Staff lines, letter names, and intervals emphasize pitch proximity: diatonic steps and semitone alternations; but fifths occupy an equally important position in the fundamental organization of pitch.
Letter names & intervals
Fifths untangle the messy order of intervals and scales.
Perfect fifths (or their inverse, perfect fourths) generate all letter names; other intervals can’t do that.
… Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# F## …
It is no surprise that intervals can be also ordered by perfect fifths. Doing so reveal the spectrum of qualities. (Since letter names and intervals can be generated using fifths, we can alternatively express intervals as combination of fifths and octave displacements.)
Scales & key signatures
Seven fifths make a diatonic scale. The tone/semitone pattern of TTSTTTS can be more efficiently described as seven fifths, which key signatures imply.
The circle of fifths show how each shift swaps one note for another, and crucially, why the tonic moves by a fifth. Because the ordering by fifths traces back to scales and letter names themselves.
Intervals & chords in diatonic scales
Perfect and diminished fifths also reveal the hidden structure behind interval and chord qualities in diatonic scales.
In a diatonic scale, all fifths are perfect except for B to F, a diminished fifth. From the perspective of fifths, this little asymmetry creates the rich variety of intervals and chords in diatonic scales. When intervals in diatonic scales are arranged in generic fifths, the quality changes whenever B transposes to F (arrows below).
Chords are different intervals combined. Chords (or any set of notes) in a diatonic scale have as many qualities as there are notes, since each of those B’s have to loop back to F’s at different points. That’s why there’s three types of triads and four types of seventh chords in diatonic scales. Of course, adding accidentals like raised leading tones in minor can further alter these chords.
TL;DR: Symbolically, abstract pitch concepts emphasizes pitch proximity (steps and semitones); but they are underpinned by a deeper organization based on fifths.
A Pythagorean Postlude
The fifth-based approach above is loosely Pythagorean. Pythagorean tuning uses pure octaves (2:1 frequency, 1:2 string length) and pure fifths (3:2 frequency, 2:3 string length) to generate all notes. I say ‘loosely’ because the pitches above can be mapped to any kind of tuning, and Pythagoreans did other things like worshipping numbers and abstaining from beans. Other tunings imply different kinds of pitch structure. For example, in equal temperament (12TET), F# = Gb, which means that music notation contains redundancies in 12TET. In Pythagorean tuning, the difference between F# and Gb is real (a Pythagorean comma). It doesn’t mean that staff notation and intervals must be tuned in a Pythagorean way, but it best correlates with the methods and structure of Pythagorean tuning.