Toward a formal study of jazz chord sequences generated by Steedman’sgrammar

M. Chemillier

Abstract This paper shows how to generate jazz chordsequences incrementally using Steedman’s grammar bytaking advantage of some formal properties. We point outthe specific role played by particular chord sequencescalled ‘‘cadential sequences’’. By precompiling them, onecould improve the implementation of Steedman’s gram-mar in a real-time improvisation system, and make it morereactive to the inputs of the user.

Keywords Formal grammar, rewrite rule, chord sequence,harmonic substitution, jazz, improvisation, real-time

1IntroductionFormal grammars are a simple way to define sequences ofsymbols. The sequences generated by a given grammar arederived from a particular symbol called axiom, using aspecific type of derivation called ‘‘rewrite rules’’. Toillustrate the derivation mechanism, consider the rewriterule a fi aa. Then abaac is derived from abac, the secondoccurrence of a being replaced by aa.

It is easy to see that the rewrite rule mechanism is wellsuited for modeling the musical notion of ‘‘variation’’.Thus a great amount of research has been made in theapplication of the grammar formalism to music (see [2, 7,9, 10, 15, 16] for a selection of references related to thissubject). One of the most relevant approaches in thisdirection is Steedman’s study of jazz chord substitutions.This is a device used by jazz musicians to create more andmore elaborate variations of the harmonic structureunderlying an improvisation.

Steedman’s grammar is designed to generatively specifythe set of possible blues chord sequences. It is based oncertain rules called ‘‘substitution rules,’’ which allow themore elaborate sequences to be derived or generated fromthe most basic form by the replacement of the originalchords by substitutes ([17] page 56).

The paper written by Steedman in 1984 has been quiteinfluential on related works by Johnson-Laird and Pachet

[8, 11, 12, 13, 14]. It has been followed by further articlesby Steedman himself on the same subject [18], but thesenew developments are more to do with making thegrammar deliver an explicit harmonic interpretation thanchanging the musical content of the syntactic grammaritself. The form of Steedman’s grammar that we shall usein the present paper is the 1984 original form, which wehave already discussed in [7].

The problem addressed in this paper is to generate jazzchord sequences incrementally using Steedman’s grammarby taking advantage of some formal properties. Our moti-vation comes from the fact that we want to make use ofSteedman’s formalism as a component of a system ofimprovisation in real-time. The real-time constraint im-plies that the sequences are generated incrementally, not asa whole. To achieve this goal, it is necessary to look moreclosely at the effects of Steedman’s substitutions rules on agiven chord sequence.

After basic definitions, we discuss some problems re-lated to the use of the grammar formalism in a real-timegeneration context. Then we recall a few aspects ofSteedman’s grammar focusing on the specific role ofdominant seventh chords. Finally, we turn to the maincontribution of this article, which is the characterization ofthe jazz chord sequences generated by Steedman’s gram-mar in terms of particular sequences called ‘‘cadential’’.

2Definition of formal grammarsThe following definitions are adapted from [3]. Let usrecall that for every set of symbols X, we denote by X� theset of finite sequences of elements of X.

A grammar G ¼ (s, N, T, R) is defined by two disjointsets of symbols, the terminals T and the nonterminals N, aparticular nonterminal s 2 N called the axiom, and a finiteset R included in (N[T)� · (N[T)� of rewrite rules. A re-write rule (u, v) 2 R is usually written in the form ufiv.

We say that a sequence g is derived from f, and we writeffig, if and only if there are factorizations f ¼ wuz,g ¼ wvz with w, u, v, z 2 (N[T)�, such that ufiv 2 R.Thus any rule ufiv means that the sequence u may bereplaced by v inside any sequence in which it occurs. Forany integer p, define

f �!p g

if and only if there exist f0, f1, ..., fp such that f0 ¼ f, fp ¼ g,and fififi+1 for i = 0, . . . , p� 1. In this case, p is called thelength of the derivation from f into g. Finally, we define

Focus Soft Computing 8 (2004) 617–622 � Springer-Verlag 2004

DOI 10.1007/s00500-004-0386-3

Published online: 3 September 2004

M. ChemillierIRCAM-UMR CNRS 9912,1 place Igor Stravinsky,F-75004 Paris-FranceTel.: +33 1 44 78 16 49Fax: +33 1 44 78 15 40e-mail: chemilli@ircam.fr

I thank Mark Steedman for his comments on this text.

