Swinging with FM: Synthesis of Swing by Means of Frequency Modulationcmmr2017.inesctec.pt/wp-content/uploads/2017/09/8_CMMR... · 2017. 9. 24. · Swinging with FM 3 movement curves

Swinging with FM: Synthesis of Swing by Means of Frequency Modulation

Carl Haakon Waadeland and Sigurd Saue,

Department of Music, Norwegian University of Science and Technology,

7491 Trondheim, Norway [email protected]

Abstract. The present paper demonstrates how a technique of frequency modulated rhythms might be an interesting tool in making synthesis of various live performances of swinging music. Dynamic, time-dependent features are introduced and implemented in a model based on rhythmic frequency modulation, RFM, previously developed by the authors of this paper. We here exemplify the potential of the new, extended model by simulating various performances of swing in jazz, and we also indicate how the computer implementation of the RFM model might be an interesting tool of electro-acoustic music.

Keywords: Rhythm performance, swing, synthesis, frequency modulation, computer implementation.

1 Introduction

´Swing´ is a concept that many are likely to associate with jazz music. Indeed, one meaning of the notion “swing” is used to denote a jazz style that developed in the United States during the 1930s, see for instance [1]. Another meaning of “swing” is related to communicative qualities of a music performance (originally, a performance of jazz music). Subject to this understanding, swing is conceived as a process through which the musicians, both individually and in an interactive context of playing together, make a musical phrase, a rhythm, or a melody “come alive”, by creating a performance that in varying degrees communicates motional aspects to the listener, thereby making the listener want to move along with the music. Or, as stated by the composer and jazz historian, Gunther Schuller; a rhythm is perceived as swinging when: “… a listener inadvertently starts tapping his foot, snapping his fingers, moving his body or head to the beat of the music”, [1], page 223. Seen as such, the qualities characterizing a swinging performance may be typical of musical performances belonging to other traditions than the jazz tradition, as well.

A swing groove is a particular rhythmic ostinato played by jazz drummers. The swing groove originated in the swing era, and has developed further in later jazz styles, like bebop and contemporary jazz. A typical swing groove is most often played on the ride cymbal. In its mots basic form a swing groove may have the written representation illustrated in Fig. 1.

Proc. of the 13th International Symposium on CMMR, Matosinhos, Portugal, Sept. 25-28, 2017

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2 C.H. Waadeland and S. Saue

Fig. 1. An often written representation of a swing groove, commonly played on a ride cymbal.

The remark “With swing feel” in the notation in Fig. 1 indicates that the experienced jazz drummer plays this groove according to his embodied understanding of swing, where the eight notes in the swing groove are not performed with equal durations, as written in Fig. 1, but rather with characteristic long – short patterns. These patterns change the ratio between the durations of successive pairs of eight notes, often called the swing ratio, from 1:1 to 2:1, or around 3:1, - or somewhere in between, - dependent on, among other things, the individual drummer and the tempo of the performance. Several studies of how different performance conditions influence the swing ratio have been conducted, see e.g., [2], [3], [4], [5], [6], [7], [8]. Different performances of long – short patterns of durations subdividing the beat are also found in some Latin – American music, as well as in funk and hip hop [9], [10], [11]. Moreover, it is interesting to know that notes inégales is a common practice in the performance of French baroque music, - similar to the use of swing eights in jazz, see [12].

Common to all the above mentioned examples of different swing performances is that the musician in his performance makes various deviations from exact note values. “Deviations from the exact” in music performance have been empirically investigated since the 1930s, and Carl E. Seashore stated that these deviations are a characteristic feature of artistic expression [13]. Various such deviations have been studied in empirical rhythm research by investigating systematic variations of durations, SYVARD, see e.g., [14], and are also discussed and investigated as participatory discrepancies, PD [15], [2], [16], [17]. Deliberate discrepancies or deviations may be seen as a process by which (more or less) conceptualized structural properties of rhythm are transformed into live performances of rhythm. Such a process of rhythmic transformation is often denoted expressive timing, see [18].

One of the authors of the present paper, Waadeland, has presented a continuous model of expressive timing [19]. A basic idea in this model construction is to represent rhythmic structure (e.g. note values) by sinusoidal movements, and to apply frequency modulation to obtain movement curves that in varying ways create deviations from exact note values. A theoretical interpretation offered by this model is to view expressive timing as a result of rhythmic structure being “stretched” and “compressed” by actions of movements. Applying rhythmic frequency modulation, RFM, as suggested in the model, it is possible to simulate characteristic features of many different empirically documented examples of expressive timing. The other author of this paper, Saue, has constructed a MIDI-based computer implementation of RFM [20], and by applying this implementation we have been able to make sounding examples of the RFM syntheses.

