2.1 General presentation

The Hardanger fiddle is a variety of the violin used in the folk music of the western and central part of southern Norway. It is traditionally played as a solo instrument for couple dancing. Its name comes from the area of Hardanger, from where it spread to many of the valleys in southern Norway in the 18th century.

It features a highly decorated body and fretboard, a short neck and a flat fingerboard and bridge, which supports the characteristic polyphonic playing-style with abundant use of drones, double stops and ornaments, as shown for instance in the transcription in Figure 1. Five sympathetic strings contribute to the richness of the resonating sound. Its impact on the precise note annotation task is discussed in Section 8.

The intricacy of the performance style makes machine transcription difficult, particularly when it comes to determining pitch, note onsets and rhythmic grouping, and discerning between melodic notes, drone notes and ornaments, and between musically essential sounds and noise.

The vast majority of Hardanger fiddle tunes are played in D tonality with the G string tuned up a whole step (A-D-A-E). The music is modal and pitch intonation patterns are characterized by great variability, often deviating considerably from those found in European art music. Rather than being derived from particular scales or chord structures (cf. harmonic tonality), pitch intonations are conditioned by shifting contextual factors, including melodic formulas, local tonal centers (which often equal the open strings) and string resonance.

2.2 Beats in Hardanger fiddle music

Historically, Hardanger fiddle music is used for dancing, recreation and ceremonial functions by rural Norwegian communities (Aksdal and Nyhus, 1993). Within the total repertoire, dance tunes are by far the most abundant. These are divided into types of triple meter (springar) and duple meter (gangar/rull/halling) types. Local styles are called dialects, indicating a parallel to the differences in spoken language between different regional districts. This conception concerns small but important (i.e. dialectal) distinctions in repertoire, playing style and dance style. These distinctions are most apparent within the springar genre, which includes differences in the basic meter between the various springar types. Some types have asymmetrical beat cycles following a short-long-medium or long-medium-short pattern, which introduces some additional analytical challenges (see below).

Since this is soloistic melody-driven music, beats are generated from within the flow of played melodic-rhythmic events rather than being represented by an accompanying instrument or similar. Structurally speaking, then, a beat typically consists of a quarter note, two eighth notes or a triplet, but which in reality are subjected to various degrees of temporal stretching and ornamentation. It should also be noted that the meter is not given from the succession of played beats as such, which is particularly evident in the springar genre. More concretely, the start of the metric cycle (the downbeat) and the subsequent accentuation pattern (how beats are experientially weighted) are often not evident from the melodic-rhythmic structuring of the tunes or how they are performed. These features are instead determined by the associated cycle of dance movements, which differs between the various sub-genres (Blom, 1981). For instance, in the so-called Tele-springar the down movement of the dancers’ center of gravity corresponding to the downbeat occurs on the long beat (long-medium-short asymmetry), while in the Halling-springar the downbeat is located on the short beat (short-long-medium asymmetry). This means that one and the same tune (melodies often traveled between regions) can be heard as having the first beat on different locations. Notably, the fiddler’s foot tapping represents this more generalized sense of “beat” (i.e. meter) that comprises both accentual (e.g. heavy-light-light) and temporal (short and long beats) properties or qualities.

Given that the relevant metrical framework (cf. above) is established, identifying played beats is in principle relatively straightforward for an experienced listener, since each beat generally is represented by a confined melodic-rhythmic figure (a triplet, two eighth notes etc.). However, Hardanger fiddle music presents some particular challenges in this regard. Firstly, the start of the beat is sometimes ambiguous due to intentionally unclear onsets or multiple competing onsets occurring more or less simultaneously. Secondly, due to how melodic-rhythmic figures and their constituents are timed, phrased (note ties through the bowing) and accentuated, it is not always obvious how notes are grouped, which in turn may imply that it is unclear which note belongs to which beat. Thirdly, at times it is not clear whether a particular rhythmic articulation should be considered as the beat onset or as a syncopation against the beat onset (before or after). The reason why this is an issue is the emergent and context specific nature of rhythmic-temporal reference structures, particularly in the springar material. Concretely, the striking variation in beat duration from one measure to the next that often occurs is generally a product of the melodic-rhythmic structuring of the tune, including the associated stylistically idiomatic phrasing (Johansson, 2017). That is, these variations are generally not syncopations, but simply beats of varying duration. Identifying the exceptions, then, cannot rely on locating onsets against a reference grid. Instead, the particular melodic-rhythmic context in its totality needs to be considered. These ambiguities (microlevel and structural) are intentional and contribute to the desired smooth rhythmic feel of the Hardanger fiddle dance tunes (Johansson, 2022). However, they naturally present some significant challenges to the analyst.

3.1 Automated note detection

Detecting notes from audio recordings is an MIR task, as part of the more general task of Automatic Music Transcription, that can be relatively well fulfilled on particular types of music using deep learning techniques (Elowsson, 2020; Hawthorne et al., 2021). Performance can reach 97% in F1 score metric for joint onset and pitch estimation in piano transcription, and 84% when adding offset estimation. For more challenging types of music, in particular styles of music that are very different from those considered in the training phase, performance decreases significantly.

Ensuring good transcription results on a specific music style requires therefore to train the machine learning models using annotated data collected from a sufficiently large sample of that music. In addition, systematic evaluation of the models requires to compare the results with a collection of transcriptions carefully established and validated by music experts.

3.2 Note annotation

Researchers have used many different techniques to create annotated datasets for polyphonic transcription in the past. One method is to record individual voices in isolation to facilitate easier annotation. Examples include the four-voiced Bach10 dataset (Duan et al., 2010), the TRIOS dataset (Fritsch and Plumbley, 2013) consisting of musical trios, a five-voiced woodwind recording (Bay et al., 2019), the audio-visual URMP dataset (Li et al., 2018), and the MedleyDB multitracks dataset (Bittner et al., 2014). For polyphonic instruments, the annotation of many simultaneous notes can be cumbersome and time-consuming.

Another method for those kinds of instruments is to generate the sounds and annotations directly from MIDI. The technique has been used for piano datasets (Poliner and Ellis, 2006; Emiya et al., 2010; Hawthorne et al., 2019), but has also been applied across the full range of the general MIDI instrument specification (Elowsson, 2020). To increase the variability and the size of the dataset, researchers can use data augmentation, varying tempo, pitch, dynamics, and timbre during synthesis (Elowsson, 2020).

Although the MIDI generation strategy is appealing because of its efficiency, synthesized MIDI often lacks the full range of variation and complexities found in real performances. Researchers can in this case instead create datasets by synchronizing sheet music with an associated recording (Thickstun et al., 2017).

Sheet music cannot be considered for Hardanger fiddle music due to the improvisatory component of this music, making each performance of a given tune a bit different to the others. So each single performance leads to a specific transcription, which therefore needs to be manually obtained in the training phase. Musicians cannot be asked to play only single monodic voices of a given piece either, so the full polyphony needs to be transcribed altogether.

3.3 Note annotation interfaces

Sonic Visualiser (Cannam et al., 2006, 2010) is a popular open-source application that offers functionality for visualizing and annotating audio, as well as the dedicated modules: Sonic Lineup for rapid visualization of repeated performances, Tony (Mauch et al., 2015) designed primarily for solo vocal note transcription, and Sonic Annotator for non-interactive audio feature extraction in batch mode. Of particular relevance is Tony, which has a refined GUI allowing a large range of navigation, playback and editing processes.

Among commercial applications, Melodyne, originally developed for tuning monophonic vocal recordings, has evolved to also enable visualization and correction of polyphonic music. The software can be used as a tool for annotation by dividing the tone curves into MIDI notes, exporting the MIDI and converting it to annotations.

None of the existing note annotation interfaces offer the features we found necessary to quickly and efficiently annotate note onset with a very high level of time and pitch precision.

