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I have some conceptual trouble understanding the differences between rhythm and melody.

I get that harmony is the vertical dimension of music (thinking in terms of sheet music or a piano roll view in a DAW) and melody is the horizontal dimension. However, the way rhythm is defined seems to place it on the horizontal dimension as well. The problem then, as a mathematician might put it, is that harmony and melody already provide a basis for the two dimensions of sheet/piano roll music. In particular, if a melody completely defines a horizontal slice of music (or maybe more geometrically precisely, a curve passing through only one note at a time), how is it still possible for the rhythm to vary while holding the melody constant?

My grade school understanding was that rhythm was the percussion's contribution to the horizontal dimension and melody was the non-percussion's contribution to it, so that a melody would not completely define the horizontal dimension, but only the role of certain instruments in occupying it, but I don't think this explanation is correct, because people attempt to illustrate the concept of rhythm on non-percussive instruments as well.

To offer another guess at how melody may not completely define the horizontal dimension, is it the case that melody is only concerned with the order of notes, while rhythm takes credit for their duration? For example, if I specify a string of notes such as A, C, E, G, F, C#, A, B, A, C, A, does this string, in and of itself, constitute a melody, even though the string might sound very different depending on the duration of each note?

How do rhythm and melody compete and cooperate to define the horizontal space of music?

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    Rhythm is a component of melody. It can exist without melody, but melody does not exist without rhythm. Asking how they compete and cooperate is kind of like asking how flour and bread compete and cooperate. You can have flour without bread, but you can’t make bread without flour. I don’t believe they do compete, and that cooperation is innate to their relationship.
    – wabisabied
    Jun 13 at 1:27
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I get that harmony is the vertical dimension of music (thinking in terms of sheet music or a piano roll view in a DAW) and melody is the horizontal dimension.

If you're really thinking about sheet music in standard notation, or a DAW piano roll, then the vertical dimension is note selection (which in itself is an abstraction of pitch), and the horizontal dimension deals with rhythm.

In this model, melody can't be thought of as a single dimension, because it needs to be defined both in terms of pitch and rhythm. That's also the case with a harmonic progression.

The instantaneous harmony of a piece - "what chord is playing now?" - could be thought of as being defined by a set of values on the vertical dimension.

It's worth noting that this just a way to view music - it isn't 'the correct' way. If you look at a piano keyboard, then suddenly note selection is the horizontal dimension. And there are other important aspects of music, such as timbre, that could be considered 'dimensions'.

how is it still possible for the rhythm to vary while holding the melody constant?

It isn't. If the rhythm of a melody is changing, then the melody is changing.

To offer another guess at how melody may not completely define the horizontal dimension, is it the case that melody is only concerned with the order of notes, while rhythm takes credit for their duration? For example, if I specify a string of notes such as A, C, E, G, F, C#, A, B, A, C, A, does this string, in and of itself, constitute a melody, even though the string might sound very different depending on the duration of each note?

It's not a crazy idea to think of the order of notes being one aspect of a musical part, and their rhythm being another aspect. In fact, that's pretty close to the dimensions we see on sheet music. However, your definition here doesn't agree with what people mean by 'melody', which refers to something defined by both note selection and rhythm.

How do rhythm and melody compete and cooperate to define the horizontal space of music?

It makes no sense to talk about rhythm and melody competing, because they're different levels of concept: melody is a higher-level concept defined partly in terms of rhythm.

My grade school understanding was that rhythm was the percussion's contribution to the horizontal dimension and melody was the non-percussion's contribution to it, so that a melody would not completely define the horizontal dimension, but only the role of certain instruments in occupying it, but I don't think this explanation is correct, because people attempt to illustrate the concept of rhythm on non-percussive instruments as well.

I think you're correct that that's not correct - again, it seems to be mixing up lower- and higher-level concepts. We also need to be careful because some percussion instruments can play melodies!

Perhaps one model we could use is that unpitched percussion, like a drum kit, navigates the dimensions of timbre and rhythm, while pitched instruments (that can play melodies) can traverse the dimensions of note selection, rhythm, and timbre.

But again, that would just be 'a' model - not 'the correct' model. You always have to be careful about statements saying that 'these are the X dimensions of music', or 'there are X aspects of music', because various models are possible. And as the saying goes, all models are wrong (but some are useful).

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I believe you're missing an important point: melody is not just a succession of notes, it is a timed succession.

As you pointed out, with the same "list" of notes you can get drastically different melodies depending on the duration of each note (and rests!) related to the others.

The horizontal/vertical orientation that is often used for harmony/melody is just a basic simplification that, in fact, is done to simplify explanation/communication.

In fact:

  1. harmony does follow and obey to rhythm too: besides the analysis of a specific "sum of sounds" at a given moment, the concept of harmony in a musical piece is based on the variation of those "sums" in a time context;
  2. harmony and melody are not distinct parts, they are closely related by their "vertical relations" and rhythmic relations;
  3. the "vertical" dimension is composed by both harmony and melody at a specific time;

Time is, in the end, the most important part of music, and for many reasons. Without time, the very concept of pitch (and, then, harmony) doesn't exist, since the pitch is the result of a frequency, which in turn is the result of a certain amount of events within a time frame.