However, even though the previously constructed RFM model is capable of constructing interesting approximations to various live performances of rhythm, an obvious limitation of the model is related to the fact that the model is static, meaning that the parameters in the model do not vary over time. This makes the modelled


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movement curves periodic. – Live performances of rhythm, on the other hand, even different swing groove ostinatos, are quasi-periodic, characterized by various local and long-term fluctuations from static patterns of systematic deviations.

In this paper we present an extended version of the RFM model with a new computer implementation. A major achievement of the present model is that we now have implemented dynamic, time-dependent features in the synthesis. To demonstrate our model we present various examples of different new syntheses of swing, and we also indicate how RFM synthesis might provide an interesting compositional tool for electro-acoustic music. In the concluding section we point at some further possible extensions of our model.

2 Rhythmic Frequency Modulation

We start by briefly outlining the basic features of the formerly developed RFM model, [19], after which we present the new computer implementation of dynamic properties in the model.

2.1 A Continuous Model of Rhythmic Structure

When moving the index finger up an down in the air in such a way that the minimal points are in perfect synchronization with the clicks from a metronome, an idealized curve describing the finger movement could be something similar to what is shown in Figure 2. To be quite explicit, this curve is given by the mathematical function:

y(t) = A[1 – cos(2πft)] , (1)

where t is time, f is the frequency of the finger movement, and A is the amplitude (2A is a measure of the finger´s maximal distance from the minimal value, 0). In Figure 2 f = 2.

Fig. 2. Graphic illustration of an idealized performance of isochronous beats with frequency f = 2. Time is displayed along the horizontal axis, and the finger´s vertical position is measured along the vertical axis.

Viviani, [21], stated that sine waves are easy to approximate by human movements, and are among the simplest predictable motions, whereas Balasubramaniam et al. conducted an empirical experiment where they found that an unpaced oscillation of the index finger tends to create a sinusoidal, close to symmetric


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movement curve, - but when the finger movements were paced by clicks from a metronome, the movement curves attained different characteristic asymmetric shapes [22]. Thus, the trajectory in Fig. 2 should be seen as an illustration of a static, robot-like (unpaced) performance of isochronous beats.

If we think of the sinusoid in (1) with frequency f as a representation of (a performance of) quarter notes, a corresponding sinusoid with frequency 2f will represent eight notes, 3f corresponds to eight note triplets, whereas (2/3)f represents dotted quarter notes. In other words, every note value may be represented by a sinusoid, and we have, thus, obtained a continuous representation of rhythmic structure. Fig. 3 illustrates the connection between a sequence of notes, a movement curve, and a representation by sinusoidal functions. In this figure quarter notes correspond to the frequency f = 1 (see also [19], page 27).

Fig. 3. An illustration showing the connections between a sequence of notes, a movement curve, and a mathematical representation of sinusoids, all related to a robot-like rhythmic performance executed in perfect synchronization with a metronome. The horizontal axis displays time, t, where the first beat occurs at time t = 0. In this figure we have made A(Δ) = Δ = 1/frequency, reflecting different amplitudes of finger movements related to the performance of notes with different durations (Δ).

2.2 A Continuous Model of “Deviations from the Exact”

In the model of rhythmic structure presented in the previous, metronomic performances of note values are represented by frequencies of sinusoidal functions. It is therefore tempting to suggest that deviations in live performances of note values should somehow correspond to some operation inducing deviations or alterations of frequencies. A well known technique of sound synthesis using various alterations or distortions of the frequency of an oscillator in order to achieve parameter control over the spectral richness of sound, is frequency modulation, FM, pioneered by Chowning [23]. The most basic FM instrument consists of two sinusoidal oscillators interacting to give the output:

y(t) = Asin[2πfct + dsin(2πfmt)] . (2)

A is the amplitude, fc is commonly denoted carrier frequency, fm is the modulating frequency, and d is the peak frequency deviation, cf. [23]. Looking at the output of the basic FM instrument, we observe that when d = 0, there is no modulation and the


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result is simply a sine wave with frequency fc. This very situation resembles, at least on a purely theoretical level, the situation we are trying to establish for syntheses of rhythm: When there is no modulation, the result is a strict metronomic (i.e., sinusoidal) performance. When, on the other hand, modulation occurs, various deviations of frequency are created, resulting in different kinds of “deviations from the exact” in the modeled performance.

Motivated by this observation, the basic RFM algorithm is defined on the basis of a combination of equations (1) and (2) above, and is illustrated in Fig. 4.

Fig. 4. Flowchart for basic rhythmic frequency modulation.

The output of this RFM algorithm is given by the function:

y(t) = A[1 – cos[2πfct + dsinn[2π(fmt + φm)]]] , (3)

where t = time, A = carrier amplitude, fc = carrier frequency, fm = modulator frequency, φm = modulator phase divided by 2π, d = peak frequency deviation = modulator amplitude = strength of modulation, n = exponent of modulating function.

It should be noted that equation (3) defines how a modulator operates on the function y(t) = A[1 – cos(2πfct)], representing a specific note value (e.g., quarter note). Frequency modulation of subdivisions and ties of this sinusoid is defined accordingly, and is expressed explicitly in [19], page 29. Example 1: Synthesis of Vienna Waltz Accompaniment. To give an example of how RFM works, we demonstrate an RFM simulation of a rhythmic characteristic documented in empirical rhythm research. – As noted by Bengtsson and Gabrielsson, [14], a well known feature of performances of Vienna waltzes occur at the beat level in the accompaniment; the first beat is shortened and the second beat is lengthened, whereas the third beat is close to one third of the measure length. Thus, the quarter note beats in the ¾ meter are characterized by a cyclic pattern of durations; short (S) – long (L) – intermediate (I). By modulating a metronomic (quantized) MIDI performance of a Vienna waltz accompaniment which is played with equal durations of the quarter notes, as written in the notation of the music, we are now able to simulate the characteristic S – L – I pattern by means of the RFM algorithm in (3).


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We obtain this by choosing: fc = 3 (3 beats to the measure), fm = 1, φm = 0.25, n = 1, and d = 1. This synthesis of Vienna waltz accompaniment is also presented in [19], pages 30-32. Fig. 5 shows the movement curve of this simulation. A sounding realisation of this synthesis may be heard at [24].

Fig. 5. Illustration of a modulated movement curve associated with a synthesis of a performance of the first two measures of a Vienna waltz accompaniment. The durations of the metronomic quarter notes are “stretched” and “compressed” by the action of the FM modulation, and the characteristic cyclic pattern of beat durations; S-L-I, is created.

2.3 Including Dynamic Parameters in the Model

In order to take dynamic features of rhythm performance into account in our FM synthesis, we need to include dynamic parameters in our RFM model. One obvious way to do this is to make some of the parameters in the RFM algorithm (3) time-dependent. The strength of modulation is in our model determined by the parameter d = peak frequency deviation. Thus, to be able to make the degree of modulation vary over time, we make d a function of t: d = d(t).

An empirically observed feature of rhythm performance is also that the proportional values of beat durations may change depending on the tempo of the performance. For instance, in the performance of swing groove in jazz the swing ratio tends to approximate 1 at fast tempi, whereas it is often closer to 3 at slow tempi [4], [7]. To include this feature in the model, we make d a function of tempo.

3 Computer Implementation of Dynamic Features

In order to investigate the implications of the theory of Rhythmic Frequency Modulation (RFM), we developed a computer application FMRhythm1 which allowed

1 FMRhythm is written in C++ and supports Windows only, but we intend to make it available

as a platform-independent, open source project at https://github.com/ssaue/FMrhythm


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interactive experimentation with modulation parameters [20].

Fig. 6. A simple diagram showing the two main structures of the computer application, MIDI song and FM song, and how there are interconnected through time references. Typical MIDI events are note-on (hatched) and note-off. An FM note represents a single time event and the modulation curve leading up to it.

At the core of the application are two parallel structures, labeled “MIDI song” and “FM song” respectively. “MIDI song” contains the musical material represented as sequences of MIDI events. It provides functions for handling file import/export and real-time playback based on MIDI and standard MIDI files2. “FM song” is an alternative view, abstracting the MIDI data into sequences of rhythmically significant temporal events (ignoring note endings and multiple notes in a chord). Each sequence is assigned a modulation operator, allowing modifications of the temporal distance between events.

The two main structures are interconnected only through time references, one for each note in an “FM song” sequence. Fig. 6 illustrates this connection. When computing movement curves the time references are modified by the modulating operators. During playback, the timing between each MIDI event in the “MIDI song” sequence is scaled by the corresponding time reference.

An “FM note” models the movement from one downbeat to the next as a single sinusoidal period, i.e. implementing equation (1). In order to achieve deviations from a strict metric timing we add a modulating operator similar to equation (3). In Fig. 7 below, we show the basic setup of a single operator and the parameters available.

