So maybe you're a little hazy on what is meant by phase, even more so regarding phase advancement, and not at all convinced I know what I'm talking about in suggesting that repeatedly multiplying the phase of a lower frequency by a positive integer can be used to generate a higher frequency. Like, how does that work?
Phase relates back to the sine wave, which itself relates back to the unit circle, but this is beginning to feel like a circular definition. What does it really mean?
Let's approach this from a different direction, using an analogy. Say you have a shaft, rotating at one degree per second. It's going to take that shaft 360 seconds to complete one rotation. Now say you have another shaft, the position of which is updated once per second according to the rule that its new position should be twice that of the first shaft. If the first shaft has moved 10 degrees, the second shaft will have moved 20 degrees. If the first shaft has moved 50 degrees, the second shaft will have moved 100.
But what happens when the first shaft has moved 180 degrees and the second shaft has moved 360 degrees? The second, faster shaft is already back where it started while the first shaft is still only halfway around. Fine, no problem, it's free to keep right on moving, starting a second rotation while the first shaft finishes its first, but because doubling the number of degrees the first shaft has turned will now result in a number larger than 360, we'll need to remove the first 360 degrees to bring the result into a range we can work with. So, essentially, when it gets to 360 degrees the second shaft resets to 0 degrees and keeps on moving.
Likewise, when the first shaft gets to 360 degrees, it also resets to zero and keeps moving.
But what if for every degree the first shaft moves the second shaft moves 5 degrees. The same principle applies, but because we're getting the position of the second shaft by multiplying the position of the first shaft by 5, it won't be enough to subtract 360 degrees after its first rotation, we'll need something that will work no matter how many rotations it has already completed. That something is modulo division.
In this example, after multiplying the position of the first shaft by 5 we'll take the result of that and apply modulo 360, to remove all of the full turns and leave only the amount by which the second shaft's new position exceeds a full turn. We could use the same approach for the first shaft, but in that case it's simpler to just subtract 360 degrees every time it completes a full rotation.
You may recall, in a previous installment I said that if you measure phase (rotation) in cycles, modulo division isn't necessary. This is because if we were to use modulo division in that case, it would be modulo 1.0, which is exactly equivalent to simply keeping the fractional portion of a decimal number and discarding everything to the left of the decimal point.
So, to ease back into more standard terminology, phase equates to how much the rotation of a shaft, at any given point in time, exceeds an indeterminate number of complete rotations. How far beyond the start/end point of a cycle it has progressed, and phase advancement equates to how much rotation occurs between one point in time and the next, one second and the next in the above example. It is a rate of change.
Note that in the above example we only applied phase advancement to the first shaft, to determine its phase at the next point in time, and used that to calculate the phase at the same point in time for the second shaft. The rate of phase advancement for the second shaft is only implied, never explicit.
Using this approach we might add a third shaft, applying the same multiplier to the phase of the first shaft as we did for the second shaft, and be confident that the second and third shafts would always be perfectly synchronized, rotating in lockstep.
A cycle is a cycle, whether it's a sine wave or a rotating shaft or the interplay of the tilt of Earth's rotational axis with its movement around the sun, creating seasons.
Phase is what portion of the next full cycle has been completed, and phase advancement is the rate of change of the phase, change/time. For a shaft, phase advancement is how fast it is turning. For a sound, phase advancement is its frequency, its pitch. For Earth's seasons, phase advancement is how quickly one passes into the next.
If you were confused before, I hope that you are now at least less confused.
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