High pressure in solid-state chemistry

Introduction

In solid-state chemistry, high pressures can be used in two different ways. The application of high pressures to existing materials leads toa decrease of the inter-atomic distances, such adecrease being correlated with the compressibility of the solid. On the other hand, high pressures can play an important role in the synthesis of new materials.

High-pressure effects on solids

With the decrease of inter-atomic distances enhancing electronic and steric repulsions, two important types of phenomenon can be induced:  electronic phenomena;  structural modifications.

Electronic phenomena induced by pressure

Different factors can affect such phenomena:

 the oxidation state M n+ ;  the stability of the corresponding electronic configuration for the M n+ cation;

 the chemical nature of the anion;  theinter-atomic spacing governed by the structure adopted by the solid.

Two types of electronic phenomenon can be considered:  the intra-atomic ones-such as the change of electronic configuration in the transition from high spin (HS) to low spin (LS);  theinter-atomic ones involving in particular: (i) charge transfer; (ii) disproportionation; and (iii) insulator →metal transition.

Under high pressures, due to the decrease of the M n+ -X distance, an increase of the local crystal field energy is induced, leading to the HS → LS transition. Such a phenomenon has been observed for CaFeO 3 at a pressure close to 20 GPa [1]. The electronic behaviour of Sr 3 Fe 2 O 7 under high pressures shows, successively, different electronic phenomena as observed using Mössbauer spectroscopy [2]. At normal pressure, the presence of two different isomer shift values suggests the disproportionation 2Fe

4+ →Fe 3+ +Fe 5+ .

When the pressure reaches 15 GPa, some modifications appear, suggesting a charge transfer. At P  21 GPa, only a single magnetic component is observed. Finally, at pressures higher than 28 GPa a metallic behaviour occurs, such a phenomenon being induced both by the strengthening of the Fe-O distance and by the instability of the HS Fe 4+ electronic configuration (t

3 2g e g 1 )
In some cases electronic phenomena such as electronic transfer can lead to the formation of achemical bond. Such a case is observed for some metal halides such as GeI 4 [3]. Under high pressures a transition is observed from a molecular state to a solid one through the formation of I •••I chemical bonds resulting from a decrease of Ge-I bonds, implying a charge transfer.

Structural modifications induced by pressure

Another important feature resulting from application of high pressure is the enhancement of the steric interactions between atoms throughout the whole lattice of the solid. Such interactions can lead to structural transformations. Such a phenomenon has been observed in the hexagonal ABX 3 perovskites [4] and the K 2 MO 3 oxides (M =Zr, Hf, Sn, Pb) [5].

High pressures and the synthesis of new materials

The densification effect

Due to the compressibility induced by high pressure,a densification is always observed between the starting materials and the resulting ones:

A+B → AB . VA/Z1 VB/Z2 VAB/Z3
If VA/Z1+ VB/Z2 < VAB/Z3(V > 0), high pressures hinder the formation of the resulting material AB. Otherwise (V < 0), the application of high pressure aid the synthesis of AB.

Consequently, high pressures can modify the phase diagram. As an example, in the Al 2 O 3 -B 2 O 3 system, while twocompositions 9Al 2 O 3 -2B 2 O 3 and 2Al 2 O 3 -B 2 O 3 are observed at normal pressure, at 4 GPa two new phases appear: Al3BO6 and AlBO 3 [6], due to their denser structures.

Nickel Copper

Ni 3+ Cu 3+ TNiO 3 [22] LaCuO 3 [24] SrTNiO 4 [ 2 3 ] L a 2 Li 0.50 Cu 0.5 0O 4 [25] SrLaCuO 4 [26] 3.2. The stabilization of the precursors Two different systems can be considered: (i) where the precursors are thermally unstable; and (ii) where the physico-chemical properties of the precursors are so different that the chemical reaction is difficult. In both cases, highpressures can make possiblethe synthesis of new materials.

It is difficult to improve the chemical reaction using different anionic systems but under high pressures, an oxyfluoride KTiO 2 Fwith the perovskite structurehas been stabilized [7]. Consequently new mixed anionic compounds could be prepared in the near future: (O, F), (O, N), (N, F), ... [8].

When one of the precursors is thermally unstable, the decomposition temperature (T D ) being lower than the reaction ones (T R ),highpressures-through the increase of T D (T D > T R )-can facilitate the chemical reaction. Such applications of high pressure have been developed for preparing new compounds using HgO [9], MnO 2 [10], PbO 2 [START_REF] Fujinaka | High Pressure Science and Technology[END_REF] as precursors.

Synthesis of new compositions through the difference in atomic compressibility

The structure adopted by materials is closely related to the ratio of the sizes of the atomic species. For example, the stability of the pyrochlore structure A 2 B 2 O 7 is limited by rA/rB ≈ 1.55 [22]. Under high pressures, this limit can be pushed to 1.8 for the series T 3+ 2 Ge 4+ 2 O 7 due to the difference in compressibility between T 3+ (T = rare earth) and Ge 4+ [13].

Stabilization of the highest valence states of transition metals under high oxygen pressures

High oxygen pressures have been developed in gas, liquid, and solid phases [14] and can help in the stabilization of the highest oxidation states of transition elements:

Asolid +Bsolid + (m/2)O2 → AB n+ Om G = G 0 -(m/2)RT log PO 2 .
During the last 30 years a lot of unusual valencies have been stabilized in oxygen lattices mainly derived from the perovskite structure (table 1).

Synthesis of superhard materials

The physico-chemical properties (in particular hardness) of materials depend on different factors, in particular:

 the inter-atomic distance;  the ionicity coefficient;  thestructure adopted (2D or 3D).

Consequently, materials containing light elements such as B, C, N, O could be worth investigating for the production of superhard materials [29] in view of the small atomic spacing d.The first materials found to have exceptional properties were diamond [30,31] and, subsequently, cubic BN [32].

The evaluation of the B-modulus through an empirical formula (B = (1971 -220λ)/d 3.5

) [33] has the development of improved high pressures for preparing new materials with light elements. Five different research areas have been investigated and could involve high pressures:

 the polymerization offullerenes [34,35];  the B-C-N system [36]- [38];  the B-O system [39,40];  the stabilization of carbonitrides, in particular C3N4 [41,42];  the Si-C-N system [43].

The development of solvothermal reactions

A solvothermal reaction can be described as a reaction in a closed system in the presence of a solvent in supercritical conditions, or near such conditions; consequently high pressures can be involved. Due to the specific properties of solvents in these conditions, solvothermal reactions improve the reactivity [44]. These reactions have been used in different areas:

 the synthesis of new materials (for example the phyllosiloxides-bidimensional oxides isostructural to the phyllosilicates [45]);  the preparation of the graphitic form of C3N4 [46];  the development of new processes, such as thesolvothermalpreparation of c-BN [47] at lower P, T than previously;  the shaping of materials (sintering at low temperatures [44] or the preparation of microcrystallites [48]).

