Sight-playing part – 2

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.

rhythmprobabilityrhythmprobability
[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
Table 1. Harmonic rhythms, values expressed in quarter notes – [6, 2] denotes a rhythm in which there are two chords, the first one lasts 6 quarter notes, the second 2 quarter notes.

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

likelihoodX=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.

likelihoodX=weight2gramp(X v IV)+weight1gram*p(X)

where:

  • p(X v IV) is the probability of the flow (IV, X)
  • weightNgram 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-gram12345
weight0.0010.010.115

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:

likelihoodX=weight3gramp(X v IV, vi)+weight2gramp(X v IV)+weight1gram*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.

Sight-playing — part 1

During their education, musicians need to acquire the ability to play a vista, that is, to play an unfamiliar piece of music without having a chance to get familiar with it beforehand. Thanks to this, virtuosos can not only play most pieces without preparation but also need much less time to learn the more demanding ones. However, it takes many a musical piece for one to learn how to play a vista. The pieces used for such practice should be little-known and matched to the skill level of the musician concerned. Therefore, future virtuosos must devote a lot of their time (and that of their teachers) to preparing such a playlist, which further discourages learning. Worse still, once used, a playlist is no longer useful for anything.

The transistor composer

But what if we had something that could prepare such musical pieces on its own, in a fully automated way? Something that could not only create the playlist but also match the difficulty of the musical pieces to the musician’s skill level. This idea paved the way for the creation of an automatic composer — a computer programme that composes musical pieces using artificial intelligence, which has been gaining popularity in recent times.

Admittedly, the word “composing” is perhaps somewhat of an exaggeration and the term “generating” would be more appropriate. Though, after all, composers create musical pieces based on their own algorithms. Semantics aside, what matters here is that such a (simple, for the time being) programme has been successfully created and budding musicians could benefit from it.

However, before we discuss how to generate musical pieces, let us first learn the basics of how musical pieces are structured and what determines their difficulty.

Fundamentals of music

The basic concepts in music include the interval, semitone, chord, bar, metre, musical scale and key of a musical piece. An interval is a quantity that describes the distance between two consecutive notes of a melody. Although its unit is the semitone, it is common practice to use the names of specific intervals. In contrast, a semitone is the smallest accepted difference between pitches (approximately 5%). While these differences can be infinitely small, it is simply that this division of intervals has become accepted as standard. A chord is three or more notes played simultaneously. The next concept is the bar, which is what lies between the vertical dashes on the stave. Sometimes a musical piece may begin with an incomplete bar (anacrusis).

Visualization of the anacrusis
Figure 1 Visualisation of an anacrusis

Metre — this term refers to how many rhythmic values are in one bar. In 4/4 metre, there should be four quarter notes to each bar. In 3/4 metre, there should be three quarter notes to each bar while 6/8 metre should have six eighth notes to each bar. Although 3/4 and 6/8 denote the same number of rhythmic values, these metres are different, the accents in them falling on different places in the bar. In 3/4 metre, the accent falls on the first quarter note (to put it correctly, “on the downbeat”). By comparison, in 6/8 metre, the accent falls on the first and fourth measures of the bar.

A musical scale is a set of sounds that define the sound material that musical works use. The scales are ordered appropriately — usually by increasing pitch. The most popular scales are major and minor. While many more scales exist, these two predominate in the Western cultural circle. They were used in most of the older and currently popular pieces. Another concept is key, which identifies the tones that musical pieces use. In terms of scale vs. key, scale is a broader term; there are many keys of a given scale, but each key has its own scale. The key determines the sound that the scale starts with.

Structure of a musical piece

In classical music, the most popular principle for shaping a piece of music is periodic structure. The compositions are built using certain elements, i.e. periods, which form a separate whole. However, several other concepts must be introduced to explain them.

motif is a sequence of several notes, repeated in the same or slightly altered form (variation) elsewhere in the work. Typically, the duration of a motif is equal to the length of one bar.

variation of a motif is a form of the motif that has been altered in some way but retains most of its characteristics, such as rhythm or a characteristic interval. musical pieces do not contain numerous motifs at once. A single piece is mostly composed of variations of a single motif. Thanks to this, each musical piece has a character of its own and does not surprise the listener with new musical material every now and then.

A musical theme is usually a sequence of 2-3 motifs that are repeated (possibly in slightly altered versions) throughout the piece. Not every piece of music needs to have a theme.

A sentence is two or more phrases.

A period is defined by the combination of two musical sentences. Below is a simple small period with its basic elements highlighted.

Scheme of the periodic structure of a musical piece
Figure 2 Periodic structure diagram of a musical piece

This is roughly what the periodic structure looks like. Several notes form a motif, a few motifs create a phrase, a few phrases comprise a sentence, a few sentences make up a period, and finally, one or more periods form a whole musical piece. There are also alternative methods of creating musical pieces. However, the periodic structure is the most common, and importantly in this case, easier to program.

Composing in harmony

Compositions are typically based on harmonic flows — chords that have their own “melody” and rhythm. The successive chords in the harmonic flows are not completely random. For example, the F major and G major chords are very likely to be followed by C major. By contrast, it is less likely to be followed by E minor and completely unlikely to be followed by Dis major. There are certain rules governing these chord relationships. However, we do not need to delve into them further since we will be using statistical models to generate song harmonies.