Each sequence (i.e. voice or part) of the “FM song” refers to a single modulator throughout, but different sequences may have different modulators. Originally, the parameters were time-invariant constants, leaving no possibilities for dynamic behavior. With the current implementation, we have added two dynamic features to the modulation index: Time-dependency and tempo-dependency.

2 Specifications for MIDI and the Standard MIDI File format are available from the MIDI

Manufacturers Association (MMA): https://www.midi.org/specifications.


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8 C.H. Waadeland and S. Saue Tempo-dependency is implemented as a simple linear function, partly inspired by

the findings of Friberg and Sundström [4]. The constants of the linear function, m_a and m_b, are user configurable:

metro_factor = m_a * m_nMetronom + m_b; Time-dependency is modelled as a stepwise linear envelope function, but could be

expanded to a wider range of functions later. Each breakpoint in the envelope function is specified as a <time,amplitude>-pair with time expressed in measures (and fractions thereof). There is no limit to the number of breakpoints. A continuous, time-dependent amplitude is calculated at a specified point in time, nowtime, through interpolation between the neighboring breakpoints of the linear envelope function:

double timediff = next->time – previous->time; if (timediff > 0.0) { double tfac = (nowtime – previous->time) / timediff; amplitude += tfac * (next->amplitude – amplitude); }

Finally, the modulation index of the modulator is equivalent to the time-dependent amplitude multiplied with the tempo factor:

modulation_index = amplitude * metro_factor;

4 Examples of RFM Syntheses of Swing

In this section we demonstrate how we can use the RFM model with its computer implementation to obtain new sounding simulations of live performances of swing. Moreover, we suggest an application of RFM which may point at some interesting possibilities for applying the RFM instrument as a tool for electro-acoustic music.

Fig. 7. A choice of parameters for swing simulation, illustrated in the basic RFM instrument.


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Example 2: Continuous Increase in Swing Ratio. As mentioned in the previous, jazz drummers will perform the swing groove in Fig. 1 with different swing ratios (SR), depending on the individual drummer and the tempo of the performance. We now construct a simulation of various performances of this cymbal rhythm in the following way: First, we make a MIDI recording of the cymbal ostinato where every MIDI event is quantized in eighth note triplets, i.e., SR = 2. Secondly, we import this recording as a MIDI file into the computer program, applying RFM with the following parameter values:

fc = 4, fm = 2, φm = 0, n = 2, d: between -1 and 1, cf. Fig. 7. We obtain: d = -1 gives SR = 3, whereas d = 1 results in SR = 1, see Fig. 8.

Fig. 8. Illustration of movement curves associated with two different modulations of the same rhythmic pattern. The upper corresponds to d = -1, in the lower d = 1. An amplitude adjustment, A = (Δ)1/2, has been made to indicate how kinesthetic considerations might be implemented.

By making d a function of time, we are now able to simulate various performances

of the swing groove where the subdivisions may fluctuate between sixteenth notes and eight notes, which, indeed, is the case in live performances of this rhythm in jazz [4], [7]. To illustrate this, we make:

d(t) = 1 – (1/16)t, t = time (in seconds) . (4)

For this linear function d(0) = 1 and d(32) = -1, which means that the synthesized swing performance starts at t = 0 with SR = 1, and during 32 seconds ( = 16 bars of 4/4 swing in tempo 120 bpm) the swing ratio increases continuously to SR = 2 (t = 16 sec.), and further to SR = 3 (t = 32 sec.) This synthesis of continuous increase of swing ratio may be heard at the URL [24].

It should at this point be underlined that with this particular choice of d(t) we do not try to simulate any specific live performance of the swing groove. Rather, we demonstrate how the model makes it possible to make dynamic syntheses of swing performances where SR varies over time. Example 3: Tempo Specific Swing Ratios. Analyzing excerpts from jazz recordings, Friberg and Sundström found that a general trend in the performance of swing was an approximately linear decrease in swing ratio with increasing tempo [4]. Another study by Honing and De Haas, [7], found no evidence for SR to scale linearly with tempo, but they did find that jazz experts adapt their timing to the tempo of their


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performance. To illustrate how our model may simulate the findings of Friberg and Sundström, we apply the same MIDI recording and modulation parameters as in Example 2 above, except that we make d a function of the tempo as follows:

d(tempo) = (tempo/100) – 2, (tempo in bpm) . (5)

