On the concept of Raga parentage in Carnatic music

The concept of rāga in Carnatic music is based on an ordered set of notes in an octave. Historically rāgas are broadly classified into two sets, namely Janaka (root/parent) and Janya (derived/offspring) rāgas. Every janya rāga is derived from a unique parent. We examine this classification critically and attempt to provide a quantitative basis for such a classification by defining a ‘distance’ between rāgas. The shortest identifies the parentage. Each rāga is defined by a pitch histogram vector in a 12-dimensional space. To achieve consensus, different distance metrics are used in the multi-dimensional space. Using a standard data set (refer to section 4.4), we carry out the distance analysis using entire compositions, which we subsequently fine-tune using only the parts of compositions that contain all the features of the rāga. We also perform an independent analysis for comparing the motif sequences in rāgas. We find that while the conventional method (refer to section 3) is fairly robust, there are exceptions, especially with pentatonic rāgas, and that these exceptions are actively debated in the public domain. Since quantitative methods find it difficult to achieve consensus, we conclude that while a rāga belongs to a family, it does not necessarily belong to a unique parent.


Introduction
Indian classical music is based on the concept of rāga in which melodic outlines and compositions are rendered (Ayyangar, 2019;Krishna, 2016;Shankar, 1999).A rāga, in general, consists of a set of śrutis ordered, usually, in increasing order of the pitch.Following the convention in Carnatic, South Indian music, we simply call these 'notes'.In a majority of rāgas in use, the ascending and descending movements are symmetric.However, this is not always the case leading to a huge combination of scales that forms the backbone of any rāga.The starting note is the tonic which is performer dependent; male and female artists have tonics in different ranges.All other notes are fixed using simple integer ratios ( > 1 and < 2) with respect to the tonic dividing an octave into 12 notes.This has been the norm for more than a century. 1 Indian classical music uses natural scale ratios, unlike the tempered scale where the ratio of one semitone to the next is a constant equal to 2 1/12 .Unlike the natural scale, this constant can never be defined as a ratio of natural numbers and hence is irrational. 2  In this paper, we confine our analysis to Carnatic music, although the main features of the analysis may be applied to Hindustani music (or the North Indian music) as well with minor modifications.In Carnatic music, the necessary, but not sufficient, condition for a rāga is a set of notes or a scale.The scale of a given rāga provides a skeletal frame on which many nuances like movement, oscillation, glide, pull etc., are superposed.In general, the scale consists of seven notes at the most and five at the least. 3There are some exceptions to the sevennote system when the notes in ascending and descending order are different.The rāga is said to have a personality of its own when the scale is embellished by the interaction among the notes (gamakas) and the progression of the notes (gati).While the enumeration of scale and the movement may be precise, the concept of gamakas (Viswanathan, 1977) is not easy to quantify since its enunciation varies depending on the performers.This freedom is the essence of Carnatic music since it enables improvisation emphasising the individual creativity of the performer.
Despite these qualitative features, Carnatic music has evolved a sophisticated system of classification of rāgas whose origins go back to a seventeenthcentury treatise by Venkatamakhi (Raghavan, 1941) called 'Caturdan ˙d ˙iprakāśikā'.In general, a rāga is either a parent (janaka) rāga or an offspring (janya) rāga with a well-defined parentage. 4Sometimes the janaka rāga is also referred to as complete or sampoorna rāga.The completeness refers primarily to the scale.The assignment of parentage, however, is entirely based on the 'closeness' of a janya rāga to its parent.The minimal criterion for closeness is that the notes of the janya rāga must be a subset of the notes of the janaka rāga and the closeness also depends on the perception by the ear.While this is subjective, over time a consensus has emerged for most of the rāgas in Carnatic music.It should be noted that many rāgas have been in use long before this classification system.
We take a critical look at this classification.One of the aims of computational music is to critically analyse ancient and existing practices in music (Clayton et al., 2003).As it turns out, the determination of the janaka and janya relation is accurate in most cases.We first describe the basis of the conventional classification system, followed by the quantitative methods we use to critically analyse this classification using earlier work on rāga recognition.
In Section 2, we present an overview of previous computational research on rāgas.In Section 3, we discuss the ancient Venkatamakhi's Melakarta classification scheme and provide some examples of the possible degeneracies in parentage.In Section 4, we discuss the methodology for obtaining the distance measure with a discussion on resolution and window size (number of pitch samples required) needed.We provide explicit examples of comparison between the conventional classification and the computational analysis in Section 5. A fine-tuned analysis to resolve ambiguities is presented in Section 6.In Section 7, we discuss an alternative method that estimates the closeness between two rāgas by finding the common pitch sequences.In Section 8, we briefly apply the method to some rāgas imported from Hindustani music.The results are then summarised in the last section. 5

