Mystery of the Minor Chord

Tony Deff · Joined: May 25, 2008 Posts: 41 Location: Suffolk, UK

Seeing the references to Musical Mathematics and Music - a mathematical offering on this forum, reminded me of a time I played detective with the mathematics of chords.

I was too ill to move around and so bored out of my skull that I got to doodling with chord ratios.
I had long known, as is common knowledge, that the major chord is (or, at least, should be) the ratio 4:5:6.
I had also known, as is not uncommon knowledge, that the minor chord is the ratio 10:12:15.

I think you would agree with me that the minor chord is the next most concordant cluster of notes to a major chord, but why should that be?
That would mean that 10:12:15 is the next simplest ratio to 4:5:6, wouldn't it?

It is easy to deduce that 5:6:7 is Nature's diminished chord (from the top of a Just seventh chord — and one not available from tempered scales).
As to what 6:7 and 7:8 sounded like, I built switchable dividers to listen to these ratios.
I could convince myself that 7:8 was acting as a whole tone, and that 6:7 was a form of minor third.

So, where does that leave 10:12:15?

A musical apple dropped on my head when I realised that a major chord upward from a note is "mirrored" downward from the same note. I calculated the reciprocal of 10:12:15. It comes to 180:150:120 = 6:5:4

That is, a minor chord is the reciprocal of a major chord! (Not many people know that.)

Minor mystery, solved.png

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On close inspection of the ratios, the mystery is easily solved.

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