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Beginner here, it seems like a scale just defines a particular set of notes that don't have to be played in any order to me. But if you look at A minor pentatonic scale you have A, C, D, E, G, yet with the same notes you get C major pentatonic with C, D, E, G, A. If they are not supposed to be played in order, how would you differentiate the two scales at all? Or does the order matter? And if you play a scale such as A minor above, can you mix in a note outside of that scale, or does that create a new scale from that point? Like if you played A, C, D, Db, E, F, G, is that just A minor with extra notes or would that be considered an entirely new scale?

Beginner here, it seems like a scale just defines a particular set of notes that don't have to be played in any order to me.

It depends. As Tim says, in an examination situation, you play the notes in order, as that's the 'study' form of the scale. But when the scale is used in a piece of music, then as you say, it's the set that's more important, and the notes could be in any order.

You wouldn't say you were playing a scale if the notes were out of order - but you could say you were using the scale, or using notes from the scale - such as when guitarists talk about which scales to use over which chords, for example.

If talking about the overall tonality of the piece, people often talk about 'keys' rather than scales. A piece of music that's considered to be in the key of C major might use notes outside the C major scale.

But if you look at A minor pentatonic scale you have A, C, D, E, G, yet with the same notes you get C major pentatonic with C, D, E, G, A. If they are not supposed to be played in order, how would you differentiate the two scales at all?

At least when talking about Western music, there is also the assumption that the "letter name" of the scale is also the "home note" or tonic of the passage of music using the scale. So you would talk about a set of notes using the C major scale if the home note or tonic was C; with the same set of notes, you could talk about the scale being A minor if the home note was A.

You can also talk about modes of a scale, which is another way to talk about notes from the diatonic scale being used with different home notes.

And if you play a scale such as A minor above, can you mix in a note outside of that scale, or does that create a new scale from that point? Like if you played A, C, D, Db, E, F, G, is that just A minor with extra notes or would that be considered an entirely new scale?

It could be either - it's up to the person analysing or composing the piece to decide what's the most helpful way to see it. Western music theory tends to try to fit everything into the diatonic scale, with a presumption towards major and minor - and because it's recognised that real-world music isn't this 'neat', people are used to the idea of using extra notes from outside of the basic scale.

Some comments elsewhere on this question have pointed out that the idea of a scale can also relate to more than just a set of notes, but also other practices, such as playing different notes when ascending than when descending (melodic minor), or bending certain notes through ranges of pitch (blues scale). Another recent question suggested that in Indian music, the term that we translate as 'Scale' has a meaning more like 'Key' in Western music. Unfortunately, the exact meaning of the word scale - like many words in Music theory terminology - depends somewhat on context.

As for how to differentiate between A minor and C major, you may be interested in the answers to When is a piece in A minor versus C major?

But I wanted to offer one clarification in terminology: the difference between a scale and a collection (sometimes called a scale collection). A scale would be the notes of A-minor penatonic in order: A C D E G A. That collection, meanwhile, would be those pitches, but not necessarily in that order, like C A G E D.

This helps clarify that the A-minor and C-major pentatonic scales are different, but their collections are the same.

A scale is a particular set of notes played ascending and/or descending. French word for scale is echelle, which also means ladder. And, as said previously, we scale a ladder. Best to use one rung at a time!

In an examination situation, particular scales are requested, and generally get played up and down, in note order.

By adding ( or taking away) any note/s, the scale is changed, so it becomes called something different. For example, C pentatonic minor, C E♭ F G B♭ becomes C minor blues when F#/G♭ is added.

Taking your example of pentatonics - A C D E G starting from A is A minor pent., while C D E G A is C major pent.This opens a can of worms with modes, which are covered in many places on this site.

So, basically, a pitch set of notes, which often constitute a key, when played in pitch order, is a scale, which can be one octave minimum, or as many octaves as the instrument will allow.

A scale is a pattern of ascending notes, and that pattern shows the distance between each note characteristic to that scale. It cannot be just any group of notes in any order, because even playing a scale down changes the distances between the notes.

For example, a major scales has the following pattern of whole steps and half steps: W W H W W W H. That is the ascending pattern. If that same pattern was used descending, we would get a completely different set of notes, which would match the phrygian mode.

So, in short, yes scales are ascending and the notes are played in order. This is particularly useful in practicing technique as the patterns of the scale, or pieces of it, show up in music all the time or can be used in improvising.

However, melodies and accompaniments will use the notes in various orders. When describing this to students I will say "This melody uses notes from the C Major scale", for example.

A piece does not have to use notes from only one scale, however. Other scales (and the chords built from them) can be brought in, but this gets into more advanced theory.

The Pentatonic collection of Am and C are the same (s. answer of Richard).

If they are not supposed to be played in order, how would you differentiate the two scales at all?

It won't make a difference (it doesn't matter) if you differentiate them or not. As long as we don't have a tune or an accompaniment (harmony) we can't determine whether they belong to Am or C.

BLACK IS BLACK (interesting title: meaning the black keys of pentatonic?)
(before the entry of the singer you can't definitely say what kind of music is ... maybe the last note of the second phrase is so la Do?

https://www.bing.com/videos/search?q=black+is+black&view=detail&mid=A1DC4F75F25304AAA86CA1DC4F75F25304AAA86C&FORM=VIRE

Yes, the intro is the pentatonic of Do, starting with the 2nd inversion of IIm (la do re, la do re, la do re do) turning to the tonic so la do: only the melody (la re fa) and the harmony will give the answer.