Philosophy 304a: Philosophy of Music [under construction]

"Music is the effort we make to explain to ourselves how our brains work. We listen to Bach transfixed because this is listening to a human mind." - Lewis Thomas

"All art constantly aspires towards the condition of music." - Walter Pater

Contents:

Introduction: Philosophy of music as a special case of philosophy of art

Part A: Music the Product

Section 1: Music and the sorts of art
Section 2: Analysis of the concept of music
Section 3: Music and the theories of the nature of art

Part B: Music the Process

Section n: Persons in music
Section n: Means to music
Section n: Values and disvalues in music
Section n: Evaluation of music

Introduction: Philosophy of music as a special case of philosophy of art

This article is based upon my article "Philosophy 304A: Philosophy of Art - Revisited". It treats the philosophy of music as a special case of the philosophy of art. A special case because of the conceptual discoveries outlined here. Those discoveries are due to the heuristic (structurally suggestive) power of my Three Groups thesis. My use of G1, G2 and G3 refers to that thesis, which is that the central concepts in each of the subject matters of philosophy can be sorted, non-arbitrarly, into three groups of concepts and doctrines.

Part A: Music the Product

Section 1: Music and the sorts of art

G1: Music as autographic art

An autographic artwork is one for which the artist (the conceiver) is necessarily also the maker or performer.

Is there any music for which the composer is necessarily also the performer? Yes, there is; it's called improvised music. The performer composes it while playing it.

G2: Music as formally allographic art

An allographic artwork is one for which the artist is not necessarily the performer. A formally allographic artwork is one that is primarily a set of formal instructions to be followed by the performer.

Almost all music is this sort of artwork. The set of formal instructions is called the score, and it is written in a language called musical notation.

G3: Music as contextually allographic art

A contextually allographic artwork is performed by the artist's (and audience's) non-human environment. The role of the artist is to design the instrument upon which the environment performs.

The music made by a set of wind-chimes is an example of this sort of music.

Section 2: Analysis of the concept of music

G1. The qualitative aspect of music is the notes, the units of sound, of which it consists in performance. Music in the Western hemisphere is usually composed of notes from several traditional sets of notes called scales and is called 'tonal'. Music that includes notes from outside those scales is called 'atonal'

G2. The formal aspect of music is by far its most complex aspect. This is, firstly, the order in which the notes are played. An example of a unit of musical order is a scale, which is a sequence of notes in ascending or descending order of the frequency of the sound-waves that are heard as the notes.

(Other sorts of musical form are discussed in Section 3 (G2): Music as significant form.)

G3. The quantitative aspect of music includes such factors as the varying time intervals between notes, and the magnitude (loundness) and duration of the performed composition.

Section 3: Music and the theories of the nature of art

G1: Music as representation

There are three sorts of ways music can represent X:

G1a: By mimicking the sound of X

G1b: By having a structure that maps the structure of X

G1c: By being performed with magnitudes and durations that map those of X

One well-known example of such music is Beethoven's 6th Symphony, the 'Pastoral', which was composed to represent the countryside and a storm. This sort of music is termed 'programmatic' music.

G2: Music as significant form

This is the theory of art which best captures the art of music.

The the sonic contents and the structural forms of music can both be described in mathematical terms. Mathematics and music are structurally related. The basic structures can be seen in a consideration of the three sorts of number scale: valency, ordinal, and cardinal.

(G1) The valency scale of positive and negative numbers. The two sides of this scale mirror each other. ...-3, -2, -1, 1, 2, 3... are numbers on the valency scale. Note that numbers on this scale are *absolutely* (not relativley) either positive or negative, greater or lesser.

The valency scale contains three basic 'valency forms' - oppostition, mapping and mirroring.

Opposition is the most basic 'valency form'. It is simply positivity-negativity.

The musical analog of opposition is the most basic musical form, sound and silence, positive and negative. (There is no such thing as negative sound, so silence is not neutral in relation to sound's positive, it is negative).