Conclusions

During the last 50 years, high pressures have been used fruitfully in solid-state chemistry as atool for studying the physico-chemical properties of materials in extreme conditions, for stabilizing new materials able to help in improving our basic knowledge of the solid state, and to lead to functional properties for industrial applications.

Table 1 .

 1 Fe, Lo, Ni, Cu: unusual oxidation states of transition metals stabilized under high oxygen pressures.

	I r o n


Claude Kanyou
email: kanyouclaude@gmail.com

Etienne Kouokam

Yves Emvudu

Structural network analysis: Correlation between centrality measures

Keywords: Social Network Analysis and Mining, Network structures, Structural analysis, Centrality Measure, Structural correlation, Infectious disease spread

Centrality measures are important in network analysis and mining. The correlation structures between the centrality measures are subject of the structural network analysis to improve the understanding of the process in that network. The aim of this paper is to extract useful hidden structural information about the structural correlations between centrality measures study. With Data Science analysis and visualization technics, we propose a structural analysis and visualization of the Pearson correlation between centrality measures. Overall experimentations, the same results are shown in 31 social networks with different topologies. The results show the existence of a strong positive and structural correlation between degree and closeness centrality, degree and betweenness centrality, degree and eigenvector centrality. However, we observed a strong positive and non-structural correlation between eigenvector and betweenness centrality, betweenness and closeness centrality, closeness and eigenvector centrality. Furthermore, we suggest some structural implications of these centrality measures in a network. Finally, we identify influential nodes and their state of evolution that build an effective minimization policy for an infectious disease spread. With the strong positive and structural correlations observed, in large networks, high complexity centrality measure can be approximated by low complexity such as degree centrality.

I INTRODUCTION

Networks are ubiquitous and network science has revolutionized research in the complex system. They are flexible tools for the modeling and studying complex systems [START_REF] Pellis | Eight challenges for network epidemic models[END_REF]. In the network, various explicit/local or implicit/global links connect the nodes at different structural levels. Thus the centrality measures have interest in the network where the understanding network structure allows to improve a better control of the micro and macro propagation [START_REF] Keeling | Modeling Infectious Diseases in Humans and Animals[END_REF], [START_REF] Danon | Networks and the epidemiology of infectious disease[END_REF] and even improve predictions. In public health, the infectious diseases spread through nodes in a contacts network. Some nodes have a highspreading capacity than others. Thus, the Social Network Analysis and Mining (SNAM) is useful for the modelling and the study of the network by typical analysis based on centrality measures. This in order to identifying influencer and structural nodes to build a policy to mitigation the propagation, [START_REF] Enright | Epidemics on dynamic networks[END_REF], [START_REF] Pellis | Eight challenges for network epidemic models[END_REF], [START_REF] Keeling | Modeling Infectious Diseases in Humans and Animals[END_REF].

The basis centrality measures like degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality have been studied in SNAM. Analysis, modeling, and interpretation of these centrality measures and their correlation has a great interest. In this study Newman-Watts-Strogatz and other social networks have been used to emphasize this interest.

The aim of this paper is to extract useful hidden structural information from social networks to maximize insight. For the degree, betweenness, closeness, and eigenvector centrality measures, the following are proposed: (i) An analysis and interpretation with an emphasis in the context of an infectious disease spread; (ii) A structural analysis and visualization of the Pearson correlation to emphasize the existence of a strong positive and structural or non-structural correlation; (iii) Some structural implications of centrality measures in a network.

The rest of this paper is organized as follows: the section 2 reviews the previous works, the section 3 presents the centrality measures study with interpretations and limits in area of SNAM. The methodology used is presented in the section 4, while the section 5 presents the empirical experimentations, results and discussion. We conclude in the section 6.

II RELATED WORK

The correlation between centrality measures and interpretation is studied from different perspectives in the literature. In [START_REF] Valente | How correlated are network centrality measures? Connections[END_REF], the authors have shown the existence of a strong correlation between eigenvector centrality and degree centrality, with a mean correlation of 0.92 over 58 networks.

Moreover, the correlation between centrality metrics and their application to the opinion model is studied in [START_REF] Li | Correlation between centrality metrics and their application to the opinion model[END_REF]. Authors have shown that betweenness, closeness centrality, and the components of the principal eigenvector of the adjacency matrix are strongly correlated with the degree, the 1st-order degree mass, and the 2nd-order degree mass, respectively, in both network models and real-world networks.

In [START_REF] Meghanathan | Correlation analysis between maximal clique size and centrality metrics for random networks and scale-free networks[END_REF], author has examined the correlation analysis between maximal clique size and centrality measures for random networks and scale-free networks. Furthermore, [START_REF] He | Correlation of eigenvector centrality to other centrality measures : Random, small-world and real-world networks[END_REF] have investigated the correlations between eigenvector centrality and five centrality measures including degree, betweenness, cluster coefficient, closeness and distance centrality in various types of networks. Their analysis shown that there was strong correlation between the degree centrality and the eigenvector centrality.

The rank correlation between centrality metrics in complex networks is studied in [START_REF] Shao | Rank correlation between centrality metrics in complex networks: An empirical study[END_REF]. Authors have investigated the correlation between centrality metrics in real networks, and find that the betweenness occupies the highest coefficient, closeness is at the middle level, and eigenvector fluctuates dramatically.

However, with Data Science analysis and visualization technics, by extracting useful hidden information, we propose a structural analysis and visualization of the Pearson correlation to emphasize the existence of a strong positive structural or non-structural correlation between centrality measures study. Moreover, we show the limits of centrality measures, the stakes, and interpretation of these centrality measures in the context of infectious disease spread. Structural implications of these centrality measures on the network structure have been suggested.

III THE CENTRALITY MEASURES, INTERPRETATIONS AND LIMITS IN AREA OF SOCIAL NETWORK ANALYSIS

In network structural analysis, centrality measures are widely used to determine which nodes are important in a network. They constitute the structural importance of nodes in the network [START_REF] Wasserman | Social Network Analysis : Methods and Applications (Structural Analysis in the Social Sciences)[END_REF]. The connectedness network is decisive for the usefulness of an centrality measure that is useful in the context of an infectious disease spread where the network is generally connected. In our application case of infectious disease spread, we are interested in the case of the non-oriented network. In the following definitions, we consider G = (V,E) a graph/network by its adjacency matrix A, with n nodes and m links.

