Techniques to Foster Drum Machine Expressivity 1 Introduction 2 Rhythm

Home , Drum machine

Techniques to Foster Drum Machine Expressivity

Je A. Bilmes

Perceptual Computing Group

MIT Media Lab oratory

Massachusetts Institute of Technology

Cambridge, MA 02139

Abstract

Electronic drum machines need algorithms to help them pro duce \expressive-

sounding" rhythmic phrases. In [Bilmes, 1992], I claim that three p erceptually separate ele-

ments characterize p ercussiverhythm: metric content, temp o variation, and deviations (for-

merly called event-shifts). Herein, I demonstrate that algorithms based on this mo del may

considerably facilitate repro duction of expressiverhythm. I describ e one such algorithm

which extracts the separate elements from a p ercussive p erformance. The p erformance is

then resynthesized with varying degrees of temp o variation and deviations. Without the de-

viations, the p erformance sounds mechanical. With them it sounds rich and alive. Conse-

quently, I claim that we should b egin a concentrated study on the separate elements char-

acterizing p ercussiverhythm, particularly deviations. To this e ect, development has b e-

gun on a graphical drum machine program with which deviations may b e explored.

2 Rhythm 1 Intro duction

There is no doubt that music devoid of b oth har- To err is human. Yet most users of drum machines

monyandmelodymay still contain considerable and music sequencers strive to eliminate \errs"

expression. Percussivemusic is a case in p oint, in musical p erformance. In fact, some computer

as anyone who has truly enjoyed traditional mu- musicians never turn o the quantize option, de-

sic from Africa, India, or Central or South Amer- stroying forever \ aws that make the p erformance

ica knows. sound sloppy." At the same time, other computer

Unsuitable for p ercussivemusic however, musicians complain ab out the mechanical quality

previous representations of expressive timing of computer music. They call for the development

([Clines, 1977], [Ja e, 1985], [Schloss, 1985], of techniques whichwould enable computers to

[Repp, 1990], [Wessel et al., 1991], [Anderson and sound b etter, i.e. more \human."

Kuivila, 1991], [Anderson and Bilmes, 1991], and There are two orthogonal criteria of p er-

[Desain and Honing, 1992]) can all (with the ex- formance. The rst is sheer technical pro ciency.

ception of [Desain and Honing, 1992]) b e reduced Clearly, computers have long surpassed humans

to temp o variation. on this axis. The other is expressivity, something

In [Bilmes, 1992], I intro duce a new mo del more elusive, something that gives music its emo-

of rhythmic expressivity. Sp eci cally, I state that tion, its feeling, its joy and sorrow, and its human-

b eat-based rhythm can b e characterized bythree ity. Music exudes humanity; computer music ex-

comp onents: metric structure, temp o variation, udes uniformity. This, I strive to eliminate.

and deviations (formerly called event-shift mo d-

Author's current address: Computer Science Division,

els). In this pap er, I argue that deviations are

EECS Department, U.C. Berkeley, Berkeley, CA 94720.

most imp ortant for p ercussive and non-Western

I use \computer musician" to refer to anyone who uses

music, and that they are indisp ensable study for

a computer to create music, and \computer music" to re-

any drum machine architect wishing to create an

fer to music created thereof.

expressive sounding pro duct.

When we listen to or p erform music, weof-

ten p erceive a high frequency pulse, frequently a

binary, trinary, or quaternary sub division of the

musical tactus . What do es it mean to p erceive

tatum . this pulse, or, as I will call it, 2000

. erceiving the tatum do es not necessarily

P .

king in the mind, likea clock. imply a conscious tic .

354

Often, it is an unconscious and intuitive pulse

Tatums/Minute

that can b e broughtinto the foreground of one's

12345678

twhenneeded.Perceiving the tatum im- though Tatum Num

(A)

plies that the listener or p erformer is judging and

ticipating musical events with resp ect to a high an +15%

frequency pulse. It is a natural p erception, p er- 0%

haps similar to the illusory contour in the well

-15%

known picture on the frontcover of Marr's b o ok

Deviation: % of current tatum duration. [Marr, 1982].

