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Aggiornamento to the second message of Feb 18, 2018

I have now bigger loudspeakers. No Hi-fi, but better than Pc hardware. They are connected to the usual amplifier. With them, I hear easily a difference between a sine and 3% mild distortion (a tanh deformation that compresses the sine crest by 3%), and 1% is the limit once knowing what to listen at. Which implies that, even at modest power, the previous Pc loudspeakers and headphones deformed the sound enough to make 3% distortion barely discernible. Ouch.

The following wav distorts mildly the 330Hz sine by 0%, 1%, 3% in 2s samples.

So maybe 1% of mild distortion is our perception limit - or my present hardware still creates as much distortion.

My sensitivity to hard clipping and to 8-bit coding has not changed.


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When you play simultaneous notes on a violin, you hear them beat, depending on the interval and the intonation. It's weak but well perceiveable at the violinist' distance. The same happens on the viola, and supposedly other polyphonic instruments with sustained sound.

To investigate the beat, I reproduce it with sounds synthesized by my software
and here are the sound samples

As a violin spectrum, I thankfully use these measures of an empty D string
and again, a spectrum synthesizes a periodic sound that fails to imitate a music instrument.

The four Beat1... wav play simultaneously a 440Hz A and a D nearly a fifth lower like violin strings. In the first 2s, the frequency ratio is 3/2, and for the next 2s, D is raised by 1Hz.

Beat1_A_DA32harm.wav uses the measured spectrum. You hear 3 beats per second when D is raised by 1Hz because the harmonics 3 of D and 2 of A interfere. This happens without intentional nonlinearity, on summed sounds, and the intensity of the beat resembles what the violinist hears. I had supposed some nonlinearity is requested at the instrument, there
and as it looks I was wrong.

Our ears perceive sound intensities on a logarithmic scale more or less, so they aren't linear. A measurement of the intensity is by nature a nonlinear process anyway. So the instrument can behave linearly, and the perception makes the interference.

Beat1_B_DA32onlyH23.wav contains a fundamental and only the harmonics 2 and 3 of the measured spectrum. It beats like the complete spectrum: 3 times per second, similar amplitude.

Beat1_C_DA32removedH23.wav uses the spectrum minus the harmonics 2 and 3. It beats more weakly and about 6 times per second, like the interference of the weaker harmonics 4 and 6.

Beat1_D_DA32sine.wav contains sine waves. I perceive no beat at medium amplitude, and only a faint one when playing loudly, when the amplifier's power supply drops. Until I improve the amplifier, I consider the power supply produces the faint beat with sines, but not the stronger beat when harmonics are present.

All is consistent with a linear interference of harmonics.

Marc Schaefer, aka Enthalpy

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How quickly do we perceive sound intensity?

My TutTrem.exe writes in TutTrem.wav a tone at frequency f whose amplitude is modulated with depth 0 <= m <= 1 by a sine at frequency g
and here are the sounds

  • Trem_1_A880sine_3Hz10Hz30Hz50Hz70Hz.wav modulates a 880Hz sine, with 0.25 depth like the others, at 3Hz, 10Hz, 30Hz, 50Hz, 70Hz. I perceive a tremolo at 30Hz but a steady sum of notes at 70Hz, the limit being around 50Hz.
  • Trem_2_C131sine_3Hz10Hz30Hz.wav. The sine C note is at 130.8Hz, about where the cello and bassoon begin their second octave. I perceive no tremolo at 30Hz already, so the ears are slower on lower notes. Well, the same happens with an amplitude detector made of electronic components.
  • Trem_3_C131bassoon_3Hz10Hz30Hz50Hz70Hz.wav plays a 130.8Hz C note with already cited bassoon spectrum containing strong harmonics. The limit is around 50Hz again, so the ears discern quick changes through the harmonics of low notes.

These observations are compatible with hair cells in our ears that measure amplitudes each over a band, supposedly wide bands with much overlap.


