What is Mathematical Music?

by John Greschak
john@greschak.com
December 4, 2002


One might say that a particular work of music is an example of mathematical music. What might lead one to say this? Perhaps one knows or believes that there is some relationship between the musical work and mathematics. But what kind of relationship might this be? Isn't it true that every musical work is related to mathematics in some way? If so, why are some musical works called mathematical while others are not? If mathematical music is a type of music, then what type of music is it?

Before considering the usage of the term mathematical music, let us ponder the meaning of the terms musical work and mathematics.

What is a musical work?

Few would disagree with the notion that a musical work is a particular type of work of art, and that works of art are created by artists. But, what is an artist? And, what is a work of art?

Here, artist shall be defined as follows:

Definition. Let T be a thing. T is an artist if and only if T has declared that T is an artist.

Here and throughout, thing means an entity, and declare means to make known (i.e. communicate) explicitly (i.e. unambiguously or clearly) and autonomously (i.e. without outside control).

In this definition of artist, the use of the word thing in place of human is deliberate. It allows for the possibility that humans are not the only artists in the universe. It is rooted in the belief that it might be shortsighted to place additional restrictions on what can be an artist. This goes against the norm. And, it begs the question: Are there any non-human things that are artists?

First, consider the male Satin Bowerbird. To attract a female, a male bowerbird constructs a large structure consisting of twigs and grass which is called a bower. Females prefer a decorated bower. So the male decorates the bower with a variety of blue objects such as feathers and plastic. Sometimes the male will use a stick to "paint" mashed blueberries on the walls of the bower. Is there an artist among these male Satin Bowerbirds? The issue is one of declaration. Unfortunately, bowerbirds and humans do not share a common language. It is not possible for a bowerbird to declare to a human that it is an artist, so we cannot determine whether or not a given bowerbird is an artist. However, it is possible that one bowerbird could declare itself to be an artist to another bowerbird. Perhaps, in the future, humans will have a better understanding of the methods by which (non-human) animals communicate. And from this knowledge, we may find that some of these animals do declare that they are artists to all those who can decipher the message.

Note: One might compare a male bowerbird to a reclusive human who secretly creates things that in some way resemble something that is a work of art. In both cases, there is no evidence that the creator declared to a human that it is an artist.

Another category of potential artists that one might consider are man-made things. Can a computer be an artist? A given computer can be programmed to sound the phrase "I am an artist!" five minutes after it is turned on and every five minutes thereafter. In addition, it can display this phrase constantly on its monitor. So a computer can explicitly make it known that it is an artist. But can it do this autonomously? For the moment, let us restrict our discussion to computers as they are today. Someone must write the program that controls a computer. Someone must turn on a computer. At first, one might not think of a computer (or any man-made thing) as something that is capable of operating autonomously. However, after a program has been written and installed, and the computer has been turned on, it can operate (and make declarations) without outside control. A computer requires electricity. But, perhaps this is analogous to the food and air that is required by a person and is not an issue of control. So, maybe a computer can be an artist.

The Voyager 1 spacecraft, which was launched on September 5th, 1977, is now over 7 billion miles away from Earth. It is departing our solar system at a speed of nearly 39 thousand miles per hour. On board is an audio recording that includes 90 minutes of music. The music was included because there are those who believe that it is unlikely that Earth is the only planet in the universe on which life exists. Are there extraterrestrial artists? Perhaps someday humans will meet and communicate with some extraterrestrial being that declares itself to be an artist.

Finally, it is possible that there might be paranormal artists. That is, there might be spiritual artists (e.g. ghosts or gods) that exist in a realm that is beyond the perception of most humans, at the present time.

Note: As used here, the word thing in the definition of artist includes entities that would normally be classified as being inanimate. Despite this, for convenience, sometimes the word who shall be used in place of that or which in statements such as "the artist, who was an inventor". Also, for convenience, the pronoun his might be used to refer to a particular artist as in the phrase "early in his career".

Next, we consider what is meant by the term work of art. Here, work of art shall be defined as follows:

Definition. Let A be an artist and let T be a thing. T is a work of art by A if and only if A has declared T to be a work of art by A. In some cases, artists collaborate. More generally, let S be a set of artists. T is a work of art by S if and only if S has declared T to be a work of art by S. A given thing T is a work of art if and only if there exists some artist A (or some set of artists S) for which T is a work of art by A (or S). The term artwork is synonymous with the phrase work of art. For a given artist A and a given work of art by A, A is said to be the creator of the work of art by A.

Let us consider some of the implications of this definition of work of art.

By this definition, a given artist A is the sole authority on whether or not a given thing is a work of art by A. If A declares that a given photocopy of Leonardo da Vinci's Mona Lisa painting is a work of art by A, then it is a work of art by A. If A captures a butterfly in a glass jar and declares that the jar (with butterfly) is a work of art by A, then it is a work of art by A. If A declares that a bean seed is a work of art by A, then it is a work of art by A.

Certainly, all individuals are entitled to have opinions and some or all individuals may believe that a particular work of art by a given artist A should not be called a work of art by A. Despite this, by the definitions given here, this would not change the fact that it is a work of art by A. Further, since it is not necessary to believe one's own declarations, even A might not believe that a particular work of art by A should be called a work of art by A. However, it would still be a work of art by A if it has been declared by A as such.

Note, these definitions have nothing to do with the notion of good. Good is an adjective that an individual might use to indicate his approval of a given work of art. Any individual may say that a given work of art is good or not. Even the artist who created the work might say that their work is not good. However, a work of art is a work of art, even if no one has said that it is good.

At times, an artist might change his mind. For example, early in his career an artist A might declare a given thing T to be a work of art by A. Later, A might recant this declaration by declaring that T is not a work of art by A. In this case, T was a work of art by A during, and only during the period of time after the declaration and before the recantation.

The definition given here of work of art permits the possibility that for two given artists A1 and A2, a given thing could be a work of art by A1 and a work of art by A2, simultaneously. For example, suppose that a group of 3 artists A1, A2, and A3 walk together in a park one afternoon. Upon arriving near a pond, A1 points to a bench and declares to the group that the bench is an artwork by A1. On the next day, the group takes a similar walk. As they approach the bench (which is now an artwork by A1), A2 points to the bench and declares to the group that the bench is an artwork by A2. After this declaration, the bench is both an artwork by A1 and an artwork by A2.

Next, we consider what is meant by the term musical work. Here, musical work shall be defined as follows:

Definition. Let A be an artist and let T be a thing. T is a musical work by A if and only if T is a work of art by A, and A has declared T to be musical. In some cases, artists collaborate. More generally, let S be a set of artists. T is a musical work by S if and only if S has declared T to be a work of art by S, and S has declared T to be musical. A given thing T is a musical work if and only if there exists some artist A (or some set of artists S) for which T is a musical work by A (or S). The phrase work of music is synonymous with the phrase musical work. For a given artist A and a given musical work by A, A is said to be the composer of the musical work by A.

By this definition, a given artist A is the sole authority on whether or not a given thing is a musical work by A. In the extreme case where all individuals believe that a given musical work does not conform to some conventional definition of the word musical (and thus do not believe that it should be called musical), it is still a musical work by virtue of the fact that an artist has declared it to be musical. For example, if an artist A writes a poem and declares it to be a work of art by A that is musical, then it is a musical work.

Of what might a musical work consist? A composer has full power to declare anything to be a musical work, so a musical work might consist of anything. While this is consistent with the definitions given here, for the most part, (human) musical works may be classified as one of two types. Generally, musical works are either abstract or concrete. An abstract musical work consists of some indirect prescription of what one should do in order to create a sonic realization of the work. This indirect prescription is called a score. For a concrete musical work, there is no indirect prescription. Instead, the work is as direct (and faithful) a representation as possible of sound itself. Such a representation might consist of a digital recording stored as a data file in some particular format. Some musical works consist of both abstract and concrete elements. For example, a score might be provided along with a recording that is to be used for generating some portion of any sonic realization of the work. One might consider such works to be part of a third class called hybrid musical works. Or, one might consider such works to be a subclass of abstract works.

In addition, most musical works contain some textual material written by the composer. This might include a title, performance instructions, an explanation of the notation used, or a description of how the work is related to other things, some of which may have been used to create the work or may have served as inspiration.