The degree centrality

The importance of a node is determined and increases with the number of its neighbors. For a node v i ∈ V it is defined by [START_REF] Freeman | A set of measures of centrality based on betweenness[END_REF]:

C d (v i ) = n j=1 (a ij ); C * d (v i ) = 1 n-1 n j=1 (a ij ).
Here C d * (v i ) represents the standard notation. The degree centrality provides information only in the local popularity. This determines which nodes can quickly and directly propagated the virus in a strictly local area. Regarding time complexity analysis of the degree centrality, with A, we obtain θ(n 2 ). However, degree centrality becomes limited in a global vision of the network analysis.

The closeness centrality

This measure defines the global "close" distance of a node to other nodes [START_REF] Freeman | A set of measures of centrality based on betweenness[END_REF]. For a node v ∈ V it is defined by: C

c (v i ) = 1 t∈V d G (v i ,t) ; C * c (v i ) = n-1 t∈V d G (v i ,t) . Here C c * (v i )
represents the standard notation, and d G (v,t) = d G (t,v) the distance between two nodes v and t. This is important to further understand information dissemination or propagation. In a social network, this measure means that an node is important if he can easily reach a large number of person with a minimum size in the paths. However, some values the closeness centrality would be undefined since geodetic distances between some nodes would not exist. Regarding time complexity analysis of the closeness centrality, with the Dijkstra algorithm, we obtain θ(n 2 ). This complexity can be reduced to θ(n + m).

The betweenness centrality

In this concept a node is most important or influential since it is necessary to cross it and go from one node to another. For a node v ∈ V it is defined by [START_REF] Freeman | A set of measures of centrality based on betweenness[END_REF]:

C b (v i ) = s̸ =v i ̸ =t∈V σst(v i ) σst ; C * b (v i ) = 2 (n-1)(n-2) s̸ =v i ̸ =t∈V σst(v i ) σst . Here C b * (v i ) is the standard notation σ st (v)
is the total of shortest paths between nodes s,t that pass through nodes v, and σ st is the total of shortest paths between nodes s and t. Nodes with higher score control the flow of transmission to impact the robustness in the network. These nodes play the role of bridges, hubs, critical links as well as cohesion between other nodes. Therefore, nodes with very high often join cohesive regions of the network, called communities. However, zero values of betweenness centrality can be observed in the case of a weakly connected network. Concerning time complexity analysis, it takes θ(n 3 ) time with the Floyd-Warshall algorithm. On a sparse graph, Johnson's algorithm taking θ(n 2 log(n) + n * m), and even better, to θ(n * m) for unweighted graphs.

The eigenvector centrality

The eigenvector centrality provides global information that an influential node is connected to others well-connected nodes in a network [START_REF] Bonacich | Some unique properties of eigenvector centrality[END_REF]. For a node v i ∈ V it is define by: λC e (v i ) = n j=1,i̸ =j a ji C e (v j ), where λ denotes an eigenvalue, a ji is an entry of A, C e is the eigenvector centrality. Assuming x = (C e (v 1 ), ... , C e (v n )); formal expression of the eigenvector centrality becomes: Ax = λx where computation implies to solve: AC e = λC e . This measure presumes that not all connections have the same importance by taking into account not only the quantity but specifically the quality of these connections. Therefore, it represents the state towards which each node v i evolves in the network. For j from 1 to n, each element a ij is the state of the links or the evolution of interactions of one node with other nodes in the network. The eigenvector is a preferred direction unchanged by A, while λ represents the stretch or compression factor in that direction. However, the eigenvalues and eigenvectors of an antisymmetric matrix can be complex. The eigenvector centrality takes time complexity at least in polynomial time.

IV METHODOLOGY

We used the classical SNAM methodology in a process of six main orderly generic steps Kazienko ( 2018 The introduction of this DataFrame facilitates extraction of hidden useful information from networks in our dataset by structural analysis, representation, and interpretation. After around 100 simulations, cross-interpretation of data visualization results has been made to be able to focus on the structural correlation and interpretation between centrality measures studied.

V EXPERIMENTATIONS, RESULTS AND DISCUSSION

Empirical assumptions and experimental configurations

(i) Nodes and links are weakly organized at the global level; (ii) By definition, degree centrality is considered as an independent variable at the local level, while closeness, betweenness, and eigenvector centrality are dependent variables at the global level. An Intel core i7, 8GB RAM; the Anaconda 3 software with the open-source NetworkX library was used and other data scientist libraries like Numpy, Pandas, Scipy, MatplotLib, and Seaborn.

Results and discussion

This study has been experimented on 31 social networks with different structural characteristics show in the figure 1 where we applied specific data analysis and visualization technics in the dataset over around 100 experimentation series. Therefore, with the main Newman Watts Strogatz network, the figure 2 shows the front portrait and the figure 3 shows the heatmap diagram.

In the figure 3 "dec" means degree centrality, "bsc" betweenness centrality, "clc" closeness centrality and "evc" eigenvector centrality. In the figure 2 and figure 3 we observe a quantitative and qualitative values of very strong positive structural correlation between the four centrality measures study. In fact, betweenness, closeness, and eigenvector centrality depend to the degree Overall the couples of centrality measure, the insight of the figure 4 show the existence of a very strong positive and structural correlation between degree and closeness centrality, degree and betweenness centrality, degree and eigenvector centrality. However, there is the existence of a strong positive and non-structural correlation between eigenvector and betweenness centrality, betweenness and closeness centrality, closeness and eigenvector centrality. The quality of the strong positive correlation between the pairs of centrality measures studied shows a strong positive structural dependence between these centrality measures. This correlation also reflects a similarity of relationships between centrality measures at the latent factors at structural levels ranging from local to global. However, the structure of correlations between centrality measures is not always orderly. These results confirm the hypothesis that the local structure strongly influence the global structures so that the global structures depend and emerge from the local structures. With the same experimentation protocol, overall the 31 social networks at different topologies, the same results were observed. Moreover, in the figure 5 we propose that the Venn diagram can be used to understand the implication of the strong positive and structural correlation between centrality measures studied. With the works in [START_REF] Daokun | Network representation learning: A survey[END_REF], by the strong positive structural or non-structural correlation between degree, closeness, betweenness, and eigenvector centrality; we propose the figure 5 to categorize the structural implications of the centrality measures studies in the network structure. Therefore, it is argued that: the degree centrality influences the microscopic structure at the local proximity; closeness and betweenness centrality influence the mesoscopic structure as the structural role proximity and intracommunity proximity; eigenvector centrality influences the macroscopic structure at the network whole properties. Hence, we observed that the local structure has a great influence on the network global structure.

Identify influential nodes insight to the critical nodes from which a virus spreads with a highspreading capacity in the network. This is useful to build an effective policy that mitigates infectious disease spread such as the priority vaccination, quarantine.

