Problem solving in math and music – what is the connection?

Take a look at these two series of numbers:

6-4-4-3-3-2 5-3-3-2-2-X

What number would you put at X’s place? Would it be 1? I think most people would come to that conclusion. What if you were asked to find the series of six numbers before the 644332 sequence? Wouldn’t you choose 7-5-5-4-4-3? And after the 533221 series, you would probably expect the combination: 4-2-2-1-1-0, right?

Now you have solved a mathematical problem. Does that feel good?

If I told you, that the two series of numbers are from a piece of music, do you think, you can guess which (hint: you probably would have to be Danish to know 🙂 )?

OK, you guessed correctly! They are from the Danish children’s tune, “Jeg ved en lærkerede” (“The lark’s nest“) by Carl Nielsen. It’s a very simple tune, and as a composition, it is brilliantly done. Here you can here the melody, made in Google Song Maker:

The Lark’s Nest – simple midi version of the song

In the melody, the two series of numbers appear in the middle of the song, in note names: C-E-C-F-D-G- -E-A-F-F-E-E-D- – -G-E-E-D-D-C– -A-F-E-E-D-D-C- – –

Looking at the melody in midi notation, we can see, that the logic of the two series can also be immediately grasped, when represented as geometric patterns:

The ‘Lark’s nest’ melody as midi/boomwhacker notation (made in Google Song Maker)

Why does it feel good to solve a math problem? You found that the missing number was 1. And you came to that conclusion via a process of looking for a pattern in the number series, and through logical thinking coming to a conclusion. How is this gratifying? Is it the fact of experiencing a confirmation that the world turned out to be, what you would expected it to be?

When you listen to or sing the melody of ‘The Lark’s Nest’, and you experience the tone C at the end of series two, was this what you expected? And if so: was the feeling of experiencing something in the world that turned out as expected gratifying? Was the feeling related, somehow to the feeling, when solving a math problem? Is listening to and playing music (also) some kind of problem solving?

Sudoku and music

I have a habit. When I need to relax my brain, I play Sudokus. I actually play a version called ‘Killer Sudoku’. In math terms, solving a killer sudoku means using basic adding/subtracting, which is of cours good training, but the satisfying part is the part where it’s actually about solving equations. I’ve been doing this for quite some time, and contrary to other time killer apps, I seem to stick to this one. It means predictability, since I know I will always solve the Sudoku, eventually; I know some tricks that will help me crack the problems; but what is more is, that it means discovering new tricks.

At a session with my composition mentor, Jexper Holmen, we talked about musical form. I am working on a series of variations for piano, and Holmen suggested that I ‘coded’ the variations with certain formal elements, something in the beginning, in the middle, and in the end, that could be easily recognized. These elements should help build a ‘contract’ with the listener, setting up something to expect; once there is expectation, expectation can be fulfilled or not, and in this way there is ‘something at play’.

In the children’s tune ‘The Lark’s Nest’, there is a build up of expectation, within the melody itself, and the fulfilling of this expectation is what makes it a good melody.

The sound of sudoku

Now, with the ideas from my mentor in the back of my head, and sitting, as usual, killing time with my killer sudoku, an idea struck me. This is what I knew:

  • solving a sudoku is basically about solving equations
  • solving equations is a gratifying thing
  • music can be build as series of numbers
  • series that follow a logical sequence, thus fulfilling the listeners expectations, it can be gratifying to listen to

So my idea was of course: What if I turned a (Killer) Sudoku into music? What kind of music would that be? As is the case with turning DNA code into music, the fact that the system used for generating music is foreign to the sound itself, makes the connection between sound and system arbitrary. Just like with letters and phonemes in language. The letters H-O-R-S-E are supposed to sound in a specific way, when spoken. In language, the relation between spoken and written words is arbitrary. However, once established, the system makes sense to the reader.

So, if I want to make a sudoku into music, I need to choose a way of translating the sudoku’s math problems and their solutions into sound. And the choice of a translation system will necessarily be arbitrary. However, there might be a chance, and this is indeed my purpose, that the audible result will make sense in a way similar to the sudoku.

Here is a sudoku, I’ve solved.