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f �!� g

if and only if f �!p g for some integer p.The language generated by the grammar is the set of

sequences of terminal symbols that can be derived fromthe axiom, using the rewrite rules of the grammar

LðGÞ ¼ fw 2 T �=s�!� wg:

3Formal grammars and real-time interactionFrom a musical point of view, this rewrite rule mechanismis well adapted to the description of harmonic substitu-tions. To illustrate this point, consider the harmonic looprepresented in Fig. 1. It is taken from a house recordentitled Rain by Kerri Chandler, a DJ from New York [5].Notice that the first three bars are very similar to bars 4–6of the well-known standard Round Midnight by Theloni-ous Monk. One can apply the following substitution rule toit

Gb7M Ab7! Ebm7 Ab7

and the result is shown in Fig. 1.We have made experiments with Steedman’s formalism

in a system of improvisation with the computer. Theseexperiments on improvisation were done at Ircam in col-laboration with Gerard Assayag and Carlos Agon andChandler’s loop has been our working example for manymonths. Besides the improvisation itself (the musicianplays phrases which are recorded by the system, andtransformed into new phrases in the same style, accordingto a model introduced in [1]), the system provides anaccompaniment part by playing voicings based on thechord sequence of the tune on which the musicianimprovises [6]. This chord sequence is elaborated usingSteedman’s rewrite rules.

There are two ways of interacting with our improvisa-tion system. The first way is to play musical phrases, as itis said before. The second way is to change the values ofsome parameters of the computation process. Forinstance, in the process of applying rewrite rules to thechord sequence, one can change the number of rules whichare applied (this corresponds to the ‘‘deepness’’ of thesubstitution process).

But there is a strong limitation in the interaction loop ofour system, due to the rewrite rule mechanism. When onechanges the number of rules applied to the chordsequence, this change has no effect until the current chordsequence is played entirely. The new value is only takeninto account in the computation of the next chordsequence to be played. Thus we have to compute the

substitutions of the chord sequence as a whole. We cannotmake the substitutions only on a small part of it, becauseotherwise we would not be able to reach all the variationsof the chord sequence that can be generated by thegrammar. This is hardly compatible with the real-timecontext where musical objects are supposed to be gener-ated on the fly.

To make it more precise, consider the rewrite rule xzfizz, and the sequence xyxxxz. The only possible successorto y in this sequence is x. If we apply the rewrite ruleseveral times, does it change the number of possible suc-cessors to y? And if so, how far in the sequence are thesymbols responsible for this change? In the initial se-quence, one can apply the rule only once, at the end of thesequence, which gives xyxxzz, and it has no effect on thepossible successors of y. On the contrary, if we apply therule two more times, we obtain xyzzzz, which introduces anew possible successor z to y. This means that in general,we cannot predict the possible successors of a givensymbol until we have applied the rules everywhere in thesequence, including its end.

We have tried to solve this problem by precompiling theset of chord sequences generated by the grammar. This setwas represented in lexicographic order as a tree, each nodecorresponding to a subset of sequences sharing the sameprefix. Thus the leaves coming out of one node gave thepossible successors for the path going from the root to thisnode. From a theoretical point of view, this tree was an‘‘automaton’’ recognizing the set of substituted chord se-quences. It was precompiled before the real-time impro-visation process begins. In this way, it became possible topredict in real-time the possible successors for a givenchord in the sequence.