VI CONCLUSION

In this paper, we address the network structural correlation analysis problem using centrality measures in the context of the infectious disease spread. We focus on the correlation structures between centrality measures in the context of an infectious disease spread. We have studied the structural correlation between degree, closeness, betweenness, and eigenvector centrality. Here, network connectedness limits were established for these centrality measures, we suggest their issues, specific interpretations, and limits in the context of infectious disease spread. By extracting useful hidden structural information, we propose a structural analysis and visualization of the Pearson correlation between centrality measures study. The results show the existence of a strong positive and structural correlation between degree and closeness centrality, degree and betweenness centrality, degree and eigenvector centrality. However, a strong positive and non-structural correlation was observed between eigenvector and betweenness centrality, betweenness and closeness centrality, closeness and eigenvector centrality. This correlation shows a strong positive structural dependence between these centrality measures that reflects a similarity of relationships between centrality measures study. However, the structure of correlations between centrality measures is not always orderly. For any node in the network, whatever the centrality measure, the higher is its score, the more important or structural influencer it becomes, the more it occupied a strategic propagation position on the network; and the more it increases the likelihood or risk of propagation. This can be used to build an effective policy that mitigates infectious disease spread. For the structural implications of these centrality measures, the degree centrality is related to the microscopic structure, closeness centrality and betweenness centrality to the mesoscopic structure, and eigenvector centrality to the macroscopic structure of the network. Further research attempts can be carried out to explore other structural properties in network distribution and segmentation. Also, we can experiment the gain insight with the use of the systemic approach to improving the general and the whole causal interpretation of the positive and structural correlation.

  ): (i) Problem definition; (ii) Data gathering and preparation; (iii) Social network modelling; (iv) Knowledge extraction; (v) Evaluation; (vi) Interpretation and deployment. We have enriched this with the use of descriptive, diagnostic, predictive, and prescriptive analysis. A specific DataFrame has been built.

Figure 1 :

 1 Figure 1: The synthesis of the structural information about the networks in the dataset

Figure 2 :

 2 Figure 2: The front portray with linear regression between degree, closeness, betweenness and eigenvector centrality

Figure 3 :

 3 Figure 3: The heatmap diagram of the Pearson correlation between degree, closeness, betweenness and eigenvector centrality

Figure 5 :

 5 Figure 5: The implication of the correlation centrality measures in the network structure


Gonzalo Romero-García
email: gonzalo.romero@ircam.fr

Corentin Guichaoua
email: corentin.guichaoua@ircam.fr

Elaine Chew
email: elaine.chew@ircam.fr

A MODEL OF RHYTHM TRANSCRIPTION AS PATH SELECTION THROUGH APPROXIMATE COMMON DIVISOR GRAPHS

We apply the concept of approximated common divisors (ACDs) to estimate the tempo and quantize the durations of a rhythmic sequence. The ACD models the duration of the tatum within the sequence, giving its rate in beats per minute. The rhythm input, a series of timestamps, is first split into overlapping frames. Then, we compute the possible ACDs that fit this frame and build a graph with the candidate ACDs as nodes. By building this graph, we transform the quantization problem into one of path selection, where the nodes represent the ACDs and determine the note values of the transcription and the edges represent tempo transitions between frames. A path through the graph thus corresponds to a rhythm transcription. For path selection, we present both an automated method using weights for evaluating the transcription and finding the shortest path, and an interactive approach that gives users the possibility of influencing the path selection.

INTRODUCTION

Many techniques have been proposed for the problem of rhythm transcription and quantization: some build a rhythm tree [START_REF] Agón | Kant: a Critique of Pure Quantification[END_REF][START_REF] Ycart | A Supervised Approach for Rhythm Transcription Based on Tree Series Enumeration[END_REF], some use probabilistic models and Monte Carlo pruning [START_REF] Cemgil | Monte Carlo methods for tempo tracking and rhythm quantization[END_REF], and others use signal processing methods like autocorrelation [START_REF] Brown | Determination of the meter of musical scores by autocorrelation[END_REF]. Here, we present yet another approach to the problem using the notion of approximate common divisors (ACDs). This mathematical object, stemming from cryptography [START_REF] Howgrave-Graham | Approximate Integer Common Divisors[END_REF], is related to the rhythmic notion of tatum, which will be used to transcribe a series of onset times into a musical rhythm. A similar technique has been used to find the greatest common divisor of an inter-onset interval histogram [START_REF] Seppänen | Computational Models of Musical Meter Recognition[END_REF]. The difference in our approach is that we consider also common divisors that are not the greatest, and we build a graph such that each candidate common divisor maps to a node in the graph.

This approach was first explored in [START_REF] Romero-García | Rhythm Transcription and Characterization for Performed Music and Arrhythmia Sequences[END_REF] to automate the transcription of a rhythmic sequence in expressive music performances and arrhythmic heartbeats into musical notation in the context of the ERC project COSMOS (http: //cosmos.ircam.fr). Here, we describe the original problem and algorithmic approach, which addresses a monophonic rhythm input, then propose an extension that can handle polyphonic input.

The main observation driving the algorithm design is that a given rhythmic input may have several transcription possibilities. We model the options using a graph where each path corresponds to a possible transcription. The choice of the path may be automated but it may also be selected interactively with a human in the loop. We will present both approaches and consider how they might coexist and cooperate within the same framework.

This paper is organised as follows: Section 2 gives an introduction to approximate common divisors, explaining how they can be used to quantize a rhythm by creating a temporal grid. After defining ACDs, we first address the problem of transcribing monophonic rhythms in Section 3, then broaden the description to the transcription of polyphonic rhythms in Section 4. While both cases can be approached using the same technique, they differ in how the input is split. Finally, in Section 5, we propose a graphical user interface that allows a user to select the path and thus influence the transcription result.

APPROXIMATE COMMON DIVISORS

The notion of an approximate common divisor (ACD) has been studied by Howgrave-Graham in [START_REF] Howgrave-Graham | Approximate Integer Common Divisors[END_REF], with integral ACDs in the context of cryptography. In this paper, we will allow ACDs to take on real values with a slightly different definition adapted to the purpose of rhythm transcription. Let us then provide an exposition on the ACD finding problem, presenting it in a form suitable for the rhythm transcription context, working our way up from the problem of finding (exact) common divisors.

