiredplate voltage swiTCH ING RECORDER for the attack and decay amplifier, a General Radio pwr.amp I •• _•-• 2high-frequency high-voltage supply was used in conjunction with I I • unit specially designedfilters. These filters consistedof :•- I •1• middle- R-C filter sectionsand voltage regulator tubes;it pro- ;_ duced a constant voltage output with a noise or "ripple" level 80 db below the output signallevel.

Audio-FrequencyOscillators

A CLUSTER OF 7 1S" SPEAKERS Since100 oscillatorswere required it was necessaryto selecta designwhich was compactand which gave a Fzo. 1. Block diagram of equipment. stable oscillation. This was accomplishedby using transistorswith printed circuits. The circuit elements acousticalwedges and 8 ft abovethe lowerset of wedges. were mountedon a panel board 5 in. long and 1« in. It will supportconsiderable weight but has little or no wide. The inductanceelement can be varied by turning reflectionfor any of the soundsin the audible range. a key which can be insertedin a small hole throughthe Chairs were supportedon this wire floor at one end of face of the panel. In this way each oscillatorcan be the chamberfor the jury of listeners.The loudspeaker turned to any desiredfrequency within about 1-octave systemwas installedat the oppositeend. range.The 100 oscillatorscan be tuned to covera range from 50 to 15 000 cps.These 100 smallpanels supporting 2. Loudspeaker System the oscillatorswere stacked together into three large This consistsof a three-channelsystem, each channel panels. Two of these large panels are shown at the transmitting, respectively,the bands of frequency20 bottom half of Fig. 4. The other panelis on the opposite to 400 cps, 400 to 4000 cps, and 4000 to 16 000 cps. sideof the portable table carrying the synthesizer. The low-frequencychannel consists of sevenAltec 803 B driversmounted close together in a circulararrangement Attenuatorsand Preamplifiers as shownin Fig. 1, with a bafflewhich was 8 ft square. The output of each oscillatoris sent through an at- This baffle was mounted so it could turn about a hori- tenuator and then to its preamplifier.The 100 atten- zontal axis. At certain anglesfrom the vertical, a more uatorswere arranged in a compactform at the back of a uniform responsefor the very low frequencieswas black panel board. White knobs (100 of them) which obtained. wereprojecting thro•tgh the blackpanel board and As indicatedin Fig. l, the medium-rangeloudspeaker which were connectedto the sliding contact of the wasa sectional-typehorn, and the high-frequencyloud- attenuator could be moved up and down in vertical speaker was of a tweeter type. The responseof this slots to control the amount of attenuation introduced system is given in Fig. 2. into eachoscillating circuit. Each attenuator covereda rangeof 50 db. They wereconstructed so that the down- 3. Tone Synthesizer ward movement of the knob producedan attenuation The tone synthesizerused to producethe synthetic tones consistedof five major parts, namely, a power supply, audio-frequency oscillators, a white noise [ ! .J-- i [ , i i , I ' sourceand filter set, attenuatorsand preamplifiers,and ;30•050 70 100 200 •00 500 1,000 ::',000 •.,000 '10,000 20,000 an attack and decay amplifier. A block diagram is FREQUENCY shownin Fig. 3. FIG. 2. Relative responseof loudspeakersystem.

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I Sou:

, ,. i HIGH VOLTAGE VOLTAG E ,, POWERSUPPLY F I LT ERS TACK& DECAY ' I WHILEN01SE DECAY FIG. 3. Block diagram for tone SUPPLYFILTERS AND synthesizer. 6.0 VOLT BAIIERY I I - AMPLIF IER I WHITENOISE

OSCILLATORS CONTROL & PREAMPLIFIERS Swiich I' I"

AUDIO 1Ameans of applying FREQUENCY attack and decay to OSCILLATORS some external source such as a continuous ATTENUATORS----•"PREAMPLIFIERStone tape recording. in db which was proportional to the distance it was by connectingthe battery voltage through a resistorand moved. A db scalewas engravedon the face of this capacitorseries circuit or shortingthrough the same panel.It is thus seenthat the relativepositions of these capacitorbut differentresistors. This functionis con- white knobsin vertical level give the relative levelsin trolled by a push-buttonswitch called the manual db of the current going from the oscillatorsto the controlswitch. The outputsignal will increasewhen the preamplifiersand showsgraphically the structureof the buttonis pushed,build up to its maximumvalue, and partials of the tone beingsynthesized. remainat that point until the buttonis released.Upon In the upper part of Fig. 4 this panel is shown.The release,the decaycircuit is in useand the decayingrate knobsin the picture are set to producethe synthetic is appliedto the signal.The rate of attackand decay is pianotone A"", the lowestnote on the piano. determinedby the adjustmentof the resistanceused in the R-C circuits.See Fig. 5. Attackand DecayAmplifier Anothercontrol used is the "dampingpedal" control whichplaces a resistorin parallelwith (by meansof a The attack and decayamplift er • functionsjust as the pushbutton)the decayresistor. Thus the rate of decay name implies--it givesa beginningand an endingor canbe changedto a morerapid oneduring the decay attack and decay to any constant-levelinput. The time and thus simulatethe actionof the dampingpedal specificrequirements placed upon the amplifier, are on the piano. listedas follows: (a) Frequencyresponse: 4-2 db from One seriouslimitation of the synthesizeris that the 50 to 15 000 cps. (b) Noise and switchingtransients: decayrate is constantand the samefor all frequencies. 50 db below the signallevel. (c) Intermodulation dis- This meansthat a curveshowing level in db vs the time tortion: 5%. (d) Harmonic distortion: 2%. (e) Time in secondswill be a straightline. The timein secondsto rate of attack and decay continuouslyvariable from decay20 db will be calledthe decaytime. This is the 4,000-6 db/sec and very nearly exponential.(f) The time for the current in the attack and decay amplifier attack and decayamplifier must operateover a rangeof to decreaseto 0.1 of its maximum value. Likewise, the 70 db (from maximum to minimum output signal). Considerabledevelopmental work was necessarybefore a circuit was obtainedwhich would fulfill theserequire- ments,particularly the last one. The circuit which was finally usedis given in Fig. 5. The conceptinvolved is that of a two-stage push-pull amplifier in which the grids of both stages(4 vacuumtubes) are biasedin accordancewith the attack and decay required. The Fro. 4. Photograph grid biasvoltage comes from a resistance-capacitance- of synthesizer. battery network which will thus provide an exponen- tially increasingor decreasingvoltage dependingupon whether the capacitoris chargingor discharging.The function(attack or decay)of the amplifieris determined

x R. N. Christensen,"An Attack and Decay Amplifier Suitable for use in a Tone Synthesizer,"thesis (unpublished),Brigham Young University, Provo, Utah (1959).

