logical representation of musical information · 2020-03-10 · is 1, that of tiar ( " ) is...

Logical Representation of

Musical InformationTetsuya Mizutani

University of Tsukuba(JAPAN)

Musical Informatics

Investigating higher level fusion of intellect and sensibility of human via music

Understanding how to play artistic music performance by computers

Importance of logical representation of musical information

Musical Information

Expression Rules and Fingering Rulesbased on

Musical Structuresand

Structural Functions of Music

Musical Structures

G1 G2 G3 G4

Motif1Phrase1

G5 G6 G7 G8 G9 G10Motif2

Phrase1 Phrase2The 1st 5 measures of Chopin’s Polonaise Militaire Op. 40, No. 1

Group: short sequence of notesMotif: 2 measuresPhrase: 2 motifsSentence: 2 phrases

Structural Functions

inceptiveanacruisis tiara

initiativedesinence

conclusive

The 1st motif of Chopin’s Polonaise Militaire Op. 40, No. 1

Inceptive: the starting point.Anacrusis: getting tenser to the initiative.Tiara: a climax area of the tension.Initiative: an ideal climax note in the structure.Desinence: diminishing the tenson.Conclusive: the end point.

Expression Curve

The 1st sentence of Chopin’s Polonaise Militaire Op. 40, No. 1,

Performed by J. Olejniczak.

measure1 2 3 4 5 6 7 8

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Agogic Rules:

The inceptive is played slowly.

The performance of the anacrusis accelerates.

The initiative is emphasized.

etc.

How slow?

How much is the rate of acceleration?

What means “emphasize”?

Musical description in natural language is ambiguous.

Fingering Rules

Fingering:

a basis for expression of performance

based on musical structures, esp. small groups (e.g. beat, semimotif, etc.)

logical representation (rule) is useful for decision of fingers.

The 1st motif of Burgmüller’s InnocenceRule:

St1 : for every adjacent pair of notes in a group, a reachable pair of

fingers must be used if they are performed legato.

St3 : two sequences of fingers for two groups must be the same if

two scores and two performance plans are both similar.

G1Group : G2 G3 G4

St1 St1 St1 St1Rules :St3 St3

Musical Structures

S[FS] : score with fingering

S : score, FS : fingering for S

S[FS]=(σ1, σ2, ... , σm) , σi : structure

σi=(σi1, σi2, ... , σil) , σil : lower structure, or

σi=(υi1, υi2, ... , υin), υij : note

υij=

<numij, ptcij, volij, lngij, itvij,

prtij, msrij, nvlij, fgnij>

numij : index number

ptcij, volij : pitch and volume

lngij , itvij : length and interval

prtij , msrij : part and measure

nvlij : note value

fgnij : finger number

Notation

υijnum=υij

1=numij , υijptc=υij

2=ptcij , etc.

S[FS](k) : kth note in the score S[FS].

σ=incp(σ)•anac(σ)•tiar(σ)•init(σ)•desn(σ)•cncl(σ)

mid(σ)=tiar(σ)•init(σ)•desn(σ)

S[FS]=(Mtf1, Mtf2)

Mtf1=(n1,1, n1,2, ... , n1,10)

n1,1=<1, A4, 80, 500, 0, 1, 1, 1/8, 5>

n1,2=<2, rest, 0, 250, 500, 1, 1, 1/16, 5>

n1,3=<3, E4, 76, 250, 250, 1, 1, 1/16, 4>

...

Example

(melody part)

Agogic RulesThe performance of the inceptive starts slowly:

Fig. 1. Structural Functions

Cross∗(〈f1, f2〉, 〈x1, x2〉) ≡(f1 < f2 ∧ x1 > x2) ∨ (f1 > f2 ∧ x1 < x2),

Cross(υ1, υ2) ≡(υptc

1 &= rest ∧ υptc2 &= rest)

∧Cross∗(〈υfgn1 , υfgn

2 〉, 〈υptc1 , υptc

2 〉).IV. Structural Functions and Agogic Rules

A. Structural Functions of Music

Each note in a musical structure could have a dif-ferent meaning and be played by its own role for eachdifferent occurrence. Structural functions of music [2]represent a certain emotion to listen or play the musicreflecting the meaning and the role. In each motif orphrase, there are 6 kinds of functions as follows:

1. inceptive: the starting point of the structure,2. anacrusis (anticipative): a range where the per-

formance could be getting tenser to the initiative,3. tiara: a couple of notes just before the initiative

in which area is a climax of the tension,4. initiative: an ideal climax in the structure which

must be emphasized,5. desinence (reactive): a range where tension is di-

minished following the initiative, and6. conclusive: the end point of the structure.