A sudoku being solved

How can this mathematical problem solving become a piece of music?

First, let’s establish some points. Solving a Sudoku takes time. For me, this makes a ‘translation’ into music meaningful, since music is a time based art form. The sequence, and maybe duration, of the problem solving in the sudoku could be used as the outset for the unfolding of the composition.

Secondly, a sudoku is based on numbers. In music, sounds and rhythms can be thought of as numbers. The sudoku uses the numbers 1 – 9 in a variety of combinations. In music, I think, it makes sense to use these combinations to generate patterns of pitch (melody, chord, interval) and rhythm. As we saw in the children’s tune, a simple sequence of numbers can generate musically meaningful sequences of pitch.

Thirdly, in a sudoku, you solve one problem at a time, but all problems are ultimately connected; solving one problems makes it possible to solve the next; in some cases, there will be an intermediary situation, where there are 2 – 3 possible solutions to a problem; reducing the problem to that, can help finding the final solution to another problem. In other words: Solving a sudoku means untangling a web of connections between connections. How can this be translated into music?

Experiments

In order to answer this question, we need to conduct some experiments. Here is a setup, I’m working on.

First of all, I think more instruments are needed. In order to establish a ‘solution’ to a problem, there is a need for something recognizable, and distributing different solutions to different instruments might do the job. The sudoku consists of 9 fields, nine vertical, and 9 horisontal lines, each with the numbers 1 – 9. A single number solution therefore affects a field, and two lines. In order to ‘translate’ this problem solving grid into musical timbre, I suggest that we divide the instruments in groups. In order to cover the whole field of problems, we need a certain number of instruments. How about a sinfonietta*?

Since the occurrence of a solved problem, in a sudoku, is represented in the space of the grid, my idea is to allocate different instruments to different parts of the grid. In order to further enhance the ‘formatting’ of the spatial distribution of the sudoku’s math problem solving, I also coded the grid with pitch differences, going from deep pitches in the bottom left, to high pitches in the top right.

Mapping the spatial logic of the sudoku’s math problem solving onto musical timbre and pitch range.

Each instrument would ‘account’ for a problem solving in a certain region, and groups of instruments would gestalt the vertical and horisontal connections.

An idea for distributing the instruments spatially, mapped on the Sudoku grid

From here to a final result: a lot of hard work….!

* A sinfonietta is a kind of ensemble , typically, consists of 1 of each instrument from the symphony orchestra. In a sinfonietta, we typically have a flute, an oboe, a clarinet, a bassoon, a French horn, a trumpet, a trombone, two percussion players, a piano, a harp, two violins, a viola, a cello and a double bass.

Has musical composition influenced the discovery of the DNA?

I am working with composition as a kind of ecosystem, and one of the central aspects of this approach is to regard the musical score as a kind of DNA. In the musical score, the notes are the code to be interpreted or translated by the musician into sounding music. This music is what the community around the music need to thrive. In the living cell, the DNA is the code for the living organism, which, once translated or interpreted, form proteins that the organism need to persevere.

From this perspective, music and organic life is basically about scripture, interpretation and communication. Genetics and musical composition have these things in common, and when I compose, I take this connection a step further, you might say, and work specifically with musical expression in a way that is parallel to how living organisms interact with eachother and the environment.

Read more about my approach in this blogpost: “Music as an ecosystem“.

Of course, the fact of setting up this relation is purely cultural. It is a product of human activity through and through. In any case, science itself can be said to be a cultural phenomenon. At least, this idea about a cultural connection between genetics and composition makes me want to ask the question if there is a deeper connection, in cultural/scientific history between these to fields.

Music as a representation of genetic data

My initial research brought a rather big amount of results where musical composition is used as a means to illustrate or conceptualise genetic code.

In 1986, a japanese scientist, Susumu Ohno translated genetic code, with the help from his wife, a professional singer, into musical scores.

As you can hear, the result is a rather conventionally sounding piece in a kind of classical style.