3.4 Automated beat tracking

State-of-the-art approaches in beat tracking are nowadays able to correctly track beats in a large range of music. But performance is generally much lower on musical content that differs from that which is contained in existing annotated datasets used for neural network training, as well as in the presence of challenging musical conditions such as rubato (Pinto et al., 2021). When considering music where the beats are not clearly accentuated and not regularly spaced in time, the estimations are often in total incongruity with the perceived beats.

Concerning the analysis of Hardanger fiddle music, we cannot expect those models to give correct results, without any training on that style of music because of the often non-isochronous and variable beat durations, and unclear or ambiguous note onsets and rhythmic groupings. In fact, for some of the more challenging styles in the Hardanger fiddle repertoire, non-expert (human) listeners themselves would typically not be able to follow the beats. This indicates, here also, the need to collect ground-truth beat annotations of Hardanger fiddle music from expert musicians, both for training and evaluation purposes. And indeed, as we will illustrate in Section 7.1.2, state-of-the-art approaches in beat tracking fail to corroborate with the collected beat annotations.

3.5 Beat annotation and interfaces

One way to record humans’ judgement of beat location for a given piece of music is by asking them to tap to the music and record the temporal position of those tappings, often using Sonic Visualiser (Holzapfel et al., 2012, 2014; Mauch et al., 2009; McKinney and Moelants, 2006). The temporal precision of this data is questioned by the reliance on the reflexes and on the reactivity of the participants (Repp, 2005; Weiß et al., 2016). The decision of tapping at a given moment is based on many factors that gives some randomness to the process, and we might suppose that a tapping decision at a given instant might be updated and possibly cancelled due to what has just been heard afterwards, for instance.

One solution is to manually correct the recorded taps in a subsequent step (Holzapfel et al., 2012; 2014). An alternative is to automate this correction step by “snapping” the taps to close-by audio cues (Driedger et al., 2019).

Another annotation method is based on inserting and editing beat positions from scratch. To guide the process, a background canvas shows a graphical representation of the sound (waveform or spectrogram) onto which annotated beat positions are shown with horizontal lines. One common software here also is Sonic Visualiser. Another method is to use an automated beat tracker as starting point, the beat positions being further corrected by human annotators (Eremenko et al., 2018; Marchand and Peeters, 2015; Davies et al., 2009).

3.6 Addressing the particularities of Hardanger fiddle music

Note detection in Hardanger fiddle music recordings cannot generally rely on reference music transcriptions; and as a matter of fact this is one objective of this study to produce music transcriptions of these recordings. The complexity of the playing style requires a versatile note annotation interface with well-thought capabilities to maximise precision, concerning the temporal position of the note onsets in particular. This motivated us to design our own graphical user interface for note annotation.

Concerning beat annotation, one particular challenge is that beats, as experienced by listeners, do not always correspond to particular individual note onsets. Instead they would be better represented by considering the totality of interacting melodic-rhythmic events by which successive beats are composed. However, the most viable solution in practical terms, as we will discuss in Section 5.1, remains to locate the onsets of the particular notes that represent the start of each beat. This implies an alternative approach for beat annotation, based on selecting and annotating notes detected in the preliminary phase of note annotation.

4.1 Methodology

In our preliminary studies, we learned that it is rather time-consuming for Hardanger fiddle musicians to produce annotations for tunes that they are unfamiliar with, and accuracy may sometimes be lacking. For that reason, the chosen methodology was to ask the musicians to record tunes they are familiar with, and to annotate notes using computer assistance tools as aid. In this way, they could base the transcription not only on what they heard from the recordings, but also on a memory of what they actually played.

In order to avoid any bias, instead of offering the annotators the possibility to correct computer-generated annotations, the annotation (of the normal version, as explained below) is carried out entirely from scratch.

