What does "function" actually mean in music?

Question

I remember when the term function was introduced in math - y=f(x) - the teacher was a substitution and still himself a student. He used the term function as if we already knew of what he was talking about. And the confusion was very big among us in the teacher training college. (Only months or years later we understood after many examples that the question was about to variables and their (inter)dependence.

Some years later when I started reading of harmony I learnt that tonic and dominant also are functions. And I remember how I asked my theory teacher Szandor Veress what is meant by function and he asked me back: "what is the function of a teacher?" and he answered himself: "the students", he said. And he asked: "what is the function of a policeman?" - "to rule the traffic."

I understood that also here must be a dependence of chords but it was not clear to me.

I also understood the term function of a tool or a job.

I only fully understood it last year when I read Ernst Kurth's book about counterpoint what functional theory is. Of course I knew the terms tonic and dominant etc. but I never had found a clear explanation of the term function in the definition of "functional harmony" before reading this book:

https://books.google.com/books/about/Ernst_Kurth_Selected_Writings.html?id=i7zAsV678sQC

People always use terms and terms of concepts without having really a concept. It's easy to say the function V-I is the function of the dominant to the tonic. But is this not just a tautology?

Many questions and answers in this SE deal about function of chords and degrees, but they all imply that the meaning of function is known.

Definition of Functional Harmony

Today I know the answer.

But I wonder how you would explain it to a curious student that I (still) am.

Answer 1

See also: Definition of Functional Harmony

In functional harmony, simultaneous notes are interpreted as chords and the analysis is based around how the chords relate to the overall key and the preceding and following chords.

The relationship any one chord has in the context around it (i.e., the key and other chords) is called the chord's function. Another way to think of it is, the function of a chord is the answer to the question "what does this chord do to the harmony?"

So in the key of C major, any G major chord or G7 would be said to have a dominant function in that context, especially if the next chord is a chord that has a tonic function (the C major chord, in this example) and even more so if the previous chord has a predominant or subdominant function (which could be II AKA V/V or IV or I6/4, etc.)

Quick reminder: all music theories, including functional harmony, are just ways to attempt to understand how and/or why the music has the effect it has on most listeners. It's just a model or tool, it's never the full story. So while it can be helpful to understand chord function and functional harmony in general, such understanding is just a small, cloudy window into one aspect of the music.

Answer 2

"Function" means "role" or "responsibility"; it's closer to the ordinary meaning, different from the specialized math term which means "mapping between two sets".

Western music theory works by identifying and classifying certain recurring patterns in music, such as certain chord cadences.

When we say that some chord has a function, we are stating (the hypothesis) that the segment of music seems to fit some well-known pattern, and the chord is an element in that pattern. It is functioning as a piece of that pattern. Each piece of the pattern has a "job", and so if we know which piece of which pattern that harmony is, we can say that it's doing that job.

Sometimes there are ambiguities: more than one pattern applies to a segment of music. Sometimes that is deliberate; patterns overlap, so that for instance the ending harmony of one pattern doubles as the start of the next one. So then the same chord might have multiple functions at the same time.

Answer 3

EDIT

I'm revising my answer, because from OP comments I now understand the question isn't simply about Riemann function.

I see potentially three definitions for "function": Riemann function, mathematical function, function meaning purpose.

...But I wonder how you would explain it to a curious student...

To the extent there is a problem, that problem seems to be with the word "it." "Explain it." Explain what exactly? Explain Riemann function? Explain a math-like function concept in music? Explain the purpose (function) of musical devices? If the word "function" is used with students, and we switch back and forth between the various meanings of the word, some students might get confused.

Personally, when I hear "function" in musical context I think either Riemann function or the general meaning purpose. I don't think of the math term which in plain English means something like procedure. If someone is confused about the meaning of "function," you probably need to clear up which definition is being used.

As far as the math meaning is concerned, I like the example sentence: "the position of a planet is a function of time." We could apply that to some musical processes and add the general meaning purpose.