  • Trem_4_C65bassoon_3Hz10Hz30Hz50Hz.wav. The 65.4Hz C is about where the cello and bassoon begin. Here the limit is at 30Hz or less, quite fast for a 65Hz fundamental.
  • Trem_5_D293A440violin_3Hz10Hz30Hz50Hz70Hz.wav. Made by the programme tutut uploaded in a previous message, the wav contains a fifth with violin spectrum: an A at 440Hz and a D above 293.33Hz so their harmonics 2 and 3 beat. Here too, the beat limit is around 50Hz.

These observations are compatible with our ears measuring the amplitude in a band that contains the harmonics of both notes and perceiving the interferences as beats.

Marc Schaefer, aka Enthalpy

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Intervals tuned to simple frequency ratios are often more pleasant to our ears. Violin and viola players tune their instrument that way. Some texts claim that "good tune" follows simple ratios. So can we generalize?


First, some harmonics sound out of tune. Here's the uncorrected scale of a natural horn, and within the west European musical culture, 7*F is badly low, 11*F isn't a note, and many higher multiples are bad. Hear Beat2_1_NaturalHorn.wav from the upload
11*F is about three-quarter-tone, which serves in Romania, Greece, Iran and many more. "Proper intonation" is a matter of culture. West European musicians can train the three-quarter-tone by whistling Beat2_2_TrainQuarterTone.wav (here equal-tempered) with equal tongue movements.


Then, they may be individually nice, but simple ratios combine to derail the intonation. Hear the small thirds in Beat2_3_ F_Ab_B_D_F.wav: tuned to 6/5 here, they sum to an octave a third-tone too big.

One goal of the equal-tempered scale is to avoid this. Trombone and bowed string players learn to follow it, not just to play together with a piano, but in their own interest.


This is the correction from simple ratios to the equal-tempered scale, in cent (0.01 half-tone). I've taken big intervals complementary to the small ones, like 16/9 rather than 7/4 for the minor seventh.

The fifth and fourth (whose sum is an octave) have simple ratios only 2 cents wrong. Bowed strings are tuned to zero beat or by sounding the harmonics, which cumulates 6 cents over four strings, an error imperceptible to most people. Beat2_4_Fifth32BachFourth43Bach.wav plays both intervals according to simple ratios (no beat) and to equal temper.

At the major and minor thirds and their complements, things get ugly. Zero-beat intonation sounds better and is tempting, but violinists and others must learn and train to follow the unpleasant equal temper to avoid other trouble. The height difference is patent even with single notes. Some guitarists tune the G-B strings by sounding their harmonics 5 and 4, which is questionable. Hear Beat2_5_ThirdMajor54BachMinor65Bach.wav.

We hear beats at the major and minor seconds even when they follow simple ratios. Other ratios nearby explain it: only 37Hz separate the harmonics 8 and 7 when this interval is 9/8 and 20Hz separate the harmonics 15 and 14 in the 16/15 interval. At least, training them equal-tempered costs less. Hear Beat2_6_SecondMajor98BachMinor1615Bach.wav. At the tempered minor second, slow beats result from the harmonics 18 and 17.

Marc Schaefer, aka Enthalpy

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  • 2 weeks later...

Here are some sounds and thoughts about our perception of contrabass notes.


We hear low frequency sines very badly. In CB_1_DescentSine.wav of this archive
if the volume is set for 370Hz, the 9th sound at 58Hz is quite weak despite my loudspeakers resonate around 50Hz. Frequencies are: 370, 294, 233, 185, 147, 117, 92.5, 73.4, 58.3, 46.3, 36.7, 29.1, 23.1Hz.

The conventional frequency limits of our ears aren't defined by a mild damping like -3dB, but as when a wave gets painful before we hear a sound. So we're very insensitive well before the often cited 20Hz and 20kHz.

