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The vibrating string is tightened using tuning keys of the guitar. Assuming the amount of tightening is constant, what will change in string (mass? tension? linear density?) when a temperature change causes expansion in the string? If an increase of 10K in temperature, will the change in fundamental frequency be significant to hear a difference in pitch?

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The change in tension, as pointed out above by alephzero, is in fact not negligible, and even ~5 degrees F change in ambient will make a metal-stringed instrument with a number of strings audibly out of tune- not only from the standpoint of absolute pitch for a single string but also for the pitch intervals between different strings.

Hilmar's point is also true which contributed to the market failure of at least two electric guitar products in the 1970's which used aluminum necks and wooden fretboards. In these instruments a change in ambient temperature would cause the exposed back side of the neck to cool off or heat up before the front side (buried under an insulating layer of wood) could follow suit, causing the neck to temporarily bow out or cup in and thereby changing the tension on the strings, throwing the guitar out of tune. So you re-tune and after a few minutes the front and back of the neck would equilibrate and the cup or bow would go away, throwing the guitar out of tune again, etc., etc.

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There is conservation of mass, so the mass doesn't change. Technically speaking the string will get a little "softer" when it gets hotter so the tension will decrease. However for most string materials, the effect of 10k change affect is too small to change the pitch perceptually.

Most stringed instruments, especially wooden ones, do change their tuning with a temperature change. This is mainly due to geometry change of the body and/or neck, not the string itself. An instrument with very rigid body such as a carbon fiber guitar, will stay in tune over a wide range of temperatures and humidity.

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    $\begingroup$ I would dispute the change in tension is negligible. Metal strings have a thermal expansion coefficient of the order of $10^{-5}/\text{K}$, so an $10\text{K}$ change in temperature gives a thermal strain of order $10^{-4}$. Compare that with the strain at the elastic limit, of the order of $2\times 10^{-3}$, and the relative change in strain, and therefore in tension, is about $5\%$ which would cause a pitch shift of the order of a semitone - unless the thermal expansion of the body of the instrument (which is usually a different material) happens to cancel out that effect exactly. $\endgroup$
    – alephzero
    Nov 13, 2018 at 15:29
  • $\begingroup$ I agree with alephzero - at the very least, the claims in this answer are not obvious, and they should be backed up with relevant calculations and/or references (or amended, as appropriate). $\endgroup$ Nov 13, 2018 at 15:36

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