What is mathematics?

Here, mathematician, mathematical work and mathematics shall be defined as follows:

Definition. Let T be a thing. T is a mathematician if and only if T has declared that T is a mathematician.

Definition. Let A be a mathematician and let T be a thing. T is a mathematical work by A if and only if A has declared T to be a mathematical work by A. In some cases, mathematicians collaborate. More generally, let S be a set of mathematicians. T is a mathematical work by S if and only if S has declared T to be a mathematical work by S. A given thing T is a mathematical work if and only if there exists some mathematician A (or some set of mathematicians S) for which T is a mathematical work by A (or S). For a given mathematician A and a given mathematical work by A, A is said to be the author of the mathematical work by A.

Definition. Mathematics is the set that consists of all mathematical works.

All that has been said here regarding works of art may be said of mathematical works by replacing the words work of art with mathematical work, and artist with mathematician. For example, by the definitions given here, the words mathematician and mathematical work are not related in any way to the notion of good. A mathematical work is a mathematical work, regardless of whether or not anyone thinks that it is good. In fact, these definitions allow for the possibility of a mathematical work that consists of an incorrect proof. Such works are not excluded from the set of works that comprise mathematics.

In the definition of mathematician given here, as in the definition of artist, the word thing has been used in place of human. This has been done to allow for the possibility that there might be non-human mathematicians. Does a beaver consider itself to be a mathematician when it roams some distance from home in search of a tree that will suit the requirements of its dam-building project? Might a computer that has been programmed to prove theorems be a mathematician? Are there other non-human beings in the universe (e.g. extraterrestrials or paranormals) who consider themselves to be mathematicians? One does not know. But, it might be shortsighted to exclude such possibilities.

Of what might a mathematical work consist? A mathematician has full power to declare anything to be a mathematical work, so a mathematical work might consist of anything. While this is consistent with the definitions given here, for the most part, (human) mathematical works consist of information written using the following types of things: the written form of some language; symbols and terms that have been defined in a previous mathematical work or are defined in the work itself; and diagrams. The information might be written in any of a number of forms such as: an entry in a notebook or diary; a private communication in the form of a message or letter; or a paper, article, essay, monograph, dissertation or book that is published (i.e. made available to the public).

To get some sense of what (human) mathematics is at any given point in time, one might examine various classification schemes that have been created to categorize mathematical works. Note, mathematics continues to grow. So, it is unlikely that any particular classification scheme would be comprehensive, indefinitely. Still, such structures cover most of what is mathematics and thus, they can serve as a valuable guide. One that is particularly useful is the Mathematics Subject Classification (MSC) created by the editors of the databases Mathematical Reviews and Zentralblatt MATH. Other useful classification schemes may be formed. For example, one might use categories defined by a particular publisher of books on mathematics (e.g. Springer-Verlag). Or, one might use the names of courses in mathematics that are offered at a particular university.

Generally, a mathematical work will be about some mathematical things. Mathematicians characterize mathematical things through definitions. Here, the term definition shall be defined as follows:

Definition. Let T be a thing. T is a definition if and only if T establishes a denotative relation between two other things S and D, whereby S denotes D. T is said to be the definition of S. D is said to be the meaning of S.

Once a definition establishes a denotative relation between two things, the things involved in the relation acquire the attribute of being a sign or designee. Here, sign and designee shall be defined as follows:

Definition. Let T be a thing. T is a sign if and only if T denotes some other thing.

Definition. Let T be a thing. T is a designee if and only if there exists a sign S that denotes T.

In mathematics, a definition is not so much a statement of the meaning of a sign as it is a mechanism for assigning a sign to a meaning. The designee of a definition is that which one would like to discuss. A sign is a compact representation used to refer to the designee in a discussion. The designees of definitions in mathematics are the objects of mathematics. Here, mathematical object and mathematical sign shall be defined as follows:

Definition. Let T be a thing. T is a mathematical object if and only if there exists a mathematical work that contains a definition by which T is established as a designee.

Definition. Let T be a thing. T is a mathematical sign if and only if there exists a mathematical work that contains a definition by which T is established as a sign. A mathematical sign that is a symbol is called a mathematical symbol. A mathematical sign that is a term is called a mathematical term.

