In the first step, we determine which of the metric positions that precede a potential syncope is the candidate syncopator that might trigger syncopation on the syncope. It will prove to be useful throughout the entire modelling effort. Let u and v be two real numbers, then. It takes three arguments:. This is the last position preceding the potential syncope:. The first condition implements the fact that only the last note preceding a syncope can be a candidate syncopator. Given the metric weights of Figure 4 , the syncopator can only be on three positions relative to the syncope: it is either the metric position preceding the syncope by one 16th, by one eighth or by one quarter note.
Now that the candidate syncopes and their respective candidate syncopators have been determined, we will evaluate in a second step whether syncopation actually takes place in a specific situation. Syncopation occurs when the syncopator is marked by an onset and has a lower metric weight than the syncope, which is not marked by an onset. The definition can be formalised as follows: Let i be the metric position of the syncope in instrument a which can be the snare drum s or the bass drum b , and j the position of the respective syncopator.
Then the value of the syncopation switch function I a at i is given by. Thus, I a , i , as defined in Equation 4 , returns 1 when syncopation occurs, and 0 otherwise.
The third step is to assign a specific syncopation weight to each occurrence of syncopation. According to the Witek et al. For the snare drum, the model proposes the following procedure to calculate the weight of a syncopation, W s :. The difference between the metric weights of the syncope w i and the syncopator w j. If a position with a large metrical weight e. Together, the weights of 1 and 4 add up to 5, which is the instrument weight for two-stream syncopation.
Based on a perceptual argument, Witek et al.
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The instrumental weight is 2 for syncopation on the bass drum compared to 1 for the snare drum. A weight of 3 is added in the case of two-stream syncopation. As a consequence, two-stream syncopations triggered by syncopators in either the snare drum or the bass drum have the same instrumental weight, namely 5.
In our implementation of the model, we treat this situation as a two-stream syncopation triggered by the syncopator on the bass drum, only. In order to discount the snare drum syncopation in this specific case, we use an inverted variant of the syncopation switch function. Finally, as a fourth step, we aggregate all components defined in Equations 4 , 5 , 6 , and 7. Model syncopation S m in the snare drum and bass drum voices of a popular music drum pattern with n metric positions is calculated as follows:. The contribution of the bass drum at position i is the product of the syncopation switch function I b , i 1 if syncopation occurs, 0 otherwise and the syncopation weight W b , i for the bass drum syncopation at i.
The contribution of the snare drum is the product of the switch I s , i and the syncopation weight of the snare drum W s , i. We implemented the model as a script in R. The Model syncopation S m values of the six experimental stimuli can be inspected in Table 3. The model presented in the sections above quantifies syncopation on the basis of the notated drum pattern.
In this section we investigate how reliably our implementation based on this model predicts the Perceived syncopation ground truth we established in the listening experiment. We consider the model to be successful if it is possible to find a link function that predicts Perceived syncopation from Model syncopation with adequate fit.
This link function must be monotonically increasing as Model syncopation increases, Perceived syncopation should also increase across the whole domain of the function. The link function does not necessarily have to be linear, any kind of increasing one-to-one function can prove to be useful.
The relationship between the experimental data and the model can be studied in Figure 6. The fitted values can be studied in Table 3. Figure 6. Logarithmic link function: Perceived syncopation S p as a function of Model syncopation S m. The red curve is the best fit logarithmic function with all stimuli. A log-based function might be a promising candidate for the link function: the relationship between Perceived syncopation S p and Model syncopation S m , as seen in Figure 6 , appears to flatten for higher values.
Note that function f log passes through the origin, regardless of the choice of q and k.
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The red curve in Figure 6 is the line of best fit using all six data points. With respect to these two stimuli, S p cannot be expressed as a strictly increasing function of S m. It is defined by. Functions of this form have frequently been called exponential approach or shifted exponential functions. Again, k determines the shape of function f exp , and the function passes through the origin, regardless of the choice of q and k. The fit of the exponential approach functions can be studied in Figure 7 and Table 4. Figure 7. Exponential approach link function: Perceived syncopation S p as a function of Model syncopation S m.
