I made my first little melody in Goattracker 2.7something 2.72? as I wanted the same version as the tutorial I found.

Anyway, I'm very pleased with the result except for one thing, I have these gaps between notes. I'd like them to be joined as if you'd simply cut away the space between them.

I tried some of the slide commands but that turns a triple into long note with decreasing volume as if a single note was played and there's no difference between the two first notes that have the same value. Imagine CCC# that's supposed to be heard as three distinct notes but with no space between them.
If you sample the music there's a peak at every start of each note before wave centers. I want these peaks to stay there - as they are now, but remove the space between the notes - of possible.

It's like it's "taking a breath" between notes.

These two pictures shows similar to what I get and what I want.
First one has gaps between notes, second one doesn't have these staccato gaps between notes.
http://users.ece.utexas.edu/~valvano/Volume1/E-Book/C13_DACSound_files/c13-image014.png
http://users.ece.utexas.edu/~valvano/Volume1/E-Book/C13_DACSound_files/c13-image015.png

I want my notes tighter together, two of the same note in a row should still be heard as two separate notes though.

why don't you post a link to your *.sng files so people can help you? it varies widely on instrument style; and possibly the phase is resetting of the waveform?