During this article we will refer to chords by anchor interval collections, we will also be able to determine the quality of any of the chords by the numbers involved in the ai collection.
One common chord substitution is Amin7 in place of a Cmaj7, this example will give us insight into when we can replace one chord for another.
Assuming our anchor note is 0*, then these two chords can be written as 9 0 4 7 and 0 4 7 11. Then straight away something we can see is that these two ai collections both share 3 ais, namely 0 4 7, in this aspect these two chords should sound similar and putting one in place of the other will work most of the time.
In that example we took two chords and by converting them into ai notation we were able to see that they are similar. This shows us some more power of ai notation that the standard notation doesn't explcitily show us, since a beginner could not look at Cmaj7 and Amin7 and tell us that they are similar without knowing the structure of major and minor chords, but anyone could tell us that 0 4 7 11 and 0 4 7 9 are similar since the symbols that they both use are similar.
Instead of taking two pre-existing chords and seeing if they are similar, we can work in the opposite way, if we consider Cmaj7, then the ai's that it uses would be 0 4 7 11, and so if we take out the 7 and replace it with other notes from 0* major we get new chords which have similarities to Cmaj7. So for example we could make 0 4 5 11, 0 4 2 11, 0 4 9 11, all of which would be suitable substitutions.
One thing that we didn't mention is that there are ai's which are more important to a chord than others, for example any ai which is 7 steps away from the root of that chord so the 7 in our 0 4 7 11 or the 4 in 9 0 4 7 isn't that important to include. The reason why it's not too important is that it creates one of the most simple chord qualities, and so leaving it out won't change it too much.
On the other hand the 4 in the 0 4 7 11 and the 0 in the 9 0 4 7 are more important to include as they are one of the defining ai's as when played against the root of each chord they have a more complex quality which define the chord.
In this way we can compare two chords by the relative interval collection used to construct the ai's. So for example, with Amin7 we know that the Xmin7 relative interval collection is 0 3 7 10 and that X is 9, so generating ai's from that yields 9 0 4 7. On the other hand the relative interval collection of Dmin7 is also 0 3 7 10, and that the ai's that generates on 2 are 2 5 9 0. So we can see that these two chords have the same quality.
From our previous analysis, it should be clear that the standard system has an advantage to be able to differentiate with qualities, as if we see Dmin7 and Fmaj7, we can immediately see that these two chords differ quality wise, but we're not sure about the ai's that it defines straight away (that's where the ai's representation shines).
As we've seen ai notation for chords allows us to easily differentiate anchor interval wise, but lacks the quality comparison which the standard notation provides readily. If we are in a situation where anchor intervals are solely being used then, we should always know that there is a way to get the quality out of it.
To do this we can take any ai collection, say 9 0 4 7, then assuming that 9 is the root, then by subtracting 9 from each, or rather adding 3 everywhere, we can get 0 3 7 10, which is the relative interval collecition which generates that ai collection.
An even faster way is that gets developed over time is to notice that if we just look at the two end points of the chord in ai notation then we see 9 X X 7, which means that the 7 relative to 9 has a gap of 10, since 7 is two less than 9. Similarly if we say 9 0 X X, we know that the interval between them is a 3 so by looking at chords using this light we can start to read ai notation and at the same time extract out the quality of the chord as well.