Distance from E to F is a half step because no other notes fall between them. Why are we using the G note as a perfect fifth in C Major chord? Shouldn't be G sharp?
I need to confess your question was not easy to me to understand. We normally use these concepts on western music without thinking, and you really made me think. Thank you! :)
I think the other way to ask your question is: If I hear a major third as a "perfect" interval, why a perfect fifth is made of a major third plus a minor third instead of two major thirds?
Well, let's go:
The concept of "perfect" is related to historical concepts, to the ratio between frequencies of two notes and to the way our brains hear these frequencies on an interval.
At the time the concept of "perfect" was forged, Western music had pratically only Pythagorean temperament to tune their instruments, and the consequence of the use of this way of tune was only 4 intervals were regarded as consonances: the perfect unison, perfect fourth (diatesseron), perfect fifth (diapente) and perfect octave (diapason). At that time and using that temperament the major and minor thirds were not as usable as they are on modern temperaments. I mean, subjectively, they were not as pleasant as they are today.
What made these intervals perfect is the way our brain hears these intervals. Since the ratios of frequencies on these intervals are respectively 1:1, 3:2, 4:3 and 2:1, our brains hear these intervals in a very pleasant way, as if the frequencies on these intervals "fit" one into another perfectly. Acoustlically speaking, these intervals had no beats or hammerings.
It's difficult today to really understand why these intervals are still called perfects on an instrument tuned on equal temperament, because since fourth and fifth doesn't hold that perfect ratio on these instuments anymore, our brains don't listen them as perfect also. All intervals on modern piano (except octaves) have beats. The perfect fourth and the perfect fifth hold their names for historical reasons. If you have the chance to hear a violin/viola/cello tuning their strings on perfect fifths (they can do that), you will understand why they are called perfects.
BONUS: if you want to understand more these concepts, you could study acoustic interference and musical temperaments. I think this will untie your questions a little bit and will make you more curious about the subject. :)
The perfect fifth's size is 7 semitones. So, let's count them from C:
If we count 7 semitones, we end up on G natural, and not G#.
Also, if we had a chord that consisted of C,E and G#, it wouldn't be C major; it would be C augmented. A major chord is built with a root, a major third and a perfect fifth. The aforementioned example is built with a root, a major third and a augmented fifth, which results in a different chord.
Counting up from a note, alphabetically, using the first as number one. C-D-E-F-G. So G is number 5. Start on C# and do the same. G# is number 5. A perfect fifth is a bit of a misnomer, as is perfect fourth. I'm sure a more apposite word should have been used, but we're stuck with it now! The space between 1 and P5 is always 7 semitones. It just is. When this space is increased by one more semitone - either way - it is known as an augmented 5th. If decreased by a semitone, it becomes a diminished 5th.
You need to know a little bit more about perfect fifths than just how many semitones they are away from the root.
Remember F double sharp and A double flat are also seven semitones away from C but are in no way perfect fifths.
Now the word perfect tries to say two things to you. It tries to firstly indicate that these intervals have a very pure and easy listening quality to them. They generally just sound good.
Secondly it tells you that a perfect interval is a note that forms both part of the Major and minor scale of the root note. This also the case in with the interval of a second but because of this intervals dissonant quality it is not called perfect.
In your example both c Major and c minor the relative key of Eb Major has a G in the scale. Hence the title perfect. This phenomenon is present at the unison, the fourth, the fifth and the octave and also the second but because of its awkward sounding nature is not included in the list.