Modes of an anchored interval collection are a group of new ai collections which are derived from the original one by "rotating" the original.
To make this idea more clear we'll start with the major modes, to start with we'll recall the generic major scale ais, they would be 0 2 4 5 7 9 11. For the next little while we'll make an aside back into real notes instead of just dealing with ai's, the reason for this is that the development of modes make the most sense when it's done in this manner.
So let's say that we chose to start our scale on the note 7*. Then, we know that the way we derive the other notes from that scale is by using the ai collection as a relative interval collection to get the notes 7* 9* 11* 0* 2* 4* 6*. From these notes we could always just "rotate" the scale and start on the 9* instead of the 7* to get the notes in a different order so we would get 9* 11* 0* 2* 4* 6* 7*.
From this note collection we can pretend our anchor note is 9* now, and then we can figure out which ai collection we have here. So by subtracting the 9* from each of these notes we get 0 2 3 5 7 9 10. This represents the second mode of the major scale which goes by the name of Dorian. We can also see that due to the manner in which we created these notes that starting a dorian on the scale on the 9* that we would get the same notes as the original major on 7*, we can also think of the major scale on 7* as a mode, which we call the Ionian mode
This is the traditional method for determining different modes based on notes, but as with most of the standard methods, we can define our own anchored interval method as well. To do this we recall our original anchored interval collection which is 0 2 4 5 7 9 11, and then we can "rotate" that, in order to rotate this, we want to make the 2 a 0, so one way we can do that is to simply subtract 2 from everyting, this would yield 10 0 2 3 5 7 9, or ordered it looks like 0 2 3 5 7 9 10 (Dorian).
Let's get the next mode which is named the Phygrian, which can be obtained by rotating the Dorian once, or the Ionian twice. We'll go with rotating the Dorian once, which can be done by again subtracting 2 from our dorian. So our dorian 0 2 3 5 7 9 10 becomes 10 0 1 3 5 7 8 when ordered it is 0 1 3 5 7 8 10. We can repeat this process as many times as we like until we get back to where we started. All summed up it looks like this table.
| Mode Name | ai's involved | ai's ordered |
|---|---|---|
| Ionian | 0 2 4 5 7 9 11 | 0 2 4 5 7 9 11 |
| Dorian | 10 0 2 3 5 7 9 | 0 2 3 5 7 9 10 |
| Phrygian |
8 10 0 1 3 5 7 | 0 1 3 5 7 8 10 |
| Lydian | 7 9 11 0 2 4 6 | 0 2 4 6 7 9 11 |
| Mixolydian | 5 7 9 10 0 2 4 |
0 2 4 5 7 9 10 |
| Aeolian | 3 5 7 8 10 0 2 | 0 2 3 5 7 8 10 |
| Locrian | 1 3 5 6 8 10 0 | 0 1 3 5 6 8 10 |
At the end of the day we can see that modes simply define relative interval collection which will have their own sound associated with them.