12345678

ays explicitly stated in The tatum is not alw Tatum Num

(B)

usic. How, then, is it implied? The

a piece of m Tempo = 300 tatums/minute.

tatum is the lowest level of the metric musical hi-

erarchy. Often, it is de ned by the smallest time

Figure 1: EquivalentTemp o Variation and Devi-

interval b etween successive notes in a rhythmic

ation Representations

phrase. For example, two sixteenth notes followed

by eighth notes would probably create a sixteenth

note tatum. Other times, however, the tatum is

memb ers of the ensemble slightly adjusting their

not as apparent; then, it might b est b e describ ed

own p ersonalized temp o. Furthermore, the temp o

as that time division which most highly coincides

change needed to represent deviations in a typi-

with all note onsets.

cal p erformance would b e at an unnaturally high

The tatum provides a useful means of de n-

frequency and high amplitude. Imagine, at each

ing twocomponents of b eat-based rhythm, temp o

successive tatum, varying the temp o b etween 2000

variation and deviations. Tatums pass byata

and 354 tatums p er minute (Figure 1A). Perceptu-

certain rate, and may b e measured in tatums p er

ally, this seems quite imp ossible. However it seems

minute. Temp o variation may b e expressed as

quite reasonable to assume that a p erson could, at

tatum duration (in seconds) as a function of tatum

a constant temp o of 300 tatums p er minute, play

numb er. Similarly, deviations may b e expressed

every other note 15 p ercent of a tatum early (Fig-

as deviation (in seconds) as a function of tatum

ure 1B). In other words, although they mightbe

numb er. That is, a deviation function determines

mathematically equivalent, temp o variation and

the amountoftimethatanevent metrically falling

deviations are di erent in more imp ortantways {

on a particular tatum should b e shifted when p er-

they are distinct functionally and conceptually.

formed. Note that temp o variation is de ned p er

The previous paragraph suggests that there

ensemble, whereas deviations are de ned p er p er-

must b e some (p er p erson) upp er limit on temp o

former.

oscillation frequency. That is, any p erformance

A common question asked is, if we assume

variation not accounted for by temp o variation b e-

temp o variation is also p er p erson, why not just

cause of its high frequency must b e owing to \de-

use temp o variation to represent deviations in a

viations." The following algorithm was develop ed

p erformance? That is, is there mathematically

with this assumption in mind.

any di erence b etween temp o variation and devia-

tions? The answer is no, there is no mathematical

di erence. Either can represent p erformance tim-

3 Timing Extraction Algo-

ing. There is, however, a p erceptual di erence.

When listening to or playing in a drum or

rithm

jazz ensemble, there are times when the temp o

In [Bilmes, 1992], I p oint out the necessityofan

is considered constant, even though memb ers are

algorithm that extracts deviations from a p erfor-

playing o the b eat. Notice, the concept of b e-

mance. Presented herein is one that extracts the

ing \o the b eat" suggests that there is deviation

quantized score, the temp o variation, and the de-

from some temp o generally followed by the ensem-

viations. The input to the algorithm is a list of

ble. There is no concept, however, of individual

attack times.

The stroke of the hand or baton in conducting, or, of-

The algorithm is given complete metric

ten, the main quarter note b eat

knowledge of the p erformer. That is, it knows the

When I asked Barry Verco e if this concept had a term,

he felicitously replied \Not until now. Call it temporal

time signature, the numb er of tatums p er b eat,

atom,ortatom." So, in honor of Art Tatum, whose tatum

the numb er of b eats p er measure, and where the

was faster than all others, I chose the word tatum.

in time, the reference instrumentinformstheen- b eginning of the measure is (the answer to \where

semble what the temp o is. The p erformance in- is one?"). There are twoversions of the algorithm;

strument, dep ending on whether it is dominant one is primarily for p ercussivemusic, the other is

in the ensemble (such as a lead drum), controls slightly more general.

