Piano Physics at Purdue

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Sound Samples

Latest modeling results (2003)


Using Calculated Tones to Study Piano Hammers

Here are some results from our piano modeling project. These are calculated piano tones, computed using only Newton's laws (i.e., F=ma). In addition, ALL of the parameters of the calculation have been determined from separates studies of hammers, strings, soundboards, etc. None of the parameters have been adjusted just to make the tones sound good.  They were all taken from independent work, so there is no fudging! Note that C4 is middle C.

These sound files are in AIFF format, so if your brouser cannot play them directly, it will probably give you the choice of downloading the files.

A full discussion of our modeling calculations is given in a paper which has been acceptted for publication in the Journal of Applied Signal Processing.

First, to calibrate you ear, here are some real piano tones recorded from a reasonably good Steinway model M grand piano.

These are just simple scales.

Real piano scale C2<->C3
Real piano scale C3<->C4
Real piano scale C4<->C5
Real piano scale C5<->C6

Now come some calculated tones.

In our initial work we modeled piano hammers using a simple power-law force function F=Kz^p  where z is the amount the felt on the hammer surface is compressed by contact with the string.  Based on previous research (including some of our own), an exponent p=3 was thought to model real hammers fairly well.  Here are some calculated tones using these power-law hammers.

Piano scale with power-law hammers C3<->C4
Piano scale with power-law hammers C4<->C5

The tones produced by these power-law hammers are not very good.  So, we have studied the hammers more carefully, and explored their properties in greater detail.  It has long been known that real hammers display some hysteresis - that is, the force-compressions (F-z) relation is different on the compression branch when compared to the decompression branch.  We have now studied this in detail, and have used a mathematical model introduced by Stulov to describe this hysteretic function.  We also devised a new way to measure the hammer-string force when the hammer stikes a real string.  When we use these results and the Stulov function for F(z) the calcualed tones are much improved.  Here are some examples.

Piano scale with hysteretic hammers C2<->C3
Piano scale with hysteretic hammers C3<->C4
Piano scale with hysteretic hammers C4<->C5
Piano scale with hysteretic hammers C5<->C6

Here are some simple melodies calculated with hysteretic piano hammers.  Please judge for yourself!

Bach minuet in G
Part of a Mozart sonata
Bach prelude in C

One of our conclusions is that modeling the hammer with a hysteretic function that is derived from measurements of a hammer striking a string is essential to producing a realistic piano tone.

Copyright © 2003 Piano Physics at Purdue Team