The recording and subsequent note annotations were made by three musicians: two music students, skilled fiddlers, from the Norwegian Academy of Music (S1 and S2) and one professional fiddler, Olav Luksengård Mjelva (P). Tables 1 and 2 list the tunes recorded and annotated by each musician. For each tune, five different versions were recorded: first playing in a normal way, and then following four distinct expressions: sad, angry, happy, and tender. For each tune, the normal version is first annotated by the musician; then the note annotations are automatically transferred to the other versions, and further checked and corrected by the musician (Elowsson and Lartillot, 2021).

The onset timing evaluation condition for polyphonic transcription is usually set to 50 ms.¹ Yet according to a study (Friberg and Sundberg, 1993), listeners can notice time displacements of just 10 ms. On the other hand, since fiddle music has rather undefined transients at onsets, a narrower margin than 20 ms is very hard to achieve. We therefore think that 20 ms is a reasonable threshold for very precise onset detection for this music. Hence, we encouraged performers to be very careful regarding onsets, and try to keep errors within 20 ms. This means that we can only allow a very narrow margin of error for the annotations to ensure that they can be reliably used for evaluation.

4.2 Note annotation interface

We developed a new Graphical User Interface in MATLAB, called Annotemus, to allow musicians to annotate each played note as easily and efficiently as possible, while maximising the degree of precision in the annotations. The software was made available to the musicians as a standalone application, which they can install and use on their own computers without MATLAB license nor IT support. Figure 2 shows a screenshot of the software.

The canvas of the annotation interface is a two-dimensional time-frequency representation, the “Pitchogram” (Elowsson, 2020), using a graphical representation of the fundamental frequencies detected in the sound as background image. This representation offers a relatively accurate representation of the temporal evolution of the pitch curves, even in polyphony. Sporadically there can be local mistakes such as octave errors. The users of the software are advised to rely primarily on the sound as heard, and to use the graphical representation only as a practical—and not necessarily truthworthy—guide.

Each note has to be indicated as a horizontal line starting at its perceived onset and ending at its perceived offset, and located at a particular frequency.² The performer could use various key commands as an aid during annotation. This includes audio playback of the current window, playback between the start and end of one or several selected notes, playback that starts prior to a selected annotated note and ends at the annotated onset position, playback with a click at each annotated onset position, and playback with a synthesized version of the annotated score played in one of the stereo channels using an additive synthesis algorithm of our own. The performers were instructed to first try the playback that ends at the annotated onset position for locating the exact onset times for the normal recording and the click and synthesized functionality for verifying annotations, but were free to use whichever method they felt most comfortable with.

All playback functionality is offered with the option of slowing it down to an arbitrary speed selected by the annotator. Since Hardanger fiddle music contains frequent sequences of very fast note successions, the slowdown functionality was used extensively during the annotation process. It is also possible to listen to specific note fundamentals, separated from the other notes in the audio recording.

4.3 Note annotation comparison

Sequences of note annotations are saved as CSV files. The software offers the possibility to automatically compare two sequences of note annotations of the same tune. One sequence is considered as the reference sequence, the one annotated by the user; the other sequence will be called the alternative sequence, corresponding to either a sequence automatically generated, or from another user.

To align the notes, two dissimilarity matrices are computed based on onset time and pitch differences between all pairs of notes. The dissimilarity values are turned into similarity values using an inversion function. A dissimilarity of 0 ms or 0 cent corresponds to the maximum similarity, namely 1, while a dissimilarity of or higher than 130 ms or 75 cents is associated to the minimum similarity, i.e. 0, effectively cutting away larger dissimilarities. Dissimilarity values below those limits are transformed into similarity values following the curve of the descending slope of a Hann function of length 130 ms and 75 cents respectively. The two similarity matrices are then summed together. The alignment pairs are detected by considering the problem as a maximum weight bipartite matching problem, picking out elements from the matrix such that each row and column gets only a single non-zero but the sum of all the chosen elements is as large as possible. The resulting alignment consists of this series of alignment pairs, each associating one note from the first sequence with another note from the second sequence.

