Optimality in musical melodies and harmonic progressions: the Travelling Musician Title: Optimality in musical melodies and harmonic progressions: The travelling musician.Author: Daniel Schell, Composer, Engineer (ULB), M.Sc. Operational Research (LSE) Topics keywords: n° 6. Combinatorial optimisation; n° 57. OR applications in other domain: musical composition.Conference of the author with musical examples played by Judith Gudor (clarinet) and Emese Mali (synthesiser). A fuller version of this paper has been published in the EJOR journal.  Abstract

A "chord" is a vertical set of notes. A "melody" is a path going through these chords considered as a graph. The author describes an algorithmic approach to the generation of tiles, group of chords, connected along remarkable paths.

The traditional harmony is based on a set of rules such as variety, connectivity, tiling, enumeration which eventually produces a 'beautiful' and 'interesting' melody/chord relationship. A set of rules and criteria are defined in order to evaluate this melody/chord relationship.

The compositional program contains two algorithms:

1 ) The first generates a list of possible sets of chords fulfilling the optimality criteria.

2) The second connects them and evaluates the criteria for each paths.

A selection of the best tiles is then made with the help of the composer. A beautiful tile - or 'beau karo' as the author names them - might show -  or not – high symmetry and remarkable connectivity. The best tiles given by the algorithms are compared to the ones in usage by the tradition.

 

Euro has commissioned the author for two musical compositions. First, 'The Travelling Musician' a brass quintet for the Hungarian Ensemble 'Brass in the Five' directed by Peter Burget and Gergely Vajda. Then  'Ten optimal studies' a set of pieces for various ensembles. Frequent references are made to this pieces of music.

The algorithms presented here were thought as helping devices for the composer, more than complete algorithms which would for instance produce Bach like music without any human intervention.

 

First definitions

1.1       Chromatic set of notes on Z-12. Traditional music accepts 12 notes per octave. We assign an integer value to each note. With (C=0; C# OR Db =1; D=2; D# or Eb = 3; E = 4; F = 5; F# OR Gb =6; G =7; G# OR Ab  = 8; A = 9; A# OR Bb =10; B =11) . MOD 12 (A) insures that note A remains in the range 0 <= A <=11IntervalsThe interval is the number of half-tones between two notes.INTERVAL: ('unison' =0; 'minor second' OR 'half tone' =1; 'major second' OR 'tone'=2; 'minor third'=3; 'major third'=4; 'fourth' = 5; 'augmented fourth' OR 'diminished fifth' 6;

'fifth'=7;  minor sixth'=8; 'major sixth'=9; 'minor seventh'=10; 'major seventh'=11; 'octave'=12)

1.2       Scales

Figure 1-1 The diatonic major scale figured by letters, then by digits from Z-12

The advantage of figuring by numerals is that we can learn about the constitution of the scale into tones (equivalent to two half-tones) and half-tones. The successive intervals are (2, 2, 1, 2, 2, 2, 1).

1.3       MelodiesA melody is a horizontal set of  connected notes perceptible as such.

Figure 1-2 The first part will be clearly perceived as a melody. The second part will be perceived a set of disconnected note.

1.3.1        Rules for an optimal melody

1.      It should be well connected. 2.      It should show a degree of variety with intervals such as  +1, -1, +2, -2,  +3, -3…     3.      Small intervals should be more frequent than large ones.4.      It  should start from a note, explore a region, then come back to a neighbour of this note.

1.3.2         The algorithmic generation of melodies. Two methods:

1           Use existing melodies. Then transform these melodies by operations such as transposition, inversion, retrograde or any type of transformation. This method has been used in the 'Two Italian Trios' from the author.

2           Generate melodies using a random or fractal generator, or any type of non musical data.

1.3.2.1       Random fractal and Brownian melodies

Figure 1-3 A Brownian or random fractal chromatic melody. The trombone solo in the first part of the Travelling Musician, bars 44 and foll. .We adopt the graph of  a Brownian motion.  With y= + or – 1 half-tone we produce the chromatic melody of Figure 1-3.  The melody is clearly well connected. However the step is always = 1 and this is against rule 2 of the optimal melody. Furthermore such melodies can go far away from their starting point and this is against rule 4.

Figure 1-4 A Brownian diatonic melody. The Horn solo in the Travelling Musician, part 1, bars 71 and foll. The same melody, given to the trumpet (bar 80) is treated as a fugue.The Brownian motion is mapped to a diatonic scale (5, 7, 9, 10, 0, 2, 3). In this case we choose y from the values (1, 2, -1, -2).

1.3.2.2        Random Normal Gaussian melodies

Figure 1-5 A Gaussian walk in a diatonic space. 'fantasy of a happy driver' is the title of the solo fluegelhorn (the Travelling Musician, part 1, bars 13 and foll.)We start from a note, say 0, and at each step we add interval from the distribution. If the melody goes to far up or down we re-adjust with an octave step (see the arrow under the note 7). If we take the same distribution, but apply it to the chromatic notes, we obtain the melody from Figure 1-6.