The inversion of a melody is a function of direction and melodic inversion is used to develop new, thematically related material. An inversion function would take a series of intervals and directions and return the same set of intervals with all the directions reversed.

A harmonic sequence is a function of transposition and sequential harmony can be used to move to a new key. A harmonic sequence function would take a two chord progression and an interval of transposition and return the original two chords followed by two more chords whose roots are located the interval of transposition away from the original two chords.

Syncopation is a function of rhythm and syncopated rhythm is used to create excitement. A syncopation function could take a melodic fragment and change metrical rhythmic patterns to contra-metric values without changing the pitches.

So from these examples, to invert, to sequence, and to syncopate are all functions in the mathematical sense that output of the process is determined by the input. In plain English these functions are all verbs.

Can Reimann function be described as a function in the mathematical sense that the output of such a function is determined by the input? I say "no."

We really understand a chord's Riemann function only after knowing where the chord goes. If the 'input' is chords C G Am ...what would a Riemann function determine is the output? Riemann function is about the succession of chords. Any sensible Riemann function process would provide the next chord as output. But, we don't know where the music is going. It could modulate. More importantly we have know way to if the next chord should be functional according to Riemann function. In other words: Riemann function is not a musical procedure - the mathematical meaning of function - but a description of existing harmony.

...What does “function” actually mean in music?

To the extent that 'function' in music will be understood by some as 'Riemann function' it is merely a description of harmony. Music can be described as functional or non-functional.

...I remember when the term function was introduced in math - y=f(x)

Riemann function is not one of those functions.

...tonic and dominant also are functions...

No, they aren't. If we use the mathematical meaning of function, tonic and dominant are not processes. They are properties of chords in the Riemann model. We could have a mathematical function called tonicize but that is something different. After applying a tonicize function to a chord we could then refer to the chord's tonic property and it being the tonic of a tonality.

So, if we really want to be mathematical about this, the common musical concept of "a chord's function" is a property, a description, an adjective. It is not a mathematical function, it isn't a process, it isn't a verb.

http://www.oxfordmusiconline.com/grovemusic/search?q=function

https://en.wikipedia.org/wiki/Function_(mathematics)

https://www.google.com/search?q=function+definition

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My original answer...

I suppose it works like this in my mind:

Example, there are chords Dm, G7, and Am...

I can place them into a key C major...

The (Riemann) functions are basically the chord root identities labelled with Roman numerals ii, V7, and vi, further I can give general labels of pre-dominant, dominant, and (I selected vi on purpose) a non-tonic chord which gets' shoe-horned into the Riemann function system as another pre-dominant. If the Am had been C we would have the tonic and very neatly fulfilled the expected Riemann functions...

Riemann function is also described as a flow of events: pre-dominant, to dominant, to tonic.

Depending on the rhythmic phrasing I could say the harmony forms a deceptive cadence.

Normally, that is the end of the Riemann analysis story.

But, as I understand the question, the issue is not about the labels, but rather "what is the purpose fulfilled by the functions?"

So, in the Riemann sense: "what is the function of harmony?" answer "to lead to and form a cadence."

Notice that I specify cadence and not chord progression.

If we take something like V6 IV6 iii6 ii6 I6/4 V7 I, the initial chords in the progression don't fulfill Riemann function, but the ending does. To me, the essential functional part is the cadential part. Something similar could be said about sequential harmony where the sequential passage is a shifting of the tonal center - indeed, pivot chords get two labels - after the sequence a cadential passage confirms tonality at which point function becomes clear.

Whether any of this is satisfying intellectually or musically is another matter. My attitude about functional harmony - and harmony textbooks - changed a lot after I read Gjerdingen's Music in the Galant Style which turned me on to figured bass and solfege. Why should Riemann function trump rhythm and metrical function, or the function of consonance/dissonance and stability/unstability? All of which get's short shrift in most textbooks compared to Roman numeral analysis. "What's the function of a chord or harmony?" Answer: "to express yourself musically!" "Why does some music lead to cadences?" Answer: "because that is the convention is those styles!"