The comparison between the two sequences is based on the following color convention:

• A couple of note annotations from each sequence that are nearly identical (with an inter-onset interval below 35 ms and an inter-pitch interval below 40 cents) is displayed by only showing, in green, the note defined in the reference sequence, ignoring the variant in the alternative sequence. Thus in the case of nearly identical notes, the reference sequence is considered as the authoritative source of truth.
• A couple of note annotations that are more different are displayed with two blue lines. By manually deleting one of the lines, the other becomes accepted and thus green.
• Any isolated note annotation from one sequence that cannot be aligned to any other annotation from the other sequence is displayed with one red line. The note can be accepted in the annotation (thus turned green) or deleted.

As aforementioned, the initial note annotation of each tune was made by the musician themselves. The last author of this article, also expert in Hardanger fiddle music, checked the complete note annotations once again and made a few minor corrections when necessary. The comparison interface for note annotation is also used as part of the subsequent beat annotation task, as discussed in the next section.

5.1 Note-based beat annotation

As aforementioned, one particularity of Hardanger fiddle music is that beats are not regularly spaced in time. Performers, listeners and dancers are not necessarily aware of this variability, implicitly feeling that variation as part of the rhythm/groove/beat of the tune (Johansson, 2022). This tolerance for fluctuations in beat durations also implies that the degree of precision in identifying the exact location of beat onsets may vary considerably without compromising musical interactions (Johansson, 2010). Beat tracking by participants tapping to the music is therefore not expected to produce a consistent output. For this reason it is preferable for our purposes to annotate the beats directly by identifying the note onsets that represent the start of each beat.

In this context it should also be noted that the experience of beats in Hardanger fiddle music is not comprehensively represented by their onset points. This is because beats comprise the complete melodic-rhythmic figures by which they are composed, and how beats are experienced is not only dependent on temporal features but also accentual qualities of performed rhythms (Johansson, 2022). In addition, the experienced temporal location of beat onsets does not always align with the played note that is associated with the beat onset. For instance, annotators have reported that there sometimes is a discrepancy between the preferred note-based beat annotation and where they experience the beat onset to be located (cf. Sections 2.2 and 6.2).

This also supports the relevance of the process of music transcription as a core principle for music analysis: considering beat tracking here as a process closely associated with turning the audio recording into a “symbolic” representation of music.

Sometimes beat onsets are not associated with any note onset: in some cases notes are tied across beats, meaning that there is no audible representation of the beat onset (melody note, ornament or bow onset). In our view, this silent beat, at the music discourse level, does not need to be associated with any particular time point, although the beat subdivisions before or after might need to be made explicit to ensure a temporal anchoring of that musical moment. Another related issue is syncopation, where the only note played around that beat onset is heard to be ahead of (or behind) the beat onset position.

The previous discussion justifies the proposed approach, based on collecting beat annotations as selection of particular notes in the annotations for which the onset of each successive note would correspond to the onset of each successive (non-silent) beat. Annotators were therefore asked to associate each successive beat with one annotated note. In the case several notes are played synchronously at a given beat onset, it does not matter which note is selected, as we do not find it necessary, at least for the current study, to address such level of temporal precision.

Annotators have been asked to skip silent beats, but with the possibility to annotate beat subdivisions at the vicinity of the silent beats. This is particularly emphasised for the tune Spretten with 3/8 time signature, as it features a particular “two against three” rhythm, where the duration of two successive bars is divided into three equal rhythmic values. In such a case, instead of indicating the first beat of the second bar, which remains silent, the annotators are tasked to indicate the ternary sub-beats, i.e., the first beat of the first bar, the third beat of the first bar, and the second beat of the second bar, as illustrated in Figure 3.

It would be even more informative to ask annotators to indicate all possible subbeats, but it would be time expensive, and does not seem necessary, as long as only subbeats in the vicinity of silent beats are indicated. We hypothesise that the other subbeats could be rather easily predicted. But if necessary, annotation on a finer grain can be carried out in future work.