Mapping is the parallels between the numbers on the two sides of the scale. Each number is paralleled by its opposite.

The musical analog of mapping is the next most basic musical form, the mapping of the opposing notes in the rising and falling versions of the same musical scale. (Keep in mind that we must use two established senses of the word 'scale' here - the mathematical and the musical.)

Mirroring is the parallels plus the oppositions. Each side of the valency scale of numbers mirrors the other.

So the third most basic musical form is the mirroring between the rising and falling versions of the same musical scale.

(G2) The ordinal scale of numbers shows a position (or range of positions) within the linear sequence of lesser to greater numbers. First, second, third, etc., are ordinal numbers. Note that numbers on this scale are only *relatively* large or small according to how close they are to the upper and lower limits of the sequence. For example, ninth is large if there are ten positions in the sequence, but small if there are a hundred.

The basic 'ordinal forms' are simple linear sequences and simple dimensional structures.

A simple linear sequence is a set of consecutive numbers taken from anywhere in the infinite sequence of numbers. A simple dimensional structure is a linear parallel between two different sequences.

In musical terms, a simple linear sequence is a musical scale, ascending or descending. And a simple dimensional structure is a parallel between two different musical scales. In musical terms such a parallel is called harmony.

So the basic ordinal forms in music are scales and harmonies.

A non-linear sequence of musical notes is called, if it is aesthetcally pleasing, a melody.

The simple musical structure that part-parallels two related melodies is called polyphony. Polyphony requires either a complex musical instrument such as a keyboard instrument or it requires the playing of two or more instruments together.

(G3) The cardinal number scale shows magnitudes and durations. One unit, two units, three units, etc., are cardinal numbers. Note that numbers on this scale are units in conventional measuring systems. As such, their sizes are relative to the unit-size of each measuring system. For example, five New Zealand dollars (units of currency) are small measured against five units of U.S. currency.

Notoriously, there are no absolute amounts on the ordinal scale. Magnitudes (sizes, amounts in space) are measured in differing and sometimes competing systems that are different because they use different unit-sizes.

The same is the case with the measurement of durations (amounts in time). The simplification and standardization of time measurement is a quite recent historical development.

Durations and magnitudes are also essential ingredients of musical sequence and structure.

There are three sorts of musical durations. The durations of the notes - note can be 'held' - depending upon the capabilities of the instrument it is played on - briefly or sustainedly. For example, an organ can hold a note much longer than a banjo can. Secondly, there are the durations of the silences between the playing of notes. Notes can be played without any silences at all between them - the length of the silence can range from zero to a point where the interval between notes would make them unrecogizable as music. Which brings us to the third sort of musical duration, the lengths of the varous different sections of a composition and the composition's total length.

The first and second sorts of durations comprise the 'rythyms' of a piece of music.

Magnitudes in music are the loudnesses with which the various notes are sounded.

Notice that it is the durations and magnitudes in a composition that are open to interpretation by the performer. The performer *must* play the sequences and structures that the composer specifies, otherwise it is not the composer's composition that is being played, it is instead a 'performed version' (as distinct from a 'composed version') of the work. And although the composer can and usually does specify the durations and magnitudes he/she requires, perfomers feel free to very them so as to give individual character to a performance. This is a major component of the perfomer's 'art'.

G3: Music as emotive expression

G3a: Emotions caused by the environed - narcissism and and self-contempt

G3b: Emotions caused by the internal environment - love and revulsion

G3c: Emotions caused by the external environment - desire and fear

Part B: Music the Process

Section n: Persons in the music process

G1. Composer
G2. Performer
G3. Audience

Section n: Means to music

G1. Musical mimesis

G2. Musical imagination

G3. Musical experiment

Section n: Musical values and disvalues

Positive musical values:

G1: Positive aesthetic value in music

G2: Positive conceptual value: Innovation

G3: Positive practical value: Technical skill

Negative musical values:

G1. Aesthetic disvalues in music

G2. Conceptual disvalue: Musical cliche

G3. Practical disvalue: Technical ineptitude

Section n: Evaluation of music

Three levels of evaluation of art:

G1: The expert level

G2: The social / historical level

G3: The personal level

Appendix A: Why do we like music?