Answer 4

Contrary to what many people seem to think here, to me the definition of "function" is exactly the same in mathematics and music (and computer science).

Let's look at the definition from dictionary.com:

function: a relation between two sets in which one element of the second set is assigned to each element of the first set

Now this may confuse people, since we're talking about "sets" here, so just to be certain we're all on the same page: (also from dictionary.com)

set: a collection of objects or elements classed together

What is slightly misleading here is that the above definition very much implies tangible concepts, but obviously in mathematics a "set" can be infinite and/or abstract etc. (E.g. the set of positive integers)

So, what does this have to do with music?

Let's start with "set". The chromatic notes are clearly a set (of frequencies). We noticed that within that set we can identify subsets, with a very interesting quality: we can actually describe a set of intervals, which we call scales. If we move the order of the intervals in that set around, we call them modes. That's one way of identifying sets withing music, let's call these the vertical sets.

But there are other sets to be found: regardless of which scale we used, we noticed that certain interval progressions lead to the same emotions. We wanted to name these (and thereby created sets) and decided upon "tonic", "subdominant", etc. These are the horizontal sets.

So in fact the word "dominant" describes a set: all the fifth elements within scales.

Alright, now let's get to "function".

Let's look at the mathematical definition again: a relation between two sets in which one element of the second set is assigned to each element of the first set

For example we have the following two sets:

1) [C, D, E, F, G, A, B]

2) [G, A, B, C, D, E, F#]

So what are these two sets? (some of) the tonics of the (major) scales are set 1. (and coincidentally it also constitutes the C major scale) And set 2 are the dominants.

I.e. what I've done here is I took the tonics of these major scales (vertical sets) and put them together into set 1 (a horizontal set) Then I took the dominants of the same major scales and put those into set 2

TL/DR; So the function here is the relation between the tonic and the dominant. In the vertical set of the C major scale, C functions as the tonic to the dominant G. (Which would not be the case for the F major, since there C functions as the dominant to the tonic F) To lead it back to the mathematical definition: the first element in set 1 is assigned to the first element in set 2. etc.

Now, how you want to describe that function is entirely up to you. (i.e. it "leads", sounds like home, etc. etc.)

Sidenote: of course the function of a teacher is "to teach". Set 1 is all the teachers, set 2 is all the students, what is the relation between those two sets? The teachers teach to the students, i.e. their "function" is to teach.

Answer 5

Just to chip in a bit on some nice answers here already: I think the question could still have some refinement.

"People always use terms and terms of concepts without having really a concept. It's easy to say the function V-I is the function of the dominant to the tonic. But is this not just a tautology?"

Not necessarily, with a bit of experimenting and challenging your ears and perception you can device a system in which for instance IV has a more dominant function and V-I would sound more like subdominant-> tonic.

"Many questions and answers in this SE deal about function of chords and degrees, but they all imply that the meaning of function is known."

It might be interesting to think about this: The idea of function, is it really something that pertains to music ? I think not, it is part of the ANALYSIS of music. It helps us qualify things we perceive, communicate concepts, observe patterns so that we may then use all of our capabilities to challenge ourselves and come up with something fresh that we like !

And while others feel like the idea of a math function does not fit, I think it does quite nicely. It would be looking somewhat like this: a function taking in some previous chords, and current tones, then spitting out a tuple of a tension (on a scale of very tonic to very dominant) and some possible expected next chords. That is what it does, in my perception. As for the objection that a math function can only have one possible output for a given input, this is how we have defined them now because that was useful. Sometime ago we would talk about single valued or multi valued functions.

On another note: we could also state that the function of a chord is to accompany a melody. The melody then becomes the a question ... is it possible o have a bunch of chords without a melody or will we perceive certain parts (the top or bottom notes for instance) automatically in a melodical sense ? Well... since I seem to go off track I'll leave it here. Nice question!