My cello students often come to college as music majors with what I jokingly refer to as 'freshmen intonation'. Freshmen intonation is not necessarily horrible intonation but it is intonation that needs much more refinement. The question at hand, therefore, is what are some basic ways we can develop better intonation on the cello.
First, let's define faulty intonation. Faulty intonation is the failure to tune intervals (the distance between pitches), both melodically and harmonically (intervals played harmonically are notes that sound simultaneously and melodic intervals are played successively). Whether it is the inability to match pitch or play a melody in tune or tune a note in a chord, all have to do with the failure to tune intervals.
Second, let's also understand that one's first line of defense in intonation is not actually the ear. Rather, it's muscle memory; it's feeling the finger spacing within the left-hand and the angle of the elbow. The ear is the second line of defense, informing the player if the muscle memory was correct. I've actually written a set of etudes along with Nick Nirva to help address the difficulty of changing finger spacing between the positions, a muscle memory issue. Feel free to check it out in my online store.
Having identified the problem, we need to come up with ways to improve intonation. This is a complex question because many factors contribute to centered intonation but there are some basic things that we can do to help us improve.
In my teaching I like to assign double stops (harmonic intervals or playing two notes simultaneously) to improve intonation. I find that the more one practices tuning double stops the better the ear and the muscle memory get, resulting in better intonation overall.
One can practice double stops through studies and playing with drone notes. Practicing thirds, sixths, and octaves have long been a staple of string pedagogy. Violoncello Technique by Mark Yampolsky edited by Gordon Epperson is an advanced book for this end. The Ivan Galamian Scale System for Violoncello edited by Hans Jorgen Jensen also has thirds, sixths, and octaves. If you would like a beginning book of double stops try Double Stops for Cello by Rick Mooney.
But one doesn't have to have a method book to practice double stops; they can and should be practiced right in the repertoire. We can tune notes to open strings and tune melodic notes as double stops. The repertoire is replete with double stop opportunities and we should take advantage of this.
We can also practice scales to drone notes. Teachers often hold drone notes for their cello students during scales and other passages. For example, when playing an E major scale it is helpful to tune each note of the scale to an E drone. We need to keep the tonic (1st note) of the scale in our ear as we are playing and not lose track of it. Playing a drone to the tonic of the scale is a great way to learn this.
Of course, this isn't the only facet of training that is important for good intonation; developing an organized method for shifting is also imperative. But a great place to start is practicing double stops within the same position.
As you practice double stops, your ear will become more and more sensitive to intonation and your left hand will develop more muscle memory regarding finger spacing.
I put together some cello intonation exercises for you to practice tuning notes to drones (held notes). Drone notes can be obtained from a drone CD or from a metronome or tuner. Here is a Cello Drone CD for download. If you use a tuner or metronome, make sure that the note sounds clear and not "buzzