From experience body weight exercises are a lot about CNS effiency, more so than lifting exercises because moving your body in space efficiencly requires a lot of fine coordination that we aren’t ware off… in that sense they are not unlike the olympic lifts that are best trained with a high amount of sub-maximal work. So you are indeed on the right track, HOWEVER what I don’t like is the “mandatory” rep increase per workout. At first it will be easy to follow because you are stating from a very submaximal weight. But it won’t be long until the planned progression cannot be sustained.
If you want to use your method I suggest the “minimal increase method” used by Doug Hepburn in which you add ONE rep per workout.
With the reps you mentioned it would look like this:
Workout 1: 3 x 7, 3 x 6, 3 x 5
Workout 2: 3 x 7, 4 x 6, 2 x 5
Workout 3: 3 x 7, 5 x 6, 1 x 5
Workout 4: 3 x 7, 6 x 6
Workout 5: 4 x 7, 5 x 6
Workout 6: 5 x 7, 4 x 6
Workout 7: 6 x 7, 3 x 6
Workout 8: 7 x 7, 2 x 6
Workout 9: 8 x 7, 1 x 6
Workout 10: 9 x 7
Workout 11: 3 x 7
Workout 12: Test max reps
And you would do pull-ups 3 days a week, so the training block shedule would look like this:
WEEK 1
Workout 1: 3 x 7, 3 x 6, 3 x 5
Workout 2: 3 x 7, 4 x 6, 2 x 5
Workout 3: 3 x 7, 5 x 6, 1 x 5
WEEK 2
Workout 4: 3 x 7, 6 x 6
Workout 5: 4 x 7, 5 x 6
Workout 6: 5 x 7, 4 x 6
WEEK 3
Workout 7: 6 x 7, 3 x 6
Workout 8: 7 x 7, 2 x 6
Workout 9: 8 x 7, 1 x 6
WEEK 4
Workout 10: 9 x 7
Workout 11: 3 x 7
Workout 12: Test max reps
NOW here’s the thing… I think that if you can do 12 legit pull-ups, the 7 reps target is too low. You should hit 80% of your max every workout if you want maximal neural gains. So MY recommendation would be:
3 x 9, 3 x 8, 3 x 7
And the progressing being:
WEEK 1
Workout 1: 3 x 9, 3 x 8, 3 x 7
Workout 2: 3 x 9, 4 x 8, 2 x 7
Workout 3: 3 x 9, 5 x 8, 1 x 7
WEEK 2
Workout 4: 3 x 9, 6 x 8
Workout 5: 4 x 9, 5 x 8
Workout 6: 5 x 9, 4 x 8
WEEK 3
Workout 7: 6 x 9, 3 x 8
Workout 8: 7 x 9, 2 x 8
Workout 9: 8 x 9, 1 x 8
WEEK 4
Workout 10: 9 x 9
Workout 11: 3 x 9
Workout 12: Test max reps
From the new max reps test you would calculate a new 80%… so if you get 14 reps your new target would be 11… so week 1 would be 3 x 11, 3 x 10, 3 x 9… you get the idea.
But I would test myself before starting… you think you can get 12 reps, but for the program to work your max reps need to be all solid, non grinding reps… so it’s not the maximum reps you can get, it’s the max reps you can dominate and do in perfect form.