Knowingly, we hear low musical sounds by their partials. A (nearly bass tuba) euphonium spectrum measured on Bb=58Hz is

 #  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16
dB 15 -1 -1  0 -1 -6 -8  0 -7 -16 -22 -13 -15 -19 -34 -40

CB_2_DescentEuphonium.wav uses it to synthesize by TutE.exe sounds well audible at D=73.4, Bb=58.3, F#=46.3, D=36.7, Bb=29.1, F#=23.1Hz. The bassoon reaches Bb=58.3Hz, the piano A=27.5Hz, the contrabassoon Bb=29.1Hz or A=27.5Hz.

CB_3_RemoveLowHarmonics.wav contains sounds at Bb=29.1 and Bb=58.3Hz with harmonics 1 to 15 equally strong and higher ones dropping. Then it removes the harmonics 1 to 6 from the 29.1Hz sound and 1 to 4 from 58.3Hz. I hear a small difference when removing the harmonic 5 of 29.1Hz or the harmonic 3 of 58.3Hz, both around 160Hz.

Maybe our lowest hair cell is centered on such a frequency, and only its residual sensitivity to lower frequencies extends our range down. Or maybe not.

The radiation resistance of a small acoustic source varies like F2, so low frequency components are hard to emit. While the contrabass tuba achieves it, the contrabassoon doesn't even try. Here a measured contrabassoon spectrum

 #   1   2   3   4   5   6   7   8   9  10  11  12
dB -19 -22 -13 -10  -8  -2   0  -2  -1  -2  -6 -19


If they have audible partials, we perceive musical sounds at very low frequency, with no clear limit, but as individual periods more than a sound if the frequency is low.

CB_4_PeriodsOrSound.wav plays a note at 10, 30, 50, 70, 90Hz. Its spectrum is wide:

 #   1  2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 23 25 27  29  31
dB -14 -4 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0 -2 -4 -6 -8 -10 -12

I hear individual periods up to some 70Hz. This is but more than the 50Hz tremolo of the 880Hz sine of Mar 18, 2018 in this thread

CB_5_PeriodsOrSound.wav plays a note at the same frequencies, but the spectrum is flat from harmonic 1 to 4 and zero above. Now I hear individual periods up to 50Hz, again but more than the 30Hz tremolo on the 131Hz sine.

The stronger modulation depth can make the difference.

So I propose (...and may not be the first) that, when hearing a contrabass sound, we notice individual periods if enough harmonics interfere within a frequency band (like the bandwidth of individual hair cells) to let the amplitude wobble there, and not too quickly.


Acoustics is a means and music the goal, so here are records of contrabass woodwinds:
contrabassoon https://www.vsl.co.at/en/Woodwinds/Contrabassoon
contrabass clarinet youtube v=d-aqcHlSFEI and youtube v=wUNZjNVFMkY

Marc Schaefer, aka Enthalpy

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The perceived beats in a contrabass sound depend on the number of peaks per period.

I use here three waveforms with the same harmonic amplitude but different phases to vary the period shape:

 #   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20
dB   0   0   0   0   0   0   0   0   0   0   0  -3  -6  -9 -12 -18 -24 -30 -40 -50
 °  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90
 °  90 180 -90   0  90 180 -90   0  90 180 -90   0  90 180 -90   0  90 180 -90   0
 °  90 -90 -90  90  90 -90 -90  90  90 -90 -90  90  90 -90 -90  90  90 -90 -90  90


In this archive:

  • CB_6_SinglePeak.wav has a single strong peak per period.
  • CB_7_TwoPeaksClose.wav has two close arches.
  • CB_8_TwoPeaksDistant.wav has two strong peaks well separated.

These three sounds play D=73.4Hz - Bb=58.3 - F#=46.3 - D=36.7 - Bb=29.1 - F#=23.1Hz, and

  • At 73Hz the phase makes no difference for me, as already observed at 440Hz.
  • Two close arches per period sound like a single one. But they help bad loudspeakers or amplifiers.
  • I hear individual periods from 58Hz down with one peak or two close arches, but two distant peaks need 29Hz to beat as clearly. Again as quick as the reaction time already observed, 70Hz or 50Hz.
  • At 29 and 23Hz, the two peaks per period beat more quickly than one peak or two close arches.

