A fun way to derive the melodic minor scale

This thought just occurred to me as I was listening to the audiobook of Frank Wilczek‘s excellent Fundamentals: Ten Keys to Reality.

• The melodic minor is a 7 note scale consisting of only semitones or whole tones, but with no 2 consecutive semitones (put another way, it is “locally diatonic“)
• The whole tone scale is a 6 note scale consisting only of consecutive whole tones
• The melodic minor contains 5 continuous notes of the whole tone scale
• To create a 7 note scale from the 5 continuous notes we need to add 2 more notes
• These 2 notes can be either whole tones or semitones
• Neither of them can be a whole tone, because that would result in a full whole tone scale
• Therefore, both added notes must be semitones
• Therefore, the melodic minor consists of 5 notes from the whole tone scale, bounded at both ends by a semitone
• Corollary: since the whole tone scale is symmetrical, the melodic minor scale is also symmetrical