Re: Non-integer tempo marking

[Aaron Hill]

On 2019-02-22 11:29 pm, Martin Tarenskeen wrote:
> I tried to create it as a \markup. For MIDI output this doesn't work
> though. You could add \midi { \tempo 8 = 145 } for that.
>
> %%%%%%%%%%
>
> \score {
> {
>   \tempo \markup {
>       \concat {
>         \smaller \general-align #Y #DOWN
>         \note #"4" #1 " = 72.5"
>       }
>   }
>   c'
> }
>
> \layout {}
> \midi { \tempo 8=145 }
>
> %%%%%%%%%%

Should \tempo be used within a \midi block?  According to a rule in 
convertrules.py, it was deprecated in favor of directly setting 
tempoWholesPerMinute, which supports being set to any rational number:

%%%%
  \midi { \context { \Score
     tempoWholesPerMinute = #(ly:make-moment 987 29) %% ~136.138 BPM
  } }
%%%%

Mind you, that rule was back in 2.9.16.  And it doesn't sound like 
anything was actually deprecated if \tempo still works in \midi.

Regardless, it should be noted that LilyPond internally only supports 
specifying an integral number of quarter notes per minute.  This comes 
from Tempo_performer::process_music casting the rational 
tempoWholesPerMinute to an integer after multiplying by four, since 
Audio_tempo::per_minute_4 is an integral type.

If we consider the crazy value in my example above, LilyPond would end 
up storing this as only 136 BPM before converting it to the value stored 
in the MIDI file.

The MIDI specification allows pretty fine granularity for tempo, letting 
you specify the number of microseconds per quarter note.  For 136.138 
BPM, that works out to 440,729 microseconds per quarter note whereas an 
exact 136 BPM is 441,176 microseconds per quarter note--a difference of 
about 447 microseconds per quarter note.  After 2,237 quarter 
notes--roughly 16.45 minutes of music--there would be a drift of one 
second.

tl;dr: \tempo 8=145 will unfortunately not result in \tempo 4=72.5.  
Depending on rounding, it would end up as either 72 or 73.


-- Aaron Hill