The ACD finding problem in R

The common divisor finding problem in N is that of finding the common divisors of a series of N natural numbers,

D = (d 1 , d 2 , ..., d N ) ∈ N N . ( 1
)
This problem is equivalent to finding the numbers a ∈ N such that ∀n ∈ {1, ..., N },

d n ∈ aN ≜ {am ∈ N : m ∈ N}.
This problem, which is well posed in N, can be extended to R with some adaptations. First of all, we will allow both a and the durations d n to be positive real numbers, i.e.: a > 0, d n > 0, ∀n ∈ {1, ..., N }. Then, we define the a-grid as

aZ ≜ {am ∈ R : m ∈ Z} . (2)
This new framework allows us to extend the problem to a series of timestamps,

T = (t 0 , t 1 , ..., t N ) ∈ R N +1 , (3)
where t 0 = 0 and ∀n ∈ {1, ..., N }, t n = t n-1 + d n . Here, we will work with timestamps, but the definitions work for both timestamps and durations.

In practice, the timestamps are expressed in some time unit, typically MIDI ticks or seconds, and we search for numbers a > 0 such that t n fits into the a-grid ∀n ∈ {1, ..., N }. In musical terms, a is the duration of a certain note value that functions as the tatum of the rhythmic sequence and the numbers m n such that

t n -t n-1 = d n = am n
are the multipliers of a to produce the d n 's.

Definition of approximate common divisors

In practice, unless algorithmically generated, durations are rarely if ever exact multiples of a non-trivial divisor. We thus introduce some flexibility into the common divisor finding problem through a threshold τ > 0. We now relax the notion of fitting the a-grid by requiring timestamps to be within the threshold τ of the a-grid, i.e.:

ϵ(t n , aZ) ≜ min m∈Z |t n -am| ≤ τ , (4)
∀n ∈ {0, ..., N }, where ϵ(t n , aZ) is the closest distance between the timestamp t n and the a-grid. Since we require all the timestamps to be within the threshold, τ , of the agrid, Equation 4 thus implies that the maximum distance between the timestamp series T and the a-grid, ϵ T (a), is also within the threshold τ , i.e.,

ϵ T (a) ≜ max n ϵ(t n , aZ) ≤ τ . (5)
Figure 1 shows the fit between a timestamp series T = (0, 0.98, 1.52), given in seconds (s), and the 0.5 s-grid with a threshold of 0.05 s. We will use the threshold τ = 0.05 s for all remaining examples. 0.0 0.5 1.0 1.5 Time (s) timestamps threshold With these definitions in place, we now formally define ACDs for a timestamp series T .

Definition 1 Let T = (t 0 , t 1 , ..., t N ) ∈ R N +1 be a timestamp series. a > 0 is an approximate common divisor (ACD) of T with threshold τ > 0 if 1. ϵ T (a) ≤ τ ; and, 2. ϵ T has a local minimum at a.

The first condition ensures that the ACD satisfies the threshold requirement for the timestamp series T . The second condition makes the set of ACDs discrete by selecting the ACD that minimizes the error within each interval {a ∈ (0, ∞) : ϵ T (a) ≤ τ }.

In practice, since a 2 is an upper-bound for ϵ T (a), we will see an increasing number of ACDs as a approaches 0, most of which are irrelevant since they are too small. This is why we will choose a lower bound for a, say 0.2 s.

Given a timestamp series T = (0, 0.98, 1.52) s, we plot the function ϵ T for the range a ∈ [0.2, 1.0] as shown in Figure 2. A horizontal line marks the threshold τ = 0.05 s and dots the ACDs. We compute ϵ T (a) at steps of 1 ms. Since the definition of the ACDs involves local minima, the resolution of the computations can influence the results; in this paper, the precision is set to the millisecond. The computational complexity of the ACDs is linear with respect to the length of the timestamp series and the number of candidate values for a. The code uses Python libraries NumPy [START_REF] Harris | Array programming with NumPy[END_REF] and SciPy [START_REF] Virtanen | SciPy 1.0: fundamental algorithms for scientific computing in Python[END_REF] and is available online1 .

Vectors associated with an ACD

The point of a number a being an ACD of a timestamp series T ∈ R N +1 with threshold τ is that there is an associated vector

M = (M [0], ..., M [N ]) ∈ Z N +1 (6)
where M [n] = arg min m∈Z |t n -am|, that consists of the integer values of the grid that fit the timestamps. Setting T = aM ∈ R N +1 , we have

|| T -T || ∞ = ϵ ≤ τ , (7)
where ϵ = ϵ T (a) will be called the approximation error.

We can think of T as the approximated timestamp series, which will never be more than ϵ off from the original. Moreover, we can deduce the integer durations

∆ = (M [n] -M [n -1]) N n=1 ∈ Z N (8)
that satisfy the inequality

|| D -D|| ∞ ≤ 2τ , (9)
where D = a∆ ∈ R N are the approximated durations.

In the example showed in the Figure 2 we get a 1 = 0.5 s a 2 = 0.312 s a 3 = 0.25 s ϵ 1 = 0.02 s ϵ 2 = 0.044 s ϵ 3 = 0.02 s with a i being the ACDs and ϵ i being the corresponding errors, ϵ T (a i ). We also retrieve the integer vectors

M 1 = (0, 2, 3) M 2 = (0, 3, 5) M 3 = (0, 4, 6) ∆ 1 = (2, 1) ∆ 2 = (3, 2) ∆ 3 = (4, 2) .
The next section will show how to use ACDs and the resulting vectors to transcribe monophonic rhythms.

MONOPHONIC TRANSCRIPTION

Let us first pose the problem of monophonic transcription in a way adapted to the formalism proposed for ACDs: when we have a monophonic rhythm, for example one produced by an instrument playing one note at a time, we can assume for simplicity that the release of each note occurs at the same time as the onset of the following note.

Moreover, since two onsets cannot occur at the same time, integer durations should not be 0. This means that the ACDs should be small enough to prevent this from occurring, say by requiring that the minimum candidate ACD be smaller than the smallest duration we wish to transcribe.

Let us consider, for instance, the rhythm

ˇ" ˇ" = ˇ" ˇ" ‰ = = ˇ" ˇ"
played at a tempo of ˇ" = 60. The exact timestamp series and durations will thus be T = (0, 1, 1.5, 2, 2.75, 3, 4) s (10) D = (1, 0.5, 0.5, 0.75, 0.25, 1) s ,

where we consider the onsets of all notes and the release of the last note.

If humans were to play this rhythm, they would deviate slightly from these timestamps. Throughout this section, we consider a human realisation of this rhythm given by T = (0, 1.018, 1.531, 2.061, 2.888, 3.179, 4.286) s (12) D = (0, 1.018, 0.513, 0.53, 0.827, 0.291, 1.107) s . (13)

Let us show how one can transcribe this timestamp series into note values using ACDs.

Frames of a timestamp series

As mentioned, the ACD definition can result in many very small ACDs, which are not very interesting for music transcription because they imply too fine of a time resolution and make the note values too large. We therefore only consider ACDs above a lower bound of 0.2 s in the example.

