Question: This is a Merriam Webster def of the word **cycle**:

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> 1: an **interval** of time during which a sequence of a recurring succession of events or phenomena is completed
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This is a Merriam Webster def of the word **interval**

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> 1a: a space of time between events or states
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At first, I thought that "interval of time" simply means "a period of time" but after referring to a dictionary I realized that there isn't a definition like that for this word. As it relates to the first definition of the word "cycle", does this mean that the meaning of that definition is "the period between each cycle that takes place"?
It's weird though because things that are happening during the duration of the cycle are also the cycle, but the definition seems to me like it only talks about the time between. Or maybe **interval** simply **means period** and I missed something.

Answer: 
So, there are three key words at play in the Merriam Webster definitions: *cycle*, *interval*, and *space*. And from my point of view, the definitions are fairly clear. However, in your commentary, you introduce a fourth word: *period*. That doesn't necessarily confuse things, but it might do. The reason is that the word "period" has a technical definition in Physics. So, avoiding that word for now, I interpret the essence of the MW definitions and words as follows:

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> A **space** of time is the simplest. It's just some time; a chunk of time, if you like; a blob; a clump; and so on. The only additional aspect of the concept of "space" is that we are talking about *contiguous* time; i.e. a *single* chunk/blob/clump.
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> An **interval** of time is just a **space** with some kind of significance associated with the start and end points. Obviously all (finite) spaces of time have a start and end point, but **interval** is used when those points are of some relevance in the context.
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And:

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> A **cycle** is an **interval** that is subject to being *repeated* in a way that is in some way, more or less, predictable, analyzable, significant, and so on.
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So, a **cycle** is a special form of **interval**, which in turn is a special form of **space** (of time).

Finally, back to **period**. In Physics, the **period** of a wave (more generally, of harmonic motion) is the reciprocal (i.e. 1 over) the **frequency**. In that sense, you are correct that "interval" is synonymous with "period". And in everyday use, they are probably safely interchangeable.
I've already given one answer, but I wanted to give another based on one thing you say which you suggest may be at the heart of what's confusing you. First, let me set up an example which may help clarify:

Consider someone working out at a gym. Their workout is based on what in gym terminology we might call a "circuit". It might look like this:

1. one set of five bench presses
2. rest for 30 seconds
3. one set of five squats
4. rest for 30 seconds
5. one set of five deadlifts

And then their gym session as a whole might consist of three of those circuits, with 90 seconds rest between each.

Now referring back to the Merriam Webster definitions, it might be tempting to say that "cycle" is simply a synonym for "circuit" in the above. But in fact the MW definition is such that "cycle" refers very specifically and only to the time itself, and not to the various exercises.

So the thing you say that may be relevant is:

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> It's weird though because things that are happening during the duration of the cycle are also the cycle, but the definition seems to me like it only talks about the time between.
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But that's not exactly correct. It is *not* the case that *"things that are happening during the duration of the cycle* [i.e. the five items in the circuit] *are also the cycle"* Applied to the gym example, the word "cycle" means nothing more than the blob/lump/space of time from the start of the first bench press in a circuit, to the last deadlift in that same circuit.

Also, it's not clear, in the above quote from your question, what you mean by *"the time between"* Do you mean the time between exercise *within* a circuit -- i.e. 30 seconds? Or do you mean the time between circuits -- i.e. 90 seconds? (Or maybe even the time between each benchpress, each squat, etc -- i.e. not specified, but usually very little if any time!)

In summary, using my example:

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> **Interval** could correctly be used to refer to any of the times I mention: the 30 seconds between exercises, or the 90 seconds between circuits. The key is that both have meaningful **start and end points**
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> **Cycle** could correctly be used to refer to any one sequence of the five items in my list. The key is that the entire sequence of five is subject to **repetition**.
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