Ohno’s experiments are described in this article: “The Ohnos and Genetic Music”, where the author, genetics professor Manuel Ruiz Rejon also mentions other examples of a musical representation of genetic codes, that are more elaborated. The basic idea is, as I understand it, to make an auditive representation of data. In this case genetic code. The thing about genetic code is, that it implies a lot of repetition. The same can be said about (certain types) of music. What these scientists have done is to represent genetic code data in sound, thus “combining this type of tool, which works “by ear” with existing visual and analytical tools to study the nucleotide sequences of genes and genomes found in the enormous databases that are currently available, where the main difficulty is to sort and make sense of them.” I did something similar, 9 months ago, when I explored how the occurence of prime numbers might sound.

In science, the most common way of representation is through visual means, and I think it makes sense to ask what we might miss? In fact, I can’t help thinking what would happen if we used sound as a means of representation in working with scientific facts.

Composition as a prerequisite for thinking genetic code

I’ve looked a little at how musical composition has been used to illustrate genetic code. How about the other way around? To which degree can it be said, that the development of the pentagram, ie. the Western representation of musical sounds in writing, might have contributed to the idea of code in living organisms?

After an initial search, I came to realize, that this is not an easy question to answer. I started by checking out who actually ‘discovered’ genetic code, and if these persons might have had some sort of epiphany by thinking musical composition. Or maybe they simply had had a musical upbringing. I had to rapidly discard this thought as too far fetched.

In order to answer the question about a possible relationship between the music notation and the ‘discovery’ of the DNA code, it makes sense to look for their respective roots.

The roots of the notational system

How did the notational system come into being in the first place? What were the prerequisites? What kind of thinking, what epistemology was needed for it to be conceived?

Well, musical notation seems to have been developed out of a necessity in the Church to streamline musical practises over large geographic distances. Put in the words of Wikipedia: “Christian monks developed the first forms of modern European musical notation in order to standardize liturgy throughout the worldwide Church”.

It also seem that musical notation was closely linked to written text, where some extra symbols, originally, were added to the text of the songs used in liturgy. I think that the close relationship to written text accounts for the left-to-right linearity of the musical score. Actually, it might be argued that music is not necessarily linear.

Gregorian Chant with quasi-musical notation, called “neumes”

When I use one of my compositions as a ‘map’ for improvisation, I ‘read’ the score non-linearly; I jump from one ‘musical event’ to the other without regard to where they are in the score’s temporal logic.

In addition, it is certain, that some basic mathematical principles had to be in place before the notational system as we know it could be developed. To take one example, the idea of a fraction is behind the way that we divide the composition into bars, the bar into beats, and the beat into quarters, eights, etc., and triplets, quintuplets and so on.

The musical score has made it possible, similar to written text, to store and manipulate musical thinking in a non-temporal medium. Life is an ongoing, temporal process. The DNA is a way of conceptualising this process in a non-temporal manner. Is this a possible link?

The roots of the discovery of DNA code

What were the prerequisites for discovering DNA code? What technological development was needed? What kind of thinking, what epistemology was needed for it to be conceived?

These are, you might say, really heavy weight questions, that I am not at all in a position to answer; my guess is, that research in the line of Science and Technology Studies (STS), in the style of Bruno Latour and others would be an interesting place to look for answers. In other words, people who research into science with the assumption that they are not only looking into something which is (also) a cultural phenomenon.

Coming to understand the genealogy of the musical notational system seems much more straightforward than the genealogy of genetic science. This still doesn’t answer the question about a possible connection.

Since going for a directly observable connection has proven, as I have mentioned in the beginning of this blogpost, to be a complete failure, how would it be possible to answer this question?

I think it would require that we take quite a few steps backwards, in order to look at the musical notational system in a broader cultural-technical perspective. And ask if there might have been, throughout history, examples of a cultural environment, where the explorations into what kinds of thinking the musical notational system might offer, and how these kinds of thinking might have influenced thinking within the realm of natural sciences.

A possible place to look, I think, would be in the periods of time, where composers and scientists engaged in exploring and experimenting in their respective fields, interacted somehow, and exchanged ideas. Where composers explored and experimented with the possibilities offered by the musical score, in conjunction with the cultural-technological developments in musical instruments, and the the interaction between composer and his/her public, and society in a broader sense. And where natural scientists, for their part, explored their field, while taking advantage of the cultural-technological developments of the time, and – of course – the interaction with society in a broader sense.

