Maths vs. Music - Tuning and harmony

[Alex K]

For the specialists:

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I thought with the

Sin(2pi * 440 * t * (1+E^-t))

I was implementing a sine wave that at each point in time had the frequency 440 * (1+E^-t) Hertz, but that's not true. The change in phase of the sine wave is dominated by the E^-t factor becoming smaller, and the result is a pitch that quickly becomes lower and then approaches 440 from below. This is easy to solve by adding a

Sin(2pi * 440 * t * (1+E^-IntegerPart[t] ))

because now within each second we have a constant note corresponding to the frequency 440 * (1+E^-IntegerPart[t] ) which does what we want, namely asymptotically approach 440 without ever reaching it, and the decrease in t now appears as a sudden phase change whenever IntegerPart jumps to a higher value, which corresponds to an audible clicking noise.

For my actual sample, i didn't use the exponentially decaying example because it so quickly approaches 440 that differneces soon becomes inaudible. Instead I used

Sin(2 pi * 440 * (1 + 1/IntegerPart[t+1])

which also starts with a'' 880 and approaches a' 440 asymptotically without ever reaching it, but not as quickly. Last but not least, I used sin^3 instead of sin in order to include some harmonics and make the sound more instrument-like:

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https://dl.dropboxusercontent.com/u/4685...ngnote.mp3

How far in do you think you still hear differences in pitch?

p.s. as proof that it keeps descending even if you don't hear the individual steps any more, you can click from say 30s to 50s while playing.