Previous research
Previous research has demonstrated that rāga specific information can be obtained from cent-folded pitch 4 The word janaka means father while janya means derived.We have taken the liberty of interpreting janaka as parent and janya as offspring in gender neutral terms. 5The figures given in this paper are shown in greyscale.The coloured plots of these figures can be found in this link: https://tinyurl.com/colorplotshistograms.In Fujishima (1999), the concept of pitchclass profile (PCP) is introduced for chord recognition.PCP is a 12-dimensional vector in a twelve-note scale.The coefficients of the vector give the intensity of each of the notes.In Gómez (2006), the harmonic pitch class profile (HPCP) is formulated by improving upon PCP for identifying tonal similarity between music excerpts.
In HPCP, the frequency of the semitones is multiplied by a weight for each semitone.The HPCP vector has a higher resolution compared to the one-semitone resolution of PCP.Template-matching algorithm on finegrained pitch distribution (FPD) is used in Bozkurt (2008), and Gedik and Bozkurt (2009, 2010) for finding the similarity between makams (similar to rāga in Turkish music).Different variants of pitch histograms are extensively utilised for rāga recognition in Hindustani and Carnatic music.Pitch-class distribution (PCD) (similar to PCP) along with Pitch-class Dyad (bigram) distributions (PCDD) are extracted as features for automatic identification of Hindustani rāgas in Chordia (2006), Chordia and Rae (2007), Chordia andŞentürk (2013), andVelankar et al. (2021).PCDD is the distribution of bi-gram probabilities of notes in a melody.PCD is a 12dimensional vector due to the 12 semitone scale, and hence the number of possible bi-grams is 12 × 12, making PCDD a 144-dimensional vector.In Chordia and Şentürk (2013), along with the aforementioned features, higher dimensional Fine-pitch distribution (FPD) and Kernel-density pitch distribution (KPD) are computed.Instead of assigning a single pitch value to a bin of the histogram, in the case of KPD, typically a windowed kernel function like Gaussian computes the probability densities based on observed samples.An extensive set of classifiers (k-nearest neighbour using L1, L2, L3 and Bhattacharyya distance; Multivariate Gaussian (MVG); Gaussian mixture model (GMM); Parzen window) are trained on these features.Multiple resolution PCP features (12, 24, 36, 72 and 240 dimensions) are computed based on the transcription of notes and weighted by the number and duration of notes in Koduri et al. (2011).PCP features are also extracted without the note transcription along with other features.K-nearest neighbour clustering with Kullback-Leibler distance (KLD) as the distance measure is employed for rāga recognition of 10 Carnatic rāga.Along with the weighted PCP features, continuous pitch histograms with multiple bin resolutions are used in Koduri et al. (2012) and perform very well for automatic recognition of 67 Carnatic and 32 Hindustani rāga.Template matching is also attempted in Koduri et al. (2012) for rāga identification, where rāga templates are obtained from the high-resolution octave folded histograms either by obtaining the values of bins at locations corresponding to the 12 just intonation intervals or by picking the most salient peaks.Similarly, PCP along with n-gram distribution performs well in identifying Carnatic rāgas using Support vector machine (SVM) classifier (Kumar et al., 2014).Likewise, note histograms as features for random forest classifier is used for structural analysis and rāga identification in Hindustani music.Recently Ganguli and Rao (2018) (2020) for finding the distance between rāgas in Carnatic music.In Belle et al. (2009), features like peak, mean, standard deviation and probability of each swara, are added along with FPD and PCP to capture the swara intonation difference between rāgas to better separate rāgas using KLD distance.These swara features along with skew and kurtosis are extracted for the peaks detected from tonic normalised pitch histograms in Koduri et al. (2014), and Koduri et al. (2012).Additionally, in Koduri et al. (2014), context-based swara distribution is also used.Rāga recognition is also done using finite state automata (FSA) (Sahasrabuddhe & Upadhye, 1992) and hidden Markov models (HMMs) (Pandey et al., 2003;Sinith & Rajeev, 2006, 2007).Melodic motifs are characteristic of rāgas.Many papers have tried to automatically discover and find the similarity between melodic motifs or phrases of different rāgas (Dutta & Murthy, 2014a;Gulati et al., 2015;Rao et al., 2014;Ross et al., 2012).Rāga recognition using automatically discovered melodic phrases was attempted by Gulati et al. (2016) and Gulati et al. (2016).Stochastic modelling has been used in rāga recognition with further inputs from the nature of the data sample, repetitive phrases in melodic contours in Ranjani et al. (2011) and Ranjani et al. (2019).While Ranjani et al. (2011) uses a semi-continuous Gaussian mixture model (SC-GMM) to represent the notes of a rāga, Ranjani et al. (2019) uses the property of repetitive sequences for rāga recognition.CNN-transfer learning is used (Sreejith & Rajan, 2021) for rāga identification without using pitch.In Dutta et al. (2015), rāga verification is performed by matching the cohorts (pallavi lines) of different rāgas using a sequence matching technique -the Longest Common Segment Set (LCSS).Rāga identification using pallavi lines of rāgas has been shown to give high classification accuracy for a large set of rāgas (Padmasundari & Murthy, 2017).We have built upon the tools, algorithms and inferences from Dutta et al. (2015), Padmasundari and Murthy (2017) and Viraraghavan et al. (2020) in this work.
In this paper, we adopt two independent approaches, used in matching melodic outlines, to determine closeness between any two rāgas as a precursor to determining the parentage of a given rāga.The first method relies on the pitch probability distribution method, while the second method uses a sequence-matching technique.In either case, the basic data is obtained using pitch contours of rāgas averaged over many performances by different performers.In the pitch histogram method, we use this data sample to construct the cent-folded pitch histogram distribution after tonic normalisation to represent a rāga as a 12 -dimensional vector.The distance may be computed in many ways, and we compare different metrics.While this may suffice for most rāgas, in some cases the distances are too close to determine the parentage.In such cases, we employ well-tested methods used earlier.
The pitch histogram is finetuned by using the refrain lines of the compositions, which contain the essential elements of the rāga (Padmasundari & Murthy, 2017).The distances are computed using the fine-tuned data sample.Alternatively, we also extract the defining sequences of the rāga from the refrain and determine the level of matching with another parent rāga using the Longest Common Segment Set (LCSS) algorithm (Dutta et al., 2015).This gives a score to compare the closeness among the rāgas.The Indian art music rāga recognition dataset6 created by CompMusic (Gulati et al., 2016;Gulati et al. (2016)) is used for analysis in this paper.In addition, where necessary, this data sample is augmented by musical sequences culled from open sources.If a rāga sample is not available, then we have used the available audio recordings in the public domain.Many representative examples of janya rāgas and their janakas as determined by the above methods are given.

Standard classification of ragas
The classification of rāgas used in Carnatic music is rather ancient and has been summarised in a seventeenth-century treatise by Venkatamakhi called Caturdan ˙d ˙iprakāśikā.The rāgas themselves precede this system.In this system of assignment, a janya rāgas has in principle all its notes embedded in the janaka rāga, it could be a subset, or it could be all seven notes but permuted from the linear symmetric order of notes.While this is a minimal requirement, possible degeneracies are resolved by the ear and by consensus evolved over time.The classification works with reasonable accuracy though in some cases when the subset has overlap with many different sets, there could be confusion in the assignment of the janaka rāga.With the advent of music retrieval systems and computational techniques, we are now in a position to take a critical evaluation of this ancient system.
The janaka rāgas are, by definition, the 72 Melakarta rāgas each of which has the same seven notes symmetrically in ascending and descending order.The janya rāgas have either all or a subset of the seven notes in some welldefined order which need not be linearly increasing or decreasing in pitch.To understand this more clearly, we first set the notation that is conventionally used: the 12 notes in an octave7 are denoted as.

