Personally, I prefer linear curves. I think we do that when we play instruments in real time, too.

I would agree with Dave, linear is a more natural tempo change,
exponential tempo changes sound more electronic, which are great for
certain electronic scenarios, but if it were me, I'd stick with linear
if I wanted it to sound more natural.

That said, I don't see why minor exponential curves couldn't work too,
or sections of slightly different linear changes.

On Fri, Mar 4, 2016 at 1:52 AM, Dave Seidel  wrote:
> Personally, I prefer linear curves. I think we do that when we play
> instruments in real time, too.
>
> On Mar 3, 2016 8:44 PM, "Michael Mossey"  wrote:
>>
>> Supposing that I'm computing the timing of notes during an accelerando or
>> rit, has anyone investigated what tempo curve sounds most natural? For
>> instance, expressing the beats per minute as B(t), should B be linear in t,
>> or exponential, or what? It's somewhat subjective, but what I want to avoid
>> is tempo changes that seem to bunch all toward one end -- for instance, the
>> tempo changes subjectively quite fast and then the rate of change
>> diminishes, as opposed to a subjectively uniform change in rate.
>>
>> Mike
>>
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el 2016-03-05 a las 14:50 Hlöðver Sigurðsson escribió:

> Very interesting and underestimated topic. 

in fact, there has been _quite_ a lot of research in the field of
computational musicology about this topic in recent years. it began in the
late 80s - early 90s with the people like sundberg, desain and honing, and
it has had a huge increase in the last decade or so (bilmes, benadon, and
many, many others). the different aspects of microtiming have been and are
being extensively analyzed, both variations in tempo like ritardando,
accelerando and rubato, and deviations of events with respect to the
isochronous metrical grid, like notes inégales in baroque music, or swing
eight notes in jazz. i would say that this topic is hardly
"underestimated". in any case, what may happen is that the results of this
research is published in books and journals not frequented by many
musicians. Henkjan Honing, to name just one author, has most or all his
papers freely available on-line. titles like "The final ritard: on music,
motion, and kinematic models", "From time to time: The representation of
timing and tempo", or "Tempo curves considered harmful" (with Peter
Desain) should give you a pretty good idea.

by the way, i'll be presenting an article about microtiming in candombe
drumming at the Fourth International Conference on Analytical Approaches
to World Music next june in NY:

http://aawmconference.com/presenters.html

anyone in the area is cordially invited.

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I have used parabolic curves with good effect. See a detailed 
description in part 4.1 of the following paper:

http://www.zogotounga.net/surmulot/High-level%20Musical%20Concepts%20in%20muO.pdf

A lot of tempo variations in the following piece were made this way:

http://www.zogotounga.net/zik/Tout%20va%20bien%20(merci).ogg


Stef

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