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

Analytical Approaches to World Musics, Vol. 13, No. 2. Published December 2025. Analysis of Unmetered Coordination in the Tōgaku Ensemble of Japanese Court Music (Gagaku) Martin Clayton and Sayumi Kamata n 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. I 2 Analytical Approaches to World Musics 13.2 (2025) Figure 1. Layout of instruments (drawn by Sayumi Kamata) Clayton & Kamata: Analysis of Unmetered Coordination 3 Section Instrument (left to right) Description Wind Ryūteki ⿓笛 transverse bamboo flute Hichiriki 篳篥 short double-reed pipe Shō 笙 free-reed mouth organ String Koto (Sō, Gakusō) 箏（楽箏） 13-stringed long zither Biwa 琵琶 four-stringed lute Percussion Shōko 鉦⿎ dish-shaped gong Taiko 太⿎ large suspended drum Kakko 鞨⿎ small barrel drum Table 1. Instruments of the tōgaku ensemble [3] The above discussions are based on examples from metrical-formal pieces,1 and nonmetrical 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. 1. By metrical-formal we intend a formal organisation built on metrical structure (e.g., sections comprise specific numbers of measures). 4 Analytical Approaches to World Musics 13.2 (2025) [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. 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).” Clayton & Kamata: Analysis of Unmetered Coordination 5 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 hichiriki3, 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) 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. 6 Analytical Approaches to World Musics 13.2 (2025) 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 Clayton & Kamata: Analysis of Unmetered Coordination 7 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. 8 Analytical Approaches to World Musics 13.2 (2025) 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. 4. Biographical sketches of the three interviewees can be found in the Appendix. Clayton & Kamata: Analysis of Unmetered Coordination 9 Piece Meter/ Jobuki Structure Recording 1_Ichikotsuchō no Chōshi 壱越調調⼦ Jobuki (no taiko) 2 takes 2-7_Gagaku Suite “Shunnōden” 春鶯囀 2_Yūsei 遊声 Jobuki (no taiko) A – B – C – B – C – B 2 takes 3_Jo 序 Jobuki (“hyōshi-jūroku” 拍 ⼦⼗六 16 strong notes on the taiko) A – B – C – B’ 1 take 4_Sattō 颯踏 Metrical (“haya-ya-hyōshi hyōshi-jūroku” 早⼋拍⼦ 拍 ⼦⼗六 8 bars5 of 4/4 make up one rhythmic cycle, which is repeated 16 times) A – B – C – B’ 1 take 5_Juha ⼊破 Metrical (“haya-mu-hyōshi hyōshi-jūroku” 早六拍⼦ 拍 ⼦⼗六 6 bars of 4/4 make up one rhythmic cycle, which is repeated 16 times) A – B – C – B’ 1 take 6_Tesshō ⿃声 Jobuki (“hyōshi-jūroku” 拍 ⼦⼗六 16 strong notes on the taiko) A – A – A’ – A’ – B 2 takes 5. It is commonly understood among musicians and researchers that the rhythmic unit called “kobyōshi” in Japanese corresponds to a bar/measure (Shiba 1971, Nelson 2008, Terauchi 2011), and we follow this convention. 16 Analytical Approaches to World Musics 13.2 (2025) 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 Clayton & Kamata: Analysis of Unmetered Coordination 17 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.11 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: i. 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) ii. Of the wind instruments, the ryūteki starts first; once the hichiriki joins in, their phrases are roughly aligned with each other. iii. Pairs of taiko strokes (dark blue), each followed by shōko strokes (green), mark the ends of Sentences. iv. 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. 11. 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. 18 Analytical Approaches to World Musics 13.2 (2025) v. 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). Clayton & Kamata: Analysis of Unmetered Coordination 19 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) 20 Analytical Approaches to World Musics 13.2 (2025) [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. Clayton & Kamata: Analysis of Unmetered Coordination 21 [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 2.12 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.) • Percussion. 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 12. Hichiriki: 8.1 vs 7.5 secs (8.5% longer in take 1); ryūteki, 7.9 vs 6.8 secs (15.8%). 22 Analytical Approaches to World Musics 13.2 (2025) 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 secs13; 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)14 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. 13. Hichiriki, mean = 6.55 secs, SD 1.30; ryūteki, mean 6.21 secs, SD 0.84 (Tesshō take 1). 14. In these figures kissho_m refers to the metrical part and kissho_jb to the jobuki part of the same piece, respectively. Clayton & Kamata: Analysis of Unmetered Coordination 23 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). 24 Analytical Approaches to World Musics 13.2 (2025) [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 Clayton & Kamata: Analysis of Unmetered Coordination 25 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. 32 Analytical Approaches to World Musics 13.2 (2025) 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 Clayton & Kamata: Analysis of Unmetered Coordination 33 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).16 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 16. Comparison with bin sizes of 20ms and 50ms revealed no significant differences in the result. 34 Analytical Approaches to World Musics 13.2 (2025) 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 beat17. 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. 1918): 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. 17. 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). 18 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. Clayton & Kamata: Analysis of Unmetered Coordination 35 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. 36 Analytical Approaches to World Musics 13.2 (2025) Piece Section Wavelet peak periods (secs) Mean beat length (secs) Sattō A 0.87**, 1.87**, 3.86**, 7.27** 2.05 B 0.76**, 1.52**, 3.14**, 6.28** 1.59 C 0.68**, 1.41**, 2.83**, 5.86** 1.44 B’ 0.64**, 1.27**, 2.64**, 5.28** 1.32 Juha A 0.76**, 1.57**, 3.36**, 6.96** 1.87 B 0.66**, 1.37**, 2.83**, 5.66** 1.44 C 0.62**, 1.27**, 2.64**, 5.28** 1.33 B’ 0.57**, 1.19**, 2.38**, 4.92** 1.24 Kisshō A 0.55**, 1.03**, 2.22** 1.28 B 0.93**, 1.87**, 3.86(ns) 0.95 C 0.84**, 1.68**, 3.48** 0.87 B’ 0.78**, 1.62**, 3.25(ns) 0.8 Table 4. Peak periods of wavelet energy vs mean beat length. Wavelet analysis carried out for range 0.5–8 secs. (** p<.05) Clayton & Kamata: Analysis of Unmetered Coordination 37 Piece Section Take 1: Wavelet peak periods (secs) Take 2: Wavelet peak periods (secs) Yūsei A 0.76**, 1.62*, 3.36** 3.6* B 0.55*, 0.87** 0.84**, 1.32* C n/a 1.80* B 0.61**, 1.37* 0.59*, 0.90* C 1.87* 3.30**, 1.19* B 1.46** 0.59**, 1.18* Tesshō A 0.52**, 1.07** 1.23* A’ 0.52** 0.55**, 2.92** A’’ 0.52**, 1.07** 1.11** A’’’ 0.52*, 1.04** 0.66** B 0.66** 0.55**, 4.59* 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) 38 Analytical Approaches to World Musics 13.2 (2025) 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. Clayton & Kamata: Analysis of Unmetered Coordination 39 Figure 19. Plots of total wavelet energy against time for all onsets, comparing the two takes of Tesshō. Section boundaries are shown in red. 40 Analytical Approaches to World Musics 13.2 (2025) 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 Clayton & Kamata: Analysis of Unmetered Coordination 41 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.