"the Craft Of Musical Composition" Root Movement Confusion

Hello

Just in case of the slim chance that someone who visits here owns Paul Hindemith's book "The Craft of musical composition, Book 1: Theory" I have a question to ask about "degree-progression" that is confusing the hell out of me and if someone here understands it better and could explain it to me it would be greatly appreciated.

Hindemith lists the harmonic value of a chord movement first by a fifth: "A progression based on the interval of a fifth between it's roots naturally has a surer foundation than one based on a minor sixth; this is the strongest of all chord progressions" and shows an example of a CMaj chord moving to a GMaj chord.

The next strongest chord movement is by a 4th: "the next best chord progression after that based on a fifth is that based on a fourth" and shows an example of a CMaj chord moving to a FMaj chord. So obviously this book differentiates the different chord movements of a fourth and a fifth by stating they have different strength.

My confusion is how do you know when the chord progression is a fifth or a fourth? In the example shown a Cmaj chord moving to a Gmaj chord is considered a "fifth". Yet another example in the book shows a G chord moving to a C chord and calls this the movement of a fifth, but the movement "G-C" is exactly the same as "C-F" which was previously said to be a fourth. Hindemith sometimes calls a progression a fifth and then the exact same movement at other points in the book a fourth and vice versa, yet it's not like these are interchangeable as he has already established the difference in strength between a a root movement of a fourth and one of a fifth. I've read it all as carefully as I can but I can't figure out when a chord movement is considered a fifth up or a fourth down/ fourth up or a fifth down.

If any understands this could you please give me some pointers?

Whether it's a fifth or a fourth is dependent on the lowest notes movement to the second. C to G is a fifth because G is 7 half steps AWAY from C. If you reverse it and G is the first note, that is called an inversion. A flipped version of the original. G to C becomes a fourth because the distance between the two notes is 5 half steps, a shorter distance. When you invert an interval, the note stay the same the distance changes

Here is a good inversion chart:

Diminished unison - augmented octave

Perfect Unison - perfect octave

Augmented unison - diminished octave

Diminished second - augmented seventh

Minor second - major seventh

Major second - minor seventh

Augmented second - diminished seventh

Diminshed third - augmented sixth

Minor third - major sixth

Major third - minor sixth

Augmented third - diminished third

Diminished fourth - augmented fifth

Perfect fourth - perfect fifth

Augmented forth (tritone) - diminished fifth (tritone)

From there it repeats. Hope this helps

in context of a degree-progression, from what i know, all the intervals should be inverted/transposed to see whoever has the most valuable intervals in the whole progression. there are many things who contribute to making a central tone. not only if it has a fifth and a fourth, but in what order (the best case is where the tone is preceded by a fifth), who has the rhythmic accent (the first bar), which tone is the lowest in pitch in reality. after the central tone is established (which is hard sometimes) then all the chords reffer to this parent.

this means that if there's a F chord going to the C chord, and C is the central tone, then the F is the forth, even if in reality it comes before the C chord, or if it's in a lower octave (lower F to higher C is has the interval of a fifth, but it doesn't matter), it's still considered 4-1 from a degree-progression's point of view. 

if F were the central tone, then it would be a 1 going to a fifth (1-5). the same notes on the staff.

hope i got it right.