Clayton_Kamata 2025

AAWM Journal 13/No. 2 (2025)

Analysis of Unmetered Coordination in the Tōgaku Ensemble of Japanese Court Music (Gagaku)

Martin Clayton and Sayumi Kamata

Gagaku, Japanese music, free rhythm, empirical analysis

The paper presents new approaches to the analysis of unmetered ensemble music, and the first detailed description of the temporal structure of unmetered pieces of a gagaku suite. We present novel analytical methods to explore the relationship between metrical and non-metrical forms of temporal organisation, including visual plotting of events to aid the understanding of structure, analysis of phrase lengths and of inter-onset intervals, including wavelet analysis. The structure of unmetered pieces is compared to that of metered pieces in the same genre, and considered in the light of interview data. We describe a form of ensemble music in which the group coheres through complex interactions between instrumental parts while the possibility of metrical structure emerging for musicians or listeners is minimised. The structure is based on sequences of long wind phrases, decorated melodically by string instruments and punctuated by the percussion section; crucially, and in contrast with metered sections, neither strings not percussion settle into metrical patterns. This novel analysis extends understanding of gagaku music while also developing analytical approaches in so-called ‘free’ or unmetered rhythm to encompass composed ensemble music.

Martin Clayton is Professor of Ethnomusicology at Durham University.

Sayumi Kamata is Researcher at Tokyo National Research Institute for Cultural Properties.

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In this paper we explore empirically the organization of instrumental gagaku music in unmetered or free rhythm sections, comparing this with metrical organization in the same genre. The term gagaku 雅楽 refers to the entire body of classical music and dance that has been performed in the Japanese imperial court, and this paper focuses on tōgaku 唐楽, a subset of instrumental ensemble music that comprises a large proportion of the gagaku repertoire. The tōgaku repertoire includes, alongside pieces organized with reference to clear metrical structures (Kamata and Clayton 2025), others apparently without any meter, known as jobuki 序吹. The organization of this music—composed, with a clear formal organization and the coordination of up to eight different parts, but apparently unmetered—has not been discussed in any depth in the musicological literature. This may be partly because their organization is even less explicitly described through notation and description than that of metrical pieces, and is even more dependent on musicians’ practical knowledge. Using empirical analysis of a published set of audiovisual recordings, the aim of this paper is both to extend research on performance practice of gagaku into these jobuki sections, and to develop that on unmetered or “free” rhythm to consider forms involving significant ensemble coordination.

[2] The instruments of the tōgaku ensemble can be grouped into three classes: wind, string and percussion (see Fig. 1 and Table 1). Previous studies of contemporary gagaku performance have indicated that the winds are responsible for the main melody and, particularly in metrical pieces, the strings “trace the melody while clearly indicating the beats” and the percussion instruments “set the tempo and rhythm of the piece by creating a series of rhythmic patterns” (e.g., Endō 2013). Three prominent characteristics relating to its time organization have been highlighted are: (a) the gradual acceleration observed as a piece progresses (Shōno 1987, Terauchi 2011); (b) a markedly drawn-out interval between the last beat of the measure and the first beat of the next (Garfias 1975, Nelson 2008); and (c) deliberately “loose” synchrony in the ensemble (Kamata and Clayton 2025). As a result of these features, it may be quite difficult for those who have never heard gagaku before to immediately recognize its metrical structure from a performance.

Figure 1. Layout of instruments (drawn by Sayumi Kamata)

Table 1. Instruments of the tōgaku ensemble

[3] The above discussions are based on examples from metrical-formal pieces,[1] and non-metrical jobuki pieces are only briefly mentioned. The main reason why the discussion so far has concentrated on metrical examples appears to be that, in the central repertoire of some eighty suites of tōgaku, only eight suites (Shunnōden 春鶯囀,  Konju 胡飲酒, Goshōraku 五常楽, Kishunraku 喜春楽, Sokō 蘇合香, Manjuraku 萬秋楽, Somakusha 蘇莫者 and Sanju 散手) contain examples of consistently non-metrical pieces. However, contrary to the impression given by the small proportion of non-metrical pieces, tōgaku performances typically feature both metrical and non-metrical rhythms. Introductory pieces—which are always played at the beginning of a performance before the suite proper—are non-metrical, and in some instances, non-metrical sections are included at the beginning or end of the metrical sections of the main piece. It is essential to consider both metrical and non-metrical aspects when exploring the temporal organization and coordination in gagaku performance practice.

[4] While non-metrical rhythms in instrumental ensembles are often described in the historical literature as “jobuki,” it should be noted that the term does not refer to the rhythms themselves, but rather refers to the act of blowing of the wind instruments.[2] Previous studies have simply labelled them as “free rhythm” (Garfias 1975, Endō 2013, Nelson 2008) or “non-metrical” (Gamō 1987, Terauchi 1996, Terauchi 2011); researchers who use the latter term may have wanted to avoid giving the impression that jobuki pieces are free from formal constraints. This is probably because the multilayered structure, which has been noted as a characteristic of Japanese music as a whole—where multiple phrases or rhythmic units form cohesive units, which in turn come together to form larger units, thereby shaping the overall structure (Yokomichi 1986, 15)—is also observed in the composition of jobuki.

[5] The differences between non-metrical and metrical rhythms, as manifested in the notation used by contemporary performers, are illustrated using the hichiriki part as an example (Figs 2 and 3). Both pieces are marked with large drum strokes (large black circles) corresponding to the convergence points of all instrument phrases, but in Fig. 3 the number of symbols in each span is constant, whereas in Fig. 2 it is not. If looking through the notation of the whole piece, one will find repetitions of phrases within the arrangement of symbols (such as those highlighted here in blue, green and yellow) and may observe how small units are stacked up to form successively larger units in constituting the musical structure. Sukehiro Shiba (1898–1982), a gagaku performer, also recognized this structural property early on, transnotating almost all current pieces into staff notation, analyzing and classifying musical formulae based on the ryūteki melodies (Shiba 1967–72; see Fig. 4 for an example of this notation). However, the three features that characterize the rhythm— gradual acceleration, a markedly extended interval before the first beat of a measure, and deliberately “loose” synchrony—are not reflected in either the performer’s notation or the stave transcription, as these are all performance conventions. In other words, there is a limit to what can be learned from such documentation alone, which is why an analysis of audiovisual data is crucial.