S R1 R2 R3 G3 M1 M2 P D1 D2 D3 N3 S G1 G2 N1 N2 2S
which gives 16 possibilities for 12 note positions.The tonic is always denoted by S, while notes P = 3S/2 and S = 2S are fixed.Therefore once the tonic is fixed, all other notes are fixed by their natural scale ratios, as mentioned earlier.The degeneracies between R, G as well as between D, N is needed for counting purposes. 8he janaka is often referred to as a sampoorna or complete rāga.The janaka rāga is then defined by the symmetric ordered sequence S, R, G, M, P, D, N, S .This has seven notes with S as the tonic, and the 8th is simply the beginning of the next octave.Given, S, P are fixed, we have two possibilities M1, M2 for M. We may choose R, G from the four possibilities and similarly D, N from the four possibilities.Thus the total number of combinations for forming such a scale is 4 C 2 × 4 C 2 × 2 = 6 × 6 × 2 = 72.These 72 scales are organised into two groups of 36 containing either M1 or M2.These 36 are further divided into 6 subgroups characterised by the choice of R, G.Each of these subgroups has 6 rāgas corresponding to the choice of D and N.These 72 scales thus form the set of all janaka rāgas from which all other rāgas are derived.The set of these 72 rāgas is called the 'Melakarta' -meaning 'garland of creators'.Each of these 72 scales has a name and an index running from 1 to 72.For example, the 29th Melakarta rāga is the well-known rāga Shankarabharana.The janaka rāgas are defined through their scale alone even though it is only a necessary condition for a rāga.They do have other features, but that is left to the interpretation (mano-dharma) of the performers within some broad set of rules.
Next, we look at the assignment of a janya or derived rāga to its parent rāga.A necessary condition is that the notes of a janya rāga must be contained in at least one of the 72 parent janaka rāgas.However, this condition alone is not sufficient to uniquely assign a parent: • If a janya rāga has all seven notes (seven being maximum) in ascending order and/or descending order, then its janaka rāga is simply one of the 72 Melakarta rāgas with which the janya rāga shares these notes.This works independently of the other features of the rāga like movement, gamaka etc.For example, consider a rāga like Aarabhi, a very pleasant rāga with many popular compositions.The rāga has the following notes:  1.There are some exceptions; for example, rāga Anandabhairavi shares all its major notes with Kharaharapriya, whereas its parent is listed as rāga Natabhairavi which has D1 instead of D2.This comes about because of the occasional use of D1 in Anandabhairavi.We will discuss this issue later.
• If a rāga has six of the seven notes, there could be one to three possible parents as shown in Table 2.If the missing note is P, then the assignment is unique since P is non-degenerate.For example, rāga Sriranjani has a unique parent since it is missing only the note P from the rāga Kharaharapriya.If the missing śruti is M, then the rāga can be assigned to a janaka rāga with either M1 or M2.On the other hand, if the missing note is one of R, G, D, N, then it can have up to three parents depending on the position of the missing note.For example, as shown in Table 2, the rāga Chandrajyoti has two nearest neighbours, both of which qualify to be the janaka rāgas by virtue of sharing all the notes of Chandrajyoti.Only one is identified as the janaka rāga.This ambiguity persists as long as only the notes are compared.Conventionally the assignment of the parent is done by comparing the melodic contours going beyond a mere scale comparison.• It becomes more ambiguous for a pentatonic rāga.In Tables 3 and 4, some examples of symmetric pentatonic rāgas are given.In general, the parentage of a Table 1.Some examples of janya rāgas with seven notes, but with non-linear movement along with their janaka rāgas which are indexed from 1 to 72 according to the scheme noted in the text.Unless the ascending (A) and descending (D) movements are different, only the ascending notes are shown.In general, the identification is unique.However, the occasional use of a note outside the list, like D1 in Anandabhairavi, is confusing for parent identification, as seen above.

Raga Scale Aarabhi
pentatonic rāga may be 2, 3, 4, 6 and 9 rāgas depending on the missing two notes as seen in the tables.In the examples shown, rāga Hindola has two possible parents, which is the minimum number.Rāgas Sunadavinodini, Mohana and Revati have 3, 4 and 6 possible parents, respectively.Rāga Amritavarshini and Gambhiranata have a maximum of 9 possible parents for a pentatonic rāga.
Interestingly, every janya rāga has a unique assignment of the parent in the standard classification, though sometimes contested. 9In Tables 1-4, the conventional choice is also identified.In what follows, we attempt to provide a quantitative measure for such a classification using distance metrics defined on a probability distribution defined for each rāga.

Definitions and methodology
In this section, we outline the methodology adopted in the paper.For any given rāga, we create a sample data set which is averaged over many audio recordings of performances so that the purely subjective element is removed.We combine many melodic outlines in a given rāga after removing the false positives in the indexed data sample.After tonic normalisation and cent folding, a normalised pitch histogram is obtained for each rāga, which at a fine resolution serves as an estimate of the discrete probability Table 2. Janya rāgas with six notes.If the missing note is P, then the rāga has a unique parent, for example, rāga Sriranjani; otherwise, the rāga can have two to three parents as shown.The standard Melakarta assignment of the parent is indicated by a star.

Raga Scale Sriranjani
Table 3. Possible parents of pentatonic rāgas Hindola, Sunadavinodini, Mohana, Madhyamavati, Hamsadwani and Revati, taking into account the common notes in their scales.The first column shows the list of possible janaka rāgas for each janya rāga.The Melakarta classification of the parent is indicated by a star.distribution of the notes that make up the rāga.A rāga is then represented as a point in a 12-dimensional space of notes in an octave.The area corresponding to each note in the pitch histogram gives one coordinate in the Cartesian basis.Some of these coordinates are necessarily zero in a perfect sample since rāgas with more than 7 notes are rather rare.The distance between any two points in this multi-dimensional space measures the kinship of rāgas.

Raga Scale
Table 4. Same as in Table 3, rāgas Gambhiranata and Amritavarshini are pentatonic rāgas.These pentatonic rāgas have maximum degeneracy in parentage of 9 possible parents.Chalanata is identified as the conventional parent of Gambhiranata.Shankarabharana is also a possible parent of the Gambhiranata due to its matching notes with Vedhandagamana.We first study the effectiveness of distance metrics and choose the measure that is suitable for the task.

Probability histogram of pitch and its resolution
Consider the 12 notes in an octave, a subset of which defines the scale of a rāga.A rāga denoted as P, say, in this 12-dimensional space is represented by a point whose coordinates are denoted by p i , i = 1, • • •, 12.These coordinates may be zero or finite depending on the presence or the absence of a particular note and p 1 is always the tonic note S. Furthermore, the set p i is created such that. 12 The probability distribution is obtained as follows: In the simplest case, we choose a sample of a melodic outline (pitch track) in a rāga.The pitch track is extracted using the Essentia toolkit (Bogdanov et al., 2013) used for music information retrieval, with a window size of 40 milliseconds and window shift of 4.44 milliseconds.
The sample is rendered in the cent scale, which helps to stretch the differences between the 12 notes approximately by about 100 cents in the natural scale.The number of occurrences of a given note within a certain resolution is counted.If for the i-th note, this count is n i , then we have where i p i = 1 and 0 ≤ p i < 1.This defines a simple probability distribution of occurrences of notes in a rāga but it does not take into account whether the melodic phrase is ascending or descending and other nuances individually.However, it preserves the integrated effect of all the nuances in the melodic outline.In fact, the probability distribution defines a rāga with the caveat that the data sample chosen is assumed to represent the rāga.This may not always be the case, as we discuss later, and may require significant improvements (Dutta et al., 2015;Padmasundari & Murthy, 2017) in some cases.
The pitch histogram defined in Equation ( 2) is a 12dimensional vector in which each dimension represents a note on the 12 semitone scale.The resolution of each histogram bin in the cent scale is 100 cents in this case.Even though the width of a note is 100 cents, the performer usually sings the note with an average variation of ± 20 cents (Peretz & Hyde, 2003) around the note position.This can increase or decrease based on the way specific notes are played in each rāga.To account for the distribution of the note in the 100-cent range, we compute the histogram distribution of each note within this range.To do this, Equation ( 2) is modified as follows: where n ij is the number of occurrences of note in the j th bin of the i th semitone and p ij is the probability of the i th note's occurrence in the j th bin The total number of occurrences is given by.
where k is the number of bins per note and the resolution of each bin is r = 100/k cents.For example, the normalised pitch histogram is shown in the case of rāga Mohana in Figure 1.The pitch histograms with different bin widths are shown in this figure.The resolution of the histogram is higher when the bin width is smaller, as seen in the figure .In Figure 2, we compare the envelope of rāga Mohana with the envelopes of its four possible parents, assigned simply by the common set of notes, shown in Table 3.