when the temp o sp eeds up and slows down. That Version I of the algorithm requires a rep et-

is, wesay the reference instrument de nes the itive reference instrument (such as a b ell, a clave,

temp o, and the p erformance instrument controls or a bass). The reference instrumentisusedto

the temp o. Therefore, when obtaining the timing extract temp o and must rep eatedly play a known

of a dominant p erformance instrument, welook pattern. The pattern p erio d mustbeaninteger

slightly ahead, and compute multiple of the measure duration. Percussivemu-

sic normally contains such an instrument, so this

n+C

is not an unreasonable requirement. The algo-

T [n]= T [k ];

rithm pro duces the expressive timing of a perfor-

C +1

k =n

mance instrument relative to the reference instru-

where C is a parameter determining howfarinto

ment. In an ensemble, any instrument other than

the future we should lo ok. C dep ends on the p er-

the reference instrumentmay b e considered a p er-

formance instrument, and could equal zero; ac-

formance instrument.

cordingly, T [i]mayormaynotbeananticipatory

The algorithm rst computes a temp o func-

measure duration estimate.

tion using the reference instrument. The temp o

Creating a continuous time function, we

function is then transformed into a tatum dura-

tion function { tatum duration as a function of next linearly interp olate

tatum number. The tatum function determines

t x[n]

a normalized metric grid; i.e. a time grid spaced

; D (t)=D [n]+(D [n +1] D [n])

x[n +1] x[n]

so that grid markers determine the time p oints

of each tatum. The metric grid is then used to

where

judge the p erformance instrument. For each p er-

formance instrumentattack, the deviation is its

n = fn : x[n] t

distance to the nearest grid marker.

It follows that D (t) is an estimate, at time t,of

Let L b e the numb er of tatums p er measure,

the measure duration.

R b e the numb er of reference instrumentattacks

Clearly, D (t) increases as temp o decreases,

p er measure, x[n]bethe n reference instrument

and 1=D (t) decreases as the temp o decreases. We

attack time, y [n]bethen p erformance instru-

want the temp o-normalized time p oints. There-

ment attack time, and let z [n]= x[n R]beour

fore, for each measure, we nd the time p oints that

estimate of the starting time of the n measure

divide the area under 1=D (t)into L equal area re-

(if reference instrument attacks do not fall on the

gions. The time p oints provide a tatum time func-

measure starting p oints, weinterp olate, add en-

tion, a function that gives the time p oint for each

tries to x[n], and pretend that it do es). y [0] must

tatum.

lie past the rst measure's starting p oint.

So, for each measure m,0 m

For n =0:::R 1, we compute

and eachtatumi in measure m,1 i< L,we

M 2

nd b [mL + i] where

1 x[mR + n +1] x[mR + n]

P [n]= ;

M z [m +1] z [m]

z [m] b [mL + i]

m=0

where M is the numb er of measures in the p er-

and

b [mL+i]

formance. P [n] is the average measure fraction of

1=D (t)dt

z [m]

= i=L:

R the time duration b etween reference instrument

z [m+1]

1=D (t)dt

z [m]

attacks n and n +1. If the p erformer is play-

ing very uniformly (i.e., nearly quantized), and we

The array b [n] is the L-tatum p er measure time

have the score for the pattern, P [n]may b e ob-

lo cation of tatum n.

tained directly from the score rather than from

We next compute the rst order forward

the attack times.

di erence by linear convolution

Next, a rough temp o function

d [n]= b [n] ( [n +1] [n]);

x[n +1] x[n]

; T [n]=

where [n] is the unit sample sequence .

P [n mo d R]

Higher order interp olation schemes could b e used,

is computed. T [n] provides an estimate, at time

but they probably would not signi cantly alter the nal

x[n], of the measure duration. Atanyonepoint

outcome.

[0] = 1;[n 6=0]=0

To lter out high frequency variation, we

convolve again, and compute

! ! ! ! ! ! ! ! ! ! !

G ~ @ @ @ @ @ ~

d[n]= d [n] h[n];

where h[n] is either an FIR low-pass lter with a

desired stop-band, or a Savitzky-Golay smo othing

Figure 2: Guagua Pattern

lter [Press et al., 1992]. This step removes high

frequency variation in d [n]. Thus, the array d[n]

The temp o variation b[n] and the deviations

is our estimate of the duration of the n tatum.

devs[n] are computed as in version I. In this case,

We next recover the tatum p ositions from d[n]by

however, the reference instrument x[n] and the

convolving with the unit step sequence u[n]

p erformance instrument y [n] need not b e di erent.

b[n]= (d[n 1] + b [0] [n]) u[n]:

Version I I essentially computes the temp o varia-

tion as the low-pass ltered p erformance variation,

The array b[n] then gives us the time p osition of

and uses the high frequency p erformance variation

tatum n.

as the deviations. The trick is to nd the desired

For each p erformance instrumentattack

stop-band frequency, something which largely de-

y [n], we nd the closest tatum. The distance from

p ends on the musical style.

y [n] to the closest tatum is the attack's deviation.

That is

devs [n]=y [n] b[j ]

4 Results

and

quants[n]= j;

Version I of the timing extraction algorithm

was applied on a p erformance given byLos

where

Munequitos ~ de Matanzas, the extraordinary drum

j = argmin j y [n] b[j ] j;

ensemble from Matanzas, Cuba. Timing data was

obtained from the following p erformance instru-

The array devs [n] is the deviation function and

ments: the quinto (high), the segundo (middle),

quants [n] is an array of tatum numb ers. There-

and the tumbao (low) drum. A new attackde-

fore, the quantized score is given (in tatums) by

tection algorithm, using only high frequency en-

quants [n], the temp o variation by b[n], and the

ergy to determine the attack, is de ned in [Bilmes,

deviations bydevs[n]. A p ositive deviation means

1993] and was used here. The reference instru-

the attack o ccurred after the tatum, and a nega-

mentwas the guagua, a thick bamb o o cylinder,

tive one means it o ccurred b efore.

ab out 4" diameter, hit with twosticks. An ap-

Version I I of this algorithm do es not require

proximation to the reference instrument pattern

the reference instrumenttoplay rep etitively and

may b e seen in Figure 2. What follows are the re-

do es allow the reference and p erformance instru-

sults of the segundo only.

menttobethesame.Itdoes,however, require

The algorithm was run with C = 3 and

the complete score or quantized representation. If

h[n]= u[n] u[n 5]. Therefore, h[n] is just a rect-

the score is known, however, and the goal is to ob-

angular moving windowaverage. b[n] is plotted in

tain the temp o variation and deviations, it may

Figure 3. Although it lo oks as if there is consid-

b e used directly.

erable high frequency, the abscissa scale informs

Here is the main di erence from version

us otherwise. Figure 4 shows the DFT magnitude

I. P [n] is not computed from the p erformance.

of b[n]. The abscissa is in normalized frequency

Rather, P [n] is obtained directly from the score.

units, where 0:5 corresp onds to the Nyquist fre-

That is, P [n] b ecomes the measure fraction of the

quency.However, the frequency units are in cy-

time duration b etween reference instrumentat-

cles p er tatum, not in Hz. The DFT magnitude

tacks n and n + 1. The measure fraction is com-

is not plotted for f>0:1 (10 tatums p er cycle)

puted using the score. So an eightnotein time

since there is no signi cantenergy. Also, notice

would pro duce a value of 1=8, and a quarter note

the p eaks at 0.0620 and 0.0630, corresp onding to

time a value of 2=7. The starting time of each in

16.13 and 15.9 tatums p er cycle resp ectively.Itis

measure, z [n], is computed from x[n] according to

probably more than coincidentalthat16isboth

the score. If, for a particular measure, no x[n] falls

the numb er of tatums p er measure and a large

at the b eginning, weinterp olate, and create an es-

comp onent in temp o variation.

timation of the measure starting time. The only

Figure 5 shows a plot of the deviations for

other di erences are the following:

this p erformance. In this form it is hard to see any

x[n +1] x[n]

structure. Although the deviation arrayisessen-

; T [n]=

P [n]

tially an unevenly sampled signal, sp ectral anal-

ysis is still p ossible. The Lomb normalized p eri-

u[n< 0] = 0;u[n 0] = 1 0.08

0.06

0.04

0.02

0 0.16 -0.02

0.155 -0.04 Deviation (seconds) 0.15 -0.06

-0.08 0.145 -0.1 0.14 -0.12 0 200 400 600 800 1000 1200 1400 1600 1800 0.135 Tatum

0.13

Tatum Duration (seconds) Figure 5: Segundo deviations 0.125

0.12 0 200 400 600 800 1000 1200 1400 1600 1800

Tatums

o dogram [Press et al., 1992] is a magnitude sp ec-

tral analysis technique sp eci cally designed for un-

Figure 3: Munequitos ~ temp o track

evenly sampled signals. It is commonly applied to

astrophysical data where regular sampling is not

p ossible. Hoping to uncover some structure, I de-

velop ed a short-time version of this algorithm, and

applied it to the deviations.

Figure 6 shows the short-time Lombnor-

malized p erio dogram for the segundo deviations in

Figure 5. The front axis is, again, in normalized

frequency units (cycles p er tatum) where 0.5 is the

Nyquist frequency. The window size is 32 tatums,

and the overlap is 24 tatums. Notice the strong

p eak at 0.25 cycles p er tatum, implying consider-

able deviation p erio dicity near 4 tatums p er cycle.

The segundo p erformance, in fact, largely consists

of a rep eating 4 tatum phrase. For larger window

1.6

sizes (order 100), this p eak signi cantly narrows

1.4

centered right on 0.25, and other small p eaks ap- , this con rms

1.2 p ear at 0.125, 0.166, 0.333. Clearly that structure do es exist in the deviations.

1 7

The p erformance was resynthesized by

0.8 triggering select samples of the original p erfor-

Power

mance. I develop ed an automatic note classi ca-

0.6

tion algorithm to obtain the drum stroketyp es

h completed the score [Bilmes, 1993]. The

0.4 whic

various resynthesis examples follow:

0.2 y triggering the samples at the ap- 0 1. Direct { b

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 propriate time.

freq (Cycles/Tatum)

2. Quantized { using a constant temp o equal

to the overall average.

Figure 4: DFT magnitude of the temp o, b[n]

3. Quantized { using b[n] as the temp o

4. Quantized { with devs[n] added to the n

attack.

7 Using Csound. power power 10 10 5 5 0 0 0 0

500 500

tatums tatums 1000 1000

1500 1500 0.4 0.5 0.1 0.2 0.3 0.3 0.4 0.5 0 0 0.1 0.2

freq (Cycles/Tatum) freq (Cycles/Tatum)

Figure 6: Lomb Normalized Perio dogram: Se-

Figure 7: Lomb Normalized Perio dogram: Ran-

gundo Deviations

dom Gaussian Deviations

5. Quantized { with random Gaussian devia-

5 Exp erimental Tatum De-

tions added to each attack time. The Gaus-

viation Program

sian pro cess had the same mean and vari-

ance as the devs [n] array.

Drum machines and music sequencers should start

providing advanced facilities for exp erimenting

6. Quantized { with p er-tatum random Gaus-

with deviations. While waiting for this to o ccur,

sian deviations added to each attacktime.

we have develop ed a deviation exp erimentation

Here, there were 16 indep endent Gaussian

program called xited (pronounced \excited" for

pro cesses, each with a di erentmeanand

eXp erimental InteractiveTatum-Editor of Devia-

variance. The mean and variance for the i

tions, see Figure 8). Currently, xited runs on SGI

pro cess was the same as the mean and vari-

IRIS Indigo workstations.

ance of devs [n mo d i].

The program consists of a control panel,

Most p eople who listen to these examples

and anynumb er of pattern windows. The main

say that numb er 4 sounds most like the original,

windowcontrols global temp o in units of normal-

observing that only 4 contains the \feel" of the

tatums p er minute, starting and stopping, and

original p erformance. In addition, numb ers 5 and

other miscellany.