Figure 1-6 A Gaussian chromatic walk. The tuba solo (The travelling musician, part 1, bars 95 and foll.).

2         Analysis of some traditional chord sequences 2.1       Harmonic and name- triads

Figure 2-1 Three major triads, figured as Harmonic Triads - H-Triads - then as relative Triads - R-Triads. Definition: an harmonic triad H-triad is an array of three  different elements, whose values range between 0 and 11. H-Triad = = (i, j, k) ; i  j k ; 0 < = i, j, k < = 11 ;A relative triad R-triad is written under the form m, n(r)m, n (r) = = ( r, m + r, m + n + r) The advantage of the R-triad figuring is that it shows the successive intervals between the elements of the triads. It brings information on the interval structure of the triad. Open and closed forms of a triad 

Figure 2-2 The three closed and three open forms of the H-triad (0, 4, 7)The six permutations of a triad contain  two categories, open and closed triads. The ambitus is the interval difference in half-tones, measured between the highest and the lowest note of a triad. Then the triad is closed if its ambitus < 12, otherwise it is open. To obtain the Name-Triad: Take a H-triad. Generate its six permutations. Choose the H-

Triad with the shortest ambitus. R-Triad. 

2.2       The 4,3 sequence from the musical traditionLet us choose the following restrictive definition: A chord is a triad on a bass note: m, n (r) /b

 

Figure 2-3 The traditional 4-chords sequence built on degrees 0, 5, 7, 0 the major scale.    Chord1

4,3(0)notes

Salto 

Chord 2 4,3(5)notes

salto Chord 34,3(7)notes

salto Chord 44,3(0)notes

Total absolute melodic salto

Voice 1 0 0 0 -1 11 +1 0 2Voice 2 4 +1 5 -3 2 +2 4 6Voice 3  

7 +2 9 -2 7 0 7 4

Triad               12Bass 0 +5 5 -10 -5 (or 7) +5 0   Table 1 lists  in integer figures the musical data of Figure 2-3 First of all, we see that we have vertical columns corresponding to the chords, then horizontal lines corresponding to the voices.  A voice is a horizontal set containing the notes of a given level of the chords.

Rules to form a good traditional 4-chord sequenceRule 1 Covering. All the notes of a given scale should be included in the four triads. Rule 2 Surprise. The triads contain different notes in order to produce surprise and variety.Rule 3 Connection. The triads are smoothly connected one to each other. Suppose chords 1 and 2; then the inversion of chord 2 which we choose is the one which minimises the number of steps or absolute salto with chord 1. The Total Absolute Salto (TAS) between the 4 triads should be small.Rule 4 Structure. All the chords of the sequence have an equal or analogue structureRule 5 Voice Pattern. One of the voices, here the bass, does not have necessarily to follow the rule of connection. In this case, it follows a kind of pattern.

3         Some beautiful tiles3.1       The major-minor 4,3-3,4 sequence.

       .     

Figure 3-1 the 2*(4,3), 2*(3,4) 'beautiful tile'. Left on the score, right on the circle.     4,3(0)

notesSalto 

 3,4(8)notes

salto 4,3(10)notes

salto 3,4(9)notes

Total absolute salto

Voice 1 0 -1 11 -1 10 -1 9 4Voice 2 7 +1 8 -3 5 +1 6 5Voice 3  

4 -1 3 -1 2 -1 1 4

Triad               13Bass 4 +7 11 -9 2 +7 9  Let us verify the optimality rules:Rule 1 Covering. A partition of the chromatic scale (0…11). Maximum information.Rule 2 Surprise. The triads contain all different notes. Maximum variety.Rule 3 Connection. TAS (Total Absolute salto) of 13, excellent result.Rule 4 Structure. There are two 4,3 and two 3,4 chords.Rule 5 Voice Pattern. The bass line follows a balance pattern +7, +7.

The voices are so well connected that one could accept them as 'melodies' going though the chords. Listen to the second part of Optimal Study N° 3 (from bar 36), built on this beautiful tiling.

3.2       Some beautiful tilesDefinition: 'a beautiful tile' is a partition of Z-12 into 4 mutually exclusive triads, having either (four identical m,n) or (two m,n and two n,m) structures. The other has named these particular tiles 'karos' from the French 'Beaux carreaux'.

3.2.1        Beautiful karo ( 2,4 - 4,2 )

 

 Figure 3-2 A beautiful karo with high connection. The bass is not shown.

   2,4(8)

notesSalto 

 2,4(9)notes

salto 4,2(0)notes

salto 4,2(1)notes

Total absolute salto

Voice 1 2 1 3 1 4 1 5 3Voice 2 10 1 11 1 0 1 1 3Voice 3  

8 1 9 -3 6 1 7 5

Triad               11Table 2The TAS of 11 is the smallest we have met so far. This highly connected chords can give a 'flowing' effect which the author has used in 'The Travelling Musician' to simulate the flow of the traffic on a road. (For instance in bars 63-70). Listen also to the first bars (1-2) of the Optimal study N° 3 'Choral'.  

