In medieval Europe several textually and musically related monophonic liturgical chant traditions existed. Most famous is the Franco-Roman chant of the Roman rite, better known as Gregorian chant. Most other rites and traditions were abolished at some point in favor of the Roman rite and its chant [1
]. In 589 the Visigothic Kingdom of the Iberian peninsula was converted to Catholicism. In the early seventh century Iberian Catholicism developed into an independent rite of Christian worship which after the Muslim conquest of 711 became known as the Mozarabic rite. In 1080 this rite was officially abolished by the Council of Burgos and replaced by the Roman rite with its Gregorian chant. In 1085 Toledo, the centre of the Iberian church, was reconquered from Islam. Only six parishes of Toledo were allowed to continue the ancient rite. In the eleventh century pitch-readable music notation gradually came in use. Most chants of the Mozarabic rite, however, are only preserved in pitch-unreadable (adiastematic) neume notation [2
]. The chants are preserved in about forty manuscripts and fragments dating from the early eighth until the thirteenth centuries. The most important manuscript is the León antiphoner (E-L 8, Catedral de León, MS 8), dating from the early tenth-century, containing over 3000 chants preserved in adiastematic neume notation.
Though the pitches of the melodies are unknown and probably lost forever, the neumes provide important information to assist in their realization: determination of a singable and plausible pitch sequence representing the neumes. The manuscripts, in neume notation with the syllables in the underlying text, provide two important pieces of information: the number of notes in each neume and the melodic contour of the pitches internal to each neume. From note to note it is usually apparent if the melody goes up or down [3
]. This contour information can be represented using six letters:
, a note higher than the previous note;
, higher or equal;
, lower or equal;
, a note with unclear and undefined relative height. Figure 1
shows a fragment of the Canticum Zachariae
for the feast of St. John the Baptist. Shown at the top of the figure are two lines from the León antiphoner. Following that is the transcription of the neumes on the bottom line to contour letters. In the contour sequence syllables are separated by dashes and words by spaces. Finally the figure shows a passage of a performance score with a generated compatible melody (see Results).
Another important feature of chant is the presence of recurring intra-opus
patterns within single chants [4
] that would seem to represent the same melodic content in the lost chant, for example, the encircled neumes and bracketed contour sequence in Figure 1
. There is a wide consensus among chant scholars that longer (i.e., 20 or more notes) intra-opus patterns do represent the same sequence of pitches [5
]. Therefore in generated pieces, an intra-opus
pattern should be instantiated by the same musical material. Repetition is ubiquitous in music and the generation of music containing repetitions is an important open topic in the area of music informatics, because it requires the solution of equality constraints between distant events in the music surface [6
The core task of the adiastematic neume realization problem is to find pitches compatible with a specified template consisting of the melodic contour and intra-opus patterns. Since a vast number of melodies will be compatible with a given template, this is a highly under-constrained problem. Therefore a position must be taken on whether the task is viewed more as restoration
or more as generation
. Some scholars have shown melodic relations with other chant traditions for some specific Mozarabic chants [8
]. In such cases chant realization may be approached as primarily a restoration
task: using long fragments of concrete pitches found in a chant with known pitches and presumably with a historical relation. This is one motivation of the method of Maessen and van Kranenburg [9
] for chant generation, which searches a corpus of preserved chants using contour descriptions of phrases from the template. If a closely matching database piece exists, it is used to overlay pitched fragments on the new chant. Remaining regions of the chant are constructed using less stringent matches. Finally manual editing of the borders between phrases will complete the melody. Even if a closely matching database chant exists, this method still requires expert intervention to fill in unmatched regions of the new chant [10
]. The explicit use of long contour patterns drawn from a corpus has also been considered to be a general model for melody generation [11
], where contour patterns specified by the composer, or selected from a predefined list, are instantiated by specific music segments drawn from a corpus.
