Mathematics of Ritardando

Beyond feel & experience, is there a rule conductors use for ritardando in terms of (a) its rate, (b) its change in rate, and/or (c) the relationship between the final tempo and the tempo of the piece?

(In getting my software to execute a ritardando, I employed over four measures a measure-by-measure decrease in tempo, and what sounded "right" to me ultimately was decreasing the tempo by 4 bpm, then a further 9, then a further 16 (pleasing pattern) to arrive at 127/156 (close to 3/4) the original tempo.)

• In Musescore, I'm normally fine with linearly changing the tempo, with the same decrement every the same number of beats. I often have to change the tempo more than once per bar to make it sound right, though. Nov 25 '18 at 15:39

"...is there a rule conductors use for ritardando in terms of (a) its rate, (b) its change in rate, and/or (c) the relationship between the final tempo and the tempo of the piece?"

Not that I'm aware of. Such a rule would be of little value, because -- unless you're practicing with a drum machine, or other device that permits varying tempo -- there's generally no good way (other than feel) to for conductors or performers to accurately measure their tempo during a performance. They're certainly not likely to be thinking "Now this beat has to be played 10% longer than the previous one, and the following beat has to be 17.4% longer than that..."

However, if I had to guess a general shape of a tempo curve in a ritard, they generally seem to be monotonically decreasing (you don't speed up again in the middle of a ritard), and concave downward (i.e., the most ritard'ing comes just before the end of the ritard). But to what degree this is true is up to the conductor or performers to decide.

On the other hand, about the most prescriptive that I've usually seen music notation get is poco rit. (just a little bit of ritardando), or molto rit. (much ritardando). And of course, the dashed line that Richard mentions, to indicate the duration of the ritard.

That said, I am interested in what the tempo curves of actual performances would look like. I know that approximate tempo maps can be created for real performances, and I'd be surprised if someone, somewhere, hasn't researched what these curves look like.

"Beyond feel and experience"? Using a mathematical formula to create a ritard in music would not be beyond using instinct, but vice versa. It's nearly impossible to create an authentic ritard in computerized music because composers don't create ritards to slow down the music, but to create an emotional reaction in the audience. The rates of ritards depend on countless criteriae such as harmony, melody, timbre, and rhythm, and what the musician has had for lunch.

Computers are thus extremely impractical music makers. It is up to you to decide what is pleasing and program the computer to follow your wishes, if possible. Whatever you decide is pleasing, it is extremely probably that someone out there will disagree with you. The best you can do is make an intelligent decision about the music.

In terms of following ritadando markings in the score:

At the end of a piece

Ritardando markings at the end of a piece should generally be greater than other ritardandos throughout the piece, unless otherwise specified. If the music was slow to begin with, musicians should be careful not to slow too dramatically.

When there is a dashed line (- - -) indicating the length of the ritard

Musicians should gradually slow the music down until the dashed line ends in the score. There may or may not be a tempo indicated to arrive at. Obey any "a tempo" markings.

When there is a ritard marking alone

Obey the ritard marking intelligently.

I don't know if there are any conductors who conduct a ritardando or accelerando mathematically (i doubt it!), but if you want to compare a growth (which means both increase and decrease) in some musical parameter with natural phenomena you could use exponential growth:

``````f(t) = a.e^kt
``````

with `a` being the initial value of your parameter (in your case the duration of a note), `k` the growth rate (`k>0` for increase and `k<0` for decrease) and `t` being your unit of measurement (i.e. time or an index number of a note in a sequence of notes). This could be implemented in the programming language Python as follows:

``````import math

def exp_growth(init_val, growth_rate, idx):
"""calc the exponential growth"""
return init_val * pow(math.e, growth_rate * idx)

def ritard(init_dur, ritard_rate, n_notes):
"""prints single durations of a ritard. and its overall dur"""
duration_samples = [exp_growth(init_dur, ritard_rate, t) for t in  range(n_notes)]
duration = sum(duration_samples)
print("Ritard.: {}".format(duration_samples))
print("Overall duration of ritard.: {} seconds".format(duration))

# With init_dur = 0.6 seconds, every note has a 10%
# increase in duration, get the ritard. durations over 5 notes:
ritard(0.6, 0.1, 5)
``````

In this example i get the durations of every note during a ritardando over 5 notes, where my first note has a duration of 0.6 seconds (say a quarter note at tempo M.M.100) and every note should have a duration increase of 10% (relating to its preceding note). The above code will print:

``````Ritard.: [0.6, 0.6631025508453886, 0.7328416548961019, 0.8099152845456019, 0.8950948185847621]
Overall duration of ritard.: 3.7009543088718546 seconds
``````