Don't be confused by the concept of "percussion". While it generally refers to a clear sound event that happens from an absence of sound, it should in fact be considered more of a variation to the previous state: a change in frequency (pitch, timbre, harmonics, etc) and amplitude (dynamics). Consider that even a sound that follows a silence is a change in frequency: from zero (no sound) to X (some sound).
Our brains "simplify" most of those events, especially when their duration spans a certain amount of time (sustain, decay of sound, crescendo/diminuendo, etc), so the actual rhythm ("percussion") is perceived only when a drastic difference happens.

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Melody and rhythm work together. For the purpose of a rigid definition, one could consider "melody" as the ordered pitches and "rhythm" as the duration of each pitch. However, the two must be taken together: a given "melody" with different rhythms will sound very different; so, too, a given rhythm with a different melody.

To borrow the curved-line analogy for melody: the x coordinates would be the time (rhythm) elements, and the y coordinates would be the pitch elements. Thus "melody" in this strict sense would be series of ordered pairs.

In fact, there is some music defined along these lines. "Serial music", and specifically "total serialism", uses mod12 arithmetic and corresponding set/group/category theory to determine pitch sequences, durations, harmonies, and any other musical element the composer chooses to define numerically.

But in most music one might commonly encounter, pitch, rhythm, and harmony all work together, and the boundaries between each can easily become blurred. They do not adhere to the kind of mathematically precise definitions that might be desired.

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Rather like when a given phrase can be said, using pauses, inflections, etc., to create a very different meaning, so a given list of pitches will give a very different melody when the same is applied. So, no, a list of pitches played in one order, but with different rhythms, could be used to produce thousands of different melodies. And that's without considering the different harmonies which could be used with that same set of notes - in the same order.

Then, there's the fact that music isn't constant sound. Rests play a big part - no sound is often as important as some sound.

Percussion. A big red herring. Leaving aside tuned percussion, there's the untuned, some of which make very short sounds (snare), some capable of long sounds (bells, cymbals), so they can easily be included in the 'horizontal' and 'vertical' concept.

I don't think 'compete' is the word - 'complement' may be better. Rhythm and melody work together, horizontally and vertically. Taking one dimension away leaves us with no music as we know it.

Reading 'horizontally', it's way removed from reading words from a book. The length of pitches is pre-determined, which is of paramount importance to each melody. If that factor wasn't there, there would be a myriad of different ways to play a line of music. Certain 'rules' are usually observed, like partitioning the music into separate boxes (bars).

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I wouldn't say that the dimensions of music are "horizontal" and "vertical". The dimensions of music are in your head, so to speak. If we forget lyrics, the usual dimensions (as defined by me personally) humans consider when perceiving sounds they classify as music are:

  • melody : the most prominent single leading idea that could be "hummed". A good example of a non-pitched melody is the drum intro of Queen's "We Will Rock You".
  • harmony : pitch relationships
  • rhythm : transients that form a pulse to follow and which create expectations of further transients to occur.
  • timbre : all sorts of things related to upper harmony structure and partials, acoustics, dynamics, ... Timbre has both a "horizontal" and "vertical" aspect to it. Timbres are very important in creating feelings.

The dimensions are not completely separate and none of them is purely horizontal or vertical. After a silence, when there is a sound, any sound, it contributes something to all dimensions. If the sound is completely unpitched, its direct contribution to harmony is negligible, but it still affects harmony via the rhythm dimension. Whatever happens in the rhythm dimension, affects how pitches are interpreted harmonically. Whichever pitches land on a "strong" beat, have a stronger impact on harmonic interpretation, so if the unpitched noises define a steady rhythmic pulse, they pave way for the interpretation for pitched sounds that might occur later on.

Pitch and harmony are not completely separate from rhythm. If you have a completely steady-pitched sine tone, and suddenly it changes to another pitch, that's a transient as well. You can create a rhythm with pitch changes. Noises and percussive sounds are not needed for rhythm.

Melody can be seen as a combination of everything, and it can be subjective. What is this song's melody really? Ask different people, and you might get different answers. In two-voice harmony, different people can pick the different voice as being the main melody. Sometimes people describe music as being "unmelodic", but I think that's partly because they're not used to hearing those kinds of sounds, and there is no single idea they could hold on to. While at the same time someone might think it's very melodic and they start humming, reproducing some parts of the ideas in some way. Another reason for a song being described as unmelodic can be that the song's supposedly leading pitch line doesn't use harmonically significant notes, so it's sort of colorless and odorless, "ambient" so to speak.

These days, people want and expect an objective mathematical world, where everything must be quantified and measured with some kind of a scientific laboratory device. What is the melody? What is the rhythm? There has to be a single definite truth about what everything really is!? But I say that music is not a natural science, it's cultural and subjective. You, as a fuzzy, changing, deficient, short-living biological subject, and a part of a social community, are an elemental part of the thing you're trying to quantify. Observe, measure and calculate less. Get involved, take part in music. Live a musical life with others? My opinions only.

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