This paper develops the alternative view of chant realization as primarily one of generation
: making no a priori existence assumptions of closely related chants. Music generation approaches can broadly be grouped under rule-based (requiring specific rules and constraints to be encoded by the composer), and machine learning methods (learning rules and models from a training corpus) [12
]. Most machine learning approaches to music generation use statistical models, originating from the earliest successful works with Markov models [14
]. Most statistical models for music generation can be considered context models
: generating the next event in a growing sequence based on the history of previously generated events. Context models encompass a wide range, including simple Markov models [15
], n-gram and variable length Markov models [16
], multiple viewpoint models [17
], and deep learning models for music generation [18
As mentioned above, a difficult problem for music generation methods is the precise control of intra-opus repetition, especially when using context models. There were some initial attempts to generate repetitive structures ab initio with context models [22
]. An alternative powerful approach is to derive the repetition structure from known pieces, either automatically with intra-opus pattern discovery [23
] or by compositional design. In this way the structure of a known piece is maintained in a newly generated piece. Things brings up the issue of how the structure is formally represented and instantiated: in this paper the method of Conklin [6
], designed for generating chord sequences with complex repetition structures, was adapted to solve the chant realization problem.
An often overlooked aspect of statistical models for music generation is the sampling of solutions. A decision must be made whether a few solutions are found by optimization of the posterior probability (given specified information such as length and desired features of the generation), or whether a diversity of possible solutions is produced through random sampling from the posterior distribution of the statistical model [20
]. For chant generation, given that there is no single “correct” realization of a template, it is important that diversity is attainable and that sampling methods are used to select from the vast space of possible sequences.
This paper described a new method for chant generation which explicitly conserves the structure present in defined templates. Templates were carefully designed using musicological considerations and a statistical model learned from presumably related musical material was used to instantiate the templates. The method was used to generate an entire concert suite of chants which was performed at a music festival in the Netherlands.
The research has opened up two interesting issues, both arising late in the process while a concert suite was in the final stages of generation. The first issue concerns high information peaks which happen when the start of an intra-opus pattern or a defined pitch is encountered during a left-to-right random walk. In these cases the sequence might have to return to an instantiated event with an unnatural leap and low probability. This issue arises with random walk on complex templates, and an exact solution is possible only for the simplest types of statistical models and templates such as first-order Markov models with unary constraints on positions [15
]. The information peak issue can produce low probability sequences because in the presence of complex templates, it is difficult or intractable to sample sequences with the same expected frequency as defined by their probability according to the statistical model. Several inexact methods were proposed, such as Gibbs sampling [19
], bi-directional LSTM models [21
], and iterative random walk as applied in the present paper [6
A second unanticipated issue that arose is that melodies had a tendency to sit in the upper range for too long. This was observed by all the singers, although only mentioned in 3 of the 22 written audience comments. The phenomenon arises due to the presence of many undefined contours in templates, combined with the very slight preference in the statistical model for upwards contours. This can be corrected by limiting the number of undefined contours in templates, for example, by replacing them by either a concrete contour to the previous neume, or contour relation to the first note of the previous neume. Indeed inter-neume contour relations can sometimes be inferred from the manuscripts [4
]. Another solution to this problem could be the generation of entire neumes rather than single notes. Here, however, it is possible that data sparsity problems would arise for model training.
A fascinating point opened up by our research is the role of overfitting in statistical models. Usually overfitting is viewed entirely negatively as the inability of a model to generalize past the known data. However in the chant realization problem there are cases where overfitting is desired. If restoration is desired and there is a closely related chant in the corpus, an overfit model should be able to retrieve long fragments from that chant whereas a model trained for generation will tend to mainly generate novel material. It is hypothesized that statistical methods can handle both sides of the spectrum, trained to fit to any degree the training corpus, including memorizing long fragments from the corpus.
Automated pattern discovery algorithms [35
] might be used to find intra-opus patterns in the template contour sequence, thus automating the laborious step of hand annotation of a template for intra-opus patterns. Interesting patterns could be determined by statistical significance measures. To create a large collection of realizations for many templates the application of automated pattern discovery seems even necessary. An important extension of this work will be to consider inter-opus
patterns, i.e., patterns appearing across different pieces within a corpus of template pieces. If the generation problem is viewed as one of generating a suite of pieces, it is desirable that the generated pieces have some inter-opus coherence. If inter-opus patterns are detected in different pieces in the manuscript they should also be instantiated with similar musical material in generated pieces.