How many seconds will each note take in this 6/8 time signature, if tempo is 120 bpm?
My analysis :
Rest in the beginning will take three 1/8 beats - 0.75 seconds in total.
The three A note will take 3/2*1/4 = 3/8 seconds each.
Is this correct?
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As Aaron writes, what a "beat" is is defined differently in different time signatures. Within regular time signatures (where every beat is the same) there are two types: simple and compound. Simple meters have beats subdivided into two parts each. Some examples are 2/4, 3/4, and 4/4. Compound meters have beats subdivided into three (usually) parts, for example as in 6/8, where there are two beats of three pulses. This is as opposed to 3/4 where there are three beats of two pulses. The general trick is to use an 8 in the bottom of the time signature when it would seem redundant (6/8 would mathematically reduce to 3/4 if it were actually a fraction).
So, in your excerpt, each beat can be assumed to have a length of a dotted quarter (crotchet). Since there are 60 seconds in a minute, 120 bpm = 2 beats per second, so an eighth note subdivision of one beat will take one-sixth of a second.
As a note, this is why tempo markings are often written e.g. ♩. = 120, instead of just 120 bpm, since that can be ambiguous.
The timings calculated are correct only if the beat is understood in terms of the quarter-note, which is unusual for sheet-music in 6/8 time, but typical for DAWs regardless of time signature.
The time depends on how the beat is represented. In 6/8 time, the tempo (BPM) is sometimes given in terms of the dotted quarter-note and sometimes in terms of the eighth-note.
In this case, each beat lasts 1/2 seconds, and each eighth-note lasts 1/3 beat. Thus, the rest lasts 1/2 seconds, and each eighth-note lasts 1/6 seconds.
In this case, the rest lasts 3/2 seconds, and each eighth-note lasts 1/2 second.
Some DAWs always specify tempo in terms of the quarter-note, regardless the time signature. In that case, the rest lasts 3/4 seconds, and each eighth-note lasts 3/8 second.