Figure 2. Hichiriki performance notation until the first four strong taiko beats of a non-metrical piece “Jo,” based on the sets of official standard notations digitally published under the title “Gagakufu” but generally known as “Meiji sentei-fu” (“Selected Scores of Meiji”), of 1876 and 1888. (https://doi.org/10.20730/100270332), Transcribed by Sayumi Kamata to fit into one figure. The notation is read down the vertical columns starting from the right; the large characters are the shōga mnemonic syllables which represent notes on the hichiriki[3], with small characters to the left showing fingerings. A large black circle is marked to the right of where the large drum stroke enters, which is in columns 1, 3, 4 and 6. It is evident that the number of shōga syllables per taiko span, i.e., the length of the phrase up to the point where all instruments converge, is not constant. Annotations have been added to indicate repeated phrases of the hichiriki (blue, green and yellow highlights) and each taiko span (red brackets)

Figure 3. Hichiriki performance notation of the first four cycles (including the first four strong taiko beats) of a metrical piece “Juha.” This example shows one metrical cycle written on a single column, with a large black circle to the right of the shōga mnemonic syllables containing the large taiko stroke, and a small black dot to the right of the syllables that correspond to the beginning of the other bars. There is approximately the same amount of shōga syllables in each cycle. Annotations have been added to indicate repeated phrases of the hichiriki (blue and yellow highlights) and taiko cycles (red brackets)

[6] Clayton proposed a definition of “free rhythm” as “the rhythm of music without pulse-based periodic organization” (1996, 329), roughly equating “free” with “unmetered.” As he noted at the time, many examples of “free” or “unmetered” rhythm can be identified around the world. Examples of such music can be found in forms of religious chant, such as Persian avaaz (Tsuge 1978, Reckford 1987, Ohriner 2016) or Jewish nusah (Frigyesi 1993), as laments (e.g., Tolbert 1988), or as soloistic melodic traditions such as Indian alap (Widdess 1994), or Turkish/Arabic taksim (Holzapfel 2013, Cholevas 2017, Roeder 2019, Abramovay 2025). There is a small but significant analytical literature on some of these genres.

[7] London’s description of meter as a “musically-specific form of entrainment, the synchronization of attention and/or other behaviors…  with periodic rhythms in the environment” (2012, 9) suggests that “unmetered” forms lack, or eschew, the kind of regularity in the music’s time organization that would allow such entrainment. The perception of meter is an active process and it is possible that any element of temporal stability or predictability, even if unintended or short-lived, might give rise to a percept of beat or meter in some listeners; thus, the labels “metrical” and “free” cannot be considered a simple binary. It is clear from the literature that “free rhythm” forms may not be entirely free of metrical elements, but elements of pulse or even meter may be observed: either intermittent, present in the mind of the performer but not made explicit in the sound (Widdess 1994), or perceived by listeners even if not intended by performers or objectively present in the sound (Will et al 2015). Such rhythm may be related to the rhythms of speech, or to those of poetic forms which are rendered in song (Tsuge 1978, Reckford 1987). While Ohriner attempts to bring free rhythm within the framework of entrainment theory (2016), Roeder applies to the phenomenon Hasty’s notion of “projection,” which like entrainment is concerned with expectations of temporal regularities based on previous events, but is less closely aligned with the idea of metrical structure (Hasty 1997, Roeder 2019).

[8] While some of these ideas may be applicable to jobuki rhythm, none of this literature tackles the key question addressed by the present paper, which is how the gagaku ensemble is coordinated temporally in the absence of metrical structure. Even if many of the forms of free-rhythm music previously analyzed may be performed by more than one person, whether in turn or overlapping, the coordination of multiple, distinct instrumental parts in the jobuki sections of gagaku seems to present a distinct case, one which has not previously been subject to detailed analytical study. We therefore explore the temporal organization of jobuki sections of a tōgaku suite using empirical analysis of audiovisual recordings; comparison between non-metrical jobuki pieces and metrical pieces is used to highlight the particular features of the former.

METHODS

Dataset

[9] The subject of analysis here is a set of recordings of the tōgaku (literally, “music of the [Chinese] Tang [dynasty]”) suite Shunnōden, recorded by an eight-piece instrumental ensemble. Audiovisual recordings and accompanying annotations are published separately (Clayton, Kamata, Takaoka & Tarsitani 2023). The recorded material includes “Ichikotsuchō no Chōshi 壱越調調子,” the introductory piece for playing in the Ichikotsuchō mode (which is roughly analogous to a mixolydian mode on D), and the tōgaku suite “Shunnōden 春鶯囀,” literally “The Warbler Sings in the Spring.” In its full form, Shunnōden consists of six pieces— Yūsei 遊声,  Jo 序,  Sattō 颯踏,  Juha 入破,  Tesshō 鳥声  and Kisshō 急声—and falls into the most prestigious taikyoku 大曲 category, which includes structurally complex compositions and the combination of non-metrical and metrical rhythms. The focus of this analysis is the jobuki sections of Shunnōden, namely “Yūsei,” “Jo,” “Tesshō,” and the last part of “Kisshō” (see Table 2), although extensive comparison will also be made with the metrical sections of the suite.

[10] In our corpus, transcriptions of interviews with some of the performing musicians are also published with audiovisual recordings and annotations. Specifically, interviews were conducted with ryūteki player Takuya Kōketsu, hichiriki player Motonori Miura and shō player, Junko Yatsuki.[4] Gagaku performers join the ensemble in various roles depending on the occasion, so it was possible to ask questions about the parts other than the wind instruments. As these musicians’ comments on how they perceive gagaku performance have already been detailed elsewhere (Kamata and Clayton 2025), the results presented below focus specifically on their comments on jobuki and the contrast between metrical and non-metrical sections within the suite.

Table 2. Outline of the suite Shunnōden, recordings and metrical structure.

[11] Our annotation of the performances loosely follows Shiba’s approach (1967–72) in breaking the main Sections (A, B, C etc.) into Sentences (a1, a2, b1 etc.); each Sentence includes one to three melodic phrases articulated by the wind instruments.[5] Based on this documentation, our empirical analysis of the audiovisual recordings sheds light on temporal features of the music as performed that are not reflected in the extant notations.

Figure 4. Extract from Shiba’s notation of the beginning of a non-metrical piece Tesshō (1971, 317). The extract covers the A section represented below in Fig. 5.  Double bar lines correspond to the breaks between a1, a2 and a3 (i.e., in this notation, a1 is notated in 3 bars, a2 in 9 bars and a3 in 4 bars). This image is reproduced with the publisher’s permission.

[12] The audiovisual data comprises multitrack audio (one microphone for each of the eight instruments) and video recordings (five cameras). The recordings were manually annotated as follows in order to generate timing data for the empirical analysis:

1. Structural divisions (as described above, referring to Shiba’s analysis of hierarchical levels)
2. In the case of metrical pieces, each measure
3. Start and end points of each wind phrase (ryūteki and hichiriki)

[13] Event onsets were marked up for the string and percussion instrument tracks using a custom onset detection script in Matlab (Eerola et al 2018); the output of this process was subject to manual checking and editing. In the wind instrument tracks, “onsets” were taken as the points at which new stable pitches are reached—either at the start of a phrase or in the middle of a phrase following ornaments. These wind “onsets” were marked up manually in Sonic Visualiser. It should be noted, however, that the “onsets” of the shō part were not marked because they were obscured in the audio recording by the volume of the hichiriki part (the hichiriki and shō players sit next to one another). Empirical analysis was carried out in RStudio, with packages used including onsetsync (Eerola & Clayton 2024) and WaveletComp (Rosch & Schmidbauer 2018).