Putting a lower bound on ACDs will imply that, if the threshold is small enough, some timestamp series may not have any ACD. In the musical context, this can be thought of as a player that deviates from a metronome, the metronome playing the role of the grid with the ACD being the time interval between two beats2 . But this can be overcame by splitting the timestamp series into smaller blocks.

Given a timestamp series T ∈ R N +1 and a frame length L ∈ N, ∀n ∈ {0, ..., N -L + 1}, the frame F n of length L given by the vector

F n = (t n , t n+1 , ..., t n+L-1 ) ∈ R L . ( 14
)
We now focus on frames of length 3 and on finding their ACDs. It is important to note that we need to shift our frame so that one of its timestamps is 0 in order to adapt the ACDs to that frame. We will then have

F n = (t n -t i , t n+1 -t i , ..., t n+L-1 -t i ) ∈ R L , ( 15
)
where i ∈ {n, ..., n + L -1} is the centering index.

For instance, if we take the first frame of the T defined in [START_REF] Bresson | OpenMusic: visual programming environment for music composition, analysis and research[END_REF], F 0 = (0, 1.018, 1.531), we have the ACDs and the integer durations

a 0 0 = 0.51 s a 0 1 = 0.255 s a 0 2 = 0.212 s ∆ 0 0 = (2, 1) ∆ 0 1 = (4, 2) ∆ 0 2 = (5, 2) ,
which give us three ways of transcribing the first two note values that are

R 0 0 = ˇ" ˇ" ( R 0 1 = ˘" ˇ" R 0 2 = ˘" ˇ" ( ˇ" ,
where the unit is ˇ" ( .

Consistency across frames

Using the approach of frame-wise transcription, we need to impose some consistency across frames. Because of the way we have defined the frames, consecutive frames have two overlapping timestamps so one duration is shared between the two frames. For a coherent transcription, the common duration should be the same.

For instance, if we take the shifted second frame of T , F 1 = (1.018, 1.531, 2.061) -1.018 = (0, 0.513, 1.043), we have the ACDs and the integer durations

a 1 0 = 0.519 s a 1 1 = 0.259 s ∆ 1 0 = (1, 1) ∆ 1 1 = (2, 2) .
Since the frames F 0 and F 1 share the second and first duration, respectively, they should satisfy the consistency condition expressed as the equation

∆ 0 k0 [1] = ∆ 1 k1 [0] (16)
for them to be consistent. This equation will be satisfied by certain pairs (k 0 , k 1 ) but not by others. This consistency requirement motivates the construction of the graph that is presented in the next section.

The ACD graph

For each frame, we model each ACD as a node. Then, we add an edge from an ACD of a frame to an ACD of the next frame if the integer durations are consistent. If we do this for the timestamp sequence from [START_REF] Bresson | OpenMusic: visual programming environment for music composition, analysis and research[END_REF], we obtain the graph shown in Figure 3.

(

(1, 5)

Figure 3. Graph of integer durations ∆.

Figure 3 shows how the consistency condition is fulfilled. It shows also how we can choose a transcription by selecting a path in the graph. Suppose we choose the path

(4, 2) → (2, 2) → (2, 3) → (3, 1) → (1, 4) , (17)
then by merging the common integer durations, we get the rhythm (4, 2, 2, 3, 1, 4). If we set the unit to be ˇ" ) , we will recover the rhythm ˇ" ˇ" = ˇ" ˇ" ‰ = = ˇ" ˇ" . We can get different transcriptions by selecting different paths, which allows transcription to be framed as an interactive task. However, we may be interested in automatic transcription and, in this sense, we may assign weights to the edges in order to have a notion of the shortest path and the ensuing notion of the best transcription. In the following section we will propose a way of assigning weights to the edges based on the notion of tempo variation.

Assigning weights to the edges

In the first instance, we may choose to assign weights to the edges by weighting them by some function of the error ϵ. This will imply that better transcriptions are the ones that have ACDs that more closely fit the timestamps. Since the error is associated to the nodes instead of the edges, and path finding problems typically have weights assigned to edges rather than nodes, we must decide if the error is associated with the incoming or outgoing edge.

Whereas this is a valid approach, we propose another way of weighting the edges based on tempo variation. To illustrate this concept, we will use Figure 4. When we go from a node to another via an edge, we are changing the duration of the ACD and in so doing, we are varying the tempo. For example, if we go from 0.255 s to 0.259 s we are decreasing the tempo from ˇ" ) = 235 to ˇ" ) = 232, thus reducing the speed by 2 %. This is not a big change, whereas if we consider instead the transition from 0.389 s to 0.241 s, the tempo increment is 61 %.

The tempo variation between two ACDs, measured in percent, may be defined as

δ(a 1 , a 2 ) = 100 a 1 a 2 -1 (18)
and we say that the speed from a 1 to a 2 has increased by δ(a 1 , a 2 )% if δ(a 1 , a 2 ) > 0 and decreased by δ(a 1 , a 2 )% if δ(a 1 , a 2 ) < 0. Notice that we divided a 1 by a 2 because we consider speed rather than duration; the two are inversely proportional. We can then set the weight associated with an edge as a function of δ(a 1 , a 2 ). In this case, we choose the logarithmic distance between a 1 and a 2 , which is defined by

d log (a 1 , a 2 ) ≜ log 2 a 1 a 2 = log 2 a 2 a 1 . ( 19
)
This has the property that it is symmetric and returns a value of one for a ratio of 2 : 1.

Figure 5 shows the ACD graph of T weighted by the logarithmic distance.

Shortest path problem

As we have a weighted graph, we can consider the shortest path problem and its corresponding transcription, the best one according to the parameters. The shortest path problem is defined by giving two nodes of the graph and finding the path that links them and has minimal weight. However, in our case, we may have multiple nodes for the first and last frames which makes ambiguous which nodes to choose. This is easily solved by adding an artificial source at the beginning that connects to all the ACDs of the first frame and an artificial sink connected to all the ACDs of the final frame. We may associate the weight 0 to these edges and then we will have a well posed shortest path problem. This is illustrated in Figure 6. Since the graph is directed and acyclic, the shortest path can be readily computed; indeed, using the topological sorting of a directed acyclic graph, we have an algorithm in linear time [START_REF] Cormen | Section 22.4: Topological sort[END_REF]. We can see that we recover the path from (17) that gives the rhythm ˇ" ˇ" = ˇ" ˇ" ‰ = = ˇ" ˇ" . As pointed out, the shortest path depends directly on the weights, and their selection highly influences the outcome. We have presented weights based on the tempo variation or on the error ϵ, but there are many other approaches and combinations thereof, like selecting weights to avoid complex durations that are inelegant to write down, like 5, 11, 13... Conversely, if we value precision above notation clarity or physical reproducibility, we can assign a large weight to the error and allow very small ACDs. The trade-off between precision and clarity then arises as a parametrization problem that could be tuned via statistics or machine learning given labeled data. The subject of human intervention will be covered in Section 5.

