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We already created the harmony of the piece in the previous article. What we need now is a good melody which will match this harmony. Melodies consist of motifs, i.e. small fragments of about 2-5 notes and their variations (transformations).

We will start by generating the first motif – its rhythm and sounds. As we did when generating the harmony, we will use N-gram statistics for musical pieces. Such statistics will be prepared using the Essen Folksong Collection base. You might as well use any other melody base, this choice will affect the type of melodies that will be generated. For each piece, we must isolate the melody, convert it into a sequence of rhythmic values and a sequence of sounds, and from these sequences extract the statistics. When compiling sound statistics, it is a good idea to first somehow prepare the melodies – transpose them all to two keys, e.g. C major and c minor. This will reduce the number of possible (probable) N-grams by 12 times and therefore the statistics will be better assessed.

A good motif

We will begin creating the first motif by generating its rhythm. Here, I would like to remind you that we have previously made a certain simplification – each motif and its variations will last exactly one bar. The subsequent steps for generating the rhythm of a motif: – we draw the first rhythmic value using unigrams, – we draw the next rhythmic value using bigrams and unigrams, – we continue to draw consecutive rhythmic values, using N-grams of increasingly higher level (up to 5-grams), – we stop until we reach a total rhythmic value equal to the length of one bar – if we have exceeded the length of 1 bar, we start the whole process from the beginning (such generation is fast enough that we can afford such a sub-optimal trial-and-error method).

The next step is to generate the sounds of the motif. Another simplification we made earlier is that we generate pieces only in C major key, so we will make use of the N-gram statistics created on the basis of pieces transposed to this key, excluding pieces in minor keys. The procedure is similar to that for generating rhythm: – we draw the first sound using unigrams, –we draw the next sound using bigrams and unigrams, – we continue until we have drawn as many sounds as we have previously drawn rhythmic values, – we check whether the motif matches the harmony, if not, we go back and start again – if after approx.

100 attempts we failed to generate a motif matching the harmony, this could mean that with the preset harmony and the preset motif there is a very low probability of drawing sounds that will match the harmony. In this case, we go back and generate a new motif rhythm.

Generate until you succeed

When generating both the motif rhythm and its sounds, we use the trial-and-error method. It will also be used in the generation of motif variations described below. Even if this method may seem “stupid”, it’s simple and it works. Although very often such randomly generated motifs don’t match the harmony, we can afford to make many such mistakes. Even 1000 attempts take a short time to calculate on today’s computers, and this is enough to find the right motif.

Variations with raepetitions

We have the first motif, and now need the rest of the melody. However, we will not continue to generate new motifs, as the piece would become chaotic. We also cannot keep repeating the same motif, as the piece would become too boring. A reasonable solution would be, in addition to repeating the motif, to create a modification of that motif, ensuring variation, but without making the piece chaotic.

There are many methods to create motif variations. One such method is chromatic transposition. It involves transposing all notes upward or downward by the same interval. This method can lead to a situation where a motif variation has sounds from outside the key of the piece. This, in turn, means that the probability that the variation will match the harmony is very low. Another method is diatonic transposition, whereby all notes are transposed by the same number of scale steps. Unlike the previous method, diatonic variations do not have off-key sounds.

Yet another method is to change a single interval; one of the motif intervals is changed, while all other intervals remain unchanged. That way, only one part of the motif (the beginning or the end) is transposed (via chromatic or diatonic transposition). Further methods are to convert two notes with the same rhythmic value to one or to convert one note to two notes with the same rhythmic value. For the first method, if the motif has two notes with the same rhythmic value, its rhythm can be changed by combining these two notes. For the second method, a note is selected at random and converted to two “shorter” notes.

Each of the described methods for creating variations makes it possible to generate different motifs. The listed methods are not the only valid methods; it is possible to come up with many more. The only restriction here is that the generated variation should not differ too much from the original motif. Otherwise, it would constitute a new motif rather than a variation. The border where the variation ends and a different motif begins is conventional in nature.

Etc., etc.

There are many more methods for generating motif variations; it is possible to come up with a lot of these. The only restriction is that the generated variation should not differ too much from the original motif. Otherwise, it would constitute a new motif rather than a variation. The border where the variation ends and another motif begins is rather conventional in nature and everyone “feels”, defines it a little differently.

Is that all?

That would be all when it comes to piece generation. Let us summarise the steps that we have taken:

1. Generating piece harmony:
   • generating harmonic rhythm,
   • generating chord progression.
2. Generating melody:
   • generating motif rhythm,
   • generating motif sounds,
   • creating motif variations,
   • creating motifs and variations “until it’s done”, that is, until they match the generated harmony

All that is left is to make sure that the generated pieces are of the given difficulty, i.e. matching the skills of the performer.

Controlling the difficulty

One of our assumptions was the ability to control the piece difficulty. This can be achieved via two approaches:

1. generating pieces “one after another” and checking their difficulty levels (using the methods described earlier), thereby preparing a large database of pieces from which random pieces of the given difficulty will then be selected,
2. controlling the parameters for creating the harmonies, motifs and variations in such a way that they generate musical elements of the given difficulty with increased frequency

Both methods are not mutually exclusive and thus can be successfully used together. First, a number of pieces (e.g. 1000) should be generated randomly, and then parameters should be controlled to generate further pieces (but only those which are missing). With respect to parameter control, it is worth noting that the probability of motif repetition can be changed. For pieces with low difficulty, the assigned probability will be higher (repetitions are easier to play). On the other hand, difficult pieces will be assigned lower probability and rarer harmonies (which will also force rarer motifs and variations).