Most mathematical signs are either symbols (e.g. +, =, and %) or terms (e.g. Pi, polynomial, and rhombus). Some mathematical signs are composed of both a symbol and a term, or a symbol and an abbreviated word. For example, one might use the sign Znonneg to represent the set of all non-negative integers. Generally, hybrid signs such as this would be classified as symbols.

What are the mathematical objects? One may find a large list of mathematical objects in most dictionaries of mathematics. All of the following would be examples of mathematical objects: "the real number that equals c/d where c and d are the circumference and diameter of any given circle, respectively", "an expression consisting of a sum of terms each of which is a product of a constant coefficient and one or more variables that are raised to a non-negative integral power", and "a polygon with four sides of equal length". Note, in these examples, the mathematical objects are not the quoted phrases themselves. The mathematical objects are the things that are characterized by the quoted phrases.

A mathematical model is one particular type of mathematical object that is especially important for any discussion about the relationship or application of mathematics to another subject area (such as music). Here, mathematical model shall be defined as follows:

Definition. Let T and S be things. T is a mathematical model for S if and only if T is a collection of mathematical objects, each of which represents some feature of S.

Here, a feature of S means one of the following types of things: a constituent element of S, a property of S, a property of a constituent element of S, or a relationship between constituent elements of S.

When is a musical work mathematical?

As stated previously, one might say that a given musical work is an example of mathematical music if one believes that there is some relationship between the musical work and mathematics. What type of relationship would one need to believe exists for one to say this?

First, one might believe that a given musical work is a mathematical work, and therefore is part of mathematics. Can a musical work be a work of mathematics? Yes, at least by the definitions given here. If the composer of a given musical work declares himself to be both an artist and a mathematician, then he can declare a given work to be both a musical work by himself and a mathematical work by himself. While this is possible, it is unlikely that this alone would be sufficient to cause anyone (other than the composer) to believe, and say that the musical work is an example of mathematical music.

Still, it is possible to imagine a musical work that one might believe is work of mathematics (even though the composer has not declared it to be so). For example, suppose a person P devises a proof for some unsolved problem in mathematics and then encodes and publishes this proof in the form of a musical work. In this case, one might believe that the musical work is a mathematical work, and as a result, one might say that the work is an example of mathematical music. Note, for this particular case, one might also say that the work is an example of musical mathematics.

Such cases are conceivable. But, for one to say that a given musical work is an example of mathematical music, it might not be necessary for one to believe that the work is part of mathematics. Perhaps, there might be less direct relationships and conditions that would be sufficient.

There are many ways by which a given musical work might be related to mathematics. In fact, one might go so far as to say that all musical works are related to mathematics in some way. To see how this might be true, let us consider abstract and concrete musical works separately.

For an abstract musical work, generally, some model of sound is used. For example, one might use a model whereby all possible sonic realizations are represented as a sequence of discrete events that are indicated by some notational system, where each event has a particular start time, duration, pitch, intensity, timbre and spatial position. The characteristics of the particular model that is used will dictate the terms in which the musical work is expressed. The model will dictate the form of the score. Further, it will dictate the probabilistic properties of the class of sonic realizations that might be generated from such a score. One might say that the model of sound that is used for an abstract musical work is nothing short of a mathematical model for sound. And that, consequently, any abstract musical work must be inherently related to mathematics.

Models of sound are used for concrete musical work as well. For example, generally the final form of a concrete musical work is specified in terms of a model that is as direct (and faithful) a representation as possible of sound itself. Here, the model is chosen in such a way that by some metric, the members of the class of sonic realizations that are likely to be generated are contained in some extremely small neighborhood, perhaps centered about some ideal. Also, in many cases, such works are composed by assembling sound events, each of which might be created by using devices that are based on parametric models for sound generation or processing. The parameters of these models are used by the composer to create concrete musical works. Again, one might say that these models of sound are mathematical models, and that any concrete musical work must be related to mathematics, inherently.

