Riptide wrote:mjcox2000 wrote:How does the intonation of each of the Hornbostel-Sachs instrument types change as temperature changes? Why do they react this way?This is a pretty interesting question. It was hard to find any conclusive answers online so most of this is just my own analysis and so some of it is probably wrong or not the correct reasoning. [b]Aerophone:[/b] Pretty straightforward. Aerophones function by vibrating the air - the speed of sound increases roughly linearly with temperature increase in Celsius (0.6 m/s for every 1 degree increase). Since the wavelength of the sound stays constant, the frequency must increase along with the increase in speed (and consequently decrease if the speed decreases). Another way this increase can be demonstrated is by the standard frequency formula [math]f=nv/2L[/math] (or 4L for closed). Since the velocity increases as temperature increases, the frequency also increases. Something to note is that as temperature increases, the instruments themselves would also expand slightly thus changing the length of the body. This would thus change the frequency as well since length is in the denominator. I'm pretty sure that the increase in velocity would be significantly larger than the increase in length (for example a steel flute that is half a kilogram will increase only .000018 meters at a 5 degree increase compared to a 3 m/s increase in velocity) however and thus the frequency would still increase as temperature increases. [b]Idiophone:[/b] Idiophones have a pretty interesting formula for calculating frequency. This equation has velocity in the numerator (like aerophones) and thus change linearly with temperature change. [b]Chordophone:[/b] This one has me stuck a bit. For calculating frequency of strings, one is supposed to use the velocity of the string, which is given by [math]f=\sqrt \frac Tu/2L[/math]. An increase in temperature would increase the length of the string, and thus the frequency would be expected to [i]decrease[/i] since the length in the denominator is not under a square root. I'm not sure if this explanation is completely valid for Chordophones but some places online state that string instruments will become lower in pitch at higher temperatures. [b]Membranophones:[/b] Membraphones are very complex in their frequency calculations as many more variables are taken into account. Humidity can have a large impact on these instruments, but holding all other things constant, it appears that frequency increases as temperature increases (and vice versa). One explanation for this is that a Timpani functions by vibrating the air inside the timpani. As already mentioned, the velocity of sound increases as temperature increases and thus the frequency would increase.terence.tan wrote:The volume of an auditorium is 12000m^3. Its reverberation time is 1.5 seconds. If the average absorption coefficient of interior surfaces is .4 Sabine/m^2. Find the area of interior surfaces.[math]t_r=.16V/sA[/math] [math]t_r= 1.5[/math] [math]V=12000[/math] [math]s=.4[/math] Solving for [math]A[/math] results in [math]A=3200 m^2[/math].
In some places, it seems like you got hung up on small length variations instead of other, more fundamental underlying reasons for the pitch variations. [list] [*]Aerophone is right -- they go sharp as they get warmer. [*]For idiophones, they actually go flat as they warm up. This is because materials are less stiff (lower Young's modulus) with temperature increases. The stiffer an idiophone is, the higher the resonant frequency(ies) -- for example, a marimba bar's fundamental frequency varies as the square root of Young's modulus. This effect is appreciable -- my experimental tests indicate that aluminum bars go flat at a rate of half a cent per additional degree Celsius, and I confirmed with theoretical calculations involving the variation of Young's modulus with temperature to get a similar result. (Unfortunately for mallet instrument makers, resonators on mallet instruments behave like aerophones and go sharp as temperature increases, so as the bars go flat, the resonators move in the other direction and go sharp.) [*]For chordophones, your result is right, but I think it's for the wrong reason. It's true that the string expands a small amount as it warms up, but the reason they go flat isn't the increase in length. Rather, the reason is that the metal strings expand more than the wood body (steel's coefficient of thermal expansion is about .000012/degree C, while wood's is about .000003/degree C parallel to the grain), which means that tension decreases as the strings expand more than the body. [*]For a typical drum, it's very hard to predict the effect temperature will have on tuning. In a typical drum, the head, lugs, rim, and shell all affect pitch and will likely all have different coefficients of thermal expansion. Without knowing the particular materials, it's very difficult to know what will happen. (Timpani's resonant frequencies are established by the head tension -- not the enclosed air -- so while I see the motivation behind your hypothesis that frequency will increase with temperature, I don't think it's necessarily right.) [*]It's understandable that you skipped electrophones -- at first sight, they might seem perfectly frequency-stable. However, they'll almost invaruably contain some crystal oscillator to establish a clock frequency, which won't hold frequency perfectly as temperature varies. The relationship is highly variable and can go in both directions -- see sample graphs on pages 9 and 10 of http://leapsecond.com/hpan/an200-2.pdf -- but barring a temperature-compensated oscillator, the frequency certainly can vary as much as a few hundred parts per million. (For reference, a cent is 578ppm.)[/list]