Histogram resolution analysis
The resolution of the pitch histogram depends inversely on the bin width.As shown in Figure 1, when the notes are quantised to 100 cents, the normalised histogram gives the probability of the notes that make up the rāga, while the peaks in the histogram do not correspond to the notes of the rāga.On the other hand, in the higherresolution pitch histograms peaks are clearly visible at the notes defined by the scale of the rāga.
To analyse the distance dependence between rāgas on the resolution, we plot the simple Euclidean distance between rāga Mohana and its likely parents for varying bin widths as shown in Figure 3a.Euclidean is the simplest distance metric and is computed as the Cartesian distance between any two points in the 12dimensional space.The figure shows that the distance stabilises around a resolution of 10 cents in bin width, while finer resolutions of histogram introduce noise.We use a resolution of 10 cents for the bin width for further analyses.
In Figure 4, the peaks of rāga Mohana's histogram are evident in the plot with a bin width equal to 10.The 5 notes used in rāga Mohana are S, R2, G3, P, D2 as seen in the high-resolution histogram plot.The collar or the window size is the width around each note in the histogram used to filter the pitch data corresponding to each note.In Carnatic Music, owing to gamakas pitch values are continuous.We use a collar of ± 20 cents around the peak to compute the density of a given note.The peaks are clearly visible with this collar.
In order to define the probability distribution, we count the number of hits n i within the collar corresponding to each note.To do this, we define the resolution and the width around each note so that the area included in this width defines the number of hits.This eliminates random hits away from the i-th note.The collar size dependence of distances between histograms is illustrated in the case of rāga Mohana and its parents in Figure 3b.It is clear from the figure that a collar of 10-20 cents around each note is sufficient to obtain the distance measure between rāgas.A finer resolution leads to noise, while a larger value may not convey the nuances around a note.In fact, it is well known that normal adults can not differentiate notes when the difference is less than 25 cents (Peretz & Hyde, 2003).
Using the resolution and collar size analyses, we choose the optimum values of bin width and collar size for the other experiments in what follows.

Distance metric analysis
As discussed in section 4.1, every rāga may be defined by a probability distribution.Given such a probability distribution, the distance between or closeness of two rāgas P and Q may be defined in multiple ways.A general overview of distances between probability distributions is given in Cha (2007).The calculations in Table 5 show how different metrics perform on the task of separating rāgas from one another.These include (1) the Euclidean distance (as defined in Hart et al., 2000) and (2) the Bhattacharyya-Hellinger (BH) distance (a combination of the metrics defined in Bhattacharyya (1943) and Hellinger (1909)).The Bhattacharya distance has previously been used for comparing pitch histograms in Chordia and Şentürk (2013), Ganguli and Rao (2018) and Gulati et al. (2016).Manhattan/City-Block (Deza & Deza, 2006), which was shown in Gedik and Bozkurt (2010) to be better than Euclidean distance as a distance metric to compare between pitch histograms.We also calculated (3) Cosine Distance (Deza & Deza, 2006) and (4) the Pearson Correlation Distance (Pearson, 1900), which was used by Ganguli and Rao (2018) and Gedik and Bozkurt (2010) for comparing the pitch histograms.Entropy based measures like (5) the symmetric version of the Kullback-Leibler divergence (KLD) (Kullback, 1959) and ( 6) the Jensen-Shannon distance (JS) (Lin, 1991) are also calculated.The results in  Table 5 show that Kullback-Leiber Divergence (KLD) and Jensen-Shannon (JS) distance give the best separation between rāgas followed by Bhattacharyya-Hellinger (BH) distance.
We compared the best-performing metrics with a Scatter Matrix, which compares the distance between samples in different rāgas with the samples in the same raga and thus provides an estimate of the covariance matrix of the distribution.A good metric for distance estimation should give a smaller value of covariance within the same rāga set and a larger value of covariance with other rāga histograms.Multi-class linear discriminant analysis (LDA) is formulated as an optimisation problem that maximises the ratio of inter-class scatter to intra-class scatter.We compute this scatter matrix so that we may choose the metrics that maximise the ratio of the sum of between-class scatter to the sum of within-class scatter.
In particular, the intra-class-scatter and inter-classscatter for Euclidean distance measure is defined as follows.
where C = number of rāgas, N = total number of samples, μ i = mean of class i and μ = overall mean.The Equation ( 6) is extended to a different distance measure as.
where dist corresponds to the distance metrics used.
From Table 5, it is observed that Kullback-Leiber Divergence (KLD) and Jensen-Shannon (JS) distance give the best separation between rāgas followed by Bhattacharya-Hellinger (BH) distance.For further analysis, we use the following metrics -KLD, JS, BH.The Euclidean metric is also included since it is the simplest, but not necessarily the best, for comparing pitch histograms.