In the first part of the article, we have learned about many musical and technical concepts. Now it is time to use them to build an automatic composer.  Before doing so, however, we must make certain assumptions (or rather simplifications):

• the pieces will consist of 8 bars in periodic structure (antecedent 4 bars, consequent 4 bars)
• the metre will be 4/4 (i.e. 4 quarter notes to each bar, accent on the first and third measures of the bar)
• the length of each motif is 1 bar (although this requirement appears restrictive, many popular pieces are built precisely from motifs that last 1 bar).
• only C major key will be used (if necessary, we can always transpose the piece to any key after its generated)
• we will limit ourselves to about 25 most common varieties of harmonic degrees (there are 7 degrees, but some of them have several versions, with additional sounds which change the chord colour).

What is needed to create a musical piece?

In order to automatically create a simple musical piece, we need to:

• generate the harmony of a piece – chords and their rhythm
• create motifs – their sounds (pitches) and rhythm
• create variations of these motifs – as above
• combinate the motifs and variations into a melody, matching them with the harmony

Having mastered the basics, we can move on to the first part of automatic composing – generating a harmony. Let’s start by creating a rhythm of the harmony.

Slow rhythm

Although one might be tempted to create a statistical model of the harmonic rhythm, unfortunately, (at least at the time of writing this article) there is no available base which would make this possible. Given the above, we must handle this in a different way – let’s come up with such a model ourselves. For this purpose, let’s choose a few “sensible” harmonic rhythms and give them some “sensible” probability.

rhythm	probability	rhythm	probability
[8]	0.2	[2,2]	0.1
[6, 2]	0.1	[2,1,1]	0.02
[2, 6]	0.1	[3,1]	0.02
[7, 1]	0.02	[1,1,1,1]	0.02
[4]	0.4	[1,1,2]	0.02

The rhythms in the table are presented in terms of chord duration, and the duration is shown in the number of quarter notes. Some rhythms last two bars (e.g. [8], [6, 2]), and others one bar ([4], [1, 1, 2] etc.).

Generating a rhythm of the harmony proceeds as follows. We draw new rhythms until we have as many bars as we needed (8 in our case). Sometimes certain complications may arise from the fact that the rhythms have different lengths. For example, there may be a situation where to complete the generation we need the last rhythm that lasts 4 quarter notes, but we draw one that lasts 8 quarter notes. In this case, in order to avoid unnecessary problems, we can force drawing from a subset of 4-quarter-note rhythms.

Then, in line with the above findings, let’s suppose that we drew the following rhythms:

• antecedent: [4, 4], [2, 2], [3, 1], 
• consequent: [3, 1], [8], [2, 2]

Likelihood

In the next step, we will be using the concept of likelihood. It is a probability not normalised to one (so-called pseudo-probability), which helps to assess the relative probability level of different events. For example, if the likelihoods of events A and B are 10 and 20 respectively, this means that event B is twice as likely as event A. These likelihoods might as well be 1 and 2 or 0.005 and 0.01. From the likelihoods, probability can be calculated. If we assume that only events A and B can occur, then their probability will be respectively:

Chord progressions

In order to generate probable harmonic flows, we will first prepare the N-gram models of harmonic degrees. To this end, we will use N-gram models available on github (https://github.com/DataStrategist/Musical-chord-progressions).

In our example, we will use 1-, 2-, 3-, 4- and 5-grams.

In the rhythm of the antecedent’s harmony, there are 6 rhythmic values, so we need to prepare the flow of 6 harmonic degrees. We generate the first chord using unigrams (1-grams). Now, we first prepare the likelihoods for each possible degree and then draw while taking these likelihoods into consideration. The formula for likelihood is quite simple in this case

likelihood_X=p(X)

where

• X means any harmonic degree
• p(X) is the probability of the 1-gram of X

In this case, we drew IV degree (in this key of F major).

We generate the second chord using bigrams and unigrams, with a greater weight for bigrams.

likelihood_X=weight_2gramp(X v IV)+weight_1gram*p(X)

where:

• p(X v IV) is the probability of the flow (IV, X)
• weight_Ngram is the adopted N-gram weight (the greater the weight, the greater the impact of this N-gram model, and the smaller the impact of other models)

We can adopt N-gram weights as we wish. For this example, we chose the following:

N-gram	1	2	3	4	5
weight	0.001	0.01	0.1	1	5

The next chord we drew was: vi degree (a minor).

The generation of the third chord is similar, except that we can now use 3-grams:

likelihood_X=weight_3gramp(X v IV, vi)+weight_2gramp(X v IV)+weight_1gram*p(X)

And so we continue until we have generated all the necessary chords. In our case, we drew:

IV, vi, I, iii, IV, vi (in the adopted key of C major these are, respectively, F major, a minor, C major, e minor, F major and a minor chords).

This is not a very common chord progression but, as it turns out, it occurs in 5 popular songs (https://www.hooktheory.com/trends#node=4.6.1.3.4.6&key=rel)

Summary

We were able to generate the rhythms and chords which are the components of the harmony of a piece. However, it should still be noted here that, for the sake of simplicity, we didn’t take into account two important factors:

• The harmonic flows of the antecedent and consequent are very often linked in some way. The harmony of the consequent may be identical with that of the antecedent or perhaps slightly altered to create the impression that these two sentences are somehow linked.
• The antecedent and consequent almost always end on specific harmonic degrees. This is not a strict rule, but some harmonic degrees are far more likely than others at the end of musical sentences.

For the purposes of the example, however, the task can be deemed completed. The harmony of the piece is ready, now we only need to create a melody to this harmony. In the next part of our article, you will find out how to compose such a melody.