Noted that a tiara may not exist in a motif or a phrase.A structure σ can be divided into 6 parts:

σ = incp(σ) • anac(σ) • tiar(σ) •init(σ) • desn(σ) • cncl(σ),

where incp etc. are functions expressing the incep-tive etc. of σ, while • is the concatenation operatorof structures. We assume that the length of init(σ)is 1, that of tiar(σ) is between 0 and 2, and those ofother elements are greater than 0. mid(σ) abbreviatesanac(σ) • tiar(σ) • init(σ) • desn(σ).

Figure 1 shows structural functions of the first motifof Chopin’s Polonaise Militaire Op. 40, No. 1.

B. Agogic Rules

The logical representation of some primitive agogicrules [9] based on structural functions are as follows:

1. the performance of the inceptive starts slowly:

ave ◦ tpbs ◦mid(σ) < ave ◦ tpbs ◦ incp(σ),

2. the performance accelerates in the anacrusis,while it decelerates in the desinence:

Dcr(aprx ◦ agc ◦ anac(σ))∧ Inc(aprx ◦ agc ◦ desn(σ)),

3. the tiara is played longer than both the initiativeand the last note of the anacrusis:

tpb ◦ nth(anac(σ), |anac(σ)|) < ave ◦ tpbs ◦ tiar(σ),

4. the initiative is played longer than the “neighbor”notes other than the tiara:

tpb ◦ nth(anac(σ), |anac(σ)|) < tpb ◦ init(σ)∧ tpb ◦ nth(desn(σ), 1) < tpb ◦ init(σ), and

5. it ends slowly in the conclusive:

ave ◦ tpbs ◦mid(σ) < ave ◦ tpbs ◦ cncl(σ),

where ave(a) = (∑m

k=1 ak)/m for a = (a1, a2,. . . , am), ai ∈ R, and Inc(f) and Dcr(f) are func-tions that a function f : R → R is increasing anddecreasing, respectively.

V. Fingering Rules based on Structure

Some fingering rules [1] are given with logical repre-sentation as follows:

1. rule St1: for every adjacent pair of notes in agroup σ, a reachable pair of fingers must be usedif they are performed legato:

St∗1(υ1, υ2) ≡ Adj(υ1, υ2) ∧ Lgt(υ1, υ2) ⊃Reach(υ1, υ2),

St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),

where Lgt : DNOTE×DNOTE is a predicate repre-senting that a performance plan for two adjacentnotes is legato,

2. rule St1G : for each pair of adjacent groups σ1

and σ2, the reachable fingers must be used on thepair of notes of the end of the preceding groupand the beginning of the following one if they areperformed legato:

St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),

3. rule St2 : a pair of fingers for every adjacent pairof notes must not cross in a group σ if it is per-formed smoothly:

St2(σ) ≡ Phrs(σ) ⊃∀i(1 ≤ i ≤ |σ|− 1 ∧Adj(σ(i), σ(i + 1)) ⊃

¬cross(σ(i), σ(i + 1)),

measure1 2 3 4 5 6 7 8

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How slow?It does not represent (future work).

The preformance accelarates in the anacrusis, while it decelerates in the desinance:



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

measure1 2 3 4 5 6 7 8

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The tiara is played longer than the last note of the anacrusis:



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

measure1 2 3 4 5 6 7 8

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The initiative is played longer than the “neighbor” notes other than the tiara



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

measure1 2 3 4 5 6 7 8

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The performance ends slowly in the conclusive:



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

measure1 2 3 4 5 6 7 8

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Fingering InformationDistance between 2 keys x and y :

Signed key distance :

where

S[FS ] is simply written as S if there is no confusion.Each element in a note υii is represented by a su-

perscript, i.e. υnumij = υ1

ij = numij , υptcij = υ2

ij = ptcij ,etc. Additionally, S(k) indicates the kth note of thescore S, i.e. S(k) = υij if and only if υnum

ij = k.

III. Functions and Predicates

Some functions and predicates useful to representagogics [9] and fingering [1] are defined.

A. Agogics Information

Definition 3: A function tpb : DNOTE → Q+ denot-ing time per beat of a note is defined by

tpb(υ) =υitv

υnvl,

and tpbs denoting a sequence of times per beat of asequence of notes, by

tpbs(τ) = (tpb(υ1), . . . , tpb(υm)),

where τ = (υ1, . . . , υm).Definition 4: For a finite subset X of R2 having the

uniqueness condition of functions

∀xyz (〈x, y〉 ∈ X ∧ 〈x, z〉 ∈ X ⊃ y = z),

aprx(X) : R → R is defined as a function satisfyingthat a map represented by

{〈x, aprx(X)(x)〉|〈x, y〉 ∈ X}is an approximation of X.

