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.

1 Answer 1


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
  • Super answer. So much easier with equal temperament. The sums can be worked out by mere mortals,,,
    – Tim
    Jun 30, 2014 at 15:41
  • 1
    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. Jun 30, 2014 at 15:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.