To make it patent, this sound contains the three waveforms at 29.1Hz:

  • CB_9_Bb29Hz_OnePk2close2dist.wav

These observations are compatible with amplitude detectors, for instance the hair cells in charge of the harmonics around 300Hz, that discern the waveform's peaks individually if their repetition is slow enough.

Marc Schaefer, aka Enthalpy

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Our perception of height for contrabass sounds is imperfect. It's a joke topic in orchestras. Some sources want to tune the piano's lowest octave stretched. The joined sounds show a more subtle picture.

  • CB_6_SinglePeak.wav in the previous message goes to 23.1Hz F#, lower than the contrabassoon and piano, and I perceive all notes in tune.
  • But CB_A_Bb29HzOnePeakTwoPeaksOddHarm.wav contains the same 29.1Hz Bb played with different harmonic spectra, and I perceive the third one too high.

Starting with a measured euphonium spectrum enriched at ranks 10 to 20:

  • The first note has phases for one peak per period;
  • The second for two peaks;
  • The third has no even harmonics (and two peaks per period).

Waveform.xls draws a period from a harmonic spectrum. The instructions in CB_Make.txt can be pasted in the running TutF.exe by a right click.

 # |   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20
dB | -15  -1  -1   0  -1  -6  -8   0  -7 -10 -12 -13 -15 -16 -18 -20 -22 -24 -27 -30
 ° |  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90  90 -90
dB | -15  -1  -1   0  -1  -6  -8   0  -7 -10 -12 -13 -15 -16 -18 -20 -22 -24 -27 -30
 ° |  90 -90 -90  90  90 -90 -90  90  90 -90 -90  90  90 -90 -90  90  90 -90 -90  90
dB | -15      -1      -1      -8      -7     -12     -15     -18     -22     -27    
 ° |  90      90      90      90      90      90      90      90      90      90    


We hear that (or at least I do):

  • Two peaks let hear faster beats than one but don't change the perceived height;
  • Without the even harmonics, the third note seems much higher.

So our hearing would not rely on the beat repetition rate to assess a contrabass note height. The difference between the harmonics, which makes their interference frequency, may become important at low notes, more so than some GCD that would have told the exact height.

The next sounds play Bb 116.5 - F 87.3 - Bb 58.3 - F 43.7 - Bb 29.1 - F 21.8 Hz with the three spectra:

  • CB_B_OnePeakDescentBbF.wav with one peak sounds in tune even at 21.8Hz;
  • CB_C_TwoPeaksDescentBbF.wav with two peaks, too;
  • but CB_D_OddHarmDescentBbF.wav seems too high at 29.1 and 21.8Hz, even at 43.7Hz a bit;
  • the perceived error is a variable amount, not a fixed octave.

Other trials not supplied here tell that the explanation isn't only an interference of harmonics. For instance at 21.8Hz, when suppressing the (even and odd pairs) highest harmonics progressively, a euphonium note seems higher without its harmonics 7 and 8 while a flat spectrum doesn't. Bad loudspeakers too influence the perceived height, logically as they deform the harmonic spectrum. More experiments and interpretations would be needed, but I have no such plan.

Since the spectrum influences the perceived height of contrabass notes, and the position of the hammer impact changes the spectrum of a piano note, maybe hammers hitting the strings at a different position would let us hear the first octave in tune. Presently the position is said to be 1/7th of the string length to reduce this harmonic - the one that must remain at the euphonium note.

Marc Schaefer, aka Enthalpy

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Wiki has an articke about absolute ear, also called absolute pitch
compiling studies made with method and on more people than the few musicians I heard, so their opinion is more reliable than the one I expressed in the present thread

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Maybe F/2, F/3... undertones help us perceive the pitch of the highest notes? Like overtones help us for contrabass notes.

The pitch of the highest notes isn't very clear at the piano but while we perceive it easily at the piccolo flute (high C like a grand piano) and the violin (higher with "harmonic" sounds). A difference is that the piccolo and the violin can produce weak undertones at these notes. An other difference: I played them both, but the piano very shortly, so it can result from practice.