Definition 5: A function loc : (DNOTE)n×N → Q+

denoting the location of the ith note in a sequence ofnotes of length n (i ≤ n) is defined by

loc(τ, i) ={

0 i = 1,∑i−1k=1(nth(τ, k))nvl i ≥ 2,

where nth(τ, i) is the ith element of τ .Definition 6: A function agc denoting the set of

pairs of the location of a note and its time per beat,i.e. the agogic of notes, is defined by

agc(τ) = {〈loc(τ, i), tpb ◦ nth(τ, i)〉|1 ≤ i ≤ |τ |},where |τ | is the length of τ and ◦ is the compositionoperator of functions.

B. Keyboard Information

Definition 7: A function kd : KN → N denoting thedistance between two keys x, y ∈ KN on a keyboardis defined by

kd∗ : KN → {1, 2},kd∗(x) =

{2 if x mod 12 ∈ {4, 12},1 otherwise

kd(x, y) =max{x, y}−1∑i=min{x, y}

kd∗(i).

A function skd : KN → N for signed key distance isdefined by

skd(x, y) ={

kd(x, y) if x ≤ y,−kd(x, y) otherwise

For example, kd(C4, C!4) = skd(C4, C!

4) = 1,kd(C4, E5) = kd(E5, C4) = skd(C4, E5) = 18, butskd(E5, C4) = −18.

The notion of the spans that two fingers can stretchwithin some limitations is useful for the investigationof the fingering rules. Some functions for spans be-tween two fingers [1], [6] are introduced below.

Definition 8: Functions maxpr, minpr : FN → Z aredefined as the maximum and the minimum practicalstretch span, respectively, between two fingers to playtwo notes simultaneously or consecutively in legato,defined by the signed key distance between these notes.

maxrel, minrel : FN → Z are regarded as the max-imum and the minimum relaxed spans, respectively,that two fingers fall without tension onto the keys.

maxconf, minconf : FN → Z are the maximum andthe minimum of comfortable spans, respectively, whichlie between relaxed and practical.

maxpr(x, y) represents the span between x and y,where x is the finger number of the first note and ythat of the second. If C4 can be played by a thumb andE5 by a little finger simultaneously with the practicallimitation, then maxpr(1, 5) = skd(C4, E5) = 18. Ifx = y then maxpr(x, y) is 0, else if x < y then it ispositive, otherwise it is equal to −maxpr(y, x). Otherfunctions are similar. And moreover, if the first fingeris a thumb, then minpr etc. are negative since the nextfinger can be passed over the thumb.

Definition 9: A predicate Adj : DNOTE × DNOTEdenoting that two notes are adjacent in the score, butsome rests may occur between them, is defined by

Adj(υ1, υ2) ≡ υptc1 .= rest ∧ υptc

2 .= rest∧ υnum

1 < υnum2

∧ ∀i(υnum1 < i < υnum

2 ⊃ S(i)ptc = rest).Definition 10: A predicate Reach : DNOTE ×

DNOTE denoting that two keys can be played simul-taneously for two fingers with a practical span, i.e. thefingers are reachable for the keys, is defined by

Reach(υ1, υ2) =

minpr(υfgn1 , υfgn

2 ) ≤ skd(υfgn1 , υfgn

2 )

≤ maxpr(υfgn1 , υfgn

2 )

∧ υptc1 .= rest ∧ υptc

2 .= rest.Definition 11: A predicate Cross : DNOTE×DNOTE

denoting that the two fingers cross, i.e. a pair of fingernumbers is opposite order to a pair of key numbers, isdefined by

Cross∗ : FN2 ×KN2,





ij = k.





tpb(υ) =υitv

υnvl,


tpbs(τ) = (tpb(υ1), . . . , tpb(υm)),



∀xyz (〈x, y〉 ∈ X ∧ 〈x, z〉 ∈ X ⊃ y = z),





loc(τ, i) ={

0 i = 1,∑i−1k=1(nth(τ, k))nvl i ≥ 2,






kd∗ : KN → {1, 2},kd∗(x) =



kd∗(i).


skd(x, y) ={











2 .= rest∧ υnum

1 < υnum2




Reach(υ1, υ2) =



2 )