The Ancient Greek philosopher Pythagoras is credited with first noting the parallel between music and mathematics and between mathematics and value. The follwing quote is from the author of the article on Pythagoras in the 6th edition of The Columbia Encyclopedia (author not named):

"Beginning with the discovery that the relationship between musical notes could be expressed in numerical ratios, the Pythagoreans elaborated a theory of numbers, the exact meaning of which is still disputed by scholars. Briefly, they taught that all things were numbers, meaning that the essence of things was number, and that all relationships—even abstract ethical concepts like justice—could be expressed numerically. They held that numbers set a limit to the unlimited—thus foreshadowing the distinction between form and matter that plays a key role in all later philosophy."

I don't know that my own discoveries about the parallels between mathematics, values and music are a rediscovery of those made by the Pythagoreans, but I too have discovered such parallels. The concept that connects the three is the concept of scales, as follows:

In mathematics there are three numbers scales:

G1. The valency scale of positive and negative numbers. The two sides of this scale mirror each other. ...-3, -2, -1, 1, 2, 3... are numbers on the valency scale. Note that numbers on this scale are *absolutely* (not relativley) either positive or negative, greater or lesser.

G2. The ordinal scale of a position or range of postions within the linear sequence of lesser to greater numbers. First, second, third, etc., are ordinal numbers. Note that numbers on this scale are only relatively large or small according to how close they are to the upper and lower limits of the sequence. For example, ninth is large if there are ten positions in the sequence, but small if there are a hundred.

G3. The cardinal scale of magnitudes and durations exclusive of position. One, two, three, etc., are cardinal numbers. Note that numbers on this scale are units in conventional measuring systems. As such, their sizes are relative to the unit-size of each measuring system. For example, five New Zealand dollars (units of currency) are small measured against five units of U.S. currency.

These number scales connect with music in (something like) the following way:

G1. The valency scale (positve mirrors negative): a set of musical notes, rising in pitch - and called a rising scale - is mirrored by the same set of notes but as a descending scale.

G2. The ordinal scale: The basic musical form is the sequence of notes. The composer chooses notes from several of the basic sequences of notes (the musical scales) to construct a melody. A well-formed sequence of notes is a melody. Two or more simulataneous sequences of notes, if musically compatible, form a harmony

G2. The cardinal

Appendix B: The Three Groups in this topic

Appendix C: (Temporary)

From the comments page of: My Approach to Art Criticism | lukeprog

I promise you I'm not being sarcastic here, but I'm really very grateful to you for the stimulus your post gave me to think further about the links between mathematics, music, and evaluation.

But before elaborating on that and attempting to convince you that your outlook is mistaken, I want to correct you on a couple of things you have apparently taken me to have said but that I didn't say at all.

First, I did not say that we can declare some (or any) music great before we have heard it. What I do say is that if it is great we can, or some of us can, recognize its greatness when we hear it (though we might not recognize it on first hearing).

Second, I did not suggest that there is only one legitimate kind of musical evaluation. You persist in ignoring the distinction lukprog and I make between three kinds of evaluation, the personal (e.g. your 'breakup songs' example), the socio-historical (the 'test of time'), and the expert (which might or might not agree with either of the other two).

Now, since reading your posts and thinking further on these matters I have discovered what I think is a quite plausible set of reasons for the conclusion that music does have objective value and that, on all three levels of evaluation, recognition of that objective value is possible, indeed likely, if the listener has sufficient aesthetic intelligence.