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Fro. 5. Attack and decay amplifier circuit.

attack time is the time in seconds for the current to db on a levelmeter. This procedurewould be straight- reach 0.9 of its maximum value. forwardif the tone were steady.But a piano tone is continuallyvarying sothe pianotone wasrecorded on a 4. Frequency Shifter continuousloop so that the tone couldbe continually The frequencyshifter consistedof a tape recorder repeated.The analyzer was set to give a maximum with a synchronousmotor whichwas driven by a com- responsefor eachpartial near the beginningof the tone. binationof an oscillatorand a poweramplifier. By vary- Then a pure tonefrom an oscillatorwas sent through ing the frequencyof the oscillator,the speedof the tape the analyzerwith this settingand the frequencyad- recordercould be increasedto « normal speed and justedto givemaximum response. The frequencyof this decreasedto -} normalspeed or to any speedwithin pure tone was then measuredon the electroniccounter theselimits. This madeit possibleto set up any partial which will measurefrequency with an accuracyof structureon the synthesizer,record it on the frequency about 0.1%. In this way the frequenciesand the rela- shifter, and then shift to any desired fundamental tivelevels of thepartials at thebeginning of th• tone frequency. were determined.The variation of the partialswith time canbe inferredfrom the sonographmeasurements. Method of Analyzing the Piano Tone In a third methodof obtainingthe partial structure and its variation with time, a level recorderwas used. During the first part of our analysiswork a sonograph The complextone from the loopwas sent through the was used. This instrument has been described in the analyzer to the recorder. All of these methodshave literatureand is in commonuse. The toneto be analyzed their advantagesand disadvantages,and one method is recorded on a rotating cylinder. The sectioneris servesto checkthe others. In our generalstudy of switched on and then the instrument draws horizontal musical sounds all of these methods have been used. lines whose lengths are proportional to the relative The generalarrangement of theseand other standard levelsof the partials.The positionof a pegon the top of instrumentsis shownin Fig. 1. the rotating cylinder determinesthe time during the As shownin Fig. 1, the current from the oscillators duration of tone that correspondsto the partial struc- goesinto the preamplifiersand then into the attack and ture beingmeasured. By movingthe pegon the rotating decayamplifier. From hereit may be switchedto the drum the chosen time can be changed in 0.04-sec loudspeakersystem or to any of the otherinstruments intervals.The sonographwill recordonly soundshaving shown. a duration lessthan 2.4 sec.Thus the partial structure With this instrumentation we can record the tones can be measuredat 60 differenttimes during the dura- from the piano. From this recordedtone we are able to tion of sucha tone. The approximatefrequency can be find the partial structureand how it varieswith time. read from this graph. Alsothe attack and decaytimes can be measured.This In the later part of our work a conventionalanalyzer analysisis thenused to producea synthetictone, which wasused which passed only a narrowband of frequency may be comparedto the real tone by judgmenttests. (approximately4 cps).The outputwas read directly in Alsonew toneshaving any partial structureand any

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attack and decay times can be created for judgment PARTIAL NLIHBER 5 i 0 • 5 20 25 •0 •5 1•0 uses. ß I I I I 501 • C"' rzß 52.7 EXPERIMENTAL 20 DECAY= .

