Each box consists of the distribution of negative log- likelihoods across the 70 leave-one-out analyses. We applied our algorithm from Section 3 to the full set of 70 songs. The resulting logistic regression model was then used to make predictions on disputed songs and song portions, and on songs known to be collaborations between Lennon and McCartney. The optimal significance level threshold for the variable screening was 0.
Conditional on selecting variables using the 0. Thus, the final logistic regression model for predictions involved an average of L 1 and L 2 penalties, but more heavily weighted towards a ridge penalty. Of the 40 features that were selected through sure independence screening, 29 were non-zero in the final model as a result of elastic net logistic regression.
The full set of 29 variables is listed in Table 4. Melodic transition: down 1 note on diatonic scale, not incl. Melodic transition: up 1 note on diatonic scale, not incl. Table 4. Coefficient estimates in the final logistic regression in the second column, and ROC analysis c -statistics in the third column. The c -statistics are computed from the 70 leave-one-out probabilities with the variable removed from the prediction algorithm; thus smaller c -statistics indicate greater variable importance.
Distinguishing song features of Lennon and McCartney authorship can be learned from the coefficient estimates of the logistic regression. These results offer interesting interpretations of musical features that distinguish McCartney and Lennon songs. One clear theme that emerges is that McCartney tended to use more non-standard musical motifs than Lennon. These chord changes also create an ambiguity about whether the song is in the major or relative minor key.
In contrast, the chord change between ii and IV coefficient of 1. This is exhibited in the negative coefficients for note transitions moving up or down one note on the diatonic scale, and the positive coefficient 1. Lennon also more often started melodic phrases at the 3rd or ended phrases at a 5th, both of which are notes on the diatonic scale.
In contrast, McCartney more often used a flat 3, and transitions from the flat 3 to the tonic in his sung melodies, both of which are notes often associated with a blues scale and not the diatonic scale. In addition to the coefficients, we report a measure of variable importance in the third column of Table 4.
Our measure has close connections to an early approach developed in the context of random forests Breiman, In particular, the importance of a variable can be assessed by randomly permuting its values across observations, and then computing an overall measure of model performance. The lower the performance measure after permuting the variable, the more important the variable. For our approach, randomly permuting the values of a musical feature across songs is effectively equivalent to having the feature removed because sure independence scanning should eliminate the feature in the first step of our prediction algorithm.
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Thus, our variable importance measure was computed as follows. First, we removed the musical feature whose importance we wanted to assess. We then applied our out-of-sample procedure from Section 4.
We performed an ROC analysis on these probabilities and the known authorship of the 70 songs and summarized the c -statistics in the third column of Table 4. Lower values of the c -statistic indicate greater variable importance. The c -statistic without eliminating any features is 0.
Generally, higher absolute values of coefficient estimates correspond to lower c -statistics. Musical features with the lowest c -statistics, all less than 0. The only feature with a Lennon leaning and having a c -statistic less than 0. Compared with the McCartney feature of a downward half-step move, upward half-step moves may correspond to particular note transitions that are distinct from the downward moves.
We applied the fit of our model to make predictions for eight songs or song portions with disputed authorship, and for 11 known to be collaborations. The prediction probabilities were derived by applying the fitted logistic regression to the songs of unknown and collaborative authorship. For each song of disputed or collaborative authorship, we computed 70 probability predictions based on leaving out each one of the 70 songs in our training sample.
It is worth noting that these intervals are conservative because one fewer song is used than with the corresponding point prediction. The probability predictions and corresponding confidence intervals are displayed in Tables 5 and 6. Table 5. Table 6. We also display the distributions over the 70 predicted probabilities for each disputed song as density estimates in Figure 4.
Figure 4. Density plots of the leave-one-out probability predictions for the eight songs of disputed authorship. For the songs and fragments of disputed authorship, all of the probabilities are lower than 0. For the songs during the study period that were understood to be collaborative, it is unclear to what extent Lennon and McCartney shared songwriting efforts.
However, it is worth noting that our model was developed on a set of songs and song portions that were of single authorship, and that applying our model to songs of collaborative authorship may result in predictions that are not as trustworthy. As with the information in Table 5, most of the collaborative songs in Table 6 were inferred to be mostly matching the style of Lennon.
One feature of the song is the predominance of the flat third. This McCartney-like motif may be responsible for the high probability that the song is inferred to be written by McCartney. Compton reported that the former was claimed to be entirely collaborative, and that the latter was initiated by McCartney even though the song was written collaboratively. Two of the collaborations are worthy of comment. However, perhaps the song might have been special to him as it had much more of his imprint.