First reason. There is a strong triple link between mathematics, music, and values. The links can be seen by considering the concept of scales. The scales I refer to are the valency scale, the ordinal scale, and the cardinal scale. Numbers, music and values can each be defined with reference to these three sorts of scale. For the detail on this you will have to read the section called 'Why do we like music?' in my article on philosophy of music, which, forgive me, I am still in the process of writing.

Second reason. As mathematics developed, two main kinds became increasingly apparent, pure math and applied math. Applied math is the math of science and technology. It has become so important to the development of science that one scientist, Paul Davies, recently referred to math as 'The Mind of God' in his book of that title. (As a philosopher, I think Davies underestimated God by limiting His mind to math, just as the religious consider that 'the God of philosophy' is an inadequate account of the God they believe in.) But, to get to my point here, just as math is central to the scientific description of the physical world, it can also provide the core of an account of such aspects of the human world as music and the evaluation of music. Music is readily describable in mathematical terms. And the 'scales' links between math, music and evaluation strongly suggest that the core of an account of musical evaluation also might be given in mathematical terms.

But where does objectivity come into this? It is generally agreed among experts in the study of knowledge that mathematico-scientific knowledge has by far the strongest claim to objectivity of any sort of human knowledge. No one thinks it odd that religous fundamentalists from a hostile foreign culture came to the U.S. to study how to fly airliners (which wouldn't exist without mathematico-science) into their enemies' iconic buildings as part of an attack on their religious and cultural enemies. They did so because they knew as well as their enemies did that mathematico-scientific technology would work as much to their ends as to those of their enemies. In other words, they knew that such knowledge is objective knowledge, or as nearly so as humans are capable of.

You can see where I'm going with this. If a math-guided accont of the world, including music, is objective, then perhaps also a math-guided account of musical evaluation can be equally objective. But, forgive me, I'm still working on that.

~ posted by bertie

Odysseus, ask yourself what, ultimately, is the nature of a work of musical art. In what does it consist? Is it a physical thing like a painting, a sculpture, or...a CD? No, a musical composition can exist wholely in the mind of its composer. In the scene in Amadeus in which Salieri mentions "those meticulous ink strokes" he is appalled at finding that Mozart's scores show no evidence of the process of composition. The scores are mere recordings in musical notation of works of music that were composed entirely in Mozart's mind. And then Mozart died, as he would eventually have done with or without Salieri's help, but his music lived on. In what form did it live on? In the forms of Mozart's original score AND accurate copies of that score. Now, Da Vinci's La Giaconda is a unique physical object located in one place at a time. Such is not the case with a work of music. Where is Beethoven's 5th Symphony? It is scattered in thousands of places all over the world, whereever a copy of the score is. Sure, we could go further and ask Is scattered papers with ink on them what we mean when we refer to Beethoven's 5th? Does not the 'real' Beethoven's fifth consist of sound?

To this latter question I say, much as it seems to contradict what we mean by music, no, music is not sound. Sound is merely the 'human's eye view' of something that ultimately exists not even in mental form but in mathematical form. And, as a thing that is ultimately mathematical in nature, it is as objective as the entities of mathematics are.

Further, since mathematics is the non-physical sub-struture of the physical world, a work of music is part of that non-physical sub-structure. A work of music is the form, and the physical score and the physical sound-waves and the mental sounds are merely different sorts of content that clothe that form.

The functioning of the physical world produces stars and planets and human brains, and the functioning of human brains produces, among other things, works of music. They enter the world first as purely mental things, produced by the functioning of the brain (the functioning called 'musical imagination'), but are expressed in such physical things as musical scores and the sound-waves we hear as sounds. Is music mere sound-waves? - is it mere ink strokes on paper? Is it even the 'sounds' our memory can replay for us in our minds? No, it is something more basic than any of those things. It is part of the mathematical sub-structure of the functioning physical world. As such it is as objective in nature as the entities of mathematics are. Ask Isaac Newton whether mathematical entities are static Platonic entities.