Hi Stephane, I agree that higher string tension intuitively implies more stiffness, hence more inharmonicity. That this is not the case is due to the following: 1. The intrinsic mechanical property of piano wire, or any other solid material, is more aptly related to its elastic modulus (E) and not to any forces applied to the material, or to its dimensions, or to any resultant stresses developing in the material under a load. E is said to be intensive and not exactly the same thing as stiffness; which is to say that a size 17 wire coiled up on your workbench is no less inherently “stiff” than one pulled to tension, or to a similar piece of steel existing somewhere in your car. E is simply a measure of the constituent material. 2. But stiffness is defined as the resistance of an elastic body to deformation <http://en.wikipedia.org/wiki/Deformation> by an applied force, and it is here that we infer that a higher tensioned string is stiffer than a slack one, and we would be correct using this reasoning. But for the purposes of inharmonicity, stiffness relates not only to the inherent mechanical properties (E) but also to the dimensions of the material and any forces applied to it. Stiffness is thus said to be extensive and thus relative to its size and application. 3. Inharmonicity (per Wikipedia) is the “degree to which the frequencies <http://en.wikipedia.org/wiki/Frequency> of overtones <http://en.wikipedia.org/wiki/Overtone> (known as partials, partial tones, or harmonics <http://en.wikipedia.org/wiki/Harmonic> ) depart from whole <http://en.wikipedia.org/wiki/Whole_number> multiples of the fundamental frequency.” Thus it is a measuring rod of the phenomenon caused by combined elements of inherent mechanical properties (E), diameter and length of the material, and forces applied to that material. So the question still begs: If stiffness is due to inherent mechanical properties, applied forces, and dimensions of the steel itself, why then does a higher tensioned string yield lower inharmonic values? Now applying a force to the material (within limits) does not change a material’s properties but does, however, affect the material’s behavior. So now a thought experiment. If you can imagine that all points of string segmentation (including the attach points, or termination points) exist as loosely tied square knots, then the actual and useful and “shorter” vibrating segments exist between the knots, since these rather large knots would interfere with pure vibrational segmentation from point to point. As the string is tightened the knots also tighten allowing “longer” segments to vibrate unimpeded. Tighten the string still further and further until theoretically the knots are so tight that they are practically nonexistent and now all segments, including the fundamental, are vibrating closest to their theoretical lengths. Since these knots can not disappear altogether there will always be some measure of inharmonicity. While this analogy isn’t perfect, keeping the visual in mind helps to imagine that fatter wire will also have fatter knots at any tension, and that thin wire will always have smaller knots at any tension. Still, the properties of the material itself (E) primarily determine inherent stiffness even if higher tensions tend to smooth out and lower inharmonicity by allowing the many segments to vibrate closer to their theoretical frequencies. Hope this helps. Best Regards, Nick Gravagne, RPT Piano Technicians Guild Member Society Manufacturing Engineers Voice Mail 928-476-4143 _____ From: pianotech-bounces at ptg.org [mailto:pianotech-bounces at ptg.org] On Behalf Of Stéphane Collin Sent: Saturday, February 07, 2009 8:18 AM To: pianotech at ptg.org Subject: Re: [pianotech] inharmonicity in piano wire (SNIP) Anyway, for a same speaking length, inharmonicity decreases as tension increases (hear the bell like sound, very inharmonic, when you release tension in a string). How this relates to that, I can’t imagine clearly. I would at first think that higher tension in the string would increase even more the difficulty to bend the string at the attach points. Anyone on this ? (SNIP) Stéphane Collin. -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://ptg.org/pipermail/pianotech_ptg.org/attachments/20090211/5445a2b8/attachment.html>
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