It is known that in a real-pianotone the partial struc- 0 ture is varying; that is, the decaycurves of the partial f, = 65.• ATTA•3K_-.O05 tones will not be straight lines. It is also known2 that DECAY_-1.26 the partials comingfrom a struck string are not strictly harmonic. At the impact of the hammer, a sound like that of hitting a boardis superimposedon the tone from the string.This soundis particularly noticeablein the upper three octavesof the piano. When the piano tone øF"x •=•6• % •'$231 50-• %,• ATTACK:.00•"X A"'00:•I •' 10•6 rz = 2095 is dampedwith the pedalit stopsfirst the low tonesand I • . DECAY:•.e•l ••0=1.1 I A-.0O} A=.'oh5 20 D•.21 D:O.8 finally the higherones giving a characteristicending of 02 the tone. The pedal itself producesa noise which is 0 • 5 10 1,25 •5 6 7 8 readily recognizedas due to its movement. If one wanted to reproduceall thesecomplicated sounds, one Fro. 6. Averagepartial structure of the studiopiano. would make a tape recordingof the actual piano tone. But we are interestedin knowing the relative impor- inharmonicand can be calculatedby the formulagiven tance of these various factors. by Ybung2 andothers. This paperdescribes experiments which were designed to increaseour understandingof theseand other factors Preliminary Judgments Tests which governthe quality of piano tones.To begin the To obtain a first approximationof the relative im- investigation,tones from a Baldwin grand piano were portanceof the variousfactors influencing the quality recorded.The piano was in a studioroom of about the of the piano tone the followingidentification test was usual characteristics. made. Synthetictones were createdhaving partial The tonesrecorded were designatedthus: structuresin accordancewith thoseshown in Fig. 6. C"' Three octaves below middle C. From thesesynthetic and the originalpiano tones, the followingprogram was recorded.[The letter in the C" Two octaves below middle C. parentheses(R) indicatesit wasa real tone.Similarly C' One octave below middle C. the letter (S) indicatesit wasa synthetictone.-] C Middle C, frequency261.6 cps. Test (1) c (s)--c (s)--c (}•)--c (s)--c (}•). C1 One octave above middle C. Test (2) c' (}•)--c' (}•)--c' (s)--c' (s)--c' (s). C2 Two octaves above middle C. Test (3) c" (s)--c" (s)--c" (s)--c" (s)--c" (}•). Ca Three octaves above middle C. Test (4) c'" (•)--c'" (s)--c'" (s)--c'" (•). The samedesignations were usedon the G-pitched,or Test (5) C1(S)--C 1(R)--C 1(S)--C1 (S)--C1 (R). any othertones, using middle G as the G a fifth above Test (6) c• (s)--c • (s)--c • (•)--c • (•)--c • (•). middle C. Test (7) About 2 secafter the key was struck, the damping c•(s)--c•(s)--c•(s)--c•(}•)--c•(s). pedal was usedto dampenthe tone. Thesetones were Test (8) c" (s)--c' (•)--c" (•)--c (•)--c'" (s)-- analyzedby useof the sonograph;three or four samples C (R)--C1 (5)--C2 (5)--C3 (R). being taken at differenttimes. The averageof these four sampleswas taken as the partial structure.The A jury offour musicians was asked to checkwhich they resultsof thesemeasurements are given in Fig. 6. consideredwere the real-pianotones. A secondjury of A careful measurementof the spacingon the fre- laymen also took the test. The musicians identified quencyscale of the tracingsmade on the sonograph 90% correctlyand the laymen86%. Both teamsscored indicated that, within the observationalerror, the lessthan 75% on thesetests for identifyingmiddle C as partialswere approximately harmonic except for those a real-pianotone, showingthe syntheticC tone was a of C'" and C". The frequenciesof the partialsof these better match for this than for the others. two low-pitchedtones were found to be definitelyhigher From listeningto the above program of tests and thanthe harmonicfrequencies. For example,for C'" the talking to membersof thesetwo juries,it was obvious 30th partial frequencywas found to be 1105cps which that therewere a numberof cluesfor identifyingthe is 134 cpsgreater than the harmonicfrequency. It will real-pianotones. Some of theseare (1) the noiseof the be seen later that the partials of any piano tone are pianohammer striking the string, (2) the noiseof the pedaldampening the tone, (3) higherbackground noise •'R. W. Young,J. Acoust.Soc. Am. 24, 267-273 (1952). for the piano tones,(4) reverberationeffects of the room

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•0- C" 30 1 20 • ß21)_ •o

0 •'

30- 2O • •o 10- • o •

0 •. 2o E •o

= . • 0 x • • 10- 20

I • •o • 20 25 30 25 30

FT. 7. Curvesshowing the changinglevel vs time for the partials FT. 8. Curvesshowing the changinglevel vs time for the partials 1, 2, 3, and 4 of the tone 5, 10, 15, 20, and 25 of the tone C'".

for the real-piano tones which were absent for the were made on each of the 30 measurablecomponents. synthetic tones. Samplesof these resultsare shownin Figs. 7 and 8. The electricalcircuit was arrangedso that the orig- In Fig. 7 the decay curvesfor the first 4 partials are inal beginningand endingof the toneswere eliminated. given and in Fig. 8 the curvesfor partials numbered This removed the clues associatedwith the starting 5, 10, 15, 20, and 25 are given. and stoppingof the tones.Then the musicianscorrectly Theseresults are rather surprisingalthough somewhat identified74% of the tonesand the laymen 75%. In similar results have been observed before. To be sure Test 8, wherethe tonesare arrangedhaphazardly as to there was no artifact, the key for C'" was struck a frequency,the scoreswere much lower, namely 63% secondtime and a separateanalysis made. The two sets correct.It will be rememberedthat a 50% scoremeans of pointsin the figuresrepresent the two setsof data. that the observeris guessing.It should also be re- It is seenthat there is goodagreement. The discrepancy memberedthat in these tests the frequencyof each at the end of the tonesmeans simply that one tone was partial in the synthetictone was an exactmultiple of dampedquicker than the other. the fundamental and the rate of decay was constant It will be seen that some of these curves exhibit a and the samefor all partials. fairly uniform rate of decay such as for partial 2, 4, and 15. However, most of the othersshow very irregular Partial Level vs Time decay time characteristics.It is obvious from these The preliminarytests showedmore clearly how to curvesthat the partial structureis continuallychanging proceedto matchreal-piano tones with synthetictones. as the tone dies away. In Fig. 9, similar data on the Beforeimproving the synthetictones, it wasdecided to piano tone C is given for the first six partials. It is make a more carefulstudy of real-pianotones. So the obviousfrom these data that the partials of the piano C" was chosenfor a more critical study to seehow the tone do not even approximatelydecay at the samerate. variouspartials change with time. By meansof the They sometimesincrease in intensity rather than de- sonograph,at a timeinterval of 0.08 sec, 30 observationscrease. Thus to give the decay rate as X db per sec,

PLANO TONE C ß I PARTIA.L • ox 32 PARTIAL •''•',.•• %-'-- •5PARTIALD 4PARTIAL J • •x • • + 6 PARTIAL

FT. 9. Relative level vs time for the first six partials of the tone C.

.2 .• .6 .8 1.0 1.2 1.• 1.• 1.8

TIME - SECONDS

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e,•k•'"•p, GPARTIAL I 0ß OFFANECHOIC ICE CHAMBER '•"'•.•.._ • HALL

2o ...... "- x,'--,,,.______,,__ __,.¾ :o:;..., -_•:>.:o•...... m 10

0 Fig. 10. Room effect on the level vs time curvesfor the partials 1 and 2 of the tone G. • 7o

•0

0 1 e 5 • 5 6 ? 8 9 TIME - SECONDS

as some authors still do, gives a rather erroneous determine which room was preferable for listening to picture. the music.The musiciansvoted 1 for Room O (studio), Sincethese tones were recordedin a live musicstudio, 1 for Room H (reverberant),and 6 for Room A (ane- it was decided to bring a piano into our anechoic choic).The laymenvoted 1 for Room O, 5 for RoomH, chamberto see if these irregular variations with time and 6 for Room A. Thus the anechoic chamber was weredue to the roomor werecharacteristic of the piano. definitely consideredbest for listening.This confirmed A Hamilton upright piano was taken into our labora- the conclusionreached some time ago that musicians tory where there were three rooms of different rever- prefer to listen to musicin a nonreverberantroom. How- beration characteristicsavailable in which the piano ever the player alwaysprefers to play in a reverberant couldbe played. Room O had a reverberationtime for room. speechof 0.6 sec (studiotype). In Room H the rever- While the piano wasin eachof theserooms, the tones beration time was 2.2 sec, that is, very reverberant. producedby playing the white keys on the piano were Room A was the anechoic chamber and the reverbera- also recordedon the tape recorder.The result of tests tion time was very closeto zero. It could not be meas- taken with the sonographon the piano tone G played ured with the instruments available. in thesethree roomsare given in Figs. 10 and 11. It is The samepiano selectionwas recordedin each of the seenthat the irregularitiesin the decaypattern exist in three rooms.Judgment tests were madefirst by a jury all three rooms. In the acousticallylive room (hall) of 8 musiciansand then by a jury of 12 laymen to they are no greaterthan in the acousticallydead room.

6O o... G PARTIAL3 ß ANECHOIC CHAMBER %•'R, 0 OFFICE 5o "---L,,•, • .^,, •o

z_ 0

Fig. 11. Same as Fig. 10 but for partials 3 and 4 of the tone G. ,., 70

0 1 2 5 •- 5 6 7 8 9 lO

TIME - SECONDS

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But are thesevariations necessary for a goodquality

20- tone or are they just accidental? To help answerthis question,it is necessaryto de- 0 TI•IE = .52 SEC termine if a tone having a constantharmonic content 50- and a constantdecay rate would be indistinguishable by most observersfrom the piano tone if the duration •0-

• 20-o of the tone was about 1 sec,corresponding to the time • TIME = .60 SEC. • •o for a half or whole note dependingon the tempo. It will be seenfrom the figuresthat during this time the decaycurve is almosta straightline, correspondingto a 0 TI•E = .8• SEC logarithmicdecay. 50- For matching the harmonic content with the syn- •0- thesizer, we were not sure that the sonographwas o sufficientlyreliable for the higherpartials, particularly TI•E • 1.16 SEC •0- for the low pitched tones.So the analyzerwas usedfor obtainingthe harmoniccontent at the beginningof the 20- tone. 10o -'" I I I I 1 I I I I i I _...... I I • • I 2 5 • 5 6 7 8 9 10 11 12 151[ 1'51• 1'718 •IAL •R Warmth as a Factor in Piano Quality Fro. 12. Partial structures at different times for the tone Synthetictones were constructedwith partial structures thus determined.For the tonesbelow middle C, It can alsobe noted that the decayrate is fasterat the there was still somethinglacking in the quality of the beginningand becomesslower after the first two sec. tone. The musicianssaid the toneslacked live-ness,or If the decay rate at the beginningpersisted, the tone warmth. The warmth is probablydue to rapid variation would go below hearingthreshold in about 3 sec, but of the partial structure.We will usethis term "warmth" actually the tone can be heard for about 20 sec. for indicating this factor of the quality of a musical Measurementson G• and G•. gave similar results. tone. To imitate exactly the varying intensity among The curvesin Fig. 12 showhow the partial structureof the partials would require a rather complicatedcontrol the pianotone G" changeswith time. It is obviousthat mechanism.It probably could be built. However, it to matchthese real-piano tones with synthetictones is a wouldgive only a tone similarto one recordeddirectly very complicatedprocess. Of coursethe easiestway to from the piano. do this is to makea tape recordingof the tone.Then all Is it possibleto warm the toneby somesimple process thesevariations are preservedand can be reproduced. rather than trying to follow the variationsof 30 or 40 partials?A methodfor doingthis was suggestedby the TaB•.v.I. Observedfrequencies of the partialsof pianotone A'"', commentof musiciansthat four or five violinsplaying as well as the frequenciescalculated for B =0.00053. in unisonproduces a muchwarmer tone than that from a singleviolin. One way of implementingthis suggestion n 27.5n obsf• calcfn db n 27.5n obsfncalcf• db is as follows.Set up the desiredpartial structureon the 1 27.5 27.5 27.51 --23 26 715 835 833 --17 synthesizer.With these settingscreate a continuous 2 55 55 55.06 --29 27 742.5 876 874 -16 3 82.5 83 82.2 -- 9 28 770 917 916 -- 7 4 110 111 110.5 --19 29 797.5 959 959 -- 9 TABLV.II. Frequenciesof partials of piano tone 5 137.5 138 138.4 0 30 825 1003 1003 --18 G'" for B =0.00028. 6 165 166 166.5 -- 3 31 852.5 1049 1047 --23 7 192.5 193 195 --15 32 880 1094 1093 --25 8 220 222 224 --28 33 907.5 1144 1140 --33 n 48.6n obsf, calcf, db n 48.6n obsf• calcfn db 9 247.5 251 253 --22 34 935 1189 1187 --16 1 48.6 49 48.6 --26 18 874.8 912 914 -36 10 275 281 282 -- 12 35 962 1237 1236 --11 2 97.2 97 97.3 --17 19 923.4 967 969 --23 11 302.5 311 312 --13 36 990 1287 1286 --17 3 145.8 145 146 0 20 972 1024 1026 --26 12 330 340 342 -- 3 37 1017.5 1336 1337 --26 4 194.4 194 195 --15 21 1020.6 1082 1083 --33 13 357.5 371 373 -- 3 38 1045 1386 1388 --28 5 243 244 244 --18 22 1069.2 1140 1139 --21 14 385 402 404 -- 7 39 1072.5 1440 1441 --31 6 291.6 292 293 --20 23 1117.8 1198 1197 --21 15 412.5 435 436 --11 40 1100 1495 1495 --33 7 340.2 340 342 --14 24 1166.4 1249 1257 --36 16 440 467 469 -- 15 41 1127.5 1550 1550--27 8 388.8 391 392 --29 25 1214 1318 1315 --40 17 467.5 501 502 -- 8 42 1155 1602 1607 --34 9 437.4 440 442 --20 26 1263.6 1379 1378 --26 18 495 534 535 -- 4 43 1182.5 1659 1664--29 10 486 491 492 -- 9 27 1312.2 1441 1441 --26 19 522.5 569 570 0 44 1210 1716 1722 --31 11 534.6 542 544 --13 28 1360.8 1504 1504 --32 20 550 605 605 + 4 45 1237.5 1776 1782 --36 12 583.2 593 595 -- 7 29 1409.4 1569 1565 --39 21 577.5 641 641 0 46 1265 1834 1842 --38 13 631.8 644 647 --10 30 1458.0 1634 1632 --39 22 605 677 678 -- 47 1292.5 1884 1904--33 14 680.4 698 699 -- 4 31 1506.6 1701 1697 --39 23 623.5 715 716 -- 6 48 1320 1940 1967 15 729 750 752 --12 32 1552.2 1763 1767 --45 24 660 753 754 --21 49 1347.51997 2031 --•'7 16 777.6 798 804 --23 33 1603.8 1832 1832 --43 25 687.5 793 793 --25 17 826.2 ... 856 .-- 34 1652.4 1902 1905 --45