6 are considered, in general, to sound \sloppy" and

The pattern windows determine the score.

\random." Accordingly, Figure 7, showing a p eri-

Each pattern window consists of a rectangular grid

o dogram for the deviations in resynthesis 5, con-

of toggle buttons (of any size), an additional row

rms that there is lack of structure. As exp ected,

of sliders, and a duration value. A pattern win-

resynthesis 2 sounds mechanical. Unexp ectedly,

dow's grid represents a rep eatedly played p ercus-

even resynthesis 3 sounds mechanical. In general,

sive phrase. The rows corresp ond to drum samples

without the correct deviations, the p erformance

or voices and the columns corresp ond to pattern-

sounds colorless and cold { with them it sounds

tatums. If a toggle is set for row i and column

rich and alive.

j , then voice i will b e triggered during pattern-

Consequently, I prop ose that, in addition to

tatum j . Each column also has a corresp onding

the ongoing studies of temp o variation, we b egin

deviation slider. The slider for pattern-tatum j

a concentrated study on p erformance deviations.

determines, in p ercentage of pattern-tatum, the

Combining b oth temp o variation and deviations

amount of time to shift all voices set to playon

could eventually pro duce the full e ect of rhyth-

that pattern-tatum.

mic expressivity.

A pattern windowalsocontains a duration

in units of normal-tatums. Therefore, di erent

patterns mayhave di erentnumb ers of pattern-

Je Foley, an undergraduate working with me at MIT,

has b een the primary implementor of this program

Figure 8: Graphical Deviation Program xited in Action

viations for a drum pattern. Indeed, some very tatums, with their absolute durations the same.

interesting rhythmic e ects may b e attained with This may b e used to express p oly-rhythms and

xited byvarying deviations and pattern dura- multi-tatum ethnic music. For example, in Fig-

tions. ure 8, the top pattern has a duration of 16 normal-

tatums and contains 24 pattern-tatums. The b ot-

tom pattern has a duration of 16 normal-tatums

6 Conclusion

and contains 16 pattern-tatums. This example

enco des, in a sense, the feeling of multiple con-

This pap er is a summary of two and one-half chap-

current tatums that is heard in African or Afro-

ters from [Bilmes, 1993]. Therein may also b e

Cuban music.

found a new drum attack detection algorithm, an

Each pattern window maintains a counter.

automatic drum stroke classi er, and the design

When the button is pressed, the counters

of a deviation learning algorithm .

cycle through their patterns mo dulo their pattern-

In this pap er, I haveintro duced the concept

tatum length. When the counter reaches a partic-

of tatum, have utilized the separate elements de-

ular tatum, anyvoices scheduled for that tatum

ned in [Bilmes, 1992] for rhythmic analysis, and

are appropriately shifted and triggered. During

have demonstrated the imp ortance of deviations

playback, deviations, toggles, and pattern dura-

for representing expressivity in p ercussivemusical

tions may all b e adjusted.

phrases. Deviations play a vital role in rhythm.

xited is thus a novel drum machine user

They should b e analyzed, comprehended, and uti-

interface. A similar suchinterface could b e used

lized. And b efore switching on that quantize op-

bymusic sequencers, or eventually,by commercial

drum machines. In [Bilmes, 1993], an algorithm is

This work, including audio examples, the timing ex-

traction algorithm, the deviation exp erimentation pro-

de ned that creates a mapping b etween quantized

gram, the drum attack detection and classi cation algo-

musical patterns and sets of deviations. This algo-

rithm, and the learning algorithm, is (or will so on b e) avail-

rithm will b e eventually incorp orated into xited.

able via anonymous ftp on amt.mit.edu:pub/bilmes-thesis,

xited provides the ability to exp eriment with de-

cecelia.media.mit.edu:pub/bilmes-

thesis, and ftp.icsi.b erkeley.edu:pub/bilmes-thesis viations and to determine the b est sounding de-

tion, we should rememb er that yes, to err is hu-

man, but to forgive divine.

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