I think there must be a good word for this, but can't think of what it might be.
Harmony and Melody are pieces of it, beat seems simplistic, Instrumentation seems too scientific and dull.
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Instrumentals — TFD
plural noun of instrumental
It's very commonly used to refer to music (minus the lyrics)
A composition for one or more instruments, usually without vocal accompaniment.
"the opening tune is an instrumental"
Performed on or written for an instrument.
"They played instrumental music at the wedding."
"an instrumental version of a song"
Or called backing tracks, karaoke, etc.
For further reading, here's a quote from a highly-upvoted and accepted answer to a similar question on ELL:
Instrumental can mean two things:
a "karaoke" version with the vocals removed; or
a song which was written without vocals in the first place.
From both the legal and writing process standpoints, the components that make up a song (by "song" I mean a musical piece that includes vocals, and may (often) include other musical instruments that accompany the vocals, with the vocals almost always including words in a specific language) are almost universally divided into music and lyrics.
When those two words are used in this sense, the word lyrics refers exclusively to any and all words that are sung or spoken or rapped as part of the song, and it does not refer to the melody or rhythm or any purely musical aspect of the vocals for the song. Often the word words is used in place of lyrics.
When used in these senses, the word music refers to everything but the lyrics (in the sense above) themselves, and therefore music in this sense does refer to the vocal melody, harmonies, and rhythms.
One visual way to separate lyrics from everything is to look at how a song is most commonly written down.
The top line of the music represents the vocal melody (and often includes vocal harmonies), and shows, in musical notation, the rhythm and pattern of notes that are to be sung. Right underneath the top line of music we see the words or lyrics written out, with each syllable aligned underneath the musical note that is to be sung at the musical time and on the pitch and for the duration indicated.
Is is clear in this case that music and lyrics (or words) separately refer to those two elements as written. In practice, especially among song writers, those words are used in the same exact way for the sounds as well.
I randomly remembered the lyrics to a "Weird" Al Yankovic parody of George Harrison's Got My Mind Set On You this morning. In the parody, "Weird" Al refers to the "music" and the "words" and "lyrics" as separate parts of making a song.
You really need words
Whole lotta rhyming words
You gotta rhyme so many words, mm-mm
To do it, to do it, to do it, to do it
To do it, to do it right, child
Oh, you gotta have-a music
You need really catchy music
This song has got plenty of music
But just six words, child
And so I'll sing' em over
And over and over and over
And over and over and over, mm-mm
And over and over and over
And over and over and over again
I'd go for 'music'.
Bing defines a 'song' as:
a short poem or other set of words set to music or meant to be sung.
The Free Dictionary describes a song as:
a piece of music, usually employing a verbal text, composed for the voice...
Oxford Dictionaries take on the word:
short poem or other set of words set to music or meant to be sung.
which is exactly the same as Bing's.
And Merriam-Webster say:
a short musical composition of words and music
So, without words, a song becomes just music.
I want to draw the distinction between the tune and the accompaniment.
If we take away the lyrics we are definitely left with one thing, the tune (or melody) and may be left with another the accompaniment.
I claim that it's not a song unless there is a tune to be sung. There are examples of pieces of music that are called songs by their creators, but which do not have words; Felix Mendelssohn's Songs Without Words being a well known example.
Not all songs have an accompaniment, many folk songs are sung unaccompanied. However in many cases the instrumental accompaniment is intrinsic to the character of the song. This seems very much the case for concert music such as Schubert's Lieder, but also in popular music the riffs, or intricate acoustic guitar parts are a formal part of the song.
Considering songs by Rogers and Hammerstein, Rogers is referred to as the Lyricist and Hammerstein as the composer. Hence we could argue that everything of the song except for the lyrics is the composition. However I don't think I've ever heard "composition" used in this way. I would say that a song has three elements: Lyrics, Tune and Accompaniment, so no single term for what's left when lyrics are removed.
As someone with a Bachelor of Music, I can tell you there is no simple answer--surprisingly, perhaps. The simplest answer would be 'accompaniment', meaning the instrumental parts that accompany the vocal line. But if we were talking about a Wagner opera, for instance, and we just took out the vocal parts, calling what remained the 'accompaniment' would seem like belittling it. If you wanted to describe what remained of a popular song after the vocal line is gone, you could talk about 'the instrumental parts' or 'the accompaniment'. Neither is totally satisfactory but they are the best available options. Or you could refer to specifics, by saying 'the orchestral part' or 'the rhythm section' or 'the keyboard part' or whatever.
Or you could refer to the structural elements, 'the harmony' or 'the chord progression'. It was common practice for Jazz musicians to discard the melody of a song and write a new melody to the chord progression, and copyright the result as their own composition. Approaches to copyright have changed, though, and you might not get away with that nowadays. That approach rested on the fact that traditionally copyright rested only in the melody and lyrics. Legal practice has changed, as indicated by the fact that Led Zeppelin were recently sued (unsuccessfully) over the instrumental introduction of "Stairway To Heaven".
As suggested in @NVZ's answer, consider composition
Composition consists in two things only. The first is the ordering and disposing of several sounds...in such a manner that their succession pleases the ear. This is what the Ancients called melody. The second is the rendering audible of two or more simultaneous sounds in such a manner that their combination is pleasant. This is what we call harmony, and it alone merits the name of composition.