Carlos Agon designed an interface for this earlier ver-sion of the system (presented at the JIM 2001 in Bourges[6]). The sequence which is currently played is shown ontop of the window Fig. 2. On the bottom of the window,there are three circles representing all the available chords(major seventh, minor seventh, and dominant seventh).The possible successors for the current chord were indi-cated in real-time to the user by lights on the circles (inFig. 2 for instance, F#M7 is the only possible successor toC#7). The user had the possibility to click on these circlesto constrain the choice of the remaining chord sequence tobe played.

This solution was acceptable for substitutions on a shortchord sequence, because in this case it was possible tocompute all the sequences generated by Steedman’sgrammar. But such a method was no longer practicable forlonger chord sequences, the number of generatedsequences being too big.

Bm7 E7Bbm7

Eb7

Abm7

Db7

Gb7M

Ab7

B7Bb7

Bm7 E7Bbm7

Eb7

Abm7

Db7

Ebm7

Ab7

B7Bb7

Fig. 1. Harmonic loop from Rain by KerriChandler, and its substitution

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In this paper, we shall see that instead of precompilingthe whole set of sequences obtained by applying Steed-man’s grammar to a given chord sequence, one couldonly precompile some particular ‘‘cadential sequences’’.The idea is to look more closely at the specific form ofthe rewrite rules introduced by Steedman, and to make aformal study of the language of sequences generated bythese rules. As we shall see, it will make more explicit therole of the cadential sequences in the substitution pro-cess. But before we turn to this more formal approach,we shall begin with a brief overview of Steedman’s 1984paper.

4An overview of Steedman’s grammar

4.1Steedman’s substitution rulesSteedman’s grammar is obtained empirically from a set ofmodern jazz 12-bar chord sequences adapted from a bookby Coker [4]. This set is considered as a wide and repre-sentative range of permissible variations of the blues basicform ([17] page 54). Steedman’s study attempts to providea small set of rules that characterize this musical ‘‘sub-language’’, and he has founded that six rules could achievethis goal.

Some of these rules are less important than the others,and we shall not take all of them into account. The lastrule, numbered ‘‘Rule 6’’ by Steedman, is dedicated to theintroduction of a particular type of passing chord named‘‘diminished seventh chord’’. We shall ignore this rulesince we do not include diminished seventh chord in oursystem (our table of voicings does not contain any voicingof this type).

The rule numbered ‘‘Rule 5’’ provides a special way ofelaborating a chord that is not part of a dominant cadence,which can be found only once in the set of blues chordsequences taken from Coker (Example (d) in Steedman’stable page 54). This rule seems to be ad hoc, and we shallignore it.

One more rule, numbered ‘‘Rule 2’’, is ignored in oursystem. It contains a substitution of a chord by its sub-dominant. Steedman points out that this rule must be usedcarefully. The replacement of a chord by its subdominantmust occur on a right branch of the metric tree of thesequence (which means a chord which is not metricallystressed). Otherwise,

‘‘since the leftmost branches of the hierarchy are themetrically stressed ones, such a substitution [would]changes the harmonic character of the sequence’’ ([17]page 61).

Thus it remains three rules in Steedman’s grammar(Rules 1, 3, 4) that we have implemented in our improvi-sation system.

Rule 1: x! x xx7! x x7xm7! x xm7

Rule 3a: w x7! Dx7 x7w x7! Dxm7 x7

Rule 3b: w xm7! Dx7 xm7Rule 4: Dx7 x7! Stbx7 x7

Dx7 x! Stbx xDx7 xm7! Stbxm7 xm7

In these rules, and so means any chord satisfyingconditions given below x is a variable over the set of chordroots. We use a common notation that conveniently re-flects chord relationships independently of any particularkey. It is based on Roman numerals to represent chords,where the numeral identifies the degree of the chord rootin the scale. The range of numerals is shown in Fig. 3. Forinstance, in the key of C major the chords correspondingto I, IV and V are respectively C, F and G.

The symbols Dx and Stbx stand for the chords withrespectively the dominant and the flattened supertonic of xas its root (for instance Dx ¼ VI and Stbx ¼ III[ whenx ¼ II).

In addition to the Roman numerals indicating thedegree of the chord relative to the key note, the notation

Fig. 2. Interface for choosing the next chordto be played (by Carlos Agon)

Fig. 3. Degrees of the chord root

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uses a system of suffixes, which specify certain additionalnotes that are played with the notes of the root chord. Thesuffix 7 means that the ‘‘dominant’’ seventh note, a tonebelow the root, is to be included. It is understood to bebased on the major chord, which gives a ‘‘dominant sev-enth chord’’, as in V7, unless explicit indication is giventhat it is based on the minor by a small m immediatelyfollowing the Roman numeral, as in Vm7 ([17] page 76).

Every chord is considered as nonterminal. Each rule istranslated into twelve different rules by giving a degree(from the circle Fig. 3) as value to the variable x (andupdating the corresponding values of Dx and Stbx). Anaxiom s is added to the grammar, together with a rule thatinstantiates the reference chord sequence on which thesubstitutions are applied. For instance, the axiom and theadditional rule associated with the blues are defined bySteedman as follows ([17] page 61):

s! I I7 IV I V7 I:

There is a convention in all Steedman’s rules that thethings on the right occupy the same total amount of timeas the things on the left. Rules 3 and 4 have the samenumber of chords on each side, so that the corre-sponding chords have the same duration on both sides.But in Rule 1, a chord can be replaced by two copies ofitself, each lasting half as long (the six basic chords ofthe blues above can thus be expanded into twelve chords,so that each of them occupies exactly one bar). Noticethat the two copies are not identical. If the original chordwas a dominant seventh chord, or a minor seventhchord, then the rightmost of its offspring is too, but notthe other ([17] page 62). It is important to point out thatthe time division process implied by Rule 1 cannot berepeated endlessly, since chord sequences generally donot include chords with a duration shorter than aquarter of a bar.

The terminal symbols of the grammar are copies of thechords themselves (recall that the sets of terminal andnonterminal symbols are supposed to be disjoint), andthere is an implicit convention that at the end of thesubstitution process, each chord is derived into a terminalcopy of itself. Steedman introduces optional rules to addsixths, ninths, and so on to these terminal copies of thechords ([17] page 69).

4.2.The role of dominant seventh chordsThe dominant seventh chords have a specific harmonicfunction which is central in Steedman’s approach, so thathis statements of the rules are very careful about thesechords:

‘‘One of the suffixes is particularly important for therules that follow. The suffix 7 [...] means that the note aflattened or ‘‘dominant’’ seventh above the root is playedwith the other notes of the chord. [...] This modificationhas the effect of making the listener expect a further chordrelated to the dominant seventh chord by having its root afifth below or a fourth above’’ ([17] page 58).

In Rule 3 above, the symbol w is a variable over chords,but there are stringent restrictions upon the chord that wmay match:

(1) w cannot match a dominant seventh chord,(2) w cannot match a chord that has had its root changed

by the previous application of a substitution rule.

As Steedman says,‘‘The restriction of w to non-dominant seventh chords

and to chords whose root has not been changed by aprevious application of a substitution rule [...] prohibits alot of ill-formed sequences that would otherwise arise’’([17] page 71).

There is a subtle distinction made by Steedman betweenthe dominant seventh chord, and a major chord which isplayed with what could be called a ‘‘bluesy feeling’’ byadding the dominant seventh note to it. In the latter one,the seventh is just an optional note added to a terminalcopy of the major chord. Steedman distinguished carefullythese chords denoted by x7¢ from the real dominant sev-enth chord x7.

‘‘There are two distinct harmonic functions that can beperformed by the chord which on the keyboard is playedwith a note a semitone below the seventh of the key note.Besides the ‘‘leading’’ dominant seventh function justdefined, it may have the function of a minor seventhchord, which does not lead anywhere in particular. [...] Inthe standard notation both are, therefore, written with thesuffix 7. Since the present rules treat these two ‘‘homo-phones’’ differently, the nonliving chords with the minorseventh are distinguished [...] with the nonstandard suffix7¢¢¢ ([17] page 58, note 7).