This archive has notes in the piano's last octave, and I believe to perceive the pitch better with undertones. Unpleasant height, adjust the volume.

  • High_C_NotesNoUndertones.wav has only sines.
  • High_D_NotesUndertones.wav adds F/2 at -50dB to the high C. This undertone is not strongly perceived as a distinct note.

The piano might add (or not?) such undertones to its highest notes, say above G, by playing them "harmonic" like on plucked and bowed strings. These strings would be an integer number of times longer and have the equivalent of a finger, possibly of silicone rubber, mildly applied at a fundamental's vibration node.

Marc Schaefer, aka Enthalpy

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Most sources claim that we discern the direction of a sound source by the relative phase at our ears, and after experimenting, I agree. Has everyone noticed?

TutSter.exe in the archive runs in Windows' Cmd.exe to create the stereo TutSter.wav that play identical sounds at right and left but with propagation delays that result from the command
p <float>
The source is 1m before the listener, the signed number is a lateral distance in metres.

The hardware matters at computers. Headphones are far better here, and connecting them directly to the sound board can be better than through the loudspeakers' amplifier, though this cuts the low frequencies. Many sound boards and amplifiers demand a balance adjustment, as we infer the source's direction much from the relative intensity. And, err, different computer hardware swaps right and left at random.

These sounds play a sine 5m on side A, centered, 5m on side B, and again:

  • Dir_A_E082sine_m5z0p5m5z0p5.wav at 82.4Hz, I perceive no direction.
  • Dir_B_A110sine_m5z0p5m5z0p5.wav at 110Hz, a direction is perceptible. The phase shift is only 0.1*lambda.
  • Dir_C_E330sine_m5z0p5m5z0p5.wav at 330Hz lets still perceive the direction. 0.3*lambda.
  • Dir_D_A440sine_m5z0p5m5z0p5.wav at 440Hz, 0.4*lambda and
  • Dir_E_E659sine_m5z0p5m5z0p5.wav at 659Hz, 0.6*lambda are unreliable. Plus or minus a half-wave is undecidable.
  • Dir_F_A880sine_m5z0p5m5z0p5.wav at 880Hz, I perceive no direction at all. It would be meaningless anyway.

Added harmonics help if some have a useful frequency.

 #  1   2   3   4   5   6   7   8
dB  0   0   0   0  -3  -8 -12 -20
  • Dir_G_E082harm_m5z0p5m5z0p5.wav at 82.4Hz, now the direction is clear.
  • Dir_H_A880harm_m5z0p5m5z0p5.wav at 880Hz (strident!), no improvement.

At at favourable fundamental frequency, we perceive a more accurate direction with harmonics. Here 0.3m on side A, 0.3m on side B, and again:

  • Dir_I_A220SineHarm_m03p03m03p03.wav at 220Hz, pure sine then with harmonics.

At least up to 330Hz, our (hair cells?) detectors must transmit the raw signal to the brains rather than the detected amplitude - unless someone has a better explanation to the directivity. I didn't expect it, because we are insensitive to the phase of the harmonics versus fundamental, and because it needs very wideband nerves.

Marc Schaefer, aka Enthalpy

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  • 7 months later...

In a renewed attempt to explain why the highest piano notes would be perceived too low, I synthesized notes shorter and shorter, at the piano's highest C and the F below.

The program is my TingPoc that already served there
PingMake.txt feeds TingPoc.exe to create Ping.wav

My ears perceive the same height for 200ms, 100ms, 50ms, 20ms, 10ms decrement times. Not an explanation for the piano's high notes. By the way, I hear a good fifth here.

Marc Schaefer, aka Enthalpy

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Again researching why the highest piano notes would be perceived too low while nobody complains about the glockenspiel and celesta pitched even higher, I synthesized notes at the piano's highest C and the F below, this time with attacks slow to snappy.