The influence of the frame length

Up to this point, we have been working with frames of length 3. This approach makes sense because we always have a common duration that constraints the paths to be consistent from one frame to the next. However, we may choose a different frame size, for instance if we have very fast rhythms. In this case, consistency will be required of all the integer durations that are common to consecutive frames.

If we have two consecutive frames F n and F n+1 of length L ∈ N, the condition for the integer durations will be

∆ n [1 : L -1] = ∆ n+1 [0 : L -2] . (20)
Here, indexing follows the Python convention where the second index is excluded.

Longer frames enforce the restrictions on ACDs and reduce their numbers. For instance, if we set the frame length to L = 4 for our previous example, we will have the graph shown in Figure 7, where the virtual initial and final integer durations are set to the null list. We see in this case that there is a single valid path that corresponds to the integer durations we had in (17).

POLYPHONIC TRANSCRIPTION

Up to now, we have only considered monophonic rhythms, i.e., rhythms formed by no more than one note at a time. This is useful in the case of singing voice, string instruments (in some cases), winds, etc. but it is insufficient in general and, for instance, for the piano. When there are several voices playing at the same time, we must adapt our method so as to have meaningful results. First, we can no longer conceive of a frame as being a subvector of the time series T of fixed size L; indeed, we may have L timestamps occurring at almost the same time and then the integer durations would all be zero. This leads us to an even deeper question: which timestamps should we consider, only onsets or also offsets? This is a very delicate question in music writing since it addresses directly the question of rests and articulation; indeed, since we no longer consider that the offset occurs at the subsequent onset, we have not only to quantize the onset but also the offset.

In order to simplify things, and acknowledging that our approach to the problem will be incomplete, we will focus on transcribing only the onsets. Then, we can account for the offsets in some sense, for instance rounding them toward the closest element in the grid. We will not tackle this problem, leaving it to future work, since it is affected by numerous factors such as performance, pedal, clean transcription and articulation.

That being said, let us consider the excerpt showed in Figure 8 We shall next attempt to transcribe a human performance of this excerpt, shown as a piano roll in Figure 9, by adapting the technique developed in the previous section.

Polyphonic frames

For polyphonic transcription, we must first define a notion of a frame that will be adapted for polyphonic rhythm transcription. Consider, as previously, a timestamp series T = (t 0 , t 1 , ..., t N ) ∈ R N . As several timestamps can now be separated by a few milliseconds, like in the case of chords, we will set the duration of the frame to L > 0 in seconds. Then, we will define the frame F t starting at t ∈ R of length L as

F t = (t n ∈ T : t ≤ t n < t + L) . (21)
In this way, each frame contains the timestamps of T that are within the time interval [t, t + L). The duration L may be tailored to each piece, or even section or segment, and may be pre-computed. It may even be time-varying according to the speed of the piece.

In order to produce a consistent transcription, we shall overlap frames such that they measure common durations. This will be done by selecting overlapping frames with a hop size of H > 0. In our case, we choose

L = 1.5 s H = L 2 = 0.75 s .
In this way, we arrive at a family of frames

{F mH ⊆ T : m ∈ N} . (22)
Even though the family is formally infinite, as time progresses, the frames will eventually be empty when the input ends, so we can consider only the subfamily of nonempty frames.

Transcribing frames

Now that we have frames, we can apply the same procedure of extracting the ACDs of the frame as in the monophonic case. By default, we will shift the frame by its first timestamp, that is F ′ = F -F [0]. However, as mentioned before, we can shift the frame on any of its timestamps, for instance the middle one, leading to potentially different results. This approach may give negative numbers in the vector M , but is completely valid from the perspective of the ACD computation, which is defined on Z, and from the transcription paradigm.

As previously, we will recover the vectors M ∈ Z Nt and ∆ ∈ Z Nt-1 whose size vary for different t = mH. It is important to emphasise that now there may be several durations in ∆ that are 0, which points to the fact that some timestamps are concurrent. We should abandon the concept of ∆ being the duration of the notes since we are only considering onsets; rather, we will think of ∆ as a consistency vector that will be used to check if two overlapping frames are consistent.

Figure 10 shows MIDI onsets fitting into a 0.218 s-grid shifted by the first timestamp of the frame.

The ACD graph

As before, we will recover the ACDs of each frame to build a graph with ACDs as nodes, linking the nodes if they are consistent. Depending on the hop size, we may have more than two overlapping frames, but then we will only connect ACDs between consecutive frames.

The consistency condition between frames will now involve timestamps that are common to consecutive frames. For two frames F mH and F (m+1)H , the consistency condition will be:

∆ mH [n i -n m ] = ∆ (m+1)H [n i -n m+1 ] (23)
∀n i ∈ N such that t ni , t ni+1 ∈ F mH ∩ F (m+1)H
, where n m ∈ N and n m+1 ∈ N are the first index belonging to F mH and F (m+1)H respectively. Now that we have established the consistency condition, we can plot the ACD graph of the onsets of the excerpt presented in Figure 9. We added a source node and a sink node to complete the path and, as seen in the Figure 11, there is only one possible path linking them. Of course, this will not always be the case. The single path solution was a consequence of the fact that the player followed the rhythm very strictly, in part due to the genre of the music. If we take the vector M built by following this path, we will recover the onsets expected by the score. We will not repeat all the considerations outlined in the previous section regarding path weighting and automating of the transcription model. Rather, in the next section, we will show how transcribing can be done interactively.

INTERACTIVE PATH SELECTION

We have modeled both monophonic and polyphonic transcription as path selection problems. Solving either of these problems can be done automatically if we assign weights to the path, but it can be interesting to select the path interactively by means of a user interface. This transforms the rhythm transcription problem into a multiple output problem, and gives humans the responsibility of selecting their preferred transcription from among a set of possibilities.

Figure 12 shows a prospective user interface for selecting the path. This interface has not been implemented; it presents only a possible graphical layout of the essential functions. It is intended, however, to be implemented in a future version of OpenMusic [START_REF] Agón | OpenMusic : un langage visuel pour la composition musicale assistée par ordinateur[END_REF][START_REF] Bresson | OpenMusic: visual programming environment for music composition, analysis and research[END_REF].

Let us go through an overview of the interface from top to bottom, then left to right.