Which periods to look into?

Even though we, in our time, have environments, where specialists of all kinds explore their own field to a detail, which in a historical perspective must seem extreme, I am not sure that own time is the best place to look for the kind of connection I am looking for. The problem is, I think, that artists, and scientists, and other specialists in our time, are too deeply dedicated to the intricacies of their little field of specialty, and that it takes too much knowledge about what others are into, in order to start to see possible connections between the fields.

What does these thoughts make you think of? Am I wrong if I say “the Renaissance man”? You know, the Leonardo da Vinci type of person?

Actually, I am thinking about two other historical periods/places. My knowledge is primarily as a composer, and from this perspective, I come to think of two historical places/periods, where musical composition were taking radically new places. One is the socalled Enlightnment (roughly 1750-1815, according to this article). My first place to look would be into Beethoven, and the environment where his musical thinking was nurtured and influential.

A rapid search online gave this result: A book about the concept of heredity, where one of the central points is, that “the concept of heredity came of age during the Enlightenment”. This makes sense, in a political sense, since it was a time where people started to question the idea, that certain people, ie kings, nobles etc., were born with certain privileges. The German composer Ludvig van Beethoven (1770 – 1827) was very preoccupied with these questions.

“Freedom and progress are our true aim in the world of art, just as in the great creation at large.” Beethoven, in a personal letter (source, cited here).

The other time and place would be Vienna in the beginning of the 20th century. Here, the music of Schoenberg and his pupils would be the place to start. A central characteristic of this school was the concept of dodecaphony, ie, the use of all twelve halfnotes in the musical system, each in their own right. Dodecaphony is an early example of what has become known as serialism, where the idea that musical parameters can be broken down into discreet entities, ie numbers (for rhythm and pitch), and terms (for dynamics, timbre etc), and that these entities can be combined in various ways, that are not necessarily fitting within the general idea of what music is supposed to sound like. This way of thinking music is, I would argue, very similar to the way of thinking needed to work with genetics, where discreet elements, ie aminoacids etc., are combined en various ways in order to generate different kinds of organisms.

The Enlightenment and early 20th century Vienna, would, I think, be two fruitful places to look for a crossover effects between musical composition and life sciences. In musicology, by the way, these two periods/places are being referred to as the first respectively second Viennese School. What happened in these places/times within the field of genetics? Well, maybe some of you have an answer, out there?

While you are thinking about these questions, I would like to invite you to listen to one of my latest improvisations, “Epilogue #0113000”; it is a (‘genetic’) spinoff of the composition “Last Dream” (#000000*)

* I use the hexadecimal system used for colors to name my compositions/improvisations. Read more about this in my blogpost here.

How to keep track of my artistic research?

Since I started working with music as an ecosystem, I ran into a problem: How can I keep track of all the different variations/spinoffs/mutations?

I needed a kind of coding system, and after some thought, I came up with the following.

Color codes as a naming system

The thought of giving the improvisations a number seemed unelegant – and boring, so I decided to use the hexadecimal system, in a way similar to the way digital colors are coded.

In the hexadecimal color code system a color is represented by 6 numbers/letters. Choosing at random, let’s take the code #11FFA6. This would give a greenish color like the one you see here :

I needed a system, where I could give a unique code to each improvisation/composition that would make it possible to

  1. Track where it comes from. IE which composition does this improvisation stem from?
  2. Tell it apart from other improvisations with the same source. IE the ‘sibling’ level.

In order to do so, I ended up with something similar to the way names are given in human societies.

The first to ciphers represent the generation. First generation is ’00’. The compositions based on anything from a generation ads a number here, and becomes next generation. Second generation thus has ’01’ here. There are 256 possible numbers in 2 hexadecimals (16×16), and that makes up for quite a few ‘generations’. In each generation, further, there are 256 possible ‘members’. Hopefully 65.536 (256×256) possible improvisations is enough for my lifetime 🙂

The next two ciphers are there to give each improvisation it’s own id, for its generation Let’s call it the ‘first name’. And the last two ciphers take the ‘first name’ from the composition it is based on, and it becomes its ‘last name’.