The red curve is the best fit exponential approach function with all stimuli. Dashed lines are the approach limits. In summary, our implementation of the syncopation model, based on the modelling work of Witek et al. In the remainder of this study, we modify the model in order to improve its fit with the experimental data. These notes act as syncopators to the rests on the subsequent eighth-note positions. The weight of these syncopes is increased by the fact that the snare drum has a simultaneous rest, so the constellation is treated as a two-stream syncopation by the model. Potentially, listeners heard the 16th notes as ornaments to the beats, rather than as syncopes.
It is the sheer number of these syncopated events on a low metric level that inflates the S m syncopation estimate for this pattern. We will use the mathematical model description presented above as a template to reformulate the syncopation model.
The parameter c regulates the relative weight between metric positions. But the weight of all the other positions is reduced as c increases.
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The h coefficient is a scaling factor, which will be explained below. A numeric optimisation process was implemented in R , which aimed to minimise the root mean square error between the link function which is separately fitted for each evaluated combination of c and d and the experimental data. A linear function was chosen as the target link function between Perceived and Model syncopation. Hence the revised model is given by.
The result of the revision and optimisation can be studied in Figure 8 and Table 5. The six data points follow a linear regression line quite closely.
Figure 8. The red line is the best fit linear regression line. In the revised model, this pattern is well aligned with the other stimuli. The revised model does not show this distortion. The accuracy and general applicability of this estimate is difficult to assess at this moment. The estimate rests on data from a small experiment with only six drum patterns. And these patterns were not randomly chosen from the population of Western popular music drum patterns.
Rather, they were selected to represent a wide range from weak to strong syncopation. A practical drawback of the revised model is that computations do not rely on integers only, as the original Witek et al. This study presented the implementation of a model for quantifying syncopation in popular music drum patterns, which was based on a method by Witek et al.
The implemented model was then empirically validated. Comparing the model predictions with the data from the listening experiment indicated that the model predicted Perceived syncopation quite well in general. However, our findings also showed that the model overemphasised some aspects of a rhythmic pattern linked to syncopation on weak metric positions, which led to inconsistencies between Perceived and Model syncopation. We revised the model and improved its fit with the empirical data by numerical optimisation.
We found model parameters such that the model expresses Perceived syncopation of the six stimuli as a linear function of Model syncopation. For the time being, the revised model seems to be our best guess at estimating syncopation in popular music drum patterns, even though many questions remain unanswered. For example, the contribution of the hi-hat to syncopation has been assumed to be irrelevant in this study, yet there is no evidence so far that the contribution of the hi-hat is indeed negligible.
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The model is based on a small-scale listening experiment with only six stimuli. A future experiment to validate and further improve the model should include more stimuli in order to better represent the diversity of popular music drum patterns and to generate more stable results. In this study we assumed, in line with Longuet-Higgins and Lee and Witek et al. This hypothetical additive property of syncopation is somewhat questionable: syncopation is based on the notion of meter.
The more syncopated events we introduce into a pattern, the less listeners will be aware of the meter, and the perceived syncopation effect will be weakened. Another problem with syncopation is that listeners with little musical expertise are unlikely to know the concept of syncopation this was the main reason we recruited only musicians for the listening experiment. It might be beneficial to replace the concept of syncopation by the concept of rhythmic complexity in the future, as suggested by Fitch and Rosenfeld Potentially, complexity can more successfully be projected on a quantitative dimension than syncopation.
More rhythmic complexity is likely to be associated with more metric confusion in the listener. Another potentially valuable metric to explore is beat strength , or, alternatively, perceptual beat salience : listeners with any degree of musical expertise will be able to answer the question whether they recognise the beat more easily in one or in another stimulus. Perceptual beat salience can be expected to be a reverse measure of syncopation or rhythmic complexity.
The authors would like to thank Lorenz Kilchenmann and Toni Bechtold for their support preparing the stimuli. Prepared the stimuli: FH. Carried out the listening experiment: FH.
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