Lawrence,
And your reply goes to the heart of my complaint about the personality type currently dictating the study of physics, where very fine details are obsessed over, but whatever context might be half an inch on either side is completely missed.
In his winning entry in the nature of time contest, Julian Barbour made the argument that the only measure of time "worthy of the name" was using the principle of least action between different configuration states of the universe. Aside all the practical and conceptual complications, it begs the question of what would we call change that isn't regular? It still effects the consequences of change and thus time.
Frequencies vary. So does time. It's called time dilation. That's why when we put an atomic clock on an airplane, it speeds up the frequency. When scientists build an accurate clock, it isn't the length of the measure that matters, but its regularity.
This regularity provides an accurate standard against which to measure actions that are not so regular, because if time was only the most regular standard, with no irregularities to mark it, there would be no "direction" of time. It would only be that metronome.
Think through this statement:
"If you have some repeated set of events, say a metronome or some cyclic process, frequency is measured as the number of those events per time they occurred."
What is "per time?" As Galileo observed, when feeling his own pulse, while watching a pendulum; With time, we are only comparing one action to another. Do I assume you are simply assuming a Newtonian absolute time, when you say "per time?"
With your musical example, your measure is the whole note. Is that some form of absolute standard, akin to Barbour's least action between different configurations of the universe?
My point in saying frequency is time, is that time is a sequential series of events, against which we measure the larger, non-linear dynamic process of change. Frequency is also a series of events, being produced by a distinct process.
"The unit of frequency is Hertz or equivalently inverse seconds ( sec^{-1} ) and so forth."
To rephrase the question; What is a second? Isn't it based on some regular action?
