Which has a greater influence on sound quality, the amp or the preamp?

Each are important but a high quality, low noise high gain preamp can bring your system to a higher level of excitement and dynamics than a basic preamp connected to a powerhouse amp.  

I have been pondering this for a while but don’t have a succinct answer yet.. but what I have goes like this.

Yes, both are important but the preamp is the heart of the system and as such is more important. However, I always remember the huge effects on bass presentation and gross change in the rest of the frequencies given by the amp.

I think it is that the preamp sets all the nuances across the spectrum… it affects much of the personality. Then the amp makes big accentuations and attenuations on top of that. While the amp makes a few big changes the preamp makes more nuanced changes across the board which add up to being more important. When you are talking similar quality. I tend to error in spending more on a preamp than amp. 

Or maybe this, if you don’t get the preamp right then the amp can’t get it right.

Both can, and do affect sound quality. Every component and interconnection in the chain affects the quality. Depending on your source you don't always need a preamp in the system. 

You may not need active preamp in the system but you will always want it, whether you know it or not.

@inna Not me, I'm very happy with my Tortuga Audio LDR Passive Preamp.

It lets me hear what`s in the recording without adding any unneeded flavoring.

Crystal clear sounding, never needs warming up either. Very easy to hear changes like rolling tubes or cables etc etc.

Also, with the numeric volume display showing 40 ( it goes up to 70), my system is playing about 85 db (too loud imo)

More people should try one and see for themselves 👍