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toneand record on the frequencyshifter. The tonefrom IV. Frequenciesof partials of piano tone G' for B=0.00005. thisis thenrecorded on the taperecorder. The speedof reproducingon the frequencyshifter is then slightly changedand a secondtone from it recordedon top of n 193.5n obsf• calcf• db n 193.5n obsf• calcf• db the first one. In the sameway a third tone is super- 1 193.5 193 194 -- 7 8 1552 1553 1553 --28 2 387 388 388 0 9 1731.5 1748 1748 --28 imposed.When these superimposedtones were re- 3 580.5 583 582 -- 2 10 1935 1942 1942 --33 producedfrom the tape recorder(2) it was found that 4 774 776 776.1 -15 11 2182.5 2137 2137 --34 5 967.5 970 970.2 --16 12 2322 2328 2331 --37 the continuoustone was much warmerthan a single 6 1163 1167 1164.4 --14 13 2515.5 2525 2526 --38 tone. This warm continuoustone was sent throughthe 7 1354.5 1362 1358.7 --17 14 2709 2720 2721 --43 attack and decay amplifierto give the synthetictone 15 2902.5 2917 2917 --44 the desiredattack and decaytimes. Later in this work a five channeltape recorderwas addedto the other instrumentsshown in Fig. 1. This rangeof pitchesmight be the causeof the warmth and made possiblea secondmethod. The tone from the be one of the factors for good quality rather than the synthesizerwas recorded on the five channels, the reverse. So a very careful measurement of both the frequencyon each channelbeing slightly different. An frequencyand alsothe level of eachof the partials in attenuatorin eachchannel made it possibleto reproduce the tonesA"", G"•, G", G•, G, G1,and G2were made. the five tones with any desiredrelative level. In this These tones were producedon a goodupright piano way a large range of warmth valueswas obtained. whichwas placed in the anechoicchamber. The analyzer A third methodof warmingthe tone was usedpar- was used to determine the partial structure at the ticularly for tones above middle C. Two adjacent beginningof the decay of the tone by the method oscillatorswere adjustedto nearly the samefrequency describedearlier in this paper. as that of the partial in order that beats would occur. The data are given in Tables I-VII. It is seenthat The loudnessof the beatswhich depends upon the rela- the inharmonicityis very largeespecially for the strings tive level of these three tones, could be controlledby in the lower frequencyrange. Young 2 and othersfound raisingand loweringthe knobson the synthesizer.This that this inharmonicity could be explained by Lord methodis similarto that usedsometimes in tuning the Rayleigh's equation for strings having stiffness.The piano where three stringsare provided for each note. magnitude of this effect dependsupon the relative The stringsare not tuned to vibrate exactly in unison amountof the two factorscontributing to the restoring but to slightlydifferent frequencies. forceof a displacedpiano string, namely, that dueto the Toneswarmed in this way wereused in identification tensioncompared to that due to the stiffness.In free tests. The judgment tests indicated that the jury of solidrods this latter effectproduces the entirerestoring musiciansmade three times as many errorsin identify- force and producespartials which are nonharmonic. ing real piano toneswhen warm toneswere used instead In stringswithout stiffnessthe restoringforce is entirely of unwarmedones. Similarly the jury of nonmusicians due to the tension and the partials are harmonicsif madetwice as many errorswith suchtones. the endsare fixed rigidly. Such an analysisgives the followingequations. Accurate Determination of the Frequency and Level of the Partials of Piano Tones nfo( + ( )