5.2 Beat annotation interface

The interface for annotating beats has been designed to be as efficient and effortless as possible. Only the first beat needs to be associated with an explicit metrical position. For instance, if the first beat to annotate is the first beat of bar 1, and if there are 3 beats per bar, its metrical position would be annotated 1 : 1/3, which can be read as bar 1, beat 1 out of 3. To simplify, a first beat of a bar can be annotated by simply indicating the bar number in the form 1 : /. All subsequent beats are annotated by simply selecting a corresponding note associated to the beat onset. The corresponding metrical annotation (such as 1 : 2/3 for the second beat of bar 1) is inferred and displayed automatically.

Silent beats, as discussed in Section 5.1, are not annotated as such, but subbeats at the vicinity can be annotated instead. For instance, in the musical example displayed in Figure 3, with a peculiar time signature 3/8 indicating one single (ternary) beat per bar, the beat of bar 1 is annotated 1 : /, and the beat of bar 2, being silent, is not annotated. On the other hand, the third subbeat of bar 1 is annotated (1 : 3/3³), as well as the second subbeat of bar 2 (2: 2/3).

When playing back the annotation of the tune, the annotated beats are sonified with short burst of noise, with a dynamic accent on the first beat of each bar.

5.3 Interface for beat annotation comparison

5.4 Campaign for beat annotation and comparison

We have so far focused on the 12 tunes (totalling 18 minutes) played by the professional musician, as shown in Table 1. To study the degree of agreement among music experts concerning beat annotations, each tune has been annotated by three experts in addition to the professional musician (P):

• a Scandinavian folk music scholar and fiddle music expert, Mats Sigvard Johansson (M),
• two music students from the University of South-Eastern Norway (S3 and S4), expert in Hardanger fiddle music.

These experts were subsequently asked to compare their annotation with the version by the musician. For each successive beat:

1. If the annotated notes are different but considered by the expert as synchronised, they are asked to ignore that divergence.
2. If the annotated notes are different and considered as non-synchronised, if the expert thinks that both note onsets offer plausible alternative beat positions, they can leave the divergence unchanged.
3. If the expert thinks that they made a mistake and that the musician’s annotation is correct, they can delete their own annotation.
4. Inversely, they can delete the musician’s annotation.

This enables to obtain beat annotations for which the variability of the expert (with respect to the musician) has been fully reflected by the expert themselves, so that divergence can be considered as valid alternative beat grids.

One of the annotators, S4, compared all the beat annotations of each tune. The superposition of multiple beat annotations, made possible by the tool presented in Section 5.3.2, helps to establish a synthetic overview of the comparison throughout the whole corpus, as presented in Section 6.2. Besides, S4 has combined the multiple versions of beat annotations of each tune into one single authoritative version, using the pairwise comparison tool, presented in Section 5.3.1, observing the following methodology:

• Two versions (for instance P and M) are compared. Any divergence in note annotations is corrected. Any divergence in beat annotations is resolved using the four rules (1–4) stated above. Rule 2 allows multiple annotations of the same beat.
• The resulting beat annotation is then further compared with another version (for instance S3) in the same way.
• Same for the fourth version (for instance S4).

6.1 Quality of the note annotations

The musicians have been encouraged to try to achieve a precise annotation of the note onsets, aiming at a precision around 20 ms, which seems to be the maximum possible precision due to the inherent imprecision of Hardanger fiddle attacks in general. The precision of the annotations can be qualitatively observed using the slowdown and multiple sonification capabilities provided by the annotation interface. Observations made by the other beat annotators confirm the high quality of the note annotations. Just a few notes needed to be corrected, mostly due to some possible bugs in preliminary versions of the note annotation interface. These corrections have been carried out as part of the beat annotation task itself.

Concerning offset annotation, due to the longer and richer resonance caused by the sympathetic strings, as discussed in Section 2.1, we did not aim for very high precision.

