If it feels like an Alberti Cipher it is one!

[Davidsch]

Did you know that there are different ways of using the Alberti formula?

1. use rings and index number
a. mention the index number in cipher
b. send the index numbers separately

2. use rings and key phrase
a. mention the key index  in cipher
b. send the key separately

3. use the plaintext n-grams and not the plaintext letters. One wheel now holds the ngrams.
...variations as above

Now consider this. Put 16 letters letters in the alphabet disc and 4 index letters are only used as key. Now, we really do not need the disc if we have no periodical shift.
We can simply make 4 tables. If there will be an automatic shift applied: just add a simple row to your table.