This may sound like quite an obvious question. Admittedly my knowledge of music theory is limited.
I'm essentially wondering if two songs can use the exact same notes while still having original melodies. For instance, the same six notes played in the same order but with different arrangement/rhythm.
If the pieces sound different, does it matter whether they share the same exact sequence of notes?
As has been described, it's not especially difficult to recompose a melody to make it unrecognizable as having been copied from some other song. I did this for the question How to write music if you already know music theory. And you'll see it pointed out in the comments that "Twinkle, Twinkle Little Star" and "What a Wonderful World" have near identical melodies at their beginnings. (I've intentionally "flattened" the rhythm of "Wonderful World" to make the correspondence as clear as possible.)
In fact, for people without musical training, it's often the case that keeping a melody exactly the same but changing the lyrics can make someone not realize they're hearing the same music twice.
For example, people are often surprised to discover that "Twinkle", "Baa, Baa Black Sheep", and the "ABC" song all share the same melody. Another example is the song "Imagine" by John Lennon. The first four lines of the song all share exactly the same three notes. But when I've helped people learn the song by ear, they're often surprised to discover how simple and repetitive the song actually is. (Learning pop music by ear is often quite disappointing for newcomers, because music they heard as intricate and complex turns out to be just two or three notes repeated. In the linked recording listen to
Imagine there's no heaven.
It's easy if you try:
no hell below us,
above us only sky.
The three notes are G, B, and A.
As to the question of "Does it matter", from a legal perspective, we can't answer here, but as a matter of some sort of musical ethics, it's no problem. It's been happening for hundreds of years. As long as the "duplicate" composition sounds truly unique, then it's fine.
True, there are quite a few songs which contain matching melody notes. It can get close to plagiarism, as several famous songs bear testament to.
Two big differences could be apparent, though. The rhythm of those notes, especially where the emphases come - which notes are on the accented parts of bars, as well as the more obvious length of each note.
The other factor here is the underlying harmony. There could be (and are!) two melodies that are identical, but with totally different harmonies attached. That will make them sound like different songs. Not as much, admittedly, as the other way round - keep the harmony and change the melody - which happens all the time in jazz.
Another way to "disguise" the same melody is through the use of tempo and rhythm. Many people know the songs 76 Trombones and Goodnight My Someone from The Music Man, but never realize that they are essentially the same melody at different tempos and rhythms (3/4 Moderato vs. 6/8 March). Kudos to Meredith Willson!
Goodnight My Someone
76 Trombones
Stravinsky's Greeting Prelude is a cleverly disguised version of a very familiar tune.
Another concept of melodic "sameness" occurs in sets of variations on a theme. These can range from a set where the melody is always exactly the same (e.g. Chopsticks where the primo part is always identical but the secondo, accompaniment part changes) to variations where great liberties are taken with the melody to the point where it can become almost unrecognizable.
A fun example of arrangements of the exact same melody is the Paraphrases on Chopsticks (actually the "Cutlet Polka," a variant) by Liszt, Cui, Borodin, Rimsky-Korsakov, Mussorgsky, Liadov and others.
Easily. The rhythm matters. I think the Marine's Hymn and the Blue Danube Waltz start with notes 1,3,5 outlining a tonic chord. Lots of pieces start with an arpeggio of some chord.
These are not complete pieces, just segments. Some segments can be quite long.
Two pieces can use all the notes of either a diatonic or chromatic scale in different orders. (I'm not sure this is what is being asked about.)
Building on some other answers already here, film music is the master of taking a tune and playing it in completely different ways for different parts of the film, allowing a motif to represent a person/thing/idea and then playing it with different emotions or styles are wanted by the film (aside, this method of composition is itself rooted in ideas and techniques that almost all classical composers of the past used).
A good example would be the Jedi theme from Star Wars that was used by John Williams variously for victories, funerals, subtle hints about characters, heroic actions and many other areas.
Compare the twin sunset theme (Episode IV):
and Qui Gon and Obi Wan lightsabering through a blast door on the enemy ship (Episode I) (at 1:02):
More subtle is the comparison between the sunset theme and Darth Vader's funeral (Episode VI), where the arrangement is very similar, but the performance is different giving a different emotion:
Rearrangement of a song means typically changing the instrumentation and/or changing the accompaniment, but normally without changing the melody. So even if the song changes character a bit, it's the same song, with the same harmony.
One of rearrangement techniques is reharmonization, when you change, sometimes quite substantially, the chords. An example could be this version of Lady Gaga's "Let's Dance" played by Dirty Loops:
It sounds quite different, but the melody feels mostly the same. In this version, even though chords are changed, the key or tonal center relative to the melody is the same as the original.
However, one can go further, and change the key, i.e. play the same notes against a different tonal center. This might be difficult for a whole melody, as good melodies often suggest a specific keys, but may work for some shorter phrases or motifs.
Then, even if you don't change the rhythm, you can place the same notes at the different part of a measure, so that some stressed notes become unstressed, some unstressed become stressed, or syncopated and non-syncopated notes switch their roles. This may change the feel of a melody by a lot.
I can't think of any examples of these two possible transformations right now, possibly because when they happen I don't recognize similarity to another melody.