Instead, we need to understand what harmonic degrees are. Keys have several important chords called triads. Their basic sound, the root notes, are the subsequent notes of a given key. The other notes belong to this key, e.g. the first degree of the C major key is the C major chord, the second degree the D minor chord, the third degree the E minor chord, and so on. Harmonic degrees are denoted by Roman letters; major chords are usually denoted by capital letters and minor chords by small letters (basic degrees of the major scale: I, II, III, IV, V, VI, VII).

Harmonic degrees are such “universal” chords; no matter what tone the key starts with, the probabilities of successive harmonic degrees are the same. In the key of C major, the C – F – G – C chord sequence is just as likely as the sequence G – C – D – G in the key of G major. This example shows one of the most common harmonic flows used in music, expressed in degrees: I – IV – V – I

Melody sounds are not completely arbitrary; they are governed by many rules and exceptions. Below is an example of a rule and an exception in creating harmony:

  • Rule: for every measure of a bar, there should be a sound belonging to the given chord,
  • Exception: sometimes other notes that do not belong to the chord are used for a given measure of the bar; however, they are then followed relatively quickly by a note of this chord.

These rules and exceptions in harmony do not have to be strictly adhered to. However, if one does comply with them, there is a much better chance that one’s music will sound good and natural.

Factors determining the difficulty of a musical piece

Several factors influence the difficulty of a piece of music:

  • tempo — in general, the faster a musical piece is, the more difficult it gets, irrespective of the instrument, (especially when playing a vista)
  • melody dynamics — a melody consisting of two sounds will be easier to play than one that uses many different sounds
  • rhythmic difficulty — the more complex the rhythm, the more difficult the musical piece. The difficulty of a musical piece increases as the number of syncopations, triplets, pedal notes and similar rhythmic “variety” grows higher.
  • repetition — no matter how difficult a melody is, it is much easier to play if parts of it are repeated, as opposed to one that changes all the time. It is even worse in cases where the melody is repeated but in a slightly altered, “tricky” way (when the change of melody is easy to overlook).
  • difficulties related to musical notation — the more extra accidentals (flats, sharps, naturals), the more difficult a musical piece is
  • instrument-specific difficulties – some melodic flows can have radically different levels of difficulty on different instruments, e.g. two-note tunes on the piano or guitar are much easier to play than two-note tunes on the violin

Some tones are more difficult than others because they have more key marks to remember.

Technical aspects of the issue

Since we have outlined the musical side in the previous paragraphs, we will now focus on the technical side. To get into it properly, it is necessary to delve into the issue of “conditional probability”. Let us start with an example.

Suppose we do not know where we are, nor do we know today’s date. What is the likelihood of it snowing tomorrow? Probably quite small (in most places on Earth, it never or hardly ever snows) so we will estimate this likelihood at about 2%. However, we have just found out that we are in Lapland. This land is located just beyond the northern Arctic Circle. Bearing this in mind, what would the likelihood of it snowing tomorrow be now? Well, it would be much higher than it had been just now. Unfortunately, this information does not solve our conundrum since we do not know the current season. We will therefore set our probability at 10%. Another piece of information that we have received is that it is the middle of July — summer is in full swing. As such, we can put the probability of it snowing tomorrow at 0.1%.

Conditional probability

The above story allows us to easily draw a conclusion.  Probability depended on the state of our knowledge and could vary in both ways based on it. This is how conditional probabilities, which are denoted as follows, work in practice:

P(A|B)

They inform us of how probable it is for an event to occur (in this case, A) if some other events have occurred (in this case, B). An “event” does not necessarily mean an occurrence or incident — it can be, as in our example, any condition or information.

To calculate conditional probabilities we must know how often event B occurs and how often events A and B occur at the same time. It will be easier to explain it by returning to our example. Assuming that A is snow falling and B is being in Lapland, the probability of snow falling in Lapland is equal to:

probability of snow in Lapland

The same equation, expressed more formally and using the accepted symbols A and B, would be as follows:

conditional probabilities formula

Note that this is not the same as the likelihood of it snowing in Lapland. Perhaps we visit Lapland more often in winter and it is very likely to snow when we are there?

Now, to calculate this probability exactly, we need two statistics:

  • NA∩B — how many times it snowed when we were in Lapland,
  • NB — how many times have we been to Lapland,

and how many days we have lived so far (or how many days have passed since we started keeping the above statistics):

  • NTOTAL.

We will use this data to calculate P(A∩B) and P(B) respectively:

Probability formulas

At last, we have what we expected:

probability formula

The probability of it snowing if we are in Lapland is equal to the ratio of how many times it snowed when we were in Lapland to how many times we were in Lapland. It is also worth adding that the more often we have been to Lapland, the more accurate this probability will be (if we have spent 1,000 days in Lapland, we will have a much better idea about it than if we have been there 3 times).

N-grams

The next thing we need to know before taking up algorithmic music composition is N-grams, that is, how to create them and how to use them to generate probable data sequences. N-grams are statistical models. One N-gram is a sequence of elements of length equal to N. There are 1-grams, 2-grams, 3-grams, etc. Such models are often used in language modelling. They make it possible to determine how probable it is for a sequence of words to occur.

To do that, you take a language corpus (lots of books, newspapers, websites, forum posts, etc.) and count how many times a particular sequence of words occurs in it. For example, if the sequence “zamek królewski” [English: king’s castle] occurs 1,000 times in the corpus and the sequence “zamek błyskawiczny” [English: zip fastener]  occurs 10 times, this means that the first sequence is 100 times more likely than the second. Such information can prove useful to us. They allow us to determine how probable every sentence is.