The piano roll panel

In this panel, the user may see the actual MIDI file and navigate the frames with the arrows. S/he may change the size of the frame and its position by dragging and dropping limits of the frame. Ideally, the user should also be able to edit the MIDI file by selecting the notes and stretching or compressing them; this way, s/he could influence the transcription by editing the timestamps.

The full graph panel

This panel shows the full ACD graph and will be related to the frame selected on the piano roll panel. Its purpose is to keep track of the path selected and to allow the user to navigate through frames so that s/he may have a global view.

The grid panel

This panel presents information from the part of the piano roll where the frame is located. However, instead of MIDI bars, the onsets (and potentially the offsets) will be shown as timestamps so that we may know exactly where each fits in the grid. Timestamps about to be transcribed are plotted in blue and the threshold shown in red. The representation helps the user understanding the scale and the degree of freedom. With the two sliders in the bottom, we can tune which timestamp will be the shift parameter and the resulting grid if we change the ACD.

The focused graph

This sub-graph of the full ACD graph serves shows the ACD selected for each frame. With the cursor, we may select an ACD. The red line connects the current to the previous ACD. Options in the previous and next frames are shown to convey the available paths when choosing an ACD.

This will be one of the main panels on which to act. Indeed, changing the ACD will trigger updates on other panels, and it will be by selecting the ACDs one by one that the transcription will be made. Nevertheless, we can initialise the graph with some optimal or near optimal path, but the interactive part of the process will be to adapt the graph by changing the ACD.

The parameters panel

The parameters panel will let the user select all the parameters needed for the ACDs computation:

1. In order to select the timestamps, the user should choose the frame duration with a slider. Also, s/he can select the overlap via a drop-down list. We may also use a checkbox to determine if the offsets are used in the computations. If that is the case, they will also appear in the grid panel. These parameters will play an important role in which ACDs are recovered.

The rhythm panel

In this panel, users can see the outcome of their choice; by selecting an ACD in the focused graph, both the ACD and the BPM (beats per minute) will be printed in non editable text boxes. The ACD corresponds to the tatum, which can be selected in a drop-down list. The beat, that will usually consist of several tatums, can also be freely chosen in another drop-down list.

In addition, when the ACD is selected, the timestamps will be transcribed into rhythms via a label in the panel. It should be noted that this rhythm corresponds to onsets and should be thought of as a musical grid rather than the durations of the notes, at least when offsets are not taken into account.

CONCLUSIONS AND FUTURE WORK

In this work, we have shown how an extension of the notion of common divisors to a continuous framework leads valuable contributions to rhythm transcription. Their linear computational complexity makes them a suitable and efficient tool for quantizing musical rhythms for large amounts of data. We have also proposed a flexible framework where we considered transcription by splitting the timestamp series into frames and computing ACDs separately. However, when this is done, adjacent frames need to be consistent, and we have chosen to model this via a graph in which ACDs are nodes and edges represent the consistency between frames.

Once this graph is set up, the transcription problem then turns into a one of path selection. Using this paradigm, we proposed two complementary ways of solving the transcription problem: by assigning weights and determining the shortest path, or by allowing humans to intervene in the process by selecting a path that results in a more desirable transcription.

Regarding this last option, we presented a prospective graphical user interface and gave an exposition of its likely elements. This interface will allow the user to directly steer the transcription and may be implemented in musicallyoriented frameworks.

In conclusion, we have proposed a framework that is simple in essence but highly parameterizable. Indeed, there are many ways in which we may affect either the ACD computation or the graph creation and weighting, for example by tuning the weights, limiting the tempo variation or by allowing only certain integer durations. A remaining challenge is the organisation of the durations into groups and measures. By considering the tatum instead of the tactus, we may enter into a low-level quantization that does not account for how the durations are grouped together. The grouping of durations to form rhythm trees has been studied by [START_REF] Ycart | A Supervised Approach for Rhythm Transcription Based on Tree Series Enumeration[END_REF] and is of major importance in OpenMusic [START_REF] Bresson | OpenMusic: visual programming environment for music composition, analysis and research[END_REF].

A way to tackle this may be to build a rhythm tree on top of the ACD graph. How this may be done is outside the scope of this paper but several possibilities can be explored, like choosing different duration groupings and evaluating if they would form a tactus. A challenge for both these approaches is the handling of ties, which could also be broken the user interface.

In the future, the tool we proposed could be incorporated into existing frameworks like OpenMusic either as a function that performs transcription automatically or as a user interface that allows composers and music editors to interact with the transcription. We would then be able to adapt criteria for weighting the graph to tune the standalone part of the algorithm. To that end, we may use several techniques, but one interesting option would be to record data from musicians transcribing rhythms in order to tap into their experience so as to make the output of the algorithm as human friendly and readable as possible.

Figure 1 .

 1 Figure1. The fit between timestamp series T = (0, 0.98, 1.52) s and a 0.5 s-grid for threshold τ = 0.05 s.

  threshold (τ = 0.05 s) ǫT

Figure 2 .

 2 Figure 2. Approximate common divisors of the timestamp series T = (0, 0.98, 1.52) s within the range a ∈ [0.2, 1.0].

Figure 4 .

 4 Figure 4. Graph of ACDs.

Figure 5 .

 5 Figure 5. Weighted ACD graph

Figure 6 .

 6 Figure 6. Shortest path outlined in red.

Figure 7 .

 7 Figure 7. Graph of T with frame length L = 4.

Figure 8 .

 8 Figure 8. Start of the Mozart's Piano Sonata No. 8 in A minor, K. 310 / 300d.

Figure 9 .

 9 Figure 9. Piano roll of a human performance of the excerpt from Figure 8.

Figure 10 .

 10 Figure 10. Timestamps of MIDI onsets fitting into a 0.218 s-grid.

Figure 11 .Figure 12 .

 1112 Figure 11. Graph of the excerpt from the onsets of Figure 9.

2 .

 2 Once the timestamps are selected, the parameters for the computation of the ACD shall be tuned with spinners; these parameters are (a) the error threshold, (b) the lower bound, (c) the upper bound, and (d) the computation resolution.

  from a Mozart sonata.

	&	oe # j	oe	oe™ oe oe	oe™	oe j oe oe #	˙oe n ™	oe
	? & { { ?	oe oe oe oe oe™ oeoe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe j # oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe	oe oe oe oe oe oe oe	oe oe oe oe ˙oe™ oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe™ ™ oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe	oe oe oe oe oe oe ‰ oe oe oe oe oe oe oe

https://github.com/Manza12/TENOR-2022

Note that a metronome usually plays the tactus rather than the tatum; here, the metronome analogy should be thought of in the sense of the tatum.

Acknowledgments

This result is part of the COSMOS project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 788960.).