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