At this point it was surmisedthat perhapsthe in- B=•-3Qd4/64l•T. (2) harmonicity of the partials particularly for the lower Q is Young's modulus,d is the diameter of the piano wire, 1 its length, and T its tension.The inharmonicity TAB•.v.III. Frequenciesof partials of piano tone constantb as definedby Young2 is related to B by G" for B=0.00015. b=865B. (3) n 98n obsf• calcf• db n 98n obsf• calcf• db For A"" and G'" the hammerstrikes a singlestring, 1 98 98 98.01 -- 1 15 1470 1494 1494 --21 which is a large gaugepiano wire which has a smaller 2 196 196 196.06 0 16 1568 1599 1598 --38 3 294 294 294.2 0 17 1666 1701 1701 --26 gaugewire wrapped around it. The G" stringsare con- 4 392 392 392.4 -- 10 18 1764 1805 1807 --27 5 490 492 490.9 -- 6 19 1862 1913 1912 --31 6 588 590 589.6 -- 2 20 1960 2019 2019 --41 TaBhr. V. Frequenciesof partials of piano tone G for B--0.0004. 7 686 689 688.5 -- 1 21 2058 2128 2126 --33 8 784 788 787.4 --25 22 2156 2334 2234 --26 9 882 889 887.4 --18 23 2252 2344 2344 --36 n 393n obsf• calcf• db n 393n obsf• calcf• db 10 980 989 987.3 --12 24 2352 2453 2453 --39 11 1078 1087 1087 --20 25 2450 2571 2565 --41 1 393 393 393 0 5 1965 1976 1976 --21 12 1176 1189 1189 --25 26 2548 2669 2675 --38 2 786 785 786.7 --19 6 2358 2380 2376 --30 13 1274 1291 1290 --11 27 2646 2778 2684 --44 3 1179 1180 1181 --13 7 2751 2798 2780 --26 14 1372 1392 1392 --10 4 1572 1577 1577 --30 8 3144 3174 3187 --44

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TABLEVI. Frequenciesof partials of piano tone TABLEVII. Frequenciesof partials of piano tone G• for B =0.0002. G2 for B=0.0002.

n 779n obsf• calcf• db n 779n obsf• calcf• db n 1568n obsf• calcf• db 1 779 779 779.1 0 4 3116 3123 3121 --36 1 1568 1568 1568.2 0 2 1558 1562 1558.6 --22 5 3895 3897 3905 --31 2 3136 3134 3137 --38 3 2337 2337 2339 --24 3 4704 4707 4709 --28

structedsimilarly but the hammerstrikes two strings. are given and under NM the percent of correctjudg- AlthoughEq. (1) wasdeveloped for a singlebare piano ments by the nonmusicians.The letter S or R in wire it was found that it would fit the data for the entire parenthesesafter the notation of the note, indicates rangeof the pianoprovided the singleconstant B was whether syntheticor real. obtained from the observeddata rather than from Q, Before discussingthe data in Table VIII, the A-B d, l, and T. preferencetests will be presentedso that these tests The value of B is givenat the top of eachtable. The and the identificationtests can be discussedtogether. numberof the partial n is givenin the first column.In It will be rememberedthat in the preferenceA-B test, the second,third, fourth, and fifth columnsare given, a tone designatedA is producedand then a tone desig- respectively,the harmonicfrequency nf0, the observed nated B of the samefrequency but of differentquality frequencyf•, thefrequencyf• calculated from Eq. (1), is produced.The observeris asked to decidewhich he and the relative level in db of the partial. prefers.These A-B judgmenttests usedthe following It will be seenthat the calculatedfrequencies are in synthetictones. For ToneA"", sevendifferent qualities goodagreement with the observedones. Consider the were considered. data for A •"• in Table I. It is the first note on the piano. It can be observedthat the 16th partial tone is a semi- (0) The tonewas taken directly from a taperecording of the piano. tone sharpfrom the harmonicseries. The 23rd partial is more than a whole tone sharp,the 33rd partial more (1) This was a synthetictone with partials having than two tonessharp, and the 49th partial is 7.3 semi- the samefrequencies and levels as found for the piano and given in Table I. tones sharperthan the corresponding49th harmonic. Similarlythe 30th partial for G"• is two semitones (2) The partial levels were the same as 1, but the sharperand the 27th partial for G" isa semitonesharper frequencieswere made harmonic. thanthe corresponding harmonic series. For thetone G • (3) This tone was the sameas 1 exceptthe funda- and for the tonesof higherfrequency, Eq. (1) explains mental was raised 38 db, that is, 15 db higher the departuresfrom the harmonicseries within the level than partial 5. observational error of measuring these partial fre- (4) The partial frequencieswere the same as 1 but quencies.However, the partialnumber is smalland so the levels were adjusted so that as the partial frequencychanged 100 cps the level decreased the departuresfrom the harmonicseries are small.It 2 db. was found that for these tones a good quality match couldbe made with a harmonicseries of frequencies. These samenotations were also usedfor the quality The partial structureas givenin Table I wasset up of the tonesat other frequenciesused in this test. on the synthesizer.The variousoscillators were tuned to the observedfrequencies and the levelsset according (5) The samepartial frequenciesas in 1 were used to thosegiven in this table.The resultwas a very good but the levelswere adjustedas follows' match for the piano tone. No warming was necessary. Number of partial 1 2 3 4 5 6 7 8 9 35 80. It is obvious that the warmth is due to the inharmon- Relative level, db 10 7 4 1 0 3 6 9 10 10. icity of the partials.This warmthgives the pianotone its distinctivepiano quality. With thesefacts in mind From partial 9 to 35, the levelswere at --10 db, new identification tests were made using synthetic and from 35 to 80 they decreased2 db per 100 tones with partial structurescorresponding to the cps change in partial structure. observed ones in Tables I-VII. The oscillators in the (6) The samepartial frequencieswere usedas in 1. synthesizerwere tuned to thesefrequencies. The levelswere adjustedas follows.The first five partial levelswere -- 18, -- 14, -- 10, --6, and --2. The levelsof the partials from 6 to 35 were all at Final Judgment Tests zero level. The partial levels above 35 were the Synthetictones made in thisway werearranged with same as 5. real tones accordingto the program shownin Table (7) The same partial frequenciesas 1 were used. VIII. The membersof the jury were asked to judge The partial levelsstarted at -- 12 db for number1, which were real and which were synthetic. Under M then roseto 0 for number6, then droppedto --18 thepercent of correctjudgments by the musiciansjury at number 14, rose to 0 at number 22, dropped