This gives: d(100) = -1, i.e. SR = 3; d(200) = 0, SR = 2; d(300) = 1, SR = 1, c.f. the findings in [4]. A sounding realization may be heard at [24] (8 bars in each tempo). Example 4: Synthesis of Participatory Discrepancies in Swing Performance. As discussed by Keil [15], and empirically investigated by, e.g., Prögler [2] and Alén [16], typical features of playing grooves and making music swing are various participatory discrepancies (PDs) in the performance of musicians playing together. We simulate such PDs between bass and drums in a jazz rhythm section by means of RFM, by first making a quantized (SR = 2), multi-track MIDI recording of a “walking” bass and a cymbal swing groove, after which we apply different frequency modulation to the bass- and drum recording. In the present example we make the cymbal perform with SR = 3.17, playing ahead of the bass, performing with SR = 2. The magnitudes of the PDs are here exaggerated compared to the findings of PDs in [4]. We do this deliberately to make the PDs in our example easy to hear, cf. [24]. Example 5: A “Weird” Two-Part Bach Invention. To give an example of how the RFM instrument might be applied to create exciting unplayable, rather “weird” performances, we import a metronomic, quantized MIDI recording of J. S. Bach´s composition “2-Part Invention No. 13 in A Minor” as a MIDI file into the RFM program. The voices played by the right and left hand are given different rhythmic modulations, in the following way:

Right hand (1st voice): fc = 4, fm = 0.0625 (=1/16), φm = 0, n = 3, d = 16 Left hand (2nd voice): fc = 4, fm = 0.0625, φm = 0, n = 3, d = -16

Observe that the modulation parameters used for the right and left hand are the same, except for the sign (+/-) of d. The result of this modulation, which may be heard at [24], is that whenever the 1st voice is making an accelerando, the 2nd voice is making a ritardando, and vice versa. The two voices “meet” every eight bar, and also at the end.

Rather than trying to simulate “real” live music performances, we have in this example created a new piece of electronic music. Seen as such, this illustrates how RFM might represent a new interesting compositional instrument for electro-acoustic music.

5 Discussion and Conclusion

In this paper we have shown how an application of dynamic frequency modulation of rhythm might be applied to obtain new syntheses of performances of swing. By introducing dynamic parameters in our RFM model we are able to simulate time-dependent typical features of live performances of rhythm, characterized by various “deviations from the exact”. As exemplified in our construction of the “weird” Bach invention, RFM synthesis may also be applied as a new compositional tool for electro-acoustic music.


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Swinging with FM 11 Several alternative computational models of expressive music performance have,

through the last 30 years or so, been developed. An overview of a large number of these models is given in [25]. An underlying assumption in the construction of these models is that there is a strong systematic link between musical structure and structure of music performances, and a common strategy in many of the model constructions has been to apply various quantitative empirical research to identify different performance rules by which expressive music performance can be modeled. – Our present model suggests an alternative approach: Rather than applying patterns of timing deviations found through measurements and analysis of timing, we present a continuous mechanism which models the systematic production of microtiming. Our idea is to apply an interaction of oscillators to achieve alterations of frequencies that create timing deviations. This is an idea which can be seen as natural on the basis of the current understanding of the role of various oscillations in neural processing of timing in the brain. At this point it is interesting to know that Mcguiness has presented a model based on interactions of oscillators to reproduce the microtiming of Clyde Stubblefield´s drum break on James Brown´s track “The Funky Drummer”, see [26]. The oscillators in Mcguiness´ model are pulse-coupled, whereas they are continuously-coupled in our model. In addition, the oscillators in Mcguiness´ model are mutually coupled, unlike our model, which utilizes one-way coupling of oscillators.

Although we believe that rhythmic frequency modulation is shown to be an interesting tool in making syntheses of various rhythmic characteristics of swinging performances of music, many aspects of the model can certainly undergo further development and change. For instance, we have so far made the strength of modulation, d, a linear function of time and tempo. An obvious improvement would be to extend the time- and tempo-dependency of d to non-linear functions. Moreover, it would be nice to implement real-time manipulation of the modulation parameters, and to develop the RFM instrument into a MIDI plugin. It should also be noted that in this paper RFM synthesis is used to simulate temporal aspects of rhythm performance, whereas in [27] frequency modulation is applied to make syntheses of movement trajectories in rhythmic behavior. It would be interesting to combine these approaches of FM rhythms to obtain a model by which aspects of timing, as well as characteristic gestures of live performances of rhythm could be simulated. Acknowledgments. The authors are grateful to the anonymous reviewers for their valuable comments and suggestions.

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Swinging with FM: Synthesis of Swing by Means of Frequency Modulationcmmr2017.inesctec.pt/wp-content/uploads/2017/09/8_CMMR... · 2017. 9. 24. · Swinging with FM 3 movement curves