The program is again my TingPoc.cpp and exe. TingBingMake.txt feeds TingPoc.exe to create a wav renamed TingBing.wav

My ears perceive the same height for 50ms, 20ms, 10ms, 5ms, 2ms rise times. Neither an explanation for the piano's high notes. And I hear again a good fifth here.

Marc Schaefer, aka Enthalpy


Still researching why the highest piano notes would be perceived too low, I synthesized notes at the piano's highest C and the F below, with varied decay times as on Nov 25, 2018 but beginning now with the shorter ones.

The program is again my TingPoc. PingBMake.txt feeds TingPoc.exe to create a wav renamed PingB.wav

When I'm not influenced first by the longer ones, my ears perceive lower and less precisely the shorter notes. Here the exponential decay time is 5ms, 10ms, 20ms, 50ms, 100ms. I hear inaccurately the 5ms and 10ms, and 20ms isn't good.

This isn't so surprising. 20ms of 4186Hz sound contain 84 periods, which is little to discern semitones 5.9% apart, and we claim to hear height far better than a semitone.

While the glockenspiel and celesta sound longer and nobody complains about their pitch, the piano's highest C may last around 20ms, and this can contribute to the imprecise and too low perception.

Marc Schaefer, aka Enthalpy


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  • 1 month later...

Here's an other way to add undertones to the piano's highest notes to help the listener perceive the height, as an alternative to the construction of Apr 21, 2018 here.


The soundboard transmits a part of the hammer's percussion to the other part of the strings whose length is a multiple of the struck one. Separate tuning is needed if the strings rub too much at the soundboard. Damping at the longer part of the string attenuates the undertone as quickly as the note.

Marc Schaefer, aka Enthalpy

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Piano hammers are heavier than the shortest strings. This slows down the rebound and may let us perceive the note too low because it is.

As measured on a Yamaha upright piano, the three strings for highest C have D=0.9mm and L=48mm so they weigh 0.72g together, of which 0.36g moves. The lighter hammer heads there have about 8mm*8mm*30mm Fagus Sylvatica (Beechwood) weighing 1.4g. The hammer is 4* as heavy as the strings and the first half-period 2.2* slower, as the strings' elasticity move more mass until they have repelled the hammer.

This affects more the high notes whose sound is shorter and strings are lighter.

In the joined archive, ToingMake.txt feeds Toing.exe to write Toing.wav that plays four times F=2794Hz and the highest C=4186Hz. F decays in 50ms and C in 10ms. C's first half-period is normal in the first two occurrences and slowed down by 2.2 in the last two.


I hear lower the C with slower half-period. Not much, but it resembles the perception of a piano.

Felt that covers the hammerhead to sound softer must slow down the rebound and first half-period further. 4µm/N would be as stiff as the strings, felt against thin strings must be softer.

Marc Schaefer, aka Enthalpy

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I claimed that the hammer's mass slows the first half-period of the piano's highest notes, altering ou perception of height. A way to check this effect is to pluck the highest strings and compare with the hammers. For instance with one half of a wooden clothes peg or some caring material, and sound fourths, fifths, octaves for easy perception.

Could someone kindly make the experiment, please? And tell us whether the plucked highest notes get a better defined height, or sound higher, than if hit by the hammers? Or ideally, install a PC microphone at the piano and bring the records here.

Thank you!

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  • 5 months later...
On 1/6/2019 at 6:42 PM, Enthalpy said:

Piano hammers are heavier than the shortest strings. This slows down the rebound and may let us perceive the note too low because it is.

Hammers heavier than the string is long known, Benade, Rossing and more known names. Treble hammers are reportedly even heavier than my estimate.

BUT they strike the treble strings near an end, not at the middle as I imagined, so the string much stiffer there rejects the hammer faster than what I computed. Though, I can't really believe hammer heads striking at 8% of a 48mm long string, so I'll give it a look next time I access a piano.

Apparently the ancestors concentrated on the effect on the timbre, not on the (perceived) sound height.

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