3.2.2         Beautiful karo 1,1

               

 

   

Figure 3-3 The four triads 1,1 This beautiful karo is difficult to connect and shows a minimum absolute salto of 29, the highest we have met so far. Given the bad connection, it will be difficult to identify the respective voices to a melody. This tile is used in Optimal Study N° 1 for tuba and trombone. It is felt that the triads themselves are melodies, as in the trombone, (bar 1) : (  7, 8, 9, 8, 7).

 

3.2.3         Beautiful karo 2,3 – 3,2

  

           

Figure 3-4 Left: the tile 2,3-3,2 represented as a melody. Right: the same on a circleWe see (Figure 3-4) , that the notes of the 4 triads placed one after each other present already a melodic character. Let us represent the same tile, this time connected: 

Figure 3-5 The same karo connected, represented without the bass.The connecting algorithm shows that this tile has a TAS of 17 which is rather good. The voices which go through the triads are made of steps of 1, 2 or 3; a good melody. This tile shows both characters of good connection and interesting melody. Optimal studies N°2 and specially N°9 are entirely based on its properties.

4         The algorithmic generation of tilesThe following algorithms have been conceived in order to fit in extremely small computer memories.

4.1       Generation of a list of tiles in lexicographic orderSuppose that we want to generate a list of all existing tiles. This list will be useful either for composing purposes, or just to check the existence of a given tile.  The total number of these partitions of Z-12 into 4 triads is 12! / (4! * (3!)E4) = 15.400.We can reduce this problem to a list of the co-sets of a given triad.Suppose that we have a triad, say (0, 4, 7) , then we have to create the list of partitions of (1, 2 , 3, 5, 6, 8, 9, 10, 11) into three triads.                                                                             (1)Replace the list (1) by (0…8) . The problem is equivalent to create  a list of partitions starting with: (0, 1, 2) (3, 4, 5) (6, 7, 8).                                                                                                        (2)We wish to choose the lexicographic order. If we take the first triad of  (2) and keep the 0 fixed, then we shall generate (C 8 into 2) = 28 combinations of the first triad.For each value of triad 1,  we have 2 triads left, as         (3, 4, 5)  (6, 7, 8) Again, if we keep the 3 fixed, we will generate (C 5 into 2) = 10 combinations,  successively:The total number of co-sets of a given triad will be 10*28= 280.The algorithm is designed in order to return a given line of lexicographic list. Example: Given a first set (1, 2, 3) (5, 6, 8) (9, 10, 11)The 116th lexicographic partition of this set would be (10, 1, 3) (2, 6, 9) (4, 3, 6)

4.2       The connection of tilesThe connection of triads one to each other, will produce smoot voice steps. Again, we wish to save the resources of the singer. Given a 4-triads sequence, the algorithm will connect the triads two by two. Suppose we have 4 triads A, B, C, D. We search first which is the best connection between A and B. We compute a TAS (Total absolute salto) for each inversion of B and choose the one with the MINTAS.We then assume that:MINTAS (ABCD) = MINTAS (AB) + MINTAS (BC) + MINTAS (CD)                               (1)We should now compute the MINTAS for all the 4!=24 Permutations of the path ABCD. As the function MINTAS is reflexive, we need only to compute it on 12 pathes.The algorithm will be further simplified by the fact that there are only (C 4 into 2) = 6 two-triads pathes, namely AB, AC, AD, BC, BD, CD.                                                                                (2)The algorithm will then be:Compute MINTAS for the 6 pairs (2).Compute the  values of the TAS on the twelve pathes (ABCD).The minimum MINTAS will be the best connected 4-triads sequence.

This problem may also be solved using a 'Travelling Salesman Problem', TSP-type algorithm. We represent the 4 chords of section 3.1 on the graph shown in Figure 6. The edges between any two vertices carry 'weights' corresponding to the TAS calculated for each 2-triad pair. We can see easily that there are two optimal solutions, both with a TAS of 11.  

Many types of algorithms can be used to solve this problem. For instance, the dynamic programming type of algorithm given in Horowitz and Sahni (1978, p232-233) gives the same results. Any minimum spanning tree algorithm can also be adapted to this type of problem. (For a discussion see Khuller & Raghavachari 1979, p 6-13 to 6-19).  

The author is presently working with Ola Rinta-Koski, on a new LISP Karo engine, which will generate, connect optimally and analyse series of 'karos'.

 

 

Thank you to Elaine Chew for her precious help in editing the text and to Philippe Van Asbroek for having organised the EURO performance.

 

                Figure 6 On the left, the graph of the connections of the karo 4,3-3,4. The solution is equivalent to the shortest path in a travelling salesman problem.  On the right are the two solutions.

 

 

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