Dr. Stuart Wright, Senior Scientist, EDAX
Since my blog about piano wires back in November 2014, I’ve continued to think about what it all means in terms of music. As I mentioned in my posting, my friend Keith Kopp provided the wires for me to look at. Keith is very generous with his time. I’ve seen him many times help out with music at various church and neighborhood socials. I’m always impressed that someone like Keith can clearly hear when an instrument is even slightly out of tune and also recognize how to fix the problem. I certainly don’t have the ear for that kind of thing. Keith mentioned to me that he can hear the difference between the two wires he supplied to me. I realized quickly I had no hope of picking that up with my insensitive ears. I then realized that Keith was saying he could hear the difference even when the two wires are tuned. I wondered what it was that he was hearing. I turned to Wikipedia for some insight and stumbled across an entry on “inharmonicity” which is “the degree to which the frequencies of overtones (also known as partials or partial tones) depart from whole multiples of the fundamental frequency”. I realized that the sound waves are travelling through the wires at slightly different rates due to elastic anisotropy coupled with grain-to-grain differences in orientation. Thus, while the average pitch of the wire will be in tune there will actually be a spread about that pitch. I might be able to estimate that spread using the principles of elastic anisotropy.
For a single crystal the elastic behavior is anisotropic as is illustrated in a plot for the elastic modulus for an iron single crystal below (courtesy of Megan Frary at Boise State University).
The elastic properties of a single crystal can be expressed in terms of a tensor. This is handy, because rotating the property tensor to reflect the grain orientation with respect to a set of samples axes is fairly straight forward (C is a fourth order elasticity tensor and g is the orientation matrix):
The next step was to simply go to a reference volume and find the single crystal elastic constants for fcc and bcc iron, plug them into OIM Analysis and then “Bob’s your uncle”. However, I learned it wasn’t nearly as straightforward as I thought. Once again, some searching on the internet led me to some papers on first principle calculations of elastic constants and I quickly discovered that estimating elastic constants at room temperature is not as simple as I would have thought. I found several papers by Levente Vitos and co-workers. Professor Vitos was kind enough to teach me a little about this field and after some correspondence with him I decided to use the elastic constants for Fe-Mn found in Zhang, H., Punkkinen, M. P., Johansson, B., & Vitos, L. (2012). Elastic parameters of paramagnetic iron-based alloys from first-principles calculations. Physical Review B, 85: 054107-1. My thinking was that while the absolute values of the constants were probably not accurate, the ratios between the components would be constant enough to at least give me a rough idea of the distribution. (I tried some of the other constants in this paper and the results were all pretty similar.)
I then calculated the distribution of elastic moduli parallel to the longitudinal direction of the wire thinking this might give me an idea of the differences in the distribution of pitches one might hear when a piano wire is struck. The results are shown below – on the left for actual elastic moduli and then on the right for how this might translate to a distribution of pitches. However, this second plot is only a schematic to illustrate my thinking – I have no idea how the elastic moduli variations would translate to pitch variations, the horizontal scale could be much wider, i.e. the flats and sharps could be much farther away from the center pitch and it may also not be a linear relationship. Also the choice of C is completely arbitrary – the actual pitch will depend on the diameter and tension on the wire.
The bad wire (in terms of breakage), which according to Keith’s ear has a clearer sound than the wire less prone to breaking, has a narrower distribution of elastic moduli. Of course, I may be completely off-base as a “fuller” sound may correspond to the broader distribution as well. Perhaps what Keith can hear is the fine balance between clarity and fullness. So if my large set of assumptions is correct then, while I may not be able to hear the difference, I can at least see the difference in the texture data.