6.2 Agreement between beat annotators

The superposition of alternative beat annotations, presented in Section 5.3.2, is used to compare between the different annotators. Generally, there is strong agreement between annotators on a structural or macro level, which includes the location of the downbeat (the start of the rhythmic cycle) and determining which notes belong to which beat. However, there are some exceptions. First, there was one instance where one annotator located the downbeat (first beat) on the second beat, thereby skewing all the subsequent beat annotations for that tune (Figure 4). This may be due to unfamiliarity with the particular style in question (Halling-springar) and since the annotator agreed that it was a mistake we do not consider this to be an example of disagreement between annotators. It should be noted, however, that the wrong interpretation makes perfect sense from the point of view of melodic/motivic organization, which highlights the need for expertise in determining the correct metrical framework.

In Section 2.2 we also anticipated that there might be discrepancies between annotators due to ambiguous rhythmic grouping. We have recorded several instances of this, one of which is illustrated in Figure 5. In this case, there is no correct alternative as both versions are stylistically viable. What can be noted is that the two interpretations may be associated with two different interpretational rationales. When phrasing and melodic organization are considered, the interpretation to the left is arguably preferable. But this results in a very long first beat (the first beat in Halling-springar is generally short), meaning that the interpretation to the right makes more sense from a metrical perspective.

On a microlevel we found a number of discrepancies between the annotators concerning the exact location of the beat onset. Most of these fall into the following two categories: beat onsets defined by or surrounded by ornamentation (third beat in Figure 6); beat onsets consisting of a double stop where the two note onsets are asynchronous, producing an effect resembling sliding into a note (second beat in Figure 6). The material contains numerous versions of ornamented beat onsets as well as various configurations of associated discrepancies between annotators that will be explored in later work.

7.1 Ground truth for computational models

7.1.1 Note detection

The 12 tunes recorded by P have been used to test a set of commercially available polyphonic note detection tools:

• Celemony Melodyne 5.3.0
• Logic Pro 10.7 Flex Pitch
• ScoreCloud

as well as the “deep layered learning” (DLL) model presented by Elowsson (2020). We also had the opportunity to train the DLL model on the rest of our dataset—i.e., the tunes performed and annotated by S1 and S2. We have tested both the original version of the DLL model from Elowsson (2020), trained on classical music, as well as the version we trained on S1 and S2’s annotations.

Both onset and pitch of each note are estimated. The output from each method, represented as a list of notes with their corresponding onset and pitch, is evaluated by aligning it with the list of notes from the ground-truth sequence, and then assessing the number of notes that can be successfully aligned. The alignment is based on the method presented in Section 4.3. All aligned notes are considered as correct, or true positives, the rest defining the false positives and negatives, leading to precision, recall and F1 scores.

Figure 7 shows the evaluation results for each algorithm, for each separate tune. Statistics across tunes are shown in Table 3. The Logic Pro module is clearly irrelevant for this task, while Melodyne gives more satisfying results, although the three remaining models are significantly more successful. Scorecloud and the original DLL model from Elowsson (2020) have relatively similar F1 values, although ScoreCloud excels in precision while DLL ensures a high recall, at the expense of precision. On the other hand, when training DLL on a subset (S) of the dataset, we can achieve much better results on the other dataset (P), reaching an F1 score of 87%.

Table 3

Evaluation of the note annotation models on the P dataset, in percentage.

Model	F1	Precision	Recall
Logic Pro 10.7 Flex Pitch	29	43	22
Celemony Melodyne 5.3.0	69	82	60
ScoreCloud	81	95	71
DLL	82	83	82
DLL trained on S	87	91	83

Figure 7 

F1 statistics of the evaluation of the following note annotation models, from left (blue) to right (green): Logic Pro 10.7 Flex Pitch, Celemony Melodyne 5.3.0, ScoreCloud and DLL, both its original version and the one trained on the S dataset. The evaluation was made on each separate tune in the P dataset, following the same order as in Table 1, represented here from left to right.