In these ways, every musical work is inherently related to mathematics. However, since most individuals would not consider all musical works to be examples of mathematical music, clearly these inherent relations alone are not sufficient for one to say that a given musical work is an example of mathematical music. Perhaps this is because these relations are ordinary, in that they exist between every musical work and mathematics. For one to classify a given musical work as mathematical music, perhaps there must exist some other extraordinary relations between the work and mathematics, which distinguish the work as being more mathematical than what is normal.

There are many types of extraordinary relations that might exist between a given musical work and mathematics. For example, one might imagine a composer that consciously used the golden ratio to determine the relative durations of the sections of a musical work. Or, another composer might have created a musical work by using musical material derived from properties of some mathematical object such as a pentagon, or from properties of a mathematical model of some object, phenomenon or emotion such as a baseball, a particular game of chess, turbulence, another musical work, or fear. Another may have frequented a local mathematics library to sit among the mathematics books while composing so as to be inspired by the totality of mathematics. These are examples of the relations "was created by using", "was derived from", and "was inspired by".

In some cases, the belief in the existence of some extraordinary relation might not be sufficient for one to say that a given musical work is an example of mathematical music. For example, suppose a music theorist reports a finding that the average written duration of notes in a given abstract musical work is approximately equal to the square root of 2 times the duration of the shortest note. Clearly, this would be an extraordinary relation between the musical work and mathematics. However, it is unlikely that this relation alone would lead one to say that the work is an example of mathematical music. Why not? Perhaps in this case, one believes that the composer did not have this relation in mind when creating the work. If one believes that a given extraordinary relation exists between a given musical work and mathematics, then the degree to which this belief contributes to one's willingness to say that the work is an example of mathematical music, might depend upon the degree to which one believes that the composer intentionally created the work in such a way that the relation would exist.

There are many ways by which one might come to have a strong belief that the composer of a given musical work acted with such intent. First, the composer may have stated this explicitly in some way. For example, the program notes (written by the composer) might describe a specific relationship between the musical work and mathematics. Second, the composer might reveal such a relationship in the course of discussing the work in an article or interview.

Even without explicit textual information or clues from the composer, there might be cases for which one would strongly believe that a composer has acted with such intent. For example, by listening to a given musical work, or by analysis, one might be able to discern some relationship between the musical work and mathematics. If one believes that such a relationship is unlikely to have occurred by chance, then one might strongly believe that the composer acted with intent. For example, consider the following scenario. Suppose a particular musical work begins with the following sequence of notes (N) and rests (R): N, N, R, N, N, N, R, N, N, N, N, N, R, N, N, N, N, N, N, N, R, etc. Even without any other information or clues, one might readily detect that this sequence is related to the sequence of prime numbers (i.e. 2 notes, then 3 notes, then 5 notes, then 7). One might believe that the probability that such a sequence was created by chance is so small that the composer must have intended for this relationship to exist.

A strong belief that an extraordinary relation between a given musical work and mathematics exists because this was the intent of the composer, might contribute to one's willingness to say that the work is an example of mathematical music. However, such a belief alone is probably insufficient. For example, consider a given musical work that consists of a sequence of two sections. Suppose that in the program notes for the work, the composer writes that the ratio between the total duration of each section is approximately equal to the golden ratio. Further, suppose one is not aware of any other extraordinary relationships that might exist between the musical work and mathematics. Contrast this with a second musical work titled Golden Ratio for which the composer states that the golden ratio was used as a determinant for a variety of elements of the work (e.g. pitch, tempo, timbre, spatial position, intensity, and the relative duration of individual notes). One probably would not say the first work is an example of mathematical music. However, it is likely that the second work would be classified as such.

For one to say that a given musical work is an example of mathematical music, perhaps one must believe that there exists a set of extraordinary relations between the work and mathematics that are pervasive. That is, perhaps one must believe that a variety of elements of the musical work are related to mathematics in some extraordinary way. Here, the phrase elements of a given musical work means the dimensions, parameters, or components of the mathematical models of sound that were used by the composer to create the musical work.

If one believes that there exists a pervasive set of extraordinary relations between a given musical work and mathematics and that the composer intentionally created the work in such a way that these relations would exist, perhaps one might then say that the work is an example of mathematical music.
 


Other Work by John Greschak

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