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The wikipedia article about diminished seventh chords mentions the following:

Fétis tuned the chord 10:12:14:17.

What does this mean? Until very recently, I had never ever seen this before, and I have no idea how to interpret this. Knowing that a diminished seventh chord consists out of a base note and three incremental minor thirds, I am at a loss as to how this can be described as 10:12:14:17.

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These are frequency ratios. The first digit is the common denominator (10). In a deminished seventh the following ratios are present:

  • 12/10 = 6/5 = minor third
  • 14/10 = 7/5 ~ sqrt(2) ~ diminished fifth
  • 17/10 : this is trickier

In a diminished seventh chord the fourth note is a diminished seventh, which is enharmonic to the major sixth (whose frequency is ~ 5/3 times the frequency of the root). To see the difference between these two frequency ratios, let's bring them to a common denominator:

  • 17/10 = 51/30, and
  • 5/3 = 50/30

As you can see, the difference is 1/30 of the root frequency between them, which is around 40 cents. 100 cents make up a semitone, so this ~40 cent difference is even less than a quarter tone, placing it a bit higher than its enharmonic, but still having them denote practically the same semitone.

Other examples for Fétis' notation:

  • dominant ninth: 4:5:6:7:9 -> 5/4, 6/4=3/2, 7/4, 9/4=9/8 -> C, E, G, Bb, D
  • dominant minor ninth: 8:10:12:14:17 -> 10/8=5/4, 12/8=3/2, 14/8=7/4, 17/8=17/16~16/15 -> C, E, G, Bb, Db
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  • Super answer. So much easier with equal temperament. The sums can be worked out by mere mortals,,,
    – Tim
    Jun 30, 2014 at 15:41
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    Except it may not be always trivial to identify certain ratios. Eg. 16/15 is a more widely used ratio for minor 2nd than the 17/16 in Fétis' dominant minor ninth. 64/45 and 45/32 are more commonly used ratios for the diminished 5th and the augmented 4th, respectively. Also, 5/3 for minor 6th / diminished 7th is also more commonly used than 17/10.
    – András Hummer
    Jun 30, 2014 at 15:47

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