Argthtjtr

If we take out the sharps and flats, I'm wondering if all scales need to be in alphabetical order in terms of their letters: A,B,C,D,E,F,G where 'A' comes after 'G'.

also..

no repeating notes: A,A#,B,C,D,E, etc.

no skipping notes: A,C,D,E,etc.

no combining flats and sharps: A,B♭,C#,etc.

Therefore the scales are always in alphabetical order and each note is different. (?)

So the scale of C minor:

C, D, E♭, F, G, A♭, B♭

The scale of F# major:

F#, G#, A#, B, C#, D#, and E#

So if we don't look at the flats then the scales satisfy the above conditions. Is this true for all scales/modes in music? I'm guessing this is some kind of agreed upon convention to make things easier to memorize because it's just alphabetical order at that point. So the only thing we have to memorize is how many sharps or flats are in each scale.

Yes, in traditional Western music theory, but there are a lot of scales/modes outside of traditional Western theory that don't follow this convention.

For instance, the pentatonic, diminished, and whole-tone scales skip certain letters or have multiple notes on some letters.

This is not always the case but would be the case for the most commonly used scales, such as major, minor, and all the standard modes. However, we can quickly find common examples of scales that skip notes, such as a pentatonic scale, where there are only 5 notes, so it wouldn't be possible to use all 7 unique letter names. The whole tone scale only has six notes, so that will also be missing a note. There are also octatonic scales, which have 8 notes, and therefore require repeating a letter name. These octatonic scales also require the use of both sharps and flats a lot times. Then we have the case of the Harmonic and Melodic Minor scales, where you can end up with mixed sharps and flats. We could also consider the chromatic scale but that tends not to be considered in these types of conversations.

So generally speaking, if you are learning or teaching scales, it's good to start with the major and minor scale and introduce their modes. These will all follow the rules you suggested. Eventually, you learn/teach that this is not entirely the case and learn the exceptions. Music theory is very much like this, where you learn a general concept, sometimes thinking of it as a rule, then you learn how that concept or rule is not always accurate or applicable.

As said many times, a scale is simply a set of notes played in order. Obviously with majors and minors, the plan works. But what about others? pentatonics won't fit that criterion, and certainly chromatic scales just can't.

But if possible, then yes. If only to make writing them out make more sense and be simpler. But something like a blues scale will have to have one repeated letter name. And diminished will be blighted in the same manner.

The word "scale" literally means "ladder" or "climb" - it's from the Latin word scala. So the notes are always going to be in order, and since we use an alphabet for the names of pitches, a scale must have the notes in alphabetical order.

But you're mistaken on the other details:

no repeating notes: A,A#,B,C,D,E, etc.

Scales can have more than one of a letter. The blues scale, the chromatic scale, the diminished scale (in either WH or HW form) are just a few examples.

no skipping notes: A,C,D,E,etc.

Scales can skip letters. Pentatonic scales are common, and in either the major or minor form they'll skip two letters.

The scales that do conform to no repeated or skipped notes are diatonic scales, which literally means "through the tones". Some theorists put additional restrictions on what constitutes a diatonic scale, but only one of each letter is a feature under any definition of diatonic.

no combining flats and sharps: A,B♭,C#,etc.

Some scales do combine flats and sharps. The D harmonic minor is a perfect example: it has both Bb and C#.

Your definitions work for the major scales and their related modes, but they don't work for all the minor scales/modes, and they won't work for any of the other dozens of scales used in music.

What you say is entirely true of how major and minor scales (and the scales in the other standard modes) are notated. It is not true of all scales, though, as several of the other answers have detailed. As an additional wrinkle, the accidentals in harmonic and melodic minor scales are sometimes double sharps, for example D# minor. This is why D# minor is usually written as Eb minor (not in music, where you'll see both, but in scales), since in Eb minor the accidental is a natural (there are two of them in the melodic minor).

Mode is grouped with key. Both being sets of tones in no particular order. A scale is playing a set of tones in ascending or descending pitch order. Don't get hung up on tone naming and ordering by the tone names like alphabetizing the letters. I think the important thing is the scale is ordered by ascending or descending pitch.

Regarding scale/mode/key. I can play in a mode or key without playing scales. Mode/key is just the set. scale means the set ordered asc/desc.