Rolling Stone magazine considered it to be the 23rd greatest song of all time Rolling Stone , Our model produces a probability of Because it is known that Lennon wrote the lyrics, it would not be surprising that he also wrote the music. Lennon claimed Compton, that McCartney helped with the bridge, but that was the extent of his contribution.
The bridge having a probability that McCartney wrote the song closer to 0. The approach to authorship attribution for Lennon-McCartney songs we developed in this paper has connections to methods used in attribution analysis of text documents. One important difference is that typical text analysis models rely on the relative frequencies of occurrence of words or word combinations. Another difference from typical text analysis problems is that songs include more than just one text stream. Songs with high probabilities of being written by Lennon or McCartney are mainly indications that the songs have musical features that are consistent with the Lennon or McCartney songs used in the development of our model.
To this end, one use of our model is to investigate whether certain sections of disputed or collaborative songs are suspected of being more consistent with particular composition styles. It is natural to ask whether that section may be more in the style of McCartney who may have had a freer hand in writing that portion of the song.
Indeed, our model applied to just the bridge section resulted in a 0. The analogous decision in a musical context is arguably much more difficult, as the complexity of choices is far greater. In our work, we needed to make many subjective decisions that influenced the construction of musical features. Such decisions included what constituted the beginning and ending of melodic phrases, whether a key change modulation should reset the tonic of the song, whether ad-libbed vocals should be considered part of the melodic line, how to include dual melody lines that were sung in harmony, and so on.
Our guiding principle was to make choices that could be viewed as the most conservative in the sense of having the least impact on the information in the data. Also, when it was not clear in cases of dual melody lines which was the main melody, we included both melody lines.
It is worth noting that the model developed here was not our first attempt. We explored variations of the presented approach before arriving at our final model, including versions that permitted interactions, alternative variable selection procedures such as recursive feature elimination and stepwise variable selection, models for the musical features as a function of authorship that were inverted using Bayes rule, random forests, as well as several others. This concern may not be apparent in the presentation of our analytic summaries, which was the culmination of a series of model investigations.
The concern of overfitting limited some of our explorations. For example, after having modest success using elastic net logistic regression without any variable pre- processing, we inserted variable screening parameterized by a p -value threshold based only on four threshold values.
Using a greater range of thresholds, especially after having learned that elastic net alone was a promising approach, and that we were tuning the model parameters based on the same leave-one-out validation data, would have had the potential to produce overfitted predictions. We suspect that our final model, however, does not suffer from overfitting concerns in any appreciable way. First, the approach we present is actually fairly simple: the removal of musical features based on bivariate relationships with the response followed by regularized logistic regression.
More complex procedures might raise questions about their generalizability. Second, we were cautious about optimizing the prediction algorithm and calibrating the predictability using out-of-sample criteria. This cascading application of cross-validation mitigates some of the natural concerns about possible overfitting. Our particular modeling approach does permit extensions to address wider sets of songwriter attribution applications. Our model assumes only two authors, but this is easily extended to multiple songwriters in larger applications by modeling authorship in a multinomial logit model, for example.
This approach would be straightforward to implement in a Bayesian setting, though implementing such a model in conjunction with variable screening would involve methodological challenges.
Several other limitations are worth mentioning. Our approach assumes that each song or more relevantly song portion contains sufficiently rich detail to capture musical information for distinguishing authorship. Shorter song fragments would have a scarcity of features, and probability predictions are expected to be less reliable. With recent improvements in technology to convert audio information into formats amenable to the type of analysis we developed in this paper Casey et al.
A justification for the musical features chosen requires an understanding of Western popular music. Middle C , often denoted as C4, has frequency The continuation of the sequence above is the same set of pitches, but at the next higher octave, that is, C5, C 5, D5, and so on.
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The 12 notes can also be visualized in a piano diagram in Figure 5. Thus, a specific note has multiple representations using this notation. Thus, D3, D4, D5 and so on, are in the same pitch class, but reside in different octaves.
One can translate or transpose the chromatic scale to start on any note given its circular structure, and to the human ear all such chromatic scales played in sequence sound essentially the same. The basis of Western music is the diatonic scale , which, starting on a given note Z [ i, j ], called the tonic of the scale, consists of the subsequence of seven notes from the chromatic scale:.
Chromatic notes that are not part of the diatonic scale are called non-diatonic. The diatonic scale permeates much of Western music, and most popular songs or portions of songs can be analyzed to be based on a diatonic scale starting at a specific note belonging to one of the 12 pitch classes; the lowest note of the diatonic scale is called the major key , or just the key , of the song, and the note itself is the tonic of that key. For our purposes, we associate, as is often done, the minor key with the major key three semitones up, as they share the same seven notes.