7.1.2 Beat tracking

The collected beat annotations have been used to test the latest available version 0.16.1 of the reference beat tracking software Madmom (Böck et al., 2016). For all the tunes, the predicted beat positions do not correspond to the annotated beats. Annotated and predicted beat onsets do coincide at isolated places, but because they do not extend along successive beat onsets, these coincidences cannot be considered as metrical convergence. Figure 8 illustrates the characteristic behavior of the beat tracker. We might hypothesise that this kind of mistake could relate somehow to the way non-expert listeners would tap beats to Hardanger fiddle music.

Figure 8 

Screenshot of the Annotemus software (cf. Figure 2) comparing the beat annotations of an excerpt of Gro Holto provided by the musician P himself (notes annotated with the red text) with the beat predicted by Madmom 0.16.1 (Böck et al., 2016) (indicated with red vertical lines).

The collected beat annotations will therefore prove useful when training machine learning models on that particular music genre, and more generally when designing and testing beat trackers dedicated to this music.

7.2 Musical analysis of asymmetrical metre

The beat annotations of the performances by the professional musician (P) enable us to study timing characteristics of Hardanger fiddle music.

The most conspicuous observation is that tempo remains relatively stable across all tunes, with a mean bar duration of 1.399 seconds, a standard deviation of 96 ms, and a mean absolute difference between successive bars of 84 ms. Indeed this is dance music and the music is expected to be played in a certain standard tempo. There is variability, still, corroborating the musician’s indication that he felt he played some tunes faster or slower than others.

A more detailed look at the temporal evolution of bar duration across each tune (Figure 9) shows that the first bar is always the longest, and much longer than any other bar, exceeding sometimes 2 s. The rest of the bars oscillate between 1.3 and 1.5 s, occasionally with an isolated bar with larger duration, up to 1.7 s. This invariance allows us to study beat duration both absolutely and relatively to bar duration, which is convenient due to the respective pros and cons of these two viewpoints.

The other, and rather contrastive, prominent observation is that beat duration is overall very irregular. The average beat duration⁷ is 466 ms — i.e., around 128 BPM — but with a standard deviation of 80 ms, and a mean absolute difference between successive beats of 103 ms, which is more than 22% of the mean beat duration. However some invariance can be observed when distinguishing the first, second and third beat of each bar. Table 4 shows their mean and standard deviation across all tunes, as well as the mean absolute difference for the same beat between successive bars, comparing it to the same statistics across all beats. With respect to overall standard deviation, the first beat of bar has much higher deviation (72 ms, compared to 65 and 51 ms for the second and third beats). But actually when considering mean absolute difference between successive bars, the second beat has slightly more variability (71 ms, compared to 69 and 56 ms for the first and third beats). This is due to the fact that the first beat of the first bar is often very long.

Figure 10 shows the histogram of beat duration, also distinguishing between first, second and third beats in bars. We notice that the distribution of first beat duration is rather multimodal, with a first cluster around 25% and a last cluster around 35% of the bar duration. This phenomenon can be analysed in more detail by observing, beyond those global statistics, the temporal evolution of beat duration ratio along single tunes. Figure 11 shows this for one particular tune, namely Godvaersdagen. The first beats of bars 1 and 2, respectively short and long, correspond to distinct modes. They actually form one 2-bar phrase, which is subsequently repeated, leading to the same short-long pattern in bars 3 and 4. Other oscillations between short and long first bars can be observed throughout the piece. In addition, we notice progressive duration changes for each separate mode across successive bars.

Beat-level variations in the asymmetrical timing patterns of springar performances seem to be related to “melodic-rhythmic” structures, in the sense that particular motivic segments are associated with particular timing profiles, suggesting that structural and other expressive features influence beat duration patterns (Johansson, 2017). Inspired by this perspective, we are conceiving software prototypes in order to offer structural and multidimensional perspectives on the complex rhythmical structuring of HF springar performances.