Data set and variance analysis
For each rāga the melodic outlines are collected from different performances including open sources.The Indian art music rāga recognition dataset 10 created by Comp-Music (Gulati et al., 2016;Gulati et al. (2016)) is mainly used for analysis in this paper.This data set contains 40 Carnatic rāgas with 12 recordings per rāga and 30 Hindustani rāgas with 10 recordings per rāga.The rāga recordings outside of this list are downloaded from public sources like Youtube. 11The overall statistics of the dataset used for analysis are given in Table 6. 12Before using the data set, we check for likely wrong labelling leading to false identification.Especially with public domain sources, this is a problem.In order to remove these false positives, we calculate the rāga histogram and the distance between compositions in each 10 Indian Art Music Raga Recognition Dataset (CompMusic): https://comp music.upf.edu/node/328 11List of songs downloaded from Youtube are available in the supplementary notes. 12More details of the data set, the list of all the rāgas, number of songs per rāga and duration of data in each rāga are available in the supplementary notes.
rāga using the chosen metrics.The songs within a rāga that are very distant from the others are considered to be false-positive data points in a rāga.This is directly verified with the data sample corresponding to the outlier rāga to make sure that it is a case of false labelling.Since the sample size is large, removing the outliers does not cause any statistical fluctuation.The histogram of each composition in a rāga is plotted along with the histogram of the rāga, and the compositions with larger distances are verified manually (by a musician), and songs that do not belong are excluded from the analyses.The tonic estimation issues may also lead to peaks not aligning with the notes and having a large distance.The tonic values are also manually corrected by a musician.The data samples mainly contain vocal music.Instrumental music like Veena, Violin and Flute are used only when the vocal recordings are inadequate for analysis.
The distance between a family of rāgas or compositions in the same rāga may be analysed through heat maps.We illustrate this using two commonly used rāgas Shankarabharana and Kalyani.In Figure 5, we have plotted the histogram corresponding to various compositions and the corresponding heat map.In the case of Shankarabharana, it is clear that all compositions are properly aligned with very few outliers.On the other hand, the sample for Kalyani has many false positives, which are seen as bright colour lines in the heat map shown on the left-hand side of Figure 5.It is also evident  from the figure that the range of distances in Kalyani (0-0.18) is almost double that of the Shankarabharana (0-0.1)heat map, due to the presence of false positives.These compositions are then removed from the list of compositions of the rāga, providing a clean sample for rāga distance analysis.

Parentage of well-known janya ragas
We begin this section with a sample of our results for distance metrics with well-known rāgas.The distance is computed using the probability distributions obtained from the full compositions in each rāga.This forms the baseline distance metric for any further improvements.
The simplest rāgas, to determine parentage as noted earlier, are the rāgas with seven notes since by matching the notes with a given janaka rāga we have almost always a unique parent as seen in Figure 6(left) where the distance between the janya and janaka rāga are given with different measures. 13 A seemingly anomalous case is that of the rāga Anandabhairavi.After examining the occurrence of notes and 13 Pitch histograms of janya rāgas with parents are available in the supplementary notes comparing them with the scales of janaka rāgas for maximum overlap, its parent should be Kharaharapriya.However, the distance analysis clearly identifies rāga Natabhairavi as its parent as seen from Figure 6(right).
Interestingly, the standard classification also identifies Natabhairavi as its parent even though the note D is different in these two rāgas.Obviously, the compositions in the performances must have involved D1 note as in Natabhairavi either consciously or otherwise.Nevertheless, it is interesting to see that both the conventional and this analysis agree on the identification of the parent of Anandabhairavi.
In Figure 7, we show some examples of rāgas with 6 notes, as shown in Table 2.All the rāgas shown are symmetric in ascending and descending order.The distance to each likely parent is shown with four metrics.As noted earlier, if the missing note is P, then the parent is uniquely assigned to the rāga with note P added on.For example, the rāga Sriranjani is missing P from Kharaharapriya.Hence its parent is unique.In fact, the pitch histogram of Sriranjani is almost identical to Kharaharapriya except for the peak corresponding to P. On the other hand, rāga Chandrajyoti can have two possible parents, one of which the rāga Pavani, is the parent according to standard classification.However, the distance metrics are not in agreement since Euclidean distance prefers Navaneetham as the parent, whereas the other three distances agree with conventional classification as seen in Figure 7. Obviously, the input data requires a closer inspection.
In general, the maximum number of parents in the 6-note rāgas is three, as shown in the example of rāga Malahari in Table 2.The parent of Malahari in the conventional classification is rāga Mayamalavagowla.These are the two rāgas introduced at the very beginning of the teaching of Carnatic music.As a result, Malahari is always associated with Mayamalavagowla.However, as seen in Figure 7, its parent is Vakulabharana according to the metrics.While there are many compositions in Mayamalavagowla, the other two rāgas have not been explored in as much detail.We shall discuss this further in the next section.
Some examples of pentatonic rāgas are displayed in Figures 8-10.The possible parentage varies from a minimum of 2 to a maximum of 9, as shown in Tables 3 and 4. 14 Because of the degeneracy arising from the possible assignments of the missing notes, there is more room for uncertainty in assigning the parent here.This gets more complicated when the distance measures do not agree.In the case of rāga Hindola, there are two 14 The corresponding pitch histograms are provided in the supplementary materials.possible parents; while the conventional classification prefers Natabhairavi, all the distance measures prefer rāga Hanumatodi as the parent.Similarly, in the case of rāga Sunadavinodini, all the distance measures prefer rāga Kosala, whereas the standard classification prefers rāga Kalyani which differs by a small margin.The pattern continues with rāgas Madhyamavati and Mohana shown in Figure 8.While the conventional parent is Kharaharapriya for Madhyamavati, the metrics prefer Natabhairavi, again by a small margin.In the case of Mohana, the metrics prefer Kalyani whereas the conventional classification is Harikambhoji with Shankarabharana not too far.The case of Mohana's parent is hotly debated among the connoisseurs of Carnatic music.It is the same case with the rāgas Revati and Hamsadwani, each with six possible parents, as seen in Figure 9 where the standard classification and the preference of distance metrics are in conflict.Rāga Hamsadwani is interesting since the closest parent is Kalyani whereas the standard classification gives Shankarabharana which is the farthest among all the six likely parents.It is to be noted that this conflict is independent of the metric used.
In Figure 10 we have given the distance measures for rāgas Gambhiranata and Amritavarshini both of which share the notes with nine possible janaka rāgas, the maximum possible for a pentatonic rāga.Gambhiranata is supposed to be janya in rāga Chalanata.The notes of Gambhiranata are shared entirely by another rāga called Vedhandagamana which is identified with a different parent, Shankarabharana.However, all distance measures prefer Sarasangi as the likely parent with the conventional parent Chalanata close behind.The rāga Amritavarshini has a similar conflict since its janaka rāga is expected to be Chitrambari whereas distance metrics prefer rāga Kosala.In fact, according to the metrics, Chitrambari is quite a distance away from other likely parents.
It is clear from the analysis above that the distance metrics capture the closeness of any two rāgas almost in unison even when they disagree with the standard classification, as is often the case.This disagreement between the conventional classification and the metrics requires a closer examination.In order to resolve the ambiguous cases as noted above, it may be necessary to have a critical look at the data sample used in the analysis.We have chosen to combine the compositions in the rāgas without further fine-tuning.Often in any performance, the rāga characteristics are brought about at the beginning itself using the pallavi (or refrain) lines repeated many times.This is true of all compositions, irrespective of the composer.The listener's mind is conditioned to the rāga through the pallavi lines.As a result, even when there are deviations, especially in the fast-paced renditions towards the end, it may go unnoticed.Thus it becomes necessary to look at the extraction of the main features of the rāga by eliminating the redundant or fast phrases, rendered towards the end of compositions.We do this analysis in the next section.

