Continuously changing pitch of stable-pitch audio sample

You're trying to do the time warp dance. If you were copying the samples point by point, your playback time (destination) and your look up time (source) would be the same. If you wanted to speed it up by a factor of two(one octave), your lookup time (source location in the array) would be moving twice as fast as your sample time (location in your output buffer). For variable speed time, you have to figure out where in the lookup table to grab the next value (function range). Where you are going to write it to is the next spot in the buffer, as in uniform time (function domain). When you calculate the location to get your signal value, it may not land on an integer. You could get some rough sound if you just use the nearest one.

So, you got two choices for better quality.

1. Increase the density of your lookup table

2. Use some method (called interpolation) to figure out what the value should be

A. Linear interpolation means draw a line between the two nearest samples and use that height whereever your fraction takes you

B. Polynomial interpolation means use the nearby samples to draw a nice smooth curve, and use that height

C. Sinc interpolation means use the sinc function as a weighted average of many nearby points to get the value

Your musician friends will tell you it doesn't sound right. Instruments are complicated and their timbre varies by frequency and the middle name of the nearest bartender.

That's really the precursor to DSP. Trying to fix the sound so it sounds right is a challenge you can have hours of fun pursuing, as others are. There is lots of material if you look for it. I am not really an expert on that at all.

In a synthesizer wavetable the source location just goes round and round and round... how much faster each step is related to the frequency.

Keep a steady tone by not changing the step size.

Linear growth in time. Add the same amount to each step.

Exponential growth in time. Add a multiple of the previous step size.
For instance,

ThisStep = 1.00001 * PreviousStep

SourceLocation += ThisStep

The step size grows exponentially and so does its accumulation.

Speeding up by a factor of $2^{1/12}$ is one semitone.

There's math problem and solution in there somewhere, I'm sure you'll spot it and solve it to get this done. Or maybe two.