To see if complex rhythms can be rescaled.
3 subjects, all musicians (1 guitarist, 1 pianist, 1 drummer), paid
for their time.
This ensures accurate timing!
Subjects will be asked to listen to some rhythms and try to tap them out. There will be several stages in the experiment:
Subjects hear the beeps and see a line showing the desired ratio
The patterns have a "base rate". This is the "1" in a ratio of X:1. So if the base rate is half a second (= 500 milliseconds, or ms), a 2:1 pattern consists of a 1 second interval followed by a half second interval.
Subjects hear the beeps and see a display showing the target ratio. They have two wooden blocks (sometimes referred to as "keys") in front of them, which they tap on using one finger from each hand. While they are hearing the beeps, they may tap along on the blocks if they like (why might this be important?). When the beeps stop, the screen flashes the word and they have to start tapping out the rhythm they just heard. After 17 taps (= 8 long/short patterns) They see the word . They then see a display which told them how well or badly they have done. This includes text, telling them what their average ratio was, how close they were to the target, etc. Although the researchers were mainly interested in what ratios the subjects produced, they wanted them to tap at about the same tempo as the stimulus. The "error of last tap" figure tells them whether they took too long or were too quick overall. This is one "trial".
The test stage is included to see whether subjects are reproducing the target ratios accurately. This time, subjects only hear 8 beeps, and then they have to produce the appropriate ratio, so for a base rate of half a second, and a target ratio of 2:1, they hear 8 beeps separated by 1.5 second intervals. This is one "trial".
The rescale stage is the same as the test stage, except that the base rate is doubled. So, for the previous example, they hear 8 beeps separated by 3 second intervals. This is one "trial".
The order in which subjects do things is important. In this experiment, the design is "within subject", which means that each subject has to do every stage for every pattern. With 4 simple patterns and 3 complex patterns, each of which has a training, test and rescale phase, this gets pretty complicated. And everything has to be repeated many times. And, subjects must be trained a lot! (remember why?)
We saw above what a trial is. A single trial is repeated several times in a row. This is called a "phase". In order to test a single ratio, 6 phases are needed, which together are called a "block". This is because we want to train, test and rescale, all on the same ratio. But testing and training use different stimuli! So, we have to train on the testing phase too!! A block looks like this:
And all this is for one single ratio! But there's lots more....
Consider one subject (JCL). She is subjected to a block on the ratio 3:2, then a block on 2:1, then 3:2, then 2:1, then 3:2, then 2:1 again. At this stage, we might agree that the experimenters have learned everything they are ever going to learn about 2:1 and 3:2 for JCL. Not so. The subject does 6 blocks on a single complex ratio (1.82:1), and then the experimenters want to know whether doing the hard task changed her performance on the easy ratios. So she does 2 more blocks on each of 3:2 and 2:1. No wonder they have to pay their subjects!
A within subjects design. 3 subjects, each of which are run on 2 simple ratios and one complex ratio, which means 1008 trials each!! Only some of these (Phase 4 and Phase 6) actually provide data.
So what did they find?
First off, each tap was done by two hands, so they had to see whether the two hands made taps at the same time. In most cases, the taps were within 15 ms (0.015 of a second) of one another, so they thought it would be ok to use the average of the two hands to get a single measurement for a tap. (What if this had not been the case?)
Sample results for a single subject:
The x-axis shows the test phase results for one subject and 3 ratios. The y-axis shows performance during the rescaling. Note that if the subjects were "perfect", the data points would all lie on the diagonal line. They are not perfect in either testing or rescaling. The fact that most points are near the diagonal for the simple ratios shows that they are pretty good on these. But look at the spread and location of the points for 2.72! During testing, this subject is not doing too badly, as his points are around 2.72 on the x-axis, even though there is a lot more variation than for the simple ratios. But during rescaling, they are all around 2, not 2.72!
Compare this subject with the subject shown in Figure 4. Here the results for the simple ratios are similar, and in testing, again, the subject produces the complex ratios fairly well, about 2.72. But in rescaling this time, they all lie around 3. This should give you a hint as to why the authors are showing the results separately for each subject.
In looking at the data, the authors make use of many quite sophisticated statistical tests. You are not expected to know these tests, but it would be a good idea to know in general what they are looking for. They look at the data in two main ways:
When doing a statistical test, for example looking for a difference in two averages, the results are said to be SIGNIFICANT or NON_SIGNIFICANT. This is NOT the same thing as IMPORTANT/UNIMPORTANT! A large difference will usually be "significant", but if it is predictable and boring, it may be unimportant. Conversely, a non-significant result may still point to something interesting, and the lack of significance may simply come from using a weak test or not having quite enough data.
Subjects showed very little if any improvement across blocks. One
subject even disimproved. This is a potentially serious flaw in the
No improvement observed for simple ratios!
The authors look at the difference between the ratio on the test phase
and the ratio on the rescale phase. They could also have chosen to
look at the difference between the rescaled ratio and the target
ratio. Why did they choose the former?
Complex ratios are, as expected, difficult to perform. Performance
may improve a little with practice.
For both simple and complex ratios, rescaling performance is always imperfect.
...to the extent that complex ratios can be accurately acquired, participants produce them using different mechanisms from those used for simple ratios.
GMP/PS and all very general relative time models cannot account for the difficulties associated with the rescaling of complex rhythms. They instead propose another class of model: clock-counter models. These also have several flavors and have been suggested before.
There are many unsolved questions remaining. Here are just some of them: