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Anharmonic Strings

May 16, 2011

When my son was young, he studied piano. When he was older, he decided to learn to play guitar. Since guitars go out of tune, one important part of guitar instruction is learning how to tune your instrument. Tuning by ear is easy. You listen for the "beats;" that is, the low frequency component you get when sine waves of different frequencies are mixed. Since musical notes in a scale are based on whole number ratios of frequencies, beats are also what make chords sound in, or out, of tune.

After a while, my son bought an electronic tuner. He would select a musical note, a microphone would detect the sound, the tuner would analyze the frequency of the plucked string, and a meter would indicate how much it was out of tune. I could design such a device, and I'm sure that any embedded systems developer could do the same, but these devices have one limitation. They may work well in tuning lightly plucked strings of guitars over a limited range from lowest to highest note; but a piano that's tuned electronically would sound out of tune. You need to tune pianos by ear.

I once worked with a physicist who did piano tuning when he was younger. There are fewer piano tuners today, since most keyboard instruments are now electronic. Piano tuners were very important in the early part of the twentieth century. Serious owners of pianos would have their pianos tuned twice each year. Concert pianists insist on a tuning before every performance. Pianos go out of tune because their design puts their materials of construction in extreme states.

Real pianos are heavy. Their major component is a cast iron frame that carries the strings. The strings are under high tension and they are expected to be repeatedly hit with a hammer, so they are made from a high strength, high-carbon steel, typically manufactured to conform to ASTM A228. High strength notwithstanding, the strings will creep, and that's why piano tuning is required at intervals. The string material is appropriately named piano wire.

Piano strings have a finite diameter, and this causes their vibration to deviate from that of an ideal string. The explanation is simple. Something thick is harder to bend than something thin, and vibration is all about bending. For a real string, the fundamental tone does not describe the piano note completely.

When you tune, you need to consider how the note's harmonics (called partials) clash with other notes. Different instruments have different harmonic structures, called timbre. One book on my bookshelf describes how to orchestrate music to prevent dissonance between the partials of various instruments.[1] This book includes such warnings such as the following:

"Owing to the complete absence of any affinity of tone quality, the combination of strings with brass is seldom employed in juxtaposition, crossing, or enclosure of parts."(Ref. 1, p. 95)

Harmonics are more important for lower frequency notes, since most harmonics of the higher notes are beyond audible range.

Piano tuning was analyzed by none other than Richard Feynman, who did it "just for fun."[2] This is all the more interesting, since Feynman was "tone deaf."[3] Feynman calculated the actual sounding frequency ("True Frequency") of a string in terms of its idealized frequency, f, and its material properties,

True Frequency = f( 1 + (B/2))

In this equation, B, the inharmonicity coefficient, is given as

B = π E A² μ f²/T²

where E is Young's modulus, A is the cross-sectional area of the wire, μ is the weight per unit length and T is the tension. The idealized frequency, f, is the frequency calculated from the simple vibrating string expression that ignores material properties; viz.,

f = (1/2L)sqrt(T/μ)

The inharmonicity coefficient, B, is about 0.00038 for a representative piano wire at note A4 (440 Hz).[2]

Because of this inharmonicity, the harmonic frequencies of piano strings deviate somewhat from simple whole number ratios to the fundamental. To compensate for this, a typical piano tuned by ear has the lowest note tuned 30 cents flat, and the highest note tuned 30 cents sharp. This compensation makes the notes more pleasing when sounded with other notes.

The other part of the piano.

Van Cliburn, September 20, 2004.

(Via Wikimedia Commons)

Nicholas J. Giordano, Hubert James Distinguished Professor of Physics at Purdue University, presented a paper on piano wire at the 161st Meeting of the Acoustical Society of America, Seattle, Washington.[3] Although piano wire properties were improved by materials advancements from iron wire in the period 1750-1820 to steel wire during 1840-1850, piano wire technology has essentially stalled at steel.

The most consonant wires are thin, but thin wires need greater strength to function in the piano application. For this reason, Giordano considered whether carbon fiber would make a better piano wire material. The best carbon fiber has a tensile strength of about 5.65 GPa, whereas a particular type of piano wire, AISI 1060 (0.6% carbon steel), has a tensile strength of about 2.20-2.48 GPa.[5]

In comparing materials, Giordano considers tensile strengths of 3.5 GPa for carbon fiber and 3 GPa for piano wire. The Young's modulus of carbon fiber is 2.30 Gpa vs 2.00 GPa for piano wire, but the most significant factor is the density difference, 1.8 g/cm³ for carbon fiber vs 7.8 for piano wire. All these material properties figure into the equation for the inharmonicity coefficient.

According to Giordano's paper, the string harmonics follow the equation,

f_n = nf₁ (1 + α n²)

where

α = (π² E r²)/(32 ρ L⁴ f₁²)

in which r is the string radius and ρ is the material density. The α term varies across the keyboard, since the string size and length changes. This is illustrated in the example of a Steinway M piano, as shown in the table.

Note	Freq. (Hz)	α (calc)	α (meas.)
A3	220	1.4	1.7
A4	440	5.0	4.4
A5	880	12	12
A6	1760	56	41

As Giordano notes, a piano with carbon fiber strings has not yet been made, but the mechanical properties of carbon fibers show that strings made from this material would have smaller deviations of harmonicity. However, there are other acoustical-mechanical effects that may be just as important. Piano manufacture is a conservative business, and the loss of market share to electronic keyboards leaves less money available for such a risky venture. The obvious thing is to arrange for big-name pianist to use a carbon fiber instrument exclusively. Every music lover knows that Vladimir Horowitz played a Steinway.

Acknowledgement:

Thanks to Prof. Giordano for sending me information on his research.

References:

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Linked Keywords: Piano; guitar; musical tuning; beats; frequency; sine wave; musical scale; just intonation; whole number ratio; chord; electronic; microphone; meter; cent; embedded system; physicist; synthesizer; twentieth century; concert pianist; cast iron; string; ASTM; creep; piano wire; diameter; vibration; fundamental frequency; fundamental tone; harmonic; partial; timbre; Richard Feynman; tone deaf; Young's modulus; tension; musical note A4; Van Cliburn; Wikimedia Commons; Nicholas J. Giordano; Purdue University; 161st Meeting of the Acoustical Society of America; Seattle, Washington; iron; steel; carbon fiber; tensile strength; radius; density; Steinway; Vladimir Horowitz; Nikolai Rimsky-Korsakov.

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