## 3 Results
### Statistical Analysis  
Cross harmonizations are performed between all song combinations in the dataset, producing \(443\times 443=196\),249 harmonizations to be examined. From each harmonization, an explanatory log file is generated in an Excel sheet, which logs information for each chord, including whether a chord is constraint, if it was selected from support or smoothing, if it is a new chord, along with some numerical values concerning transition probability and melody-chord matching, including the melodic rPCP of the segment that each chord harmonizes and the rPCP expected by the chord. Each piece takes part in 443 re-harmonizations as Song A and an equal number as Song B; the aforementioned attributes are collected for each piece, in both cases, acting as Song A and as Song B. The first goal of this part of the analysis is to examine the overall characteristics of applying support and chord grouping as described above. Table 2 analyses the examined attributes for all pieces in average, as well as for the pieces with higher (\"Straight, No Chaser\") and lower (\"Falling in Love with Love\") ratios of transitions that came from the support space, over the total number of transitions in the piece. First of all, regarding the average, it appears that support was required in about 42% of the generated chord transitions on average, i.e., an average of 16.12 chords were a result of support transitions on an average number of 38.16 transitions in total, in all cross-harmonization sessions. Additionally, 11.81 of those support transitions come either before, after or over constraint chords. This indicates that constraints are often responsible for the need to employ information from the support space. Figure 1: In a simplified example, a “universe” of three chords (equivalent to groups in the presented approach) is considered. \(T\) includes the transitions of Song B, which incorporate only chords 0 and 1, and transitions are only possible from one to the other (not to themselves). The “support” space includes all three chords, and transitions are possible between all chords (\(S\) also includes the probabilities of \(T\)). Some melodic segments are compatible with some chords at time step \(t+i\) (as in \(O^{N\times K}\)); compatible chord-melody matches are indicated in grey, incompatible in white. In case a single support transition probability is used, \(S\) is only employed to facilitate a transition between chords 0 and 1, which is not possible in \(T\); all other paths will be eventually erased by backtracking. If double (or multiple consecutive) supports are employed, then all supports, except from the final in the series, could possibly incorporate chord 2, which is not available in Song B. \"Straight, No Chaser\", while having only two constraints (beginning and ending chords), needs support for above 10 out of 12 transitions (0.90 ratio) and has the worst melody-to-chord matching with an average of 0.13 (compared to the overall average of 0.26). The increased demand for support in \"Straight, No Chaser\" is explained by the fact that the melody is highly-chromatic, rendering all 443 pieces (when they act as Song B) incompatible with at least one melodic segment. An exception is also notable with this piece: the average times that smoothing was necessary (where even support failed to provide a solution) is exactly 1. Analysis shows that this happens because of the exceptionally chromatic melodic part in bar 9, which forces the employment of a very specific chord (\(\flat\)VI major) to appear, which appears only twice in the support space as the second chord in a transition, with first chords that are incompatible with the melody in the previous bar. Actually, many of the top-10 pieces in support-per-transition are within the bottom 10 in melody matching.