Is this mentally computable password system, cryptographically secure? And anything like this out there I can find?
Firstly, you shouldn't really invent your own password manager, except for fun/experimentation/learning purposes.  Especially if you are a beginner.
Regarding your comments, I don't really see how managing your own in your head can be less hassle than managing a standard one on a machine.  I use google/chrome's (BOO!) so can use across devices and could give away passwords to certain accounts, for whatever reason.
You may find this interesting reading.  It seems very similar to your idea.  It uses a pin as well as a cue for each site.  It also goes into detail about the reduction in security as you use the system to &quot;generate&quot; more passwords.
Your scheme gives away bits of the master key in each password, and has virtually no real entropy.  There is a 23.5% chance the letters in your passwords will be 'g'.
If we change the lyrics to something tasteful like the opening verse of Dylan's Subterranean homesick blues, we still have 0% chance of a 'z'.
There are 256 possible values in a random byte.
Imagine another scheme I invent which uses a cryptographically secure salted hash, where the salt is the website address/master key/time added to database, and the input to the hash is the master key. My master key is &quot;never gonna give you&quot;.

Each of my website passwords might be anything.  They are not all based on the letters in my master key.
If websiteX password is breached, it gives an attacker nothing about the password for websiteY, or the master key.
When I change my password for websiteX, I don't need to change my master key.
The basic security of my system relies on a cryptographically secure hash that wasn't designed by me.

Obviously (as mentioned) I wouldn't use my own scheme because there are already better existing schemes designed (and implemented) by people who know better.