Intervals refer to the distance between 2 different notes. Generic intervals are a simple way of referring to these distances between notes and are the foundation upon which more complex terminology is based.
The interval between 2 notes that are on the same level is called a first, or prime.
The above image shows 3 different examples (shown in different colours) of intervals that are firsts. The thing to note here is that these all count as firsts, irrespective of the accidentals.
If 2 notes are separated by one level on the staff, the interval is called a second. A third refers to the interval between 2 notes that are separated by 2 levels, and the rest of the generic intervals follow on in a similar fashion. These are shown in the illustration below:
As I mentioned before keep in mind that accidentals do not matter for generic intervals.
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