Is there a way to determine whether a player is online or offline with command blocks?

I have some redstone circuitry in place (see attached image), involving a hopper clock, command blocks, and the scoreboard. The basic purpose is to play a sound with /playsound, and then once a certain time has elapsed (490 ticks of the hopper clock, or ~245 seconds), play the music again if the player is in range.

Scoreboard Objectives: timer (dummy)

Command Block #1 (closest to the hopper clock):

scoreboard players add @a[r=100] timer 1

Command Block #2 (adjacent to the first one):

scoreboard players set @a[score_timer_min=490] timer 0

Command Block #3 (use the process of elimination):

playsound records.mall @a[score_timer=0] ~ ~10 ~ 2.0

^{mall.ogg is not, in fact, my favorite choice. I selected it because the actual music being played is very nearly the exact length of what mall.ogg is replaced with in my resource pack.}

The system works perfectly, with one exception. When a player relogs, or logs out and then logs back in at a later time, the music (obviously) does not resume playing, and does not start again until the timer reaches 490 and resets to 0. The way to fix it (from what I see) would be to detect when a player is not logged in, and set their timer score to -1. When they logged back in, they would hear the music begin one second after.

Is there a way to detect a player's absence and then modify their score, even though they are not online?