This song by local musical treasure John Osebold—one of the main musical pistons in the machine that is "Awesome"—has been doing loops in my head for days:

John Osebold - "Apples and Rain"

Osebold's songs always seem to have a secret geometry, and this song repeats itself like an Escher staircase. So I asked John to explain its math:

ask a nerd and ye shall receive more than ye need:

GENERAL
the repeating 4-position progression (summarized by the guitar in the first 20 seconds or so) is an infinite staircase involving a bassline and a pairing of 2 notes.

BASSLINE
the song is in the key of D. the bassline simply ascends with each new position: open D, up to F, up to G, and ending at the top with A. A is the fifth below D, which melodically suggests returning to the key as the next logical move...like an escher staircase - at the top, you return back to the bottom stair to begin climbing again.

2-NOTE PAIRINGS
there are 4 positions in the song, with 2 chords in each position (8 chords total). the 2-note pairings have their own logic when considered in 4-chord sequences. in the first half (the first 4 chords), the only difference between the chords is a half-step (1 fret). chords 1 and 3 are the same fingerings, but chords 2 and 4 differ by dropping one note by a half-step: chord 2 drops the bottom note, chord 4 drops the top note. the fingerings create a mirror image on the fretboard:

chord 1 chord 2
X X
X X

chord 3 chord 4
X X
X X

in the second half (the other 4 chords), we get into whole-steps (2 frets). once again, chords 5 and 7 are the same fingerings, but chords 6 and 8 differ by bringing up the bottom note by a whole step. the anomaly (like the unique 2-line coupling at the end of an olde english poem) is chord 8, which breaks the pattern that the second half of the progression - chords 5-8 - creates:

chord 5 chord 6
X X
X X

chord 7 chord 8
X X
X X

chords 6 and 8 share the same relationship that chords 3 and 4 share - the dropping of the bottom note by a half-step. melodically, this anomaly (or misstep) drops us back down to chord 1 to begin the sequence over again.

TA-DAH!

And there you have it.