Feature extraction using pallavi lines
In the previous section, the input data consisted of entire compositions by many performers with no fine-tuning of the data.The pitch histogram consisted of averaging over the whole collection of compositions in a rāga.No ambiguity in the decision was observed across all distance metrics especially when there was disagreement with conventional classification.In this section, we look at possible improvements in the quality of the input data by extracting the salient features of the rāga.This makes the whole process involved more effective, and also computationally faster.
It is well known in Carnatic music that the pallavi or refrain of any composition is used to introduce all the features of the rāga including many intricate gamakas.Though the pallavi line is a refrain, the pallavi itself is repeated many times, usually about 8-10 times, introducing all the nuances of the rāga in the process.After the pallavi is presented in its various hues in the first couple of minutes, the rest of the composition follows.While the rāga contours are strictly adhered to in rendering the pallavi, some freedom or even deviations are exercised by musicians in rendering the rest of the composition.This is so especially when it is rendered at a fast tempo.In order to eliminate these inaccuracies, (Padmasundari & Murthy, 2017) used just the pallavi itself as a motif for rāga identification resulting in excellent matches.
Taking inspiration from this, we extracted the pallavi lines by manually identifying the beginning and the end of the repetitions.In order to eliminate any error, this sample was further verified using techniques used in Ref. (Krishnaraj, 2019) to extract each pallavi line separately, including all the repetitions automatically.Typically each pallavi line lasts about 5-10 s and about a minute on average including all the repetitions.The entire composition in Carnatic music rendered without much flourish, lasts not more than about four minutes on average.This segmentation was carried out on all the rāgas in the database and in particular on those given in the previous section both for janaka and janya rāgas.
Figure 11, shows that the histogram obtained using the pallavi lines of the composition in the rāga Mohana is almost matching with the full-duration histogram.Figure 12 shows the pitch histogram of rāga Mohana along with its likely parents.The histogram is plotted using the full song in Figure 2 and plotted using the pallavi lines of each song in Figure 12.It is clear from the plots that the pallavi part of the songs carries adequate information to compute the rāga histograms.Even though the distribution of rāga histograms looks similar, on closer examination of Figure 12 the pallavi part contains more subtle nuances compared to the full length of compositions. 15rmed with the confidence that the pallavi lines are a more accurate representation of the rāga, we reexamine  the distance measures between janya and janaka rāgas given in Figures 6-10, especially focusing on the anomalous cases.We remove the janaka rāgas which are distant and not part of the anomalous identification in the following analysis.As far as rāgas with 7 notes and nonlinear movements, there is no change as expected since there is not much room for confusion here.We begin with the rāgas with 6 notes as shown in Figure 13.It is clear that the rāga Chandrajyoti is close to both Pavani and Navaneetham while the conventional classification uniquely identifies Pavani.Except for the Euclidean metric, all other metrics differ in the third decimal place.The ambiguity is similar to that of Figure 7.However, all the three preferred metrics, BH, KLD and JS, are numerically very close to each other, indicating the closeness of Chandrajyoti to either of these two parents.Notice that the parent of rāga Malahari was identified as Vakulabharana instead of the conventional Mayamalavagowla by all the distance metrics in the earlier method using the full composition.Interestingly, the distance metrics also confirm the same based on pallavi lines.Closer examination reveals that both Vakulabharana and Mayamalavagowla differ only in the third decimal place, again indicating that either could be a parent within the errors induced by the performances.
Next, we consider the rāgas in Figures 8-10.All the pentatonic symmetric rāgas shown in Figure 8 were in conflict with standard classification.Leaving out the distant rāgas, keeping only the ones which are part of the confusion, we show the same set of rāgas analysed through the pallavi lines in Figure 14.Except for rāga Hindola, all the other three rāgas have the same set of parents as indicated in the full composition analysis given in Figure 8.All three are different from the conventional assignment of parent.In the case of Hindola, the pallavi analysis restores the conventional parent, whereas the full  composition analysis was in conflict.The conflict is evident since the metric for the two likely parents differs only in the third decimal place.Furthermore, in the case of Mohana, musicians have historically debated whether it should be Harikambhoji or Shankarabharana or Kalyani.This analysis unambiguously confirms Kalyani as the parent while Harikambhoji is not too far.
In Figure 15, the results of pallavi line analysis are given for two rāgas Revati and Hamsadwani with 6 likely parents each.Except for the Euclidean metric, there is agreement on the parentage among other metrics, and they do confirm the full composition analysis.However, in both cases, the conventional assignment is different from the preference shown by the metrics.In the case of rāga Hamsadwani, even though the conventional classification identifies Shankarabharana as its parent, in actual practice it is well-recognised that Kalyani is a better fit; this is confirmed by the metric analysis.
In Figure 16, we have redone the pentatonic rāgas with the maximum possible parents using the fine-tuned input data.In the case of Gambhiranata, the metrics show a preference for rāga Shulini.This is different from the full composition analysis where the metrics preferred Sarasangi.Both of these are different from the conventional classification, which identifies Chalanata.However, we may note that Shulini and Chalanata are close enough to consider either of them as parents.Even while listening, they sound very similar to avid listeners and hence the confusion is genuine.In the case of Amritavarshini, metrics confirm Kosala as the parent without any ambiguity even though Chitrambari is the conventional choice.