EMPIRICAL ANALYSIS

[14] In order to explore the structure of the music, including the possible presence of periodic elements—both in continuous sound durations and in intervals between events—we carried out a series of analytical processes on the dataset described above.

Visual Inspection of Synoptic Event Plots of Each Section

[15] Complementing the visual representation of Shiba in score form, we explore the structure of the music by visualizing events (onsets) and durations (wind phrases) alongside the Sentence boundaries, without having to make arbitrary decisions over rhythmic values. Where two takes are available of the same piece, we compare the two takes visually, in order to distinguish consistent aspects of the structure from accidental alignments between events and parts. (This is the case for “Yūsei” and “Tesshō,” two of the three most substantial jobuki sections.) This helps to clarify aspects of the rhythmic structure of the piece as a whole and the relationships between instrumental parts which are not revealed by the part-by-part notation currently used by performers, or by Shiba’s transcription alone. Jobuki and metrical pieces are compared to highlight the special features of the former.

Analysis Of Wind Phrase Durations

[16] In the second phase of analysis, we explore the hichiriki and ryūteki phrases by plotting phrase duration against time and calculating the proportion of time each instrument is playing for each piece. We know from the interview data that these players are focused on the individual characteristics of phrasing associated with their particular instrument (discussed in detail below).  Therefore, we explore the hypothesis that—in the absence of metrical structure—the time taken by the wind players to perform the melodic phrases is one of the main drivers of the temporal structure of the music. We look again at comparisons between takes and compare phrase durations in jobuki pieces with the metrical pieces.

Onset Interval Analysis

[17] In the third phase of analysis, we consider also the percussion and string sections. These instruments are played using strikes and plucks; thus the timing of event onsets is an important element of their contribution to temporal structure, although as noted in Kamata & Clayton (2025), preparatory movements offer an element of continuous timing data to complement these moments.[6] This analysis has two parts:

1. We analyze each percussion and string instrument in turn, looking for evidence of periodicity (consistency of interval durations) and/or acceleration, comparing results for jobuki and metrical pieces.
2. We combine “onset” data for all instruments (including ryūteki and hichiriki but not shō, where we do not have reliable data) and explore periodic components using wavelet transform analysis,[7] again comparing jobuki and metrical pieces.

RESULTS

Interviews

[18] Before describing our analysis of the audiovisual recordings, relevant information on how these musicians feel about jobuki is summarised here.[8] When we asked the musicians to talk about the recorded suite “Shunnōden,” the contrast between metrical and non-metrical jobuki sections was mentioned in all the interviews. A common emphasis of the interviewees was that the feeling was “totally different” between metrical and jobuki pieces (Yatsuki 00:21:03), and that the latter is more challenging. Although they can participate in the ensemble playing different instruments, the challenges of playing jobuki were mainly discussed from the perspective of the wind players.

[19] According to shō player Yatsuki, both the metrical and non-metrical sections may seem difficult from the outside, but the former can be “surprisingly viable” if a sense of beat and the flow of the music is not disturbed. In the latter, however, the shō may have to precede other wind phrases in some pieces, or may have to respond to the expansion or contraction of other wind phrases—in other words, wind instruments have a wider range of flexibility in jobuki, so it is necessary to “establish” one’s own playing style and at the same time have “extra energy to cope” with the other parts. In particular, to coordinate with the other instruments the shō player memorizes the details of a jobuki piece using shōga mnemonics of both the hichiriki and the shō before playing it (Yatsuki 00:22:42).  It is remarkable that other musicians also refer to both initiative (as in “establishing” one’s own style) and reactivity (as in “coping” with what others play). However, the hichiriki player Miura and the ryūteki player Kōketsu appear more conscious of the need to show their independence to the other parts, by clearly demonstrating the “katachi (lit. form)” they have acquired (Miura 00:50:13, Kōketsu 00:28:34). Regarding the wind instruments, then, there is an emphasis on each instrument’s independence, but at the same time an awareness that this places a greater load on the musicians to follow and respond to other players.

[20] Notably, the presence of order and intention behind jobuki, which is not based on the metrical-formal structure, was highlighted. Kōketsu describes this as the point of convergence where the “form” of each part moves together, i.e. when the taiko enters, only then does “beat” or “order” occur instantaneously (00:28:34). (The taiko has an important role in marking the long cycles in metrical pieces, and—as we will see—also in marking the structure of some of the jobuki pieces.) Miura also points out that, even up to this convergence point, there is a kind of “regularity” between the hichiriki and ryūteki (00:51:49): when the melody of the hichiriki starts moving, the melody of the ryūteki also starts moving, and when the ryūteki plays a technique hataku 叩く[9], the melody of the hichiriki also moves. Through such complex synchronicities, “a chaotic world is deliberately created” (Miura 00:55:08). The comment suggests that one clue to understanding the melodic progression is to listen to the detailed playing techniques of the other winds. This is more or less true of metrical pieces as well, but the musicians emphasize that there is a lot of detailed interaction between the hichiriki and the ryūteki in jobuki pieces, with each waiting to hear the other’s note before moving on to the next note.

[21] In contrast to the detailed comments on the wind parts, very little was said in the interviews about the string and percussion parts in jobuki. One of the reasons for this seems to be that the position of the string and percussion parts in the ensemble also seems to differ considerably from that of the metrical pieces. According to Miura, in jobuki—in contrast to the metrical pieces—the wind players in the back row rarely consciously observe the arm movements of the biwa and taiko. Rather, he said, both string and percussion parts need to be fully aware of the wind phrases in order to insert patterns as they progress (personal communication, 11 September 2023).

[22] In jobuki, the biwa and koto parts are defined in terms of the amount of material that should be played between each taiko point; however, the amount they are required to play in each taiko interval is not constant, which means that some spans feel rhythmically dense while others are sparse. Performers find that fitting the required material into the allotted span requires careful attention, especially when the rhythmic density varies considerably. Whereas in metrical pieces they work together to mark the beat position, in jobuki, the biwa and koto parts are less coordinated—except at the taiko point. It is stated that the koto is rarely taken into account by the other instruments, and that the musicians themselves may find that some koto passages contain too many notes. In percussion, complex patterns on the kakko are added to the performance based on the ‘katachi形 (lit. form or shape)’ created by the winds, and finally the taiko and the subsequent shōko enter (personal communication, 11 September 2023). The deliberate matching at the taiko point in jobuki (Kōketsu 00:28:34) does not seem to be limited to the winds, and seems to be an increased awareness of “moving towards the convergence point” throughout the ensemble. It should be noted, however, that the above statements are about kangen instrumental performance; in the case of bugaku dance accompaniment, musicians have to respond to the dancer’s movement timing, for example by playing longer up to a certain taiko point where there is a lot of dance movement in between.

