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There are many different scales. As far as I understand, a scale is defined by intervals between notes. For example, for major scale it should be: tone-tone-semitone-tone-tone-tone-semitone
I know that there are also scales with 6 or 5 tones.
Is it the case that any set of tones can be used as scale? Why some scales used more frequently than others? Is it just a cultural thing or the are some (semi-) objective rules why some notes can be used as scale and other not?
As you say, a scale is purely a set of notes in ascending/descending order.
Any set of notes. Every note in semitones (chromatic scale). Notes in specific spacing (major, minor scale). Notes a tone apart (whole tone scale). The list goes on - and on.
I doubt if you could come up with a set of notes that hasn't been used as a 'scale' already, but it's an interesting exercise.
The TTSTTTS pattern you mention is ubiquitous in the Western world, as so many tunes are going to sound good using it, both in melody and harmony. Another (over)used one is the pentatonic - both major and minor. Both useful, as the two notes from the major or minor scales which could be contentious are removed.
There are many other scales which are used in other styles of music (forget not the modes!), so, as the song says, Anything Goes!
Is it the case that any set of tones can be used as scale?
Yes, in theory. Literally any set of notes can be a scale. Sky is the limit.
Why some scales used more frequently than others? Is it just a cultural thing or the are some (semi-) objective rules why some notes can be used as scale and other not?
Yes. Both. It is certainly a cultural/historical thing. In the west, major and minor scales (or more generally the diatonic collection) is dominant, while in Indonesia, for example, you have slendro and pelog which are both very different from western scales -- so different that you can't represent them accurately in western music notation -- and quite different from each other even. So which scales you're familiar with is certainly a function of the culture in which you're raised.
Cultural/historical factors aside, within the western music tradition there is fairly good reason why some types of scales are used more than others. It really boils down to the intervallic content of the scale. In pitch class set theory this is called the set's vector, which is a succinct way of writing all the possible intervals within a set of notes -- all the intervals, from each note to every other note, categorized by their size. For example, the diatonic collection {0,1,3,5,6,8,10} has a vector of 254361. That means that within that set of notes you will find:
(Note that intervals which invert to each other are grouped together as equivalent.)
That's quite a rich variety of intervals which allows for a variety of melodic patterns as well as triadic harmony, arguably the most salient feature of western music. Not all sets would do that. Compare that to the whole tone scale {0,2,4,6,8,10} whose vector is 060603, which will sound very different because of the different intervallic content. Because it lacks semitones, and is completely symmetrical (ie all notes are functionally equivalent) it sounds static and directionless, and the only triad possible is the augmented triad. This is why it was not traditionally used before the 20C, when functional harmony ruled supreme. But it was its unusual sound which attracted composers of the 20C, most notably Claude Debussy.
In the 20C, many composers developed a fascination with symmetric scales, eg the whole tone and octatonic (aka diminished) scales. And Messiaen assembled a collection of what he called 'modes of limited transposition'.
IMO, I think the hallmark of a "useful" scale is a roughly flat or bell-shaped distribution of intervals in its vector, because this is indicative of a collection which is spread out over the whole octave, instead of all the notes being clumped together. But really, like I said at the top, anything goes, and a creative composer can make just about any scale sound interesting.
I would add to Tim's answer the following to address your comment on it being cultural.
Any string of intervals might work but typically notes repeat after an octave and most cultures (but NOT ALL) tend to keep the scale structure within the octave. There are some very noteworthy exceptions. For example a scale (or melodic pattern) commonly used in flamenco music changes from minor to major as you pass through one octave to another. Though relative to western music this may just be seen as using two distinct scales fused together.
The second thing to consider is that you have more options for scales in Just tuning as compared to equal tempered tuning. The standard being the 12TET, or 12 tone equal tempered used to define the piano key tones.
As an example of how intricate you can get with just 12 tones you might want to investigate the Carnatic scales in Indian music. They are based on a 12TET or chromatic scale and the basic standard scales are defined within one octave (the starting and ending note and the same letter name one octave apart). For one set of these scales the perfect 5th is held fixed, but the other notes are free to choose as long as one does not choose the same note twice. With some other rules of engagement this system gives rise to 72 distinct scales. Among them are the modes of Western music (except for the Locrian). The "altered scales" are derived from the standard ones by shifting combinations of notes by a half step (Locian emerges from here).
I might have misquoted the combinatorics but the point is that Western music is a pretty small subset of all the cultures on earth. From the point of view of western music the Carnatic scales might be deviations from the modes with accidentals included. This is not how Indian musicians would see it. Each Carnatic scale is truly unique and used in isolation from the others in constructing melodies.
There are a lot of good books that offer cut and dry formulas for understanding Carnatic melody and rhythms.
An other example are scales from Turkey and other eastern European countries (or West Asian depending on your point of view). One of the common scales used in ancient Turkish music has a quarter tone interval in it. I do not recall the name of the scale but you can google Turkish music microtonal and some other key words and find it.
The point of the last example is that even within the context of a piano we barely cover the spectrum of tones available for the human ear to distinguish. Indian Ragas (not the same as Carnatic) use intervals smaller than a 1/2 step too.
More recently musicians have been playing with things like 24TET chromaticism or N-TET where N is any integer, say 7, 9, 11, 22, etc. To build scales using 7TET would not be in any way equivalent to taking 7 intervals from the 12TET scale as those frequencies would not even be available to you. There are some piano pieces written for these microtonal TET scales. In my opinion these really push the limit in terms of being "musical".
Many guitarists make use of scales form other cultures especially in Jazz. However, since some of the original scales use micro tones (quarter steps etc) the western equivalent is an approximation. A decent list can be found in a book call Jazzology. To tell you the truth I do not really like the book in terms of how it presents theory but there is some good info in it. A guitarist can bend strings and with some practice get an authentic version of these scales.
Caution: Heretic answer.
Today's music "theory" is not a scientific theory. It is a large and complicated set of definitions and conventions that doesn't provide a toolkit for understanding how music works. Harmonies are just not built form semitones, but from ratios.
All music builds on only two scales: The overtone scale (O-scale) and its inverse, the undertone scale (U-scale). Any (musically meaningful) scale in the classical sense combines tones from different overtone- and undertone scales with different fundamental notes. Typically you will find those notes only approximated by the equal temperament tuning.
There are lots and lots of different classical scales, and knowing all of them won't tell you how to build melodies and chords from their notes or how to modulate between different keys and types of scales. However, if you save your head from all the conventions and just aim to truely understand the O- and U-scale and how they can relate to each other, you will understand all of those things. You don't need to know any classical scale and will still have much more creative freedom, as you don't have to stick to a frame which is not "real".
So to answer your main point: A scale in the classical sense is just a set of notes. How notes act will depend on the musical context, some will only have a harmonic function, others will only have a melodic function, some notes will be compatible with one fundamental note, others won't. So you could build any scale you want, but it wouldn't get you any further in using them in a musical way.