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againto --26 fornumber 33, rose again to --10 Forthe tone G', thefour qualities described for A'"' at number40, and finally droppedagain to were used. --34 at number 48. Fortones G,. and G3, only qualities 1 and4 wereused, but thenoise of thestriking hammer was not considered For thetone G"', thequalities 1, 2, 3, and4 havethe partof thepiano tone and was not in thesynthetic samesignificance asfor the similar qualities for tones. exceptthe fundamental was at 3 dbabove the level of thirdpartial instead of 15 db above it. Qualities5 and 6 Programfor Real vs SyntheticTones werespecial, as follows: Theprogram of theA-B testwas recorded as indi- (5) For thisquality of Gm, the levelsof the com- catedin Table IX. Let us considerthese data from the ponents were: identificationtests in Table VIII and A-B tests in Numberof partial 1 2 3 4 5 6 7etc. TableIX,. First considerthe judgmentdata for A'm, Level,db -- 14-- 7 0 -- 1 -- 2 -- 3 --4. the firstnote on thepiano. To a layman,the soundof Abovethe fourthpartial, the level dropped 1 db thisnote is verymuch like a noisewithout any pitch. So it wasthought that two improvementsmight be per partial. made. The first was to make the frequenciesof the (6) For this quality, the partial structurewas as follows: partialshave harmonic ratios rather than those with inharmonic ratios. Number of partial Noneof the syntheticqualities was preferred to the 1 2 3 4 5 6 7 8 9 10. pianoquality. Qualities 1 and 3 wereconsidered nearly Level, db equalto the pianoquality. The identificationtests --18 --15 --12 --9 --16 --3 0--1--2--3. (shownin TableVIII, tests1, 12,and 31) also confirm Abovethe eighthpartial, the level dropped 1 db per partial. TABLe.IX. Preferencetest on piano tones.

For the toneG", the qualitiesdescribed for A'"' were Prefer- Prefer- usedexcept for quality5 wheref• was+4 db.A fifth Test Qualityence Test Quality ence qualitywas used with partial structure as follows: No. AvsB A B No. AvsB A B Number of partial 1 0 1 16 5 G' 37 1 0 17 4 2 2 0 2 19 38 0 2 14 7 1 2 3 4 5 6 7 8 9 10. 3 3 0 7 14 39 3 0 13 8 Level, db 4 0 4 18 3 40 4 0 14 7 -10 --8 -6 -4 -2 0 --2 --4 -6 -8. 5 1 2 20 1 41 2 1 13 8 6 3 1 6 15 42 1 3 8 13 Abovethe eighthpartial, the levelsdecreased 7 4 1 3 18 43 1 4 10 11 8 3 2 10 11 44 3 2 12 9 2 db per component. 9 2 4 16 5 45 2 4 7 14 10 4 3 5 16 46 3 4 10 11 11 5 1 5 16 VIII. Syntheticand real tones used in identification 12 1 6 18 3 G 47 1 0 16 5 testsand percentof correctjudgments. 13 1 7 16 6 48 0 2 6 15 49 0 4 6 15 50 1 2 17 4 Tone Note Percentcorrect ill 14 1 0 12 9 Tone Note Percentcorrect 15 0 2 13 8 51 4 1 12 9 no. M NM no. M NM 16 0 3 15 6 52 2 4 12 9 1 AIII(S) 50 23 19 Gi(S) 63 92 17 4 0 7 14 18 2 1 9 12 G• 53 0 1 13 8 2 GI(R) 75 92 20 GI(R) 100 85 54 2 0 4 17 3 A.... (R) 88 92 21 G(S) 88 77 19 1 3 13 8 20 4 1 7 14 55 4 0 4 17 4 G(R) 50 54 22 G"(R) 100 85 56 1 2 16 5 5 Gi(R) 100 69 23 GI(S) 63 69 21 2 3 9 12 22 4 2 9 12 57 4 1 11 10 58 2 4 6 15 6 G'I(R) 100 85 24 A.... (R) 88 100 23 4 3 9 12 7 GI(R) 63 69 25 Gi(R) 100 69 24 1 5 15 6 25 1 6 19 2 G•. 59 0 4 15 6 8 GIII(S) 37 77 26 GIII(R) 50 46 60 0 4 14 7 9 GII(S) 50 67 27 Gi(S) 75 85 61 0 5 16 5 10 G'(S) 25 54 28 GII(S) 88 62 I! 26 1 0 9 12 27 0 2 13 8 62 5 4 12 9 28 3 0 10 11 11 G(S) 75 92 29 Gi(R) 75 85 63 0 4-5 db 14 7 12 A.... (S) 75 31 30 G'"(R) 63 92 29 0 4 14 7 G3 30 1 2 11 10 64 0 4-10 db 14 7 13 G(R) 63 69 31 A.... (S) 100 77 65 0 5-10 db 13 8 14 G"(S) 63 67 32 G"'(S) 30 54 31 1 3 7 14 15 G(S) 88 67 33 G•(S) 100 85 32 4 1 5 16 33 2 3 5 16 16 A.... (R) 88 92 34 G'(S) 88 54 34 4 2 9 12 17 G'"(S) 63 54 35 G"(R) 88 85 35 3 4 15 6 18 G(R) 75 77 36 G'"(R) 50 77 36 1 5 20 1

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this conclusionas the synthetictone of quality 1 was RANGE OF ATTACK AND DECAY FOR PIANO-LIKE TONES mistaken for the real tone about half of the time. The A-B on G"' (see Table IX) indicated that no Judgmentsof the attack time weremade from syn- synthetic tone was definitely preferred over the real thesizedG", G, and G2 tones.A decaytime wasgiven piano tone. However,quality ! and possiblyquality 2 each tone somewhatnear the decaytime of the piano were consideredto be about equalin quality. The tests toneshaving the samepitch. The attacktime was then alsoshow that quality 1 waspreferred above any of the varied. other qualities. The attack for G" tones must be between the limits The identificationtests (Table VIII, tests8, 17, 26, of 0 to 0.09 secto be piano-like,0.09 to 0.14 secto be 30, 32, and 36) showthat the syntheticquality ! of questionable.Any toneswith an attacktime whichis G"' couldnot be distinguishedfrom the real pianotone. greaterthan this makes the toneno longer piano-like. Thesetwo setsof preferencetests (for A"" and G"') The attack time for G tones had similar limits which seemto indicate that any quality that departsmuch were 0 to 0.05 sec,0.05 to 0.12 sec,and greaterthan from real piano quality is discriminatedagainst by 0.12 sec.For the higherpitched tones, the attack time either jury. was about 25% smallerthan thosegiven above. Considernow G" and we will seethat qualities! and A determinationof decaytime wasmade by givinga 3 wereabout equal to the realpiano tone (TableVIII). piano-likeattack to the synthesizedG", G, and G• Alsothis synthetictone of quality ! couldnot be identi- tones,and thenvarying the decaytime. The following fied from the real piano tone as indicatedin the data in limitsproduced undampened piano tones: 5-9 secfor Table VIII. Although quality ! was preferred about G", 2-5.5 secfor G, 1-4 secfor G•. Decaytimes slower equallywith the piano,quality 3 wasjudged to be some- than these sounded like sustained-tone instruments. what better (seeTable VIII, tests26, 28, and 31.) Fasterdecay times produced an unnaturaldecay. To Next, considerG'. Here, wherewe wouldleast expect soundlike a dampedpiano tone, the decaytime canbe it, there was a preferencefor qualities1, 3, and 4 over between0.8 and 7 sec, and must be dampedby the the real piano tone. Of these,3 and 4 were preferred dampingbutton (a controlon the attack and decay over 1 and alsoover 2. In the identificationtests (Table amplifierwhich produces a changein the decayrate VIII, tests10, 23, and 34) the synthetictone couldnot duringactual decay. This changecan be madelarge, be identified from the real tone. thus adjustingthe decayto a very fast rate, and for For middle G the qualities1, 2, and 4 werepreferred simulatingthe dampingpedal on the piano). to the real piano tone. The other resultsof the A-B EFFECT OF HARMONIC CONTENT ON THE PIANO tests were inconclusive.For instance,test 50 (Table QUALITY OF MUSICAL TONES IX) indicatesthat 1 is better than 2, but tests 51 and 52 indicate the opposite.Thus it would be safe to say Sincethere are an infinite numberof arrangementsfor that qualities 1, 2, and 4 are nearly equally preferred the harmonicsof a toneof givenfrequency, it is difficult and all are preferred over the piano. However, the to circumscribethose that are piano-like.An attempt to identificationtests (Table VIII) 11, 15, and 21 clearly do this was made in the followingway. The partials show that the synthetic tone of quality ! could be wereset up on the synthesizerso that the levelof each identifiedfrom the real piano tone. successivepartial was a constantnumber of decibels For G• no synthetictone waspreferred to a real piano lessthan that of the partial just belowit in frequency. tone but quality 1 wasclose. It waspreferred above the As an exampleof thistreatment applied to G" (when the differenceis 2 db), this harmoniccontent was other two qualities tried. Due to the absenceof the hammer noisein the synthetic tone, it was identified produced. correctly85% of the time. Nevertheless,in the identifi- Partial number 1 2 3 4 5 6 7 etc. cation tests the jury missedthe real piano tone 74% Relativelevel, db 0 --2 --4 --6 --8 --10 -12 etc. of the time. To our great surprisethe jury preferredthe real piano toneswith the high-impactnoise for G2 and Judgmenttests, made with suchtones, indicated the Ga rather than any synthetictones without the impact following.Tests were for themost part centeredaround noise. G", G, and G• with attackand decayapproximating These seemto indicate that most personsare satis- thoseof the piano. fied with the quality of pianotones and that any large Tone G" departuresfrom this quality seemto be disliked.Com- ments indicated that some of these tones would be (1) Thelimits for a piano-likequality extended from 2.5 interestingif they were to comefrom a musicalinstru- to 1.5 db per partial. ment different from the piano, but not as a piano (2) When the level differencebetween components was tone. large,that is whenessentially only the fundamental We will now try to describein an objectiveway what waspresent, the piano-likequality was gone and is meant whenone says the tone sounds"piano-like." the tone tended toward that of a kettledrum.

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(3) From this point until the differenceapproached 2.5 Tone G•. db, the quality approachedthat of a piano but (1) Piano-likequality had limits from 40 to 7.5 db per would be referred to by musiciansas dead, hollow, partial. or having no edge. (2) The fundamentalalone has a piano-like quality. (4) A differenceof 1 db per partial produceda tonethat However, adding the next partial in the above had too muchedge. This quality of toneapproached limits, improvesthe tone quality. that of a harpsichordrather than a piano. (3) Lessthan 7.5 db per partial producesa tonewhich is (5) Differencesless than this producedtones that were too edgy. entirely too edgy. (6) The first five or six componentscould be changed The midpointson the limits of G", G, and G•.are, aroundin any positionand the tone wasstill piano- respectively,2, 8, and 32 db per partial, each being like providedthe remainingcomponents conformed four times that of the precedingtone. A singlepartial to the limits given above. above the fifth or sixth could be eliminated without

Tone Cs producing any noticeable effect. However, if it were raised 4 or 5 db from its position in the series, it (1) Limits for a piano-like quality were from 13.0 to was distinctly noticeableand the resulting tone was 5.0 db per partial. lesspleasing. (2) Whena singlepartial, namelythe fundamental,was These conclusionsabove were based on synthetic used,the tone could not be calledpiano-like. tones whose componentswere harmonic. When the (3) Differencesgreater than 13.0db resultedin dead or componentshad inharmonicfrequencies equal to those dull tones. in the piano, results obtained were approximatelythe (4) Differencesless than 5.0 db producedtones with too same as those stated above with one very important much edge. exception.The lack of beingharmonic gives rise to the (5) The first 2 or 3 componentscould be changedand peculiar quality known as piano quality, namely, the the tone was still piano-like provided requirement live-nessor warmth. This is very important for the first (1) was fulfilled. three octaveson the piano.

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