2 )









ij = k.





tpb(υ) =υitv

υnvl,


tpbs(τ) = (tpb(υ1), . . . , tpb(υm)),



∀xyz (〈x, y〉 ∈ X ∧ 〈x, z〉 ∈ X ⊃ y = z),





loc(τ, i) ={

0 i = 1,∑i−1k=1(nth(τ, k))nvl i ≥ 2,






kd∗ : KN → {1, 2},kd∗(x) =



kd∗(i).


skd(x, y) ={











2 .= rest∧ υnum

1 < υnum2




Reach(υ1, υ2) =



2 )


2 )





4C 4E3A

Signed Key Distance

VirtualBlack Keys

4C 4E3A

4

skd(C4, E4)=4

Signed Key Distance

4C 4E3A

-4

skd(C4, E4)=4

Signed Key Distance

skd(C4, A3)=-4

Spans between two fingersMaximum Minimum

Practical maxpr minpr

Relaxed maxrel minrel

17

maxpr(1, 5)=17




-5

minpr(1, 5)=-5




15

maxrel(1, 5)=15

(Predicate) Two notes υ1 and υ2 are

adjacent in the score, but some rests may occur between them:





ij = k.





tpb(υ) =υitv

υnvl,


tpbs(τ) = (tpb(υ1), . . . , tpb(υm)),



∀xyz (〈x, y〉 ∈ X ∧ 〈x, z〉 ∈ X ⊃ y = z),





loc(τ, i) ={

0 i = 1,∑i−1k=1(nth(τ, k))nvl i ≥ 2,






kd∗ : KN → {1, 2},kd∗(x) =



kd∗(i).


skd(x, y) ={











2 .= rest∧ υnum

1 < υnum2




Reach(υ1, υ2) =



2 )


2 )





(Predicate) Two notes υ1 and υ2 can

be played simultaneously for two fingers with a practical span, i.e. the fingers are reachable:





ij = k.





tpb(υ) =υitv

υnvl,


tpbs(τ) = (tpb(υ1), . . . , tpb(υm)),



∀xyz (〈x, y〉 ∈ X ∧ 〈x, z〉 ∈ X ⊃ y = z),





loc(τ, i) ={

0 i = 1,∑i−1k=1(nth(τ, k))nvl i ≥ 2,






kd∗ : KN → {1, 2},kd∗(x) =



kd∗(i).


skd(x, y) ={











2 .= rest∧ υnum

1 < υnum2




Reach(υ1, υ2) =



2 )


2 )





(Predicate) Two notes υ1 and υ2 is

played simultaneously for two fingers with oppisite order, i.e. the fingers are crossing:



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

where



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

Rule St1: If a group σ is played legato,

then reachable fingers must be used:

where

Fingering Rules



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

Performance plan: they are played legato.

Rule St1G: If adjacent notes of adjacent

group is played legato, then reachable fingers must be used:



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

Rule St2: If a group σ is played smoothly,

then fingers must not cross:



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),



















B. Agogic Rules


















St1(σ) ≡ ∀i(1 ≤ i ≤ |σ|− 1 ⊃St∗1(σ(i), σ(i + 1))),




St1G(σ1,σ2) ≡ St∗1(σ1(|σ1|), σ2(1)),



¬cross(σ(i), σ(i + 1)),

Performance plan: it is played smoothly, emphasizing its pharasing.

Rule St3: For two groups σ1 and σ2,

if both the scores and the performance plans are similar, then these fingers must be the same:

Performance plans are similar.

where Phrs : DNOTE is a predicate representingthat a performance plan for a group is smooth,i.e. emphasizes its phrasing, and

4. rule St3 : two sequences of fingers for two groupsσ1 and σ2 must be the same if two scores and twoperformance plans are both similar:

St3(σ1, σ2) ≡ |σ1| = |σ2| ∧Sim(σ1, σ2) ∧ Smxp(σ1, σ2) ⊃∀i(1 ≤ i ≤ |σ1| ⊃ σfgn

1 (i) = σfgn2 (i)),

where Smxp : DNOTE × DNOTE is a predicaterepresenting that two performance plans for twogroups are similar, while Sim : DNOTE ×DNOTEis a predicate denoting that two sequences ofpitches are similar, defined by whether a certaindistance between two sequences is less than athreshold or not.

Figure 2 is a score of the first 2 measures ofBurgmuller’s Innocence with groups and fingeringrules to be applied. By these rules, we have some can-didates of its fingering sequences including the correctanswer

(4, 3, 2, 1︸︷︷︸G1

, 4, 3, 2, 1︸︷︷︸G2

, 4, 3, 2, 1︸︷︷︸G3

, 4, 3, 2︸︷︷︸G4

).

