Play "Taps"​​​​

JavaScript, 203 198 196 195 bytes

with(new AudioContext)for(t=i=0;n="301093301396202346202346202396331699464390301093"[i++];)with(createOscillator())i%2?l=n/2:(frequency.value=392+n*44,connect(destination),start(t+.1),stop(t+=l))

5 bytes saved thanks to Dendrobium and 1 thanks to @PatrickRoberts.

Explanation

with(new AudioContext)        // use HTML5 audio
  for(                        // iterate through the note pitches and lengths
    t=i=0;                    // t = current time to place the note
    n=                        // n = note pitch (or temporarily length)

    // This string contains the values of each note alternating between length and pitch
    //     (l1, p1, l2, p2, etc...)
    // Length value is in 16th notes (1 = 1/16th note, 2 = 1/8th, etc...)
    //     The longer notes are limited to 9 but I think it still sounds OK
    // Pitch value 0 = G4, 3 = C5, 6 = E5, 9 = G5 (multiples of 3 saves 1 byte)
    "301093301396202346202346202396331699464390301093"

  [i++];)
    with(createOscillator())  // create the note oscillator
      i%2?                    // alternate between length and pitch characters
        l=n/2                 // l = length (stored for use in the next iteration)
                              // dividing it by 2 sets the speed to 60 beats per minute
                              //     and casts it to a number
      :(
        frequency.value=392   // base note = G5 (392hz)
          +n*44,              // there is (conveniently) roughly 132hz between each note
        connect(destination), // send the note's sound through the speakers
        start(t               // schedule the note to sound
          +.1),               // short delay to distinguish two notes of the same pitch
        stop(t+=l)            // schedule the end of the note and increment the time
      )

Test it here in the browser! Works on any browser that supports the HTML5 Web Audio API.

Mathematica, 361 287 285 bytes

I went for accuracy here. Output is exactly as described in the score, played with the trumpet. You can find the file here.

"G"
e="E5";c="C5";EmitSound@Sound[SoundNote[#,5/#2,"Trumpet",SoundVolume->#3/17]&@@@{%,8,17,%,24,20,c,2,23,%,8,26,c,24,29,e,2,32,%,12,35,c,12,38,e,6,41,%,12,44,c,12,47,e,6,50,%,12,53,c,12,56,e,2,59,c,8,62,e,24,65,"G5",3,68,e,6,170/3,c,6,136/3,%,2,34,%,8,34,%,24,34,c,2,34}~Partition~3]

Thanks to @MartinBüttner for golfing suggestions.

Sonic Pi, 899 bytes

The timing is a little off, but I think it's ok.

Lightly golfed:

use_synth:blade
use_synth_defaults sustain:0.70,release:0.0
play:G4,release:0.05
wait 0.75
play:G4,sustain:0.25
wait 0.25
hold=rrand_i(3,4)
play:C5,sustain:hold,release:0.5
wait hold+0.5
play:G4,release:0.05
wait 0.75
play:C5,sustain:0.25
sleep 0.25
hold=rrand_i(3,4)
play:E5,sustain:hold,release:1.25
sleep hold+1.25
play:G4
sleep 0.70
play:C5
sleep 0.70
2.times do
  play:E5,sustain:1,release:0.25
  sleep 1.25
  play:G4
  sleep 0.7
  play:C5
  sleep 0.7
end
hold=rrand_i(3,5)
play:E5,sustain:hold,release:0.75
sleep hold+1
play:C5,release:0.05
sleep 0.75
play:E5,sustain:0.25
sleep 0.25
play:G5,sustain:2.45,release:0.05
sleep 2.5
play:E5,sustain:1,release:0.25
sleep 1.25
play:C5,sustain:1,release:0.25
sleep 1.25
hold=rrand_i(3,5)
play:G4,sustain:hold,release:0.5
sleep hold+0.5
play:G4,release:0.05
sleep 0.75
play:G4,sustain:0.25
sleep 0.25
hold=rrand_i(3,5)
play:C5,sustain:hold,release:1.5