[23] What emerges from the musicians’ comments is that the absence of a metrical framework further increases the need for the three wind instruments to understand and cooperate with each other’s detailed playing, and for strings and percussion to participate in the ensemble based on the winds’ phrase progression. So, what aspects of timing and coordination can be found empirically in jobuki? Let us now turn to the results of the analysis of audiovisual data.

Event Plots

[24] It is difficult to approach the task of analyzing the time organization of the jobuki sections given the absence of a metrical structure. However, given the pieces are composed—and repeatable with a high degree of similarity—it is possible to describe their formal structure by observing occurrences of repeated or similar material (which are also observable in the performers’ notation). The distribution of musical events performed by the various instruments can be visualized by plotting the annotation data acquired using the methods described above, and this can enable us to explore the relationships between parts. This approach has some advantages compared with traditional transcription-based approaches: in the latter it is possible to observe melodic detail, but this information tends to obscure the temporal relationships between different instruments. This is also the case for the performers’ notation. Subtle beat stretching and contraction are not documented in their written notation, although they can be observed in the performers’ shōga-singing practice.[10] It can be said that rhythmic details, including deliberate asynchrony with other parts, are acquired as embodied knowledge for the performers and not something to be documented in a fixed form.

[25] Event plots are created in RStudio using annotation data relating to the structure (e.g., sections divided into ‘Sentences’; wind phrase start and end points; and rhythmic events or ‘onsets’). Fig. 5 illustrates the first A section of the jobuki Tesshō. From top to bottom, this plot visualizes the following information:

1. The section is divided into three sub-sections or Sentences, of which the longest is the middle one, a2 (figures in blue indicate the duration in seconds)
2. Of the wind instruments, the ryūteki starts first; once the hichiriki joins in, their phrases are roughly aligned with each other.
3. Pairs of taiko strokes (dark blue), each followed by shōko strokes (green), mark the ends of Sentences.
4. The accelerating patterns of the kakko, marked in grey, indicate another division of the sentences which is not always closely aligned with the wind phrases.
5. Plucks on the biwa (pink) and koto (orange) punctuate the second and third sentences; their alignment with or relationship to the other parts and with each other is not immediately obvious.

Figure 5. Visual representation of the first A section of the piece Tesshō (take 1). The top row (‘Structure’) shows the structural division into Sentences; the next two rows the extents of the phrases of hichiriki and ryūteki (all with times in seconds written in blue). The fourth row shows the strokes of the percussion instruments (kakko in grey, taiko blue, shōko green). The bottom row shows the plucks of the string instruments (biwa in pink, koto in orange).

Figure 6. Visual representation of a part of the piece Juha. Key as Figure 5.

[26] If we compare this with a section of the metrical piece Juha (Fig. 6), in the latter case we see much more regularity. For example, the sentence b1 can be divided symmetrically into two halves at around 225 secs; the shōko strokes (green) divide the duration between the second taiko stroke (blue) into three parts; the biwa strokes (pink) are aligned with those of the taiko and shōko; and so on. In short, we can see on this plot how the strings and percussion mark out the metrical structure, with a regularity that is not present in the jobuki piece Tesshō.

[27] As is summarized in Table 2 above, these plots show how each piece in the suite, whether metrical or jobuki, features a clear structure, including the repetition of one or more sections. In terms of temporal durations, this hierarchical structure in jobuki pieces looks roughly as follows (timings for Jo are given separately as the scale is rather greater than the other pieces):

“Jo”:

Piece (15 mins)  > Sections (3–4 min) > Sentences (35–75 secs) > Phrases (3–13 secs)

“Yūsei,” “Tesshō,” “Kisshō” final section:

Piece (5 mins)  > Sections (40–80 secs) > Sentences (14–30 secs) > Phrases (3–13 secs)

[28] Since we have two takes each of Yūsei and Tesshō we can compare the timings and distribution of events between takes to get an idea of the reproducibility of the the jobuki pieces’ temporal structure. Here (Figs 7 and 8) we take as an example the first B section of “Yūsei.”

Figure 7. Visual representation of a part of the piece Yūsei (take 1). Key as Figure 5.

Figure 8. Visual representation of a part of the piece Yūsei (take 2). Key as Figure 5.

[29] The two plots are clearly very similar, reflecting a high degree of alignment between two takes of the same piece. The section is divided into two sentences of approximately 20 secs each, divided into 2 and 3 wind phrases respectively, with the two instruments closely aligned. The durations are similar, but there appears to be some flexibility: the first wind phrase is somewhat longer in take 1 than take 2.[11] The kakko part is divided into three phrases with different characteristics (each accelerating, but with different profiles). The biwa plucks are fairly evenly distributed, while the koto is sparser during the second wind phrase. The alignment between the instrument groups shows more differences between takes. The first kakko pattern ends during the second wind phrase and the second kakko pattern in the third wind phrase, for example, but they overlap considerably more in take 2 than take 1. The string parts are distributed similarly in the two takes and appear to be aligned with the winds rather than the kakko (given the melodic element this is logical); where there is inconsistency between the takes it is in the alignment between the biwa and koto (in some cases the two strokes fall close together on one take but not on the other).

[30] Similar features can be observed in the other jobuki sections (for a complete set of plots see the Appendix). They suggest the following observations about the structure and ensemble coordination.

• Wind phrases are closely aligned between the two instruments; the hichiriki tends to start fractionally before and end fractionally after the ryūteki. (There are a few exceptions: sometimes a single phrase in one corresponds to two phrases in the other.)
• As we can see in the first plot, the taiko, where present, marks the end of sentences with a double stroke; the shōko strokes are closely aligned with the taiko. The kakko has its own sequence of patterns that divide up the sections; although these often align roughly with the wind phrases, they may overlap the latter significantly. Faster or denser patterns tend to occur at the start of Sentences.
• String phrases. Both biwa and koto parts relate to the wind phrases, although their precise temporal relationship to the rest of the group, and to each other, is not obvious. In metrical pieces where the strings indicate the beat, their onsets often fall alongside those of the taiko and other percussion (Fig. 6). In jobuki, however, with the exception of the taiko point, the temporal relationship with the percussion becomes less clear. In particular, in Yūsei (Figs. 7 and 8), which was used as an example of reproducibility, the only percussion instrument is kakko, but the string parts seem to add the same ‘fixed number of notes’ to it in the two takes, using the wind instrument phrase rather than the kakko as a cue. This is also consistent with statements made in interviews.

[31] Having made some basic observations about the relationships between parts of the ensemble, we now explore the temporal characteristics of the different parts to search for evidence of temporal regularities: recurring phrase lengths or intervals between rhythmic events.

