I could use a clearer definition and explanation of the following terms:
Or, could anyone recommend a book which clearly explains more of the theory behind tuning pianos with a digital device?
Stretch - The difference required of a musical tuning to make the same named notes sound "good" - as opposed to a simple doubling of the frequency of the octaves. You might think that A 440 would perfectly match up with the octave below at 220 or the octave above at 880..
BUT, there's this inharmonicity thing. In tunelab, while you use the measure function, it displays numbers for a bunch of partials. (many times the first partial is missing) Those numbers represent the cents away from a simple math multiplication that that single string vibrates. Even though we may hear "C", that string actually sounds:
C, C+ 1 octave, C + 1 octave + 5th, C + double octave... (look up partial series of a vibrating string)
SO, those numbers you got in tunelab displaying the difference of the specific partials... The progression of the numbers for that single string gets represented by the fudge factor called the inharmonicity constant - a single number that is a multiplier that allows one to predict the location of other partials of one string based on knowing the location of one partial.
And all of this is important to predict the stretch - the musical "good" sounding of a single or double or triple octave. The machines can't hear "good", so they try and line up matching partials from multiple notes to come up with a solution. (Using inharmonicity constant to manipulate the tuning curve)
Ah, the tuning curve... It is the plotting of the difference in cents from the simple math doubling of the octaves - showing each note of the piano - Above the middle, notes are plotted sharp, below flat to try and make a musical "good" blend of the octaves. Notice I'm not even talking about temperament yet!
In the tuning curve display, look in the middle. There is an octave that is nearly a straight line representing the temperament - everything else is calculated from these notes using the preferences for certain partial matches, or single or double octaves, or octave + 5ths....
Not sure about the Deviation curve - I could hazard a guess, but I'll have to go look in tunelab to see what's being referenced.
Hope that helps!