Adding the notions of penalty point, difficulty of fin-gering and weight of rules [1], [6], the correct answerwill be obtained. The penalty point is assigned to a fin-gering sequence if it violates a rules. Each rule has itsown value for the penalty point. The difficulty meansthat if a fingering is more difficult technically thanthe other fingering, e.g. a larger span of fingers is un-derstood as more difficult than a smaller one, then alarger penalty point is assigned to the former than thelatter. The weight of rules is the priority over rules.Especially, the similarity rule St1G is understood asthe most important, thus if a fingering violates thisrule then the larger penalty point is assigned. To ob-tain the correct answer, the sequence of fingers whichtotal penalty points is the minimum is sought.

VI. Conclusions

This paper is our first step that we attempt to rep-resent artistic expression of music in a formal logicalnotation. Since logical representation is a higher levelabstraction of information, it must be quantified whenone want to perform music along with the logical rules.The quantification is one of interesting and profoundfields of informatics, especially AI.

We have already designed, implemented, verifiedand analyzed an automatic ensemble program [5], [8]by @-calculus [3] which is a smallest formal systemapplicable to verification and analysis of cooperativereal-timing systems on natural number time. The au-tomatic ensemble program is an example of higher

Fig. 2. Groups and Fingering Rules to be Applied

level intellectual and realtime control systems, but thisanalysis did not deal with any specific expression ofmusic. Therefore, we will design and analyze an en-semble program with artistic in our further work.

Another important and interesting point of our fur-ther study is investigating North African music for thelogical representation, and moreover, for the music in-formatics.

Acknowledgment

The author wish to express his gratitude to Ms.Michiyo Matsuzaki (Pf.) who gave many useful ad-vice for music.

References

[1] Hayashida, N. : A Fingering Decision System for a Pianobased on Musical Structures, Master’s Thesis in Science andEngineering, University of Tsukuba, 2004 (in Japanese).

[2] Igarashi, S. : Science of Music Performance, Yamaha Music

Media，2000 (in Japanese).[3] Igarashi, S., Mizutani, T., Ikeda, Y. and Shio, M. : Tense

Arithmetic II: @-Calculus as an Adaptation for FormalNumber Theory, Tensor, N. S., 64 (2003), pp. 12–33.

[4] Lerdahl, F. and Jackendoff, R. : Generative Theory of TonalMusic, The MIT Press, 1983.

[5] Mizutani, T., Shio, M., Igarashi S. and Ikeda, Y. : Verifi-cation of an Ensemble Program in @-Calculus, Joint Con-ference on Applied Mathematics (2003), pp. 43 – 48 (inJapanese).

[6] Purncutt, R., Sloboda, J. A., Clarke, E. F., Raekallio, M.and Dasain, P. W. M. : An Ergonomic Model of Key-board Fingering for Melodic Fragment, Music Perception14 (1997), pp. 341–382.

[7] Riemann, H. : Musikalische Dynamik und Agogik –Lehrbuch der musikalischen Phraserung, Breitkopf und Har-tel, 1884 (in German).

[8] Shio, M. Igarashi, S., Mizutani, T. and Ikeda, Y. : In-tellect and Sensibility in Informatics III: An Example ofVerification Based on @-Calculus – an Ensemble Program,Journal of Saitama Junior College (2004), pp. 119–130 (inJapanese).

[9] Tanaka, T. A study On Performance Rules based on Struc-tural Functions, Master’s Thesis in Science and Engineer-ing, University of Tsukuba, 2001 (in Japanese).

Scores are similar.

From these 4 rules, we have candidates of fingering including the correct answer (4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2).

If the correct answer want be obtained, then the notions of penalty point, difficulty, weight, etc. are needed.

G1 G4

(similar)

St1(legato)

G2 G2

St1 St1 St1

Rules : St2(not cross)

St2 St2 St2

St1G(legato)

St1G St1G

St3 St3

Group :

The 1st motif of Burgmüller’s Innocence

ConclusionsSome logical representations are given.

The first trial for musical expression to describe in logical language.

Quantification is unfinished.(eg. “how much is the rate of the acceleration?”)

An automatic ensemble system has already been represented logically, implemented, verified and analyzed.

Future Work

Application of the research of the musical informatics to North African music.

logical representation of musical information · 2020-03-10 · is 1, that of tiar ( " ) is...

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