Wind Phrase Durations

[32] In the Tesshō plots introduced above we can observe that the final phrase of each sentence tends to last around 6–6.5 secs[12]; these wind phrases are essentially the same, so this temporal regularity is simply a result of the repetition of melodic material. For the parts of the hichiriki and ryūteki, the following analysis compares the wind phrase durations across each piece, both metrical and jobuki sections. We can get a more comprehensive view of the wind phrase durations by plotting them against time for all of the pieces (Fig. 9). This shows a clear difference between the jobuki (Yūsei, Jo, Tesshō, Kisshō_jb) and the metrical pieces (Sattō, Juha and Kisshō_m)[13] In the latter we see a clear pattern, which is especially clear in the hichiriki parts: there are two clear lines of decreasing duration, matching the acceleration of these pieces. In the jobuki pieces on the other hand there is no visible pattern of acceleration, although in Tesshō we can a line reflecting see the consistent durations around 6 secs noted above.

Figure 9. Wind phrase durations of all pieces. Jobuki sections are highlighted. The metrical and jobuki parts of Kisshō are plotted separately (7_Kissho_m the metrical, 7_Kissho_jb the jobuki).

[33] The boxplot of wind phrase durations (Fig. 10) shows clearly that the mean lengths are lower for the metrical pieces Sattō and Juha (although less so for Kisshō) than for the jobuki pieces (mean 5.65 +/– 2.14 secs for metrical vs 8.04 +/– 2.7 secs for jobuki sections). Hichiriki phrases are on average a little longer than Ryūteki for all pieces (mean 7.57 vs 6.92 secs).

[34] A further observation on the wind phrases is that the proportion of the total time in which these instruments are playing is lower in the metrical sections (82%) than the jobuki (91%). In other words, they rest for a longer proportion of time in the metrical sections (Table 3, see Appendix).

Figure 10. Boxplot of wind phrase durations for all pieces. Horizontal lines indicate the mean durations. Jobuki sections are highlighted. The metrical and jobuki parts of Kisshō are plotted separately (7_Kissho_m the metrical, 7_Kissho_jb the jobuki)

[35] This analysis of the wind phrases shows a clear distinction between metrical and jobuki sections. In the metrical sections (especially Sattō and Juha), we see a predominance of two duration classes and an accelerating pattern – both of which phenomena are related to the metrical structure – and these instruments rest up to 20% of the time. In the jobuki sections, there is neither acceleration nor a bimodal distribution of durations, and the resting time is less than 10%: in Tesshō, however, a frequently repeated phrase gives rise to a predominant phrase duration of around 6 secs, which is observed at the end of Sentences. The mean phrase duration is longer in the jobuki sections and is longer for hichiriki than ryūteki in all pieces.

[36] This seems to suggest that while the melodic content of the wind phrases is similar between jobuki and metrical pieces, their duration is constrained by the metrical structure articulated by the strings and percussion – which dictates that they must end, or reach a point of emphasis, by a specific point in time. This would also explain the longer pauses between wind phrases: the wind players enter the next phrase only after the appropriate point in the metrical structure that is created by all the parts, whereas they have more independence in relation to the progression of the jobuki pieces.

ONSET INTERVALS

Onset Intervals by Instrument

[37] In this next stage of analysis, we look for evidence of consistency of time intervals (inter-onset intervals, IOIs) in the onset data, again comparing metrical and jobuki sections. To do this, we select onsets for each instrument in the string and percussion sections, and calculate the intervals between successive events. Since we are interested only in intervals likely to have a rhythmic significance, we discard intervals over 6 seconds. It is clear from Fig. 11 that the kakko and shōko intervals tend to be shorter (median 175 and 130 ms respectively), the koto favours a medium interval (median = 1015 ms) and the biwa and taiko have longer intervals (medians 3414 and 3195 respectively). We now explore the patterns of onset differences by instrument, comparing also the metrical and jobuki pieces.

Figure 11. Histogram of inter-onset intervals up to 6 seconds for wind and percussion instruments (plotted with a log scale to avoid it being distorted by the higher numbers of kakko onsets than the other instruments). Vertical dashed lines are computed median values for each instrument.

Taiko

[38] As already noted, in jobuki pieces where it is used, the taiko is generally used to mark the ends of Sentences; in metrical pieces, it also marks significant points in the metrical structure. The common pattern is a pair of strokes – a left-hand strike followed by a louder right-hand strike – although variations with additional strokes, known as kuwaebyōshi 加拍子, may be observed in the latter part of the piece. Fig. 12 shows the intervals between these pairs of strokes. Of the jobuki pieces, in Jo these gaps range between 3.5–4.5 secs, showing a slight trend to shorten; in Tesshō they are between 3–4 secs and lengthen towards the end. In the metrical pieces, these intervals range between 2.5–4 secs with an accelerating trend; in Kisshō (on the plot, the second, unhighlighted part of 6_Tessho_2), which uses a different pattern of taiko strokes, there are two accelerating lines and an overall range of intervals between 1.5–4 secs.

[39] Overall while the taiko intervals in the metrical pieces reflect their metrical structure, the intervals in the jobuki sections cover a similar range with either no (Tesshō) or more limited (Jo) acceleration. In the latter case, since the intervals are around 3 secs or above and are single intervals interspersed with much longer gaps, rather than a sequence of similar intervals, there is no possibility that they could in themselves generate a sense of periodicity.

Figure 12. Inter-onset intervals for the taiko. Left pane: all IOIs (in seconds) up to 5 secs plotted against time, separately for each piece. Jobuki sections are highlighted. Right pane: histogram of all IOIs across the corpus under 5 secs.

Shōko

[40] The shōko strikes are closely linked to the taiko’s. The plot of all intervals (Fig. 13) shows the different patterns used in each piece, and also the clear acceleration in the metrical pieces, which is evident in the longer intervals. In the shorter intervals between pairs of shōko strikes we see little variation or acceleration, although the range is somewhat slower in the jobuki than it is in the metrical pieces. These isolated, short IOIs would not be expected to afford a metrical perception.

Figure 13. Inter-onset intervals for the  shōko. Left pane: all IOIs plotted against time, separately for each piece (intervals in secs). Right pane: histogram of all IOIs under 200 ms, illustrating the clustering around 100–150 ms

Kakko

[41] The chart for the kakko onset intervals (Fig. 14) shows that the patterns are similar in each piece, comprising accelerating sequences with different gradients. Zooming in to the first 180 secs of each piece these patterns become clearer.[14] It does not seem plausible that this pattern could induce a sense of meter, given the constantly changing intervals involved; there are differences between the deployment of the patterns by piece, but not such a clear difference between metrical and jobuki categories.

Figure 14. Inter-onset intervals for the kakko. Left pane: all IOIs below 1 sec plotted against time, separately for each piece (intervals in secs). Right pane: the same data, showing up to 150 secs only to make the different acceleration patterns clearer

Biwa

[42] Turning now to the string section, the biwa onsets show a clear difference between the metrical and jobuki pieces (Fig. 15). In the former, Sattō, Juha and Kisshō show lines reflecting two interval ranges, which accelerate gradually through the pieces. This is not surprising, since many biwa onsets fall on beats 1 or 3 of the measure and given that these pieces speed up. The jobuki pieces show a more normal (rather than bi-modal) distribution, peaking between 2.5–5 secs. Again, IOIs in this range are unlikely to give rise to a perception of a regular beat.