This naming system results in a kind of genealogical tree, as you can see here:

Here, you see four ‘generations’ of improvisations, where three impros are derived from one ‘mother’ composition, and each of the three have two ‘offsprings’ each with their unique generational ‘first name’ (00 – 06). In the last row, “generation 03”, I have put an improvisation, which is supposed to be the hexadecimal 11th, ie 17th improvisation over the composition #020100.

If you look VERY closely, you might see, that the boxes are shaded differently; I wanted to make this post just A LITTLE entertaining, so I made the boxes the color that their name would give, were it a color code. (This is why I added the 17th generation 03 impro…, so it would be just A LITTLE different in its greenness 🙂 )

This system allows me to identify

  • which composition an improvisation is based on (its ‘parenthood’),
  • when it is done (generation combined with ‘first name’)

Creating a system, a format, to order different elements in a practice, like I am trying to do here, will inevitably also influence the way these elements are being created.

There are aspects of the practice that are not taken into account in the naming/ordering system. In my case, what the system doesn’t allow to know, is for example which part(s) of a composition has been used for the impro, and to what extend. This can however be shed light upon through an analysis of the involved compositions.

What it also doesn’t allow to know, implicitly, is if an improvisation has more than one parent. This is somewhat more problematic, since it will nudge me to NOT use different compositions, side by side, for example, as a basis for one improvisation. I actually had the idea of taking only endings of various compositions, and use for an improvisation. I think the naming problem sort of deterred my from going forward with this idea. A possible workaround would be to give 1 impro more than one name, that is one, for each ‘parent’.

What this naming system DOES allow is for a practice, where I relate compositions to each other in way, that I have described here, and it thus helps me stay faithful to the conceptual framework of music as an ecosystem, an organic, interdependent, developmental, experimental, and at the same time open, improvised and free approach.

Invitation – explore with us*!

I would like to invite you to a place. It is a place, I have invented. It’s a musical composition called “Last Dream”. I made this composition by improvising on the piano.

Improvising on the piano

Listen to the improvisation here:

Regarding this improvisation as an ecosystem, a landscape, I made made it into a score. I use the score as a map. I put the map/score in front of me, at the piano, and explore the lanscape/ecosystem, improvising, regrouping, repurposing the elements of the landscape. Here is one of my explorations/variations:

Variation 10 over the composition Last Dream

As you might hear, I am exploring mostly the first part of the ecosystem/landscape/composition “Last Dream”. The variations/explorations seems to add new dimensions, new organisms, new connections.

Please listen to these variations/field studies, and ask yourself: How can I contribute? Where shall we go from here? Maybe we should explore one of the variations? Make a map/composition/score from it, and use this map as a place to explore? We can have a chain or branchings of such sequences: [exploration –> map of exploration –> exploration of map of exploration –> mapping of exploration of map of exploration], etc.

Maybe you can make an improvisation, using the score as a map? Or you can improvise over the improvisations? And share it with us?

Come with your ideas. Put them in words, sound, video, or whatever format suits you… Looking forward to receiving your contributions 🙂

Upload sound on Soundcloud, and share the link. Or send us an email: caspcordes@gmail.com

* “us” – well to be honest, sofar ‘us’ are me, my hands, mind, the piano.

Music as an ecosystem?

What happens if we create and experience music as a kind of ecosystem?

Improvising on the piano, using what I would call intuitive intelligence, I am developing a method for using my body, hands, arms, and mind to create a kind of sound ecosystem.

This is how this improvisation came into existence:

Next step is to make a score, a transcription of the improvisation. Check it out here.

The score as a map of the ecosystem. A map that can be explored, researched, studied, expanded, etc. Putting the score in front of me on the piano, I made ‘journeys’ into the ecosystem, resulting in a number of variations on the composition.

A traditional format – theme and variations. In nature, we are talking organisms interacting in an ecosystem, and evolving, mutating, adjusting, etc.

Listen to the variations here:

Please share your thoughts about the variations. Help me decide which (parts) to develop further. Comment on the tracks, in the tracks. Thanks! 🙂

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