Feature extraction using motifs
The oldest known documentation of rāga parentage by Venkatamakhi (Raghavan, 1941) in the seventeenth century was essentially based on the musical similarity between rāgas determined by musically trained ears.Among various components, in essence, musicians use the significant notes and the characteristic sequences of notes (characteristic motifs) while rendering a rāga.Thus, for musically trained ears, these two components are important features for comparing rāgas.So far, we have used pitch histograms of rāgas that capture the significant notes (indicated by the frequency of the repeated notes).Although for some rāgas the pitch histogram analysis agrees with the conventional janaka rāgas, the confusion persists for most rāgas.Therefore, next, we check if matching the sequences of swaras 16 helps in reducing these ambiguities.
Here, we use the algorithm called 'longest common segment set (LCSS)' originally developed to find the closeness between two rāgas based on the similarity between sequences of swaras (phrases) in the pitch contours (Dutta et al., 2015).LCSS performs a comparison between two-time series of pitch contours based on the density of the common phrases.In LCSS, each common segment (phrase) has a similarity value.When an adjacent pitch value matches the pitch contours of two rāgas, that pitch value is included as a part of the common segment (phrase), and the similarity value of the common segment is increased accordingly.Due to noises in the pitch contour and the improvisational nature of Indian classical music, two similar phrases can have mismatches (or gaps) in the pitch values (Dutta & Murthy, 2014b;Ishwar et al., 2013).In order to allow for these mismatches, LCSS incorporates a penalty value (ρ) when an adjacent pitch value doesn't match.The penalty reduces the similarity measure to account for this error.A penalty 16 sequence of notes value of 0 allows consecutive mismatches of all lengths resulting in one common segment (phrase).A positive penalty allows consecutive mismatches of intermediate lengths depending on the similarity value, resulting in multiple phrases.A penalty of ∞ also results in multiple phrases, but it does not allow mismatches of any length (see Appendix).The resulting similarity value is referred to as the LCSS score.This method has been successfully used for rāga identification-like tasks (Dutta et al., 2015).In some sense, the LCSS method mimics the conventional method of assigning the parent.In the conventional method, the ear matches the acoustical passages to determine the closeness, in a way similar to the LCSS method, which matches segments analytically.
We don't expect the presence of identical phrases between janya rāgas and their janaka rāgas; rather, we expect minute portions of phrases to be matched.Hence, we select the penalty value of 0, allowing mismatches of all lengths.This is illustrated in Figures 17 and 18, where a single pallavi pitch track of rāga Mohana is compared with the pallavi pitch tracks of Harikambhoji, Shankarabharana, Vachaspathi and Kalyani.The LCSS algorithm finds matches in the pitch tracks, and a score is assigned to it.The two figures refer to two different performances hence the differing scores.It is easy to see the preference or bias of the performer in these two sets of figures.Thus at the level of individual performances, it would be hard to pick a unique parent since different performers and different schools of training bring in subjective elements.Therefore, in principle, to start with any one of the four could be construed as a parent.It turns out that in a majority of them, Harikambhoji has the highest score; that is, the black segments contributing towards LCSS scores are better matched in Harikambhoji.This indicates that the pitch values in the pallavi line of Mohana corresponding to the black segments are closer to Harikambhoji than to the other three rāgas.Sometimes Shankarabharana comes a close second, at other times Kalyani comes a close second.It is this fact that lies at the origin of many debates on the parentage of rāga Mohana, and many others too, in the public domain and in actual concerts.
Since the individual scores vary, to achieve a consensus here, we use the m pallavi lines of the janya rāga and compare them against n pallavi lines of the candidate janaka rāga resulting in m × n LCSS scores.From the m × n LCSS scores, we take the mean of the top 20 values to find the similarity between the janya and the candidate janaka rāga.
The mean LCSS score values between janya rāga and the candidate janaka rāgas are displayed in Figure 19.Larger the score, the better the matching between two rāgas.Since the sequence matching method is very different from the pitch histogram method, in Figure 19, we provide the sequence matching scores for all the janya rāgas considered earlier except those with 7 notes, where only one parent is possible.For each janya rāga, we include all the likely parents, not just those with ambiguities as noticed earlier.This provides an independent assessment of the parentage of the janya rāgas.Notice that in the case of LCSS, the higher the score, the better the match, unlike the case of distance metrics where the lower is the better match.
We find that for rāgas Anandabhairavi, Chandrajyoti, Malahari, Madhyamavati, Mohana, and Hamsadwani, the LCSS scores agree with the conventional choice of the janaka rāgas.However, for rāgas Hindola, Sunadavinodini, Revati, and Amritavarshini, LCSS scores clearly prefer rāgas different from the conventional choice.For Gambhiranata, LCSS scores prefer Shankarabharana as the parent, which interestingly is the conventional parent of Vedhandagamana, a rāga that has the same notes as Gambhiranata.The conventional parent of Gambhiranata is Chalanata which is the second preference based on LCSS scores but very close to the first preference.In fact, there is an indication here that when the scores are very close and not easily distinguishable and, as such, a unique parent assignment is not a valid proposition.Instead, it is hinting at an association with a family with more than one possible parent.
Overall, we find that LCSS scores from sequence matching agree with the conventional choice of parents for most of the rāgas and help in reducing the level of ambiguities from the pitch histogram analysis.However, rāgas with high degeneracy show ambiguous parentage as the pitch histogram scores and LCSS scores disagree with each other and prefer different parents from the conventional choice.

Hindustani ragas used in Carnatic music
Before we end, we examine the parentage of some Hindustani classical rāgas which are popular in Carnatic music concerts 17 in recent years.These rāgas are not part of the standard classification though some of them are identified with the parent rāgas of the standard system in recent listings.This discussion, therefore, is more an 17 The classification of rāgas in the Hindustani system is different from Carnatic music and is of recent origin (Jhairazbhoy, 1995).The 'Thaat' system uses the same 12 note system, but R, G, M, D, and N all have two-note positions giving a total of 25 = 32 possible combinations instead of 72 in the Carnatic system.
application of our method rather than a comparison with the standard classification.In Figures 20 and 21

Summary
To summarise, we began this work with the aim of critically examining the conventional Venkatamakhi classification of rāgas in Carnatic music and making an attempt at providing a computational basis for the classification of the rāgas as janaka and janya.The basic idea is very old.The standard or conventional classification uniquely assigns a janaka rāga to a janya rāga.The necessary, but not sufficient, condition is that the notes of the janya rāga must be a subset of the janaka rāga.This allows multiple possibilities for the parentage, out of which one is uniquely assigned as the janaka.This assignment heavily relies on the acoustical matching of the melodic outlines.This method works in most of the janya rāgas especially if they have seven or six notes.However, in the case of pentatonic rāgas, there are ambiguities as pointed out by many performing musicians as well as informed listeners.Thus the method which has evolved over a long period of time needs a critical examination.
In this paper, we started with the aim of critically examining the conventional classification and if possible finding a quantitative basis for the unique assignment of a janaka rāga for every janya rāga using the methods used in the area of computational music technology.This examination is done using two distinct methods.
We first use the pitch histogram method, where a rāga is represented by a probability distribution.A rāga may be represented as a point in the 12-dimensional octave space.Distance between any two rāgas in this space, not necessarily confined to janya or janaka rāgas, defines how close they are.The probability distribution is first obtained using compositions in each rāga averaged over many performers.This is then fine-tuned by restricting to the pallavi lines or refrain in each rāga since the essence of the rāga with all its nuances are captured by the pallavi lines.The probability distribution contains all the features of a rāga in a macroscopic sense where many intricate nuances are averaged out.
In contrast, the sequence matching method, LCSS, presented in section 7, microscopically analyses the common features as well as the differences between any two rāgas.This is done by checking the degree, and the density of common pitch sequences between the renditions of two rāgas.The resultant LCSS score increases with the closeness between the rāgas.
Our results show that even after using quantitative methods available now it is difficult to assign a unique parent for all janya rāgas.For example, • Consider the very popular rāga Mohana: From the pitch histogram analysis, we conclude that its parent is Kalyani whereas the LCSS method tells us that it is actually Harikambhoji.The LCSS method agrees with the conventional classification.• Similar ambiguity emerges in the case of another popular rāga Hindola.Where the pitch histogram method with pallavi lines chooses Natabhairavi as its parent, the LCSS method identifies Hanumatodi as the parent.• Overall, the LCSS method is closer to the conventional classification compared to pitch histogram method, though not by much.
In many cases, as shown in the tables using either method, the distinction is too close to assign a unique parent.Nevertheless, the analysis also confirms the robustness of the conventional classification since where there are disagreements, the metrics are too close to understand the basis of disagreement.In very few cases, the conventional classification is far off the mark.These are also the cases which are often discussed in musical fora.
We are then forced to conclude that a rāga in general belongs to a family rather than just a single parent.A critical examination also leads us to the conclusion that the underlying system that we are modelling is so complex that every approach to its quantitative classification has its own limitations -be it the conventional classification or the statistical methods used in this analysis.
While the motivation for the present analysis came from the concept of janaka and janya rāgas, the method may be used to probe the closeness of any rāga to any other rāga or to assign a family to a new rāga.The methods may also be used, as has been done before, to recognise a rāga when compared with a standard list of rāgas.More generally, the method allows for a determination of the closeness of a melodic outline with any other melodic outline.

