So I played trombone for 7 years or so back in grade school. For those who don’t know how trombones work, you can play a B-flat with the slide all the way in (known as first position). Bend the note up a little, and it’ll jump up to an F, then a high B-flat (this is generally how most brass instruments work). Bend the note down from the original B-flat, and you go all the way down a full octave to a low B-flat, skipping the F.
So what causes the instrument to skip notes and “lock” to a different one? If you buzz through just the mouthpiece, it doesn’t do this.
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Instruments based on a length of tube resonate at some lowest note, but you can also get them to resonate at some whole multiple: 2, 3, 4, 5, … The first factor is 2 which is exactly an octave in music. Going from 2 up to 3 is a factor of 3/2, which is a perfect fifth or 7 semitones. The factor of 4 is two doublings, so two octaves above where you started and a major fourth or 5 semitones up. Now 5 is 5/4 above that which is a major third or 4 semitones. 6 is an octave above 3 which is also a step up of 3 semitones or a minor third.
Because the ratio between consecutive whole numbers gets progressively smaller, the number of notes skipped reduces with each new harmonic. Music pitches are all ratios; notes at simple whole ratios sound good to us.
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