How long should this song last?
Enter the world of sheet music. A composition (the musical piece, which may or may not be a song) is divided into bars. The length of a bar is defined by the time signature.
The time signature states how the bar is divided into beats, and what length of note carries the beat.
Note lengths are always powers of 2. 4 means a quarter note, 2 a half note, 8 an eighth note (or a quaver if you're a snob), etcetera. A half note (2) is twice the length of a quarter note (4), which is itself twice the length of a quaver (8), and so on.
A time signature may look like this: 3/4. The 4 means that the quarter note carries the beat, and the 3 means that there are 3 of them in one bar. 3/2 means there are three half-notes in a beat, 7/8 means there are seven quavers, and so on.
Now, the actual speed at which a piece is to be performed depends on the tempo. That is usually expressed in beats per minute (bpm). The tempo also defines the note carrying the beat (usually the same as the one in the time signature but not always). So, you can have the time signature be 8=150, meaning there are 150 quavers in a minute (in the sheet music it would be notated ♪=150).
Both tempo and time signature can change throughout the composition.
Challenge
Use the following format for your input (or something very similar). It is a list of events:
[1,"4/4"],
[1,"4=120"],
[521,""]
This is the simplest form. It is a list of integer-string pairs (you're obviously free to go with string-string pairs if it makes your program simpler). The integer defines at which bar the event happens (starting from 1), and the string defines what happens there. If it is in the form of x/y, then it is a new time signature. If it is in the form of x=y, you have a new tempo. Lastly, an empty string designates the end of the score (exclusive, so the above example has 520 bars).
With changes, the format may look like this:
[1,"4/4"],
[1,"2=120"],
[46, "4=155"],
[67, "5/4"],
[68, "4/4"],
[152,""]
The output of the program should be the duration of the entire piece in "xm ys" (where x is the number of minutes and y is the number of seconds. You can leave out the "ys" part if there's no spare seconds, but it is not necessary).
This is code-golf, so shortest code wins!
Important note!
Real artists do not follow the tempo exactly. Only beginners use a metronome to match the exact number of seconds as notated; more experienced musicians know to dynamically speed up or slow down depending on the mood, their personal preference, etcetera. Therefore, it is perfectly acceptable for your answer to be up to 34% higher or lower than the "correct" answer. Also, the minimum length of a composition is 2 minutes and 30 seconds.
Test cases
1.
[1,"4/4"],
[1,"4=120"],
[521,""]
The time signature has 4 quarter notes a bar, and 520 bars, so 4*520=2080 quarter notes. There's 120 quarter notes per minute, so 2080/120=17.333 minutes, or 17m 20s.
2.
[1,"4/4"],
[1,"8=120"],
[46, "4=155"],
[66, "5/4"],
[76, "6/8"],
[152,""]
For the first 45 bars, there's 4 quarter notes a bar, so that's 45*4=180 quarter notes. Now the tempo is 120 8th notes per minute, which is 60 quarter notes per minute, meaning the first bit lasts 180/60=3 minutes.
Then a tempo change: from bar 46 to 67 there's 20 bars of 4/4 (thus 4*20=80 quarter notes), and at 155 quarter notes per minute you add 155/80=1.94 minutes = 1m 56s. Total is 4m 56s.
Then a time signature change: 10 bars of 5 quarter notes per bar = 50 quarter notes. 155/50 = 3.1 minutes = 3m 6s. Total is 8m 2s.
Then another time signature change. 6 eighth notes per bar for 76 bars is 76*6=456 eighth notes. Tempo is still 155 quarter notes per minute, which is 310 eighth notes per minute. 456/310=1.471 minutes = 1m 28s. Total comes down to 9m 39s.
3.
(The third movement of Shostakovich's second piano concerto, you get 5 bytes off if you listen to this while programming (not really but more people need to listen to Shosty dammit))
[1, "2/4"],
[1, "4=176"],
[75, "7/8"],
[102, "6/8"],
[103, "7/8"],
[106, "3/8"],
[107, "7/8"],
[109, "2/4"],
[112, "3/4"],
[113, "2/4"],
[116, "3/4"],
[117, "2/4"],
[120, "3/4"],
[121, "2/4"],
[124, "3/4"],
[125, "2/4"],
[155, "7/8"],
[160, "2/4"],
[175, "7/8"],
[180, "2/4"],
[181, "6/8"],
[182, "2/4"],
[186, "7/8"],
[188, "2/4"],
[222, "7/8"],
[225, "2/4"],
[286, "7/8"],
[308, "9/8"],
[309, "7/8"],
[314, "2/4"],
[317, "3/4"],
[318, "2/4"],
[321, "3/4"],
[322, "2/4"],
[325, "3/4"],
[326, "2/4"],
[329, "3/4"],
[330, "2/4"],
[356, ""]
Calculation is too long to show here, but it comes down to 4m 47s. The video is 5m 24s, proving my point that this is not an exact rule but rather a guideline.
Tags
code-golf number music
Sandbox
Does this look like an interesting puzzle? Any tags I miss? Is the +/-34% allowance large enough to matter, or should it be more?