Figure 1 .
Figure 1.The normalised pitch histogram of rāga Mohana in different bin widths.The envelope is also shown in the higher resolution (smaller bin width) histogram.

Figure 2 .
Figure 2. Comparison of normalised pitch histogram of rāga Mohana with its possible parents based on the shared notes of Mohana with the parent rāgas.

Figure 3 .
Figure 3. Plots showing the resolution analysis and collar size analysis.

Figure 4 .
Figure 4. Peaks in the rāga Mohana histogram are more pronounced in the windowed histogram shown on the left hand side.Peaks are more pronounced for collar size ± 20 cents compared to ± 50 cents.

Figure 5 .
Figure 5. Top-left: Heat map of distances between songs in the rāga Kalyani.Top-right: Histogram of the songs in rāga Kalyani.It is clear both from the heat map and histogram that there are outliers that should be understood as false positives.The same is repeated for the rāga Shankarabharana below.However, this shows little scatter, unlike Kalyani.

Figure 6 .
Figure 6.Janya rāgas with seven notes having only one possible parent is shown in Figure 6(left).Though single parent, Sriranjani is an exception with only six notes.Though Anandabhairavi has seven notes, due to the occasional use of note D1 its parent identification is resolved by the distance metrics as seen in Figure 6(right).The conventional janaka rāga is always indicated by a solid bar in the histogram and a ( * ) in the legend.

Figure 7 .
Figure 7. Distance metric for janya rāgas with six notes.The Melakarta assignment of a parent is indicated by a star in the figure legend and a solid black bar in all the figures.

Figure 8 .
Figure 8. Distance metrics for pentatonic rāgas Hindola, Sunadavinodini, Mohana and Madhyamavati.The conventional or Melakarta classification of the parent is indicated by a star.All the examples shown are either fully or partially in conflict with the conventional classification; this will be analysed in-depth in later sections.

Figure 9 .
Figure 9. Distances to likely parents of pentatonic rāgas Revati and Hamsadwani taking into account the common śrutis in their scales.The conflict with conventional classification (solid black) is clear.

Figure 10 .
Figure 10.Distance metrics for rāgas Gambhiranata and Amritavarshini.Once again there is no agreement with the conventional classification though the metrics are in agreement amongst them.

Figure 11 .
Figure 11.Histogram of rāga Mohana showing the full duration envelope as well as the pallavi lines separately.

Figure 12 .
Figure 12.Plot showing the histogram of rāga Mohana and possible parent rāgas using the pallavi lines alone.

Figure 13 .
Figure 13.Similar to Figure 7, notice that all the distance measures except Euclidean are close enough to cause confusion for Chandrajyoti and Malahari.

Figure 14 .
Figure14.Pentatonic rāgas with distance measures computed using the pallavi lines alone.Notice that the distance measures agree with each other even if they differ from standard classification.

Figure 15 .
Figure 15.Pentatonic rāgas Revati and Hamsadwani with distance measures computed using the pallavi lines alone.

Figure 16 .
Figure 16.Pentatonic rāgas with maximum possible parents.Here the confusion persists even after the pallavi lines are extracted and used in the distance analysis.

Figure 17 .
Figure 17.Comparison of a single pallavi line in rāga Mohana with a single pallavi line of its likely parent.The x-axis denotes the time (in seconds) profile of a single pallavi line of rāga Mohana; the y-axis gives the time profile of a pallavi line of the janya rāga.The grey-black line depicts the longest common segment set (LCSS) with the associated score value on the top of each figure.The diagonal segments (black) indicate matches that contribute towards LCSS scores.Here, in spite of the mismatches (grey segments), a score of 1 can be achieved if the black segments are perfect matches because of the '0' penalty.

Figure 18 .
Figure 18.Same as Figure 17, but with the pallavi lines taken from a different performance to indicate variations among pallavi lines and performers.Notice that the pallavi in Mohana includes four renderings of the same line with variations.Individual scores show the bias of each performer towards a particular parent rāga or rāgas.The scores given in Figure 19(a) to (d) are averaged over many performers indicating an average score.

Figure 19 .
Figure19.LCSS scores of select janya rāga.Note that in these cases the larger the score the better the match, unlike the distance measure where it is the opposite.
, we have given some examples of the imported rāgas from Hindustani music and their possible parent as indicated by distance measures.Here again, some rāgas are confusing.Rāga Jog is shown to have a parent in Harikambhoji.However, metrics prefer Ragavardhini as the likely parent.Almost all the assignments shown in Figures20 and 21have consensus.

Figure 20 .
Figure 20.Rāgas imported to Carnatic from the Hindustani system in recent times.The assignment of parent, where it is available, lacks a conventional classification approach and is of recent origin.
developed a computational representation of tonal hierarchy and closeness of melodic phrases in allied rāgas pairs in the context of Hindustani music by computing the distance between pitch histogram representations of different rāgas.Distance measures like L1, L2, Bhattacharyya distance and correlation, between different types of rāga histograms (similar to the types inKoduri et al. (2012)), are used to evaluate the closeness between aligned rāgas.Pitch histograms of different types of note transcription are used byViraraghavan et al.

Table 5 .
Ratio of scatter for different metrics for distances.

Table 6 .
Statistics of data used.