Frozen Inaugural Quartet & The Moral Standing of Instruments

D    ue to the mistake in administering the oath of office], if in fact Barack Obama is not the President, who is? Okay, the Constitution says that the President takes office at noon on Inauguration Day, oath or no. So I’m pretty sure that means the Presidency goes to whoever was on-camera at noon. The new President is Yo-Yo Ma! ... It’s still pretty darn historic to have the first Asian-American President and the first Vice President who is a cello.”
  —  Steven Colbert, Comedy Central , 21-JAN-2009.

S    upposedly, Oistrakh once recorded Paganini’s 17^th Caprice in one take, in an unheated, freezing-cold studio in Russia in wintertime. Impressive.”
  —  Carl Fulbrook, Cambridge (Churchill College Music Society).

C ellist Yo-Yo Ma, violinist Itzhak Perlman, clarinetist Anthony McGill, pianist Gabriela Montero performed at the Obama Inauguration on Tuesday. The quartet played a piece by composer Williams called ‘Air and Simple Gifts’ immediately before Obama was sworn in. It was 31 °F (-1 °C). The wind was blowing at about 3 mph (5 km/h). Attendees seated nearby were bundled up in scarves and thick winter coats and knit hats pulled down over their ears. How is it possible to play in such conditions? Those of us who have performed outdoors in freezing weather—in a marching band or brass band, say—dread it. But violin? Cello? Piano? Inconceivable! At least Gabriela had some fingerless gloves on…

H ere is a YouTube of the performance:



T hink of the cello! Think of the violin! (And the clarinet and the piano, too, although those are not in such clear and present physical danger as the strings are.)

T hese are not mere inanimate objects. They merit care and ethical concern—in somewhat the way that animals do. If pets have rights, then instruments do as well. The violin and cello deserve a loving, respectful home. My moral character and virtue as a human being depend on how I treat them.

H ell, I take my violin into the supermarket rather than leave it in my car’s back seat, even if only for 30 min. Swaddling the instrument in a blanket for extra insulation is good. What musician knowingly mistreats her/his instrument? Well, don’t answer that...


S o never mind the human agony of playing in freezing conditions in Washington, D.C., this Tuesday. Never mind the aesthetic tuning and articulation issues that the humans are subjected to when performing in extreme temperatures. My hopes and prayers are for the instruments.

M y ethical feelings toward instruments do come with a distinction, not unlike that of 18^th-Century Scottish philosopher, David Hume. Since the instruments are inanimate and cannot reciprocally behave virtuously toward me, they cannot have exactly the same moral standing as a person or an animal. My sense of an instrument’s ‘valor’ and ‘utility’ has to be combined with what philosophers would call ‘aprobation’—a moral calibration that takes into account its inanimateness: the fact that it is not a sentient creature. Just the same, I care about its wellbeing...

M y questions, though, are these: (a) is the physical risk of damage to the violin-family instrument mainly due to the temperature excursion or instead mainly due to the effects of sudden change in humidity? Is it (b) mostly a matter of stress between the top plate vs. saddle or instead a matter of temperature/humidity changing how the soundpost is fitting? And (c), what parts change dimension at the fastest rate? Will I (d) hear the fatal cracking of the top plate while the instrument is still cooling down, or will the distaster be more likely to happen when I bring it inside and warm it up? Finally, (e) after exposure to cold, what rate in degrees per hour would be best to allow the instrument to warm up, to prevent the instrument from cracking or the gluelines from separating upon thawing?


T he thermal expansion coefficient of dry wood parallel to the grain (as for the ebony saddle) is essentially independent of wood species and density. For ebony, the thermal linear expansion coefficient α_l is 3.1 x 10^-6/°C. But the thermal expansion coefficient across the grain (radially, as for a sitka spruce top plate, or tangentially) are strongly dependent on wood species and density (gm/cm³).

   α_r = (9.9 + 32.4ρ)  x 10^-6/°C

S ince sitka spruce ρ = 0.42 gm/cm³, α_r = 23.5 x 10^-6/°C. In other words, the chilling top plate contracts about 7.6 times as much as the ebony saddle for a given temperature excursion.


If we take the violin from at 68 °F (20 °C) room outdoors onto the inauguration platform at 31 °F (-1 °C) a contact area between the saddle and the top plate that is about 50 mm long will begin shrinking—the ebony will shrink -3.26 μm and the sitka spruce will try to shrink -23.5 μm, but will be prevented from doing so by the saddle. The woods and the glue between the spruce and the ebony will be under considerable shear and tensile stress while this is going on.

T he modulus of elasticity for the delicate sitka spruce is 9.86 GPa (1,430,000 psi), and the modulus of rupture is 65 MPa (9,430 psi). Since we don’t know—for any specific instrument, or for stringed instruments in general—the precise area A₀ to use in the force equation (below), nor do we know how spatially uniform or variable the bond between the top plate and the ebony saddle is, it is not possible to calculate the stress-strain relationships with certainty. We would have to build a special violin to be able to measure it. But ‘ballpark’ calculations I have done indicate that the stresses can easily exceed the maximum tensile strength of 2.48 MPa (360 psi) for the sitka spruce in the top plate.

W ood that is not totally dry reacts differently to varying temperature than does dry wood. When wood with non-zero moisture content is warmed, it initially expands because of normal thermal expansion but after a few hours begins to shrink because of loss in moisture content. Unless the wood is very dry initially (4% moisture content or less, which is unrealistic for an instrument), shrinkage caused by moisture loss on warming will be greater than any thermal expansion, so the net dimensional change on warming will be negative. Wood at intermediate moisture levels (say, about 8% as for a violin in its humidified case) will expand when first warmed, then gradually shrink if the warm, low-humidity condition persists. The net dimensional change will often be near zero if enough time is allowed to equilibrate at the new conditions. But at intermediate times the dimensions are expanded compared to the initial temp. For the wind conditions and high outdoor relative humidity at the Obama inauguration, I think we can safely bet that the humidity-generated effects were negligible compared to size of the thermally-generated dimensional changes and forces.

H ow rapidly do the strain and stress develop in the top plate, especially in the area that is constrained by the lower-expansion-coefficient saddle? Well, if the movement is sudden, from inside to outside, and if convection is efficient (e.g., wind at 3 mph or greater) then it will occur within 10 to 15 min.

T o fully model this process mathematically, we would have to solve the partial differential ‘Heat Equation’ for the conduction and convection of heat (cold) into the wood, from the cold-exposed surface inward. Readily doable using available software like MATLAB, but more than is reasonable to put in a CMT blog post.

A  simple Excel spreadsheet will give you some sense of the stress (force per unit area) that the top plate has to bear. Click on the screenshot below to open or download a copy to play with.


O bviously, stringed instruments do occasionally experience unintended temperature excursions that are this wide or wider—and yet a split top plate is amazingly rare. How can this be? Well, the rarity of it must be due to most such occurrences’ having a relatively slow rate of change of temperature—the interior of a closed violin case (with the instrument wrapped in cloth) in the rear seat of a car, for example, will take several hours to drop the first 10 °C when exposed to a new ambient temp that is 21 °C lower. And, in a slow cooling or slow warming excursion, the glue that bonds the various pieces of the instrument together has time enough to accomodate the shearing forces that develop. In a fast cooling or warming, the macromolecules in the glue don’t have time to relax and, as a consequence, the full stress developed (by the thermal expansion coefficients differential between the different woods and cross-grain orientation) is imposed on the joined parts immediately—with a much higher probability of catastrophic failure beyond the material’s maximum tensile strength.

W hich brings up a possibility. If you know in advance that you will be required to perform at a freezing inauguration, you might take your least-favorite instrument to a cold storage facility the day before, to incubate it at a temperature that is close to the outdoor temp that is predicted during the performance the following day. Retrieve it from the cooler, put the whole thing (the instrument in its case) in a big padded Igloo® portable insulated cooler, and schlep it to the performance venue, only pulling it out and tuning up when you are outdoors in the cold. Then when you are done, allow the instrument to warm up slowly in its still-cold case, over a day or so before opening it up at room temperature again.

A  simple example of physics / mechanical engineering in service of instrument health!

N ear the end of the film, ‘The Pianist’, the piece that Szpilman plays for the German officer is Chopin’s G minor Ballade. Yes, we suspend our disbelief when we think of the freezing room and prison-camp starvation and no opportunity for practice—yet, despite these things, the performance is excellent. But everybody thinks of the humans and their suffering and implausible virtuosity in the face of impossible odds; nobody thinks of the poor, suffering musical instruments and their impossible odds.

N ear the end of Tuesday’s inauguration, we had to suspend our disbelief at the remarkable style and grace that Ma, Perlman, McGill, and Morteno mustered. Would that their instruments had been weather-impervious fiberglass/plastic ones. I hope they are doing okay and none was injured...

T he ethics of how we treat instruments is, I want to tell you, just a ‘hop, skip, and a jump’ from the ethics of how we treat our environment and public goods. The world and its resources are ‘useful’ to us; they are ‘instrumental’ in serving the needs of humankind. But the ethics of environment is predicated on the notion that the world is a collection of ‘public goods’, not only for the human public but for animals and plants and other ‘publics’. In the case of privately-owned musical instruments, it could be argued that old, rare and historic instruments are indeed a sort of public good—with value to broader and future society beyond the scope of needs and utility of its present owner. But what about cheaper, commodity-grade instruments? Is it ever justifiable to recklessly abuse them or destroy them, or for politicians to command that they be abused/destroyed? (When a rock guitarist smashes a fine instrument on-stage, must one be a ‘virtue ethicist’ to say he is wrong to do so? S’pose so... )

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Forensic Tuning-ology: Haydn’s Fork and 423 Hz vs. 415 Hz, Back in the Day

H  aydn tells us in his letter that he had to deal with more than one pitch. It seems that ‘Viennese Pitch’ was a recognized standard and that it was considered relatively high. The date of Haydn’s letter suggests that the pitch rose in Vienna in the late 1750s or early 1760s ... One thing is clear: there was no such thing as ‘Classical Pitch’. Different standards existed at different places.”
  —  Bruce Haynes, Mozart and the Oboe, p. 47.

T  oday most players think in 440-415-392 Hz pitch categories, not knowing that the so-called baroque pitch 415 is not older than 30 years, when two harpsichord builders introduced the transposing keyboard. There is a strong evidence that most early 18th century bassoons are much lower than 440 Hz, but I found as many on 408 or 421 Hz as on every other pitch around there. The 415 Hz is a good compromise, since one has to agree on something in a mostly free-lance music scene. But if you get to build a strong, stable ensemble, you can do it differently. A good example is Harnoncourt’s Concentus Musicus. They are playing on 421 Hz since 35 years, just because Jurg Schaetflein’s oboe was happy there.”
  —  Michael Dollendorf, Bonn and San Francisco, International Double Reed Society.

Following my recent post that wondered about Dallapiccola’s tuning, two CMT readers asked questions about baroque tuning, both specifically inquiring about the historical appropriateness of A = 415 Hz for works by Haydn. A₄₁₅ is about a semitone lower than the modern A₄₄₀.

One of the question-askers went further and opined that the matter is moot anyhow inasmuch as the answer would depend on the ambient temperature, and we couldn’t possibly know what temperature the rooms were where Haydn composed and performed.

I am no historian; some links below (to Bruce Haynes’s book and others) should prove useful to those of you who care. Haydn’s tuning fork does indeed exist. So does Handel’s. Careful measurements have been made, of their dimensions and of their pitches—at unspecified temperature, presumably ordinary ‘room temperature’ (about 20 °C).

I do know a thing or two about physics and engineering. And I can say without doubt that, unless Haydn were playing either on Pluto or on Mercury, the pitch of that stationary tuning fork would have been (to a likewise stationary listener) imperceptibly different from its nominal 422.5 Hz. The pitch of a fork simply doesn’t change very much, even through wide excursions in temperature.

Here is the equation that very closely approximates the dependence of frequency upon the dimensions, for a tuning fork that has cylindrical tines:


where
  l is the length of the tines;
  E is the Young’s modulus of elasticity of the material the tines are made of (210 • 10⁹ N/m² for tempered steel at 20 °C);
  ρ is the density of the material of the tines (7.8 gm/cm³ at 20°C); and
  R is the radius of the tines.

And here is the equation for forks that have ‘square’ tines (like the Kohl fork in the photo above), where A is the cross-sectional area of the tines looking down their shafts:


These equations are simplifications. They totally ignore the many harmonics that are emitted when the metal is struck. (Those overtones rapidly die out and only the fundamental is left.) They also ignore any multi-axial anisotropy and any lengthwise or cross-sectional inhomogeneities that result from the particular method of manufacture or heat-treatment or finishing of the fork. They ignore any nonlinear or anelastic behavior.

Finally, here is the equation for how the length changes as a function of temperature in ‘normal’ temperature ranges:


where the thermal coefficient of expansion α is 1.2 • 10^-5 °C⁻¹ at 20 °C.

Once tuned, a tuning fork’s frequency does vary slightly with changes in E(T), l(T), and ρ (T), according to these equations. But if Haydn’s rooms were at 10 °C, the fork would only have been about 1.4 cents sharp, and if Haydn’s rooms were at 30 °C, the fork would only have been about 1.4 cents flat—too small a difference to hear.


Besides historical tuning forks, a lot of what we know of historical tunings comes from measurements of wind instruments—organ pipes, oboes, flutes. Yes, of course, the tuning of these is adjustable, but the scratches and markings on the organ pipes—the traces of forensic evidence as to the dimensions these were adjusted to when actually in use—and the comparative dimensions of the set of pipes (or, in the case of the flutes, oboes, and bassoons, the distances between the holes) provide ample guidance as to the actual historical pitch. Tuning forks used by orchestras between 1750 and 1820 mostly had a frequency of A = 423.5 Hz, although there were many forks, many slightly different pitches.

• 1711: John Shore’s fork, A_423.5
  
• 1714: Strasbourg Cathedral organ A_391.0
  
• 1715: English tenor fork, A_419.9
  
• 1722: Dresden’s Catholic church organ, A_415.0
  
• 1750: London, inscribed ‘Common music shop fork.’ A_424.3
  
• 1751: Handel’s tuning fork; the box which contains the fork bears the inscription: ‘This pitchfork was the property of the Immortal Handel and left by him at the Foundling Hospital, when ‘The Messiah’ was performed in 1751.’ A_422.5
  
• 1772: Gottfried Silbermann Catholic church in Dresden, A_415.0
  
• 1780: Organ builder Schulz, A_421.3
  
• 1780: Stein’s fork, A_422.6
  
• 1780: Stein’s fork for Mozart, A_421.3
  
• 1780: Organ builder Schulz, A_421.3

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