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.