This remark is quite important since the dominantseventh chords are the foundations of the substitutionprocess described by Steedman, which can be viewed as‘‘extending the authentic cadence’’. In fact, Rule 3a

‘‘has the effect of ‘‘extending’’ an authentic cadencebackward in the sequence’’ ([17] page 63).

As we shall see, these dominant seventh chords are thekey of the formal property studied in the next section.

5A formal property of chord sequences generatedby the grammarThe main contribution of this article is to provide a formalcharacterization of the sequences generated by the gram-mar in a particular case. If we denote by d a dominantseventh chord, and by u a chord sequence which does notinclude any dominant seventh chord, the problemaddressed in this section is to study the sequences that canbe derived from ud. The surprising fact that we shall pointout is that in some sense, these sequences only depend ond, and do not depend on u.

To make it clearer, we represent in Fig. 4 the derivationof the sequence ud. The derivation process described bySteedman as an ‘‘extension of the authentic cadence’’generates a new sequence z which replaces a suffix of ud(denoted by u00d), and which does not depend on thespecific content of u00. In this newly generated sequence vz,there may appear a surprising transition (from a musicalpoint of view) between the end of v and the beginning of z,but this effect is attenuated by the fact that afterward, thereis a logical progression through z leading to its final chord,which is the dominant seventh chord d (the sequence z

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being what we call a ‘‘cadential sequence’’, resulting fromthe extension of the authentic cadence). In Fig. 4, we havedrawn z as a curve, to emphasize this logical progression.The fact that z does not depend on u00 is related to thecumulative use of variable w introduced in Steedman’sRule 3.

We shall give a more formal statement to this simpleproperty. But to make the things easier, we shall restrictourselves to major chords and dominant seventh chords,and their associated rules:

Rule 1: x! x xx7! x x7

Rule 3a: w x7! Dx7 x7Rule 4: Dx7 x7! Stbx7 x7

Dx7 x! Stbx x

thus excluding the minor seventh chords xm7 from ourstatement.

We denote by A the set of all chords (x or x7), and by Dthe subset of dominant seventh chords (denoted by x7).The three rules of Steedman’s grammar are denoted by R1,R3, R4 (to keep his numbering). If u is a sequence of A�, wedenote by R(u) the set of sequences that can be derivedfrom u by applying any of the three rules R1, R3, R4, and wedenote by R1(u) the set of sequences that can be derivedfrom u by applying rule R1.

Definition 1. A cadential sequence is a sequence z of A�

such that there exists a dominant seventh chord d 2 Dsuch that z 2 R(d).

Lemma 1. For any x, y, z in A�, if y 2 R(x) and z 2 R(y),then z 2 R(x).

Lemma 2. If a sequence contains no dominant seventhchord, the application of R1 to it cannot introduce any newdominant seventh chord.

Proposition 3. If d is a dominant seventh chord, d 2 D,and u a chord sequence with no dominant seventh chords,u 2 (A\D)�, then for any sequence y derived from ud, y 2R(ud), there exists a factorization u ¼ u¢u00 such that y 2R1(u¢)R(d), which means that a suffix of y is a cadentialsequence which does not depend on u00.

Proof. By induction on the length p of the derivation ofud, assume that ud is derived into vz satisfying v2 R1(u¢),z 2 R(d). Thus z is a cadential sequence. We now consideran additional rule applied to vz which gives y

ud�!p�1vz �! y:

There are three possible ways of applying the last rule tovz.

(i) First, the rule is applied to the left factor v. ByLemma 2, R1(u¢) is included in (A\D)�, thus v contains nodominant seventh chord, which implies that R1 is the onlyrule which can be applied to v. It follows that y ¼ v¢z withv¢ 2 R1(u¢).

(ii) Assume now that the last rule is applied to the rightfactor z. Then y ¼ vz¢ with z¢ 2 R(z), thus by Lemma 1, z¢is a cadential sequence, z¢ 2 R(d).

(iii) Finally, assume that v ¼ v¢a and z ¼ bz¢, with therule being applied to ab. This rule cannot be R1, which hasonly one symbol in its left part. Since v contains nodominant seventh chord, then a j2 D. It follows that therule cannot be R4, and thus must be R3, and we shalldenote it by

R3 : ab! cb:

The derivation from vz into y can be written v¢abz¢ fiv¢cbz¢.

Since a j2 D, then a must be a major chord, so that theapplication of R1 to it is just a duplication

R1 : a! aa:

It follows that the last symbol of u¢ is a, with u¢ ¼ u¢1a. Ifthe derivation from u¢1a into v¢a does not involve the lastsymbol, then v¢ 2 R1(u¢1). If it does, then a has beenduplicated, so that v¢ ends with the symbol a, and in thiscase v¢ 2 R1(u¢). In both cases, v¢ is derived from a prefix ofu (whatever it be u¢ or u¢1).

On the other side, the symbol b can be derived

R1 : b! b0b;

R3 : b0b! cb;

where b¢ is the major chord with the same root as b. Thuscbz¢ 2 R(bz¢), and since by induction bz¢ ¼ z 2 R(d), itfollows from Lemma 1 that cbz¢ is a cadential sequence,cbz¢ 2 R(d).

Finally, since y is factorized into y ¼ (v¢)(cbz¢), one haseither y 2 R1(u¢)R(d) or y 2 R1(u¢1)R(d), which achievesthe proof.

The sequence z introduced in the proof of Proposition 3is a cadential sequence. The fact that z only depends on d(and does not depend on the suffix u00 of u), as shown inProposition 3, means that one could make a precomputa-tion of such cadential sequences (the variable d rangingover the twelve dominant seventh chords).

Furthermore, Proposition 3 could be the basis of furtherinvestigations on the general form of the chord sequencesderived by Steedman’s grammar. This simple propositionis in some sense general, since for any sequence u, one canfactorize it into

u ¼ u1d1u2d2 . . . ukdkukþ1

such that each di is a dominant seventh chord, and each ui

does not include any dominant seventh chord (with per-haps some ui being empty). This means that any chordsequence can be factorized into sequences satisfying theconditions of Proposition 3.

du"v

z

Fig. 4. The derivation from ud into vz does not depend on asuffix u¢¢ of u

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6Conclusion and perspectivesThe fact that the derivation only depends on particularchords of the given chord sequence (the dominant seventhchords) means that one could make a segmentation of thesequence in such a way that each segment contains exactlyone of these chords. Then a new implementation of thederivation process could consist in applying the substitu-tion rules independently to these segments. Therefore thechord sequence would not be treated as a whole, but rulescould be applied locally, so that the process would be moreadapted to a real-time situation. While the current seg-ment is being played during the performance, the usershould have the opportunity to modify the kind of sub-stitution that will be applied to the next one (in its‘‘deepness’’).

This could make the improvisation system more reac-tive to the inputs of the user. Moreover, we have seen thatthe derivation of segments leading to a dominant seventhchord introduces particular chord sequences calledcadential. The fact that they do not depend on the contentof the original chord sequence allows us to precompilethese cadential sequences. The derivation of each segmentthus becomes more efficient in this way.

The main problem that remains unsolved is that theremay exist rules which apply at the boundary between twosegments. Actually, considering the segmentation of u intouidi defined in the previous section, an application of R4 atthe beginning of diui+1 is possible in some cases. If we treatthe segments independently, we cannot reach the substi-tuted chord sequences that would result in applying therule at this point. What kind of substituted chordsequences can we obtain in that way? Are they interestingor not from a musical point of view? Can we predict insome sense when this happens? Can we group together thesegments involved in this situation, in such a way that theoriginal chord sequence would be factorized into segmentsthat are really independent? These questions and manyothers are the matter of the further developments of thisresearch.

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