Figure 15. Inter-onset intervals for the biwa. Left pane: all IOIs below 15 secs plotted against time, separately for each piece (intervals in secs). Right pane: histogram of all IOIs under 12.5 secs, separately for each piece

Koto

[43] The koto plots (Fig. 16) again show a clear difference between metrical and jobuki pieces: the latter show a preponderance of intervals around 1 sec, while the former seem to be more structured, with many intervals in the 2–7 sec range. The peak of the distribution in all cases is below 1 sec, however. According to the musicians, in the metrical sections the koto has the role of “clearly indicating the beats” (as noted in the Introduction), whereas in jobuki it “puts in a set number of notes” as appropriate to each wind phrase. In the latter case, it is stated that the musicians themselves may find some passages contain too many notes (see Interviews). In other words, this part in jobuki shows the most consistently fine note values among the percussion and string sections, but this does not necessarily create a sense of beat shared by the whole ensemble and may have more of an ornamental function in relation to the wind phrases.

Figure 16. Inter-onset intervals for the koto. Left pane: all IOIs below 20 secs plotted against time, separately for each piece (intervals in secs). Right pane: histogram of all IOIs under 5 secs, separately for each piece

Combined Onset Periodicity Analysis

[44] In the onset data for individual instruments, we have seen some evidence for consistent time intervals, for example around 1 and around 3 secs, even in jobuki pieces. These could in principle be related to a sense of beat or metrical structure, although there is no evidence that they are consistent enough for this to happen. We should not assume, however, that cues for metrical perception (such as regular time intervals between rhythmic events) are found only in the patterns of single instruments. Indeed, in Kamata & Clayton 2025, we showed how the percussion and string instruments combine together to mark out the metrical structures. In order to look for such metricity in combinations of instruments we use a different method, however (since simple differencing across the full set of onset times would bias the analysis to very small, insignificant intervals).

[45] We devised the following method to explore evidence of periodicity in combinations of instruments. All onsets are combined into a single CSV file per piece (in this case ryūteki and hichiriki are also included, with moments of clear pitch change marked as ‘onsets’ as they could be treated as rhythmic events). This data is converted into a histogram (i.e., a count of the number of onsets per time unit, in this case 100ms).[15] The resulting time series is analyzed using wavelet transforms to reveal its periodic structure.

[46] For example, in the metrical piece Sattō, Fig. 17a shows the all-onset histogram for measures 9–24 inclusive, with the metrical downbeats shown in red for reference (the main taiko strokes fall on points 15 and 23). Many of the onset histogram peaks lie close to the metrical downbeats, which reflects the fact that multiple instruments tend to play close together at these moments. The plot of total wavelet energy against time for this section is shown in Fig. 17b, and the wavelet plot showing the distribution of energy across periodicities in Fig. 17c. For the same section Fig. 17d summarises the wavelet energy across different periods between 0.5 and 10 secs, which shows four peaks (the red dots indicate statistical significance, p < 0.05). These peaks (0.90, 1.93, 4.00 and 8.00 secs) clearly correspond to the metrical levels in the piece at this point (the ½-beat, beat, 2-beat and measure levels; the mean beat length calculated from manual meter annotations for this section is 2.08 secs, with a range of 1.76-2.42 secs across the measures). These results are not surprising – we already know the metrical structure – but they demonstrate that by using this method, the hierarchical structure that makes up the meter is recoverable from the onset times. If a similar pattern were to be observed in the jobuki sections, it could suggest the possibility that a metrical structure may be perceptible.

[47] Comparing the wavelet energy plots shows up a clear difference between the metrical and jobuki sections (Jo and Sattō are compared in Fig. 18 – the other movements are included in the Appendix). In Sattō, Juha, and Kisshō (excepting the last section of Kisshō, which is jobuki), the wavelet energy is concentrated in a set of clear lines of gradually reducing periodicity, which is not the case in the jobuki pieces: these lines correspond to the different metrical levels as the piece accelerates (see Table 4). What is clear from the peak periods is that they lie very close to a hierarchical pattern (in Sattō and Juha, in the ratio 1:2:4:8); comparison with the calculated mean beat length shows the fastest or second-fastest level to be close to the beat[16].  We can also see a variation in energy levels as we look along the time (x) axis: the peaks here correspond to the metrical downbeats, when instruments tend to play together.  In the jobuki pieces, on the other hand, no such patterns are visible in the periodicity of the areas of higher energy (y-axis); this is confirmed when we compare the peak periods across the pieces with two takes each, Yūsei and Tesshō. In this case the results look quite different between takes, even if there are points of similarity in some sections (see Table 5). There does however appear to be a clear similarity between the energy vs time plots for different takes of the same jobuki piece, even if these are not reflected in robust periodicities which might have indicated the emergence of a beat or metrical structure. This can be seen comparing plots of total wavelet energy against time for the two takes of Tesshō (see Fig. 19[17]): there is a clear similarity in the distribution of wavelet energy peaks in relation to the structural markers, even if the relative height of the peaks varies significantly. As with the metrical pieces, the tendency for structural markers to coincide with wavelet energy peaks in the onset data is a reflection of the tendency for more instruments to ‘converge’ and play close together near the taiko strokes.

Figure 17. Wavelet analysis of Sattō, measures 9–24 inclusive. Panel a shows the all-onset histogram for this section. Panel b plots the total wavelet energy against time for the same section. In both cases metrical boundaries are indicated with red vertical lines. Panel c is a wavelet plot showing the distribution of energy across periodicities (0.5 to 10 secs). Panel d summarises the wavelet energy across different periods between 0.5 and 10 secs.

Table 4. Peak periods of wavelet energy vs mean beat length. Wavelet analysis carried out for range 0.5–8 secs. (** p<.05)

Table 5. Comparison of wavelet energy peak periods between takes of Yūsei and Tesshō. Wavelet analysis carried out for range 0.5–8 secs.  (* p< .1, ** p<.05)

Figure 18. Jo and Sattō, plots of wavelet energy against time for all onset data (plots for all movements are included in the Appendix). Sattō, the metrical movement, shows the emergence of clear bands of periodicity and a pattern of acceleration as these periods become shorter.

Figure 19. Plots of total wavelet energy against time for all onsets, comparing the two takes of Tesshō. Section boundaries are shown in red.

Summary

[48] In this section we have explored the onset timing data, searching for possible indications of regular beat or metrical structures in the jobuki sections. Instead we find a clear distinction between the metrical and jobuki parts: in the former the event structure as evidenced by the combined onsets clearly marks out the metrical structure, while in the latter, periodicities seem to emerge – as shown by the wavelet analysis which shows statistically significant energy peaks at certain periods in most sections –  but they are not sustained, and nor are they generally replicated between takes. There is evidence of particular IOIs being favored for specific instruments: for example, around 1 second for koto and 3 seconds for biwa and taiko. This could be linked to the timing of particular playing movements: the carefully controlled a stylized nature of the preparatory movements of taiko, shōko, biwa and koto may help to fix the timing of certain passages within a narrow range. This does not, however, seem to generate a sense of beat or metrical structure.

DISCUSSION

[49] As outlined in Kamata & Clayton 2025, metrical pieces in this dataset display a clear hierarchical time structure, from beat through measure and taiko cycle to Sections and whole pieces. The music is characterized by slow but gradually accelerating tempo, by an extended final beat, and by very loose synchronization. The metrical structure is unambiguously marked out by string and percussion instruments, each of which has its own individual role. In jobuki pieces we hear a superficially similar musical surface, but the metrical level of the time organization is not present, and thus neither is the gradual acceleration. (Since there are few instances when all instruments are expected to coincide, it is not possible to quantify ‘synchronization’.) Yet these composed pieces are reproducible with a high degree of similarity, and display a clear formal hierarchy of phrases, Sentences and Sections, with taiko and shōko typically marking the ends of Sentences. The coordination of strings and percussion to articulate a metrical structure is not present in jobuki pieces: their gestures, without metrical significance, can be heard more as articulating and supporting the melodic content presented by the wind section.

[50] One way of helping us understand the unmetered form of gagaku is to summarize what meter does when present. As we have seen in this analysis, meter has a strong impact on the wind phrases: we can think of the melodic gestures as being embodied with a natural duration which is revealed in the jobuki pieces, while in the metrical piece a common frame with subtle acceleration is put in place. Strings and percussion instruments play the role of indicators in this common frame, i.e., the metrical cycle. In jobuki pieces, described by interviewees as showing more independence, the winds play their phrases and take relatively short breaths, but in metrical forms they may naturally compress phrases or take longer pauses to maintain the ensemble coordination. In jobuki pieces, the need to stay coordinated with the wind phrases is an important constraint on the timing of strings and percussion. The stability of the phrases means that we do see some consistency in IOIs across pieces, rather than IOIs related to the meter which shorten as the piece accelerates.

[51] In the jobuki, then, wind phrases mark out the melodic progressions and combine to form Sentences, Sections and pieces. The length of these phrases depends mostly on their melodic content, and there is no acceleration through a piece. Hichiriki and ryūteki phrases are closely aligned with each other. The shō, an essential wind instrument, could not be included in the empirical analysis based on ‘onsets’, but the interview results highlighted the fact that performers could not play the part without a detailed understanding of the relationships between the winds. Taiko strokes, where present, mark the ends of Sentences, and shōko strokes closely follow the taiko, but kakko patterns also divide up the Sentences, being loosely coordinated with the wind phrases. Biwa and koto patterns punctuate or decorate the melody but in terms of timing are not coordinated with each other or with the percussion. Each has its own loose preferred periodicity. In short, our empirical analysis has elaborated the picture described by the musicians, in which each player has a relatively high degree of independence, but each must concentrate hard to maintain the coherence of the ensemble without relying on a metrical timing structure.

[52] Even in metrical pieces, subtle gestures and phrases of all instruments are combined, and one can feel a sound world which is superficially similar to the jobuki music. In this case, however, the emphasis is placed on all the parts working together to create a gradually accelerating metrical grid. As a result, the wind phrases no longer display the same level of independence as in jobuki, as they often have to end, or reach a point of emphasis, to coincide with a metrical event. As the metrical grid accelerates gradually, the wind phrases also become compressed.

[53] As we have seen in the analysis, it is not the case that the jobuki music is completely without periodic elements: we have observed these in various instrumental gestures. Whereas in metrical music such periodicities tend to be sustained and align with each other, through entrainment, to produce beat and meter, in these jobuki pieces they remain intermittent and largely limited to single instruments. We know from the entrainment literature that mutual attention tends to lead to entrainment, even when not intended; we also know that visual attention can be an important contributory factor (e.g., Richardson et al., 2005; Lucas et al 2011); we also know, however, that people’s ability to synchronize with periodic rhythms drops with intervals over about 2 seconds (Repp & Doggett 2007). We may hypothesize then that the successful avoidance, or management, of entrainment in this case relates to two factors: the management of attention and the control of time intervals. As for the first, the musicians are clearly paying great attention to each other. However, focusing more on the long continuous gestures of the winds rather than the koto phrases, for example, may limit the likelihood of falling into a regular beat. For the latter, many of the key time intervals used to structure the music – the wind phrases, the intervals between biwa or taiko striking gestures for example – are well over 2 seconds, while the faster intervals of the kakko are never stable (they always accelerate). The music seems almost to have been designed to limit the possibility of accidentally falling into a metrical pattern. This idea recalls the emphasis in the interviews on the “order” and “intention” behind jobuki.

[54] Hasty’s idea of projection is concerned with the possibility that a listener evaluates the duration of musical events with reference to previous durations (1997). Roeder has shown how this idea may be applied to a wide variety of free rhythm, monophonic musical examples, and how variations in duration may be an integral part of wider musical processes (2019). Applying this to jobuki is not straightforward for two reasons: first, that we are discussing ensemble music, and second, that the music is composed and reproducible. As for the first, this may not be insurmountable, since the wind phrases as performed on hichiriki and ryūteki are closely aligned in both melodic content and duration. As for the second, any sensation related to a particular phrase being longer or shorter than the preceding one must factor in – for the experienced listener or co-performer – the fact that such phenomena are highly expected; we might reasonably ask whether the durations are being compared to the previous phrase or to prior knowledge of the current phrase. For this reason, we hesitate to extend Roeder’s application of projection to these examples. However, it is clear that the tracking of phrases, and of events such as ornaments or pitch changes within phrases while they are ongoing, is key to the experience of the musicians, and that these musicians must have a well-practiced memory of the typical duration of the phrases.

[55] What the music presents is a complex whole made up of continuously trackable durations – phrases and preparatory movements – and events that punctuate those durations without generating metrical expectations. An observation that may have wider applicability is that unmetered group coordination may be easier when the whole group takes its cue from the timing of irregular wind phrases. Other instruments produce sharper attacks and are more likely to spontaneously develop beat and meter, but they are constrained by their dependence on the wind phrases; the latter offer continuous melodic trajectories of over 3 seconds that may give rise to a form of projection but not to a consistent entrained beat structure. This paper has described a form of unmetered time organization in composed ensemble music and proposed an interpretation of the processes that allow such organization to be sustained and reproduced. There is no doubt more to be discovered about this type of time organization, both in tōgaku and in other genres which may be comparable in this respect.

APPENDIX: INTERVIEWEE BIOGRAPHIES 

Motonori Miura has been familiar with gagaku since his childhood and was taught to play the hichiriki by his father Susumu Miura. He further studied hichiriki and court songs under Masasue Tōgi, umai dance under Tadaaki Ōno (both from the Music Department of the Imperial Household Agency), and received a Bachelor of Music degree in gagaku performance from the Tokyo University of the Arts in 2005. Miura is a member of Tokyo Gakuso, a professional gagaku group, and belongs to Pro Musica Nipponia, the Naoyuki MANABE GAGAKU Ensemble, Ensemble Muromachi, and the GAGAKU Association, and has been performing in Japan and abroad for 20 years. In addition to performing traditional repertoire, his activities include world premieres of contemporary works, music supervision of stage productions, recordings, media appearances and productions. Miura was an Adjunct Instructor at Tokyo University of the Arts from 2015 to 2023 and continues to teach younger students. 

Born into a family of Shinto priests serving the Tsutsukowake Shrine in Tanagura, Fukushima Prefecture, Junko Yatsuki was introduced to gagaku. She studied shō under Tadaaki Ōno, court songs under Masasue Tōgi, samai dance under Takaaki Iwanami and Seiichi Masuyama, and koto under Seiichi Masuyama (all from the Music Department of the Imperial Household Agency), and received a Bachelor of Music degree in gagaku performance from the Tokyo University of the Arts in 2010. Yatsuki is a member of Tokyo Gakuso, a professional gagaku group, and has been performing in Japan and abroad for over 15 years. As a member of the Shinto priesthood, she is also committed to the transmission of gagaku as ritual music. At her alma mater, Tokyo University of the Arts, Yatsuki was an Education and Research Assistant from 2013 to 2023 and an Adjunct Instructor from 2015 to 2023 and continues to teach younger students. 

Takuya Kōketsu studied ryūteki under Kenji Ue, Takaaki Iwanami, and Chiaki Yagi, court songs under Seiichi Masuyama, and umai dance under Tadaaki Ōno and Hokuto Matsui. He graduated with a Bachelor of Music degree in gagaku performance from Tokyo University of the Arts in 2014 and was awarded the Acanthus Music Prize. In addition to being a member of the professional group Tokyo Gakuso, Kōketsu has performed on the recordings and world tours of Tim Hecker’s “KONOYO” and “ANOYO,” “Yuzuru Hanyu Programme Concert – Music with Wings,” the anime “The Tale of the Heike” (Fuji TV), the historical dramas “Kamakura-dono no jūsan-nin,” “Dousuru Ieyasu,” “Hikaru-kimi” (NHK), “Monster Hunter Rise” and the 50th anniversary of the establishment of Japanese diplomatic relations in the UAE, among others. At Tokyo University of the Arts, he has been training younger students as an Education and Research Assistant since 2016 and as an Adjunct Instructor since 2024.

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Endnotes

[1]. By metrical-formal we intend a formal organisation built on metrical structure (e.g., sections comprise specific numbers of measures).

[2]. The character “序 jo” means “beginning,” “sequence,” “order,” etc., but does not explicitly include the concept of non-metrical rhythm; according to Terauchi (1996:18), terms such as “序吹 jobuki” and “序弾 jobiki”  are clearly derived from the name of the piece “Jo”, which is based on non-metrical rhythm. However, not all non-metrical pieces are necessarily marked with the character “序.” The character “吹,” read as fuku, sui, etc., means “to blow (sound an instrument).”

[3]. An oral mnemonic system known as shōga 唱歌 (literally meaning “singing song”) is an indispensable means for grasping the melodic details and the flow of the music as a whole, as a musician’s learning process begins with shōga in one of the wind instruments.

[4]. Biographical sketches of the three interviewees can be found in the Appendix.

[5]. Shiba’s division is modelled on seven levels of hierarchy, from the smallest unit “gakusei” to the whole piece “gakushō:” In his explanation of the hierarchical model, a unit of a structural level consists of two or more units of the lower levels. However, his analytical examples also include cases where a piece is divided in the same way on two adjacent levels. This may be partly due to the fact that gagaku pieces vary in length and include short pieces that are not suitable for the seven-level subdivision. Our sectional divisions correspond to the fourth “gakusetsu” and/or fifth “gakudan” levels listed by Shiba, depending on the piece.

[6]. By ‘preparatory’ movements we refer to movements made in preparation of sounding an instrument, which in many cases are deliberate and closely controlled by the performer.

[7]. The wavelet transform is a method for decomposing a signal into several components at different frequency/ period bands. Analogous to the Fourier transform in this respect, it has advantages when identifying periodic components which are localised, i.e., that do not persist throughout the signal. As Toivianen and Hartmann put it, “The [discrete wavelet transform] decomposes a time series into components that are localized in time and frequency by performing a number of convolutions between a time series and dilations/contractions of a function called mother wavelet. Since each wavelet (i.e., each dilation and contraction of the mother wavelet) represents a different frequency interval in the frequency domain, wavelets are comparable to bandpass filters” (2022:3). Cross-wavelet transform analysis, in which two wavelet transforms are combined, has proved an effective tool with which to explore coordination between individuals (see e.g., Clayton et al. 2019, Toiviainen & Hartmann 2022). In this analysis we use wavelet transforms of the instrumental onset data in order to extract information about the music’s periodic structure.

[8]. Semi-structured interviews were conducted individually so each musician’s comments would not be influenced by those of their peers. Time codes in the published transcripts are given in the text. In the case of supplementary personal communications with the musicians outside of the recorded interviews, references include the date.

[9]. Known as “tataku” or “hataku” (both literally meaning “tapping”). This is an ornamental technique for a sustained note, in which one fingerhole is lightly tapped to produce an adjacent lower note momentarily.

[10]. An oral mnemonic system known as shōga 唱歌 (literally meaning “singing song”) is an indispensable means for grasping the melodic details and the flow of the music as a whole, as a musician’s learning process begins with shōga in one of the wind instruments.

[11]. Hichiriki: 8.1 vs 7.5 secs (8.5% longer in take 1); ryūteki, 7.9 vs 6.8 secs (15.8%).

[12]. Hichiriki, mean = 6.55 secs, SD 1.30; ryūteki, mean 6.21 secs, SD 0.84 (Tesshō take 1).

[13]. In these figures kissho_m refers to the metrical part and kissho_jb to the jobuki part of the same piece, respectively.

[14]. One-handed techniques are referred to as “katarai” and double-handed techniques as “mororai.” It is difficult to annotate these in Figure 14, however, as there is often a transition from mororai to katarai and vice versa in the middle of a pattern.

[15]. Comparison with bin sizes of 20ms and 50ms revealed no significant differences in the result.

[16]. The fact that the mean beat duration and the shortest period in the wavelet analysis do not coincide exactly is not surprising, given the much more complex mathematics used to calculate the latter (the M ean beat length is simple average of the means for each measure in the section).

[17] While the figures given in the tables, and the wavelet energy plots, are based on analysis of the data with a period range of 0.5–8 secs, for these figures the range has been set to 2–8 secs to improve readability.