Within an octave there are four perfect intervals: perfect unison (P1), perfect fourth (P4), perfect fifth (P5), and perfect octave (P8).
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Five half-steps make a perfect fourth.
The distance between one note and another is called an interval. There are various types of interval, including major intervals, minor intervals, perfect intervals, diminished intervals and augmented intervals.
A tetrachord is a series of 3 intervals which collectively span the interval of a perfect fourth.
It's a perfect interval that is called a diminished interval when reduced by half step; there is no such thing as a perfect note. Minor intervals are also called diminished intervals when reduced by half step. If you listen to a perfect fourth and a diminished fourth, for example, they clearly have very different sounds, so they need different names.
A perfect 5th is an interval that covers the range of 7 half steps. The name Perfect is used to distinguish it from other intervals, like 3rds and 7ths, that have major and minor versions.