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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 conventions.

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.