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When you speak of 1/infinity what you are really describing is a simplification of a mathematical concept called a limit. What you are insinuating by the term, since you cannot actually use a number called infinity, is that as the limit of the denominator (in this case you are calling it infinity) approaches the theoretical value of infinity, that is, as the expression becomes so large in the denominator compared to the numerator(the 1,) that the overall expression approaches zero.

In fact in mathematical terms, you use a limit to define infinity. That is, 1/(limit of x->zero), that is 1 divided by a variable x that gets smaller and smaller (approaching zero) makes the overall fraction become larger and larger, or approach infinity. Because you cannot actually divide anything by zero (in mathematical terms it is undefined) you can only approximate the value using limits. In fact, this represents something called a singularity, which is the mathematical way of representing infinity (not the symbol, but the actual mathematics.)

When these occur in mathematics, they can represent such real things in nature as ,say, a black hole in outer space. Scientists are reasonably sure they exist and photos of certain regions in space seem to confirm their existence. Scientists predicted them many years ago, mathematically expressed as a singularity where gravity accumulates so much mass that the mass approaches infinity sucking in everything in its proximity including light itself resulting in an absence of light (or black hole).

In terms of music, as Mike and others mentioned, between any two notes there are an infinity of tones. Indian music uses elements of the infinity between notes frequently. It is in the nature of instruments like the sitar that has fretted notes but is played by not only striking the fretted notes, but by pressing down on the string to cause the notes to vary sometimes as much as the entire range between two frets or even beyond. Since it is producing an analog sound it essentially is playing everything in between. Blues and jazz frequently use a "blue note," that is the sound of a tone as it is "bent" to go flat or sharp from a fixed note. Slide guitars and other instruments that use slides often take advantage of those in between tones, so it is way more common than you might suspect even in pop music.