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I've read somewhere that tetratonic scales have been extensively played in jazz from Lester Young to Mulgrew Miller. I understand they are derived from the melodic minor scale. Could anyone explain to me how they are built and which chords they can be played over?

UPDATE: I think I have answered my own question:

The tetratonic scales should contain the tonic, the minor third, the fifth and the sixth of any melodic minor scale.

For example, the tetratonic scale of C would be C-Eb-G-A.

Now one needs to figure out the only major scale which has these notes; in this instance, it is Bb MAJ.

When you play the C melodic minor tetratonic scale over the Bbmajor scale, you highlight the 9th, the 4th, the 6th and the 7th of that scale.

It's interesting because you leave the tonic and the fifth aside.

I took a step further with a very simple progression (eddie harris's "listen here") Bb7 and Eb7. Now I decided that the key is Ab; in Roman numerals that would make the song a II7 - V7 The tetratonic scale for Ab is Bb-Db-F-G and it works nicely on Bb7 because Db is the b5 of the G blues scale and also over Eb7 because Db is the b7 of that chord. So the tetratonic scale can work on the whole cadence and is very useful to add approach notes. You can also spot the third of the chord easily if you want it.

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You are almost right :)

In fact any substructure (smaller scale contained) of any scale can be used to improvise over the current chord's chord-scale

The fewer the notes in your substructure the better defined the sound is. Triads are great to use as substructures, and also Tetratonic scales, Pentatonic and even HexaTonic scales. In praticular bi-triadic hexatonics are great for improvisation.

Look, here's an example of using substructures to improvise over a 7b5 chord when considering the Enigmatic Scale as it's chord's scale

Using Substructures For Improvisation

And here is an explanation on how to use bi-triadic hexatonics for composition and improvisation Bi-Triadic Hexatonics

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tetratonic scales aren't necessarily derived from the melodic minor scale. They can be arbitrarily constructed from the chromatic scale since a tetratonic scale simply means to have four notes in the scale. Though of course some choices are going to inherently sound better than others and will be more versatile. e.g. if you constructed a tetratonic scale made of C, Db, D, Eb (theoretically meets the requirements of being a tetratonic scale) it probably wouldn't sound as good as C, Eb, G and A as you mentioned above and would be far less versatile because not all chords would be able to accommodate for the chromaticism.

In terms of which chords you can play them over, that in a way is arbitrary as well and depends on the scale that you have made. So the two aspects are interdependent of each other. You may want to increase the tension of the music by playing a scale that doesn't have many of the chord tones of the chord in it. Or by making a scale whose notes are a semitone away from some of the notes in the chord that you are playing it over. onversely you may want the music to be less tense (dissonant) so then you may opt to create a scale that uses some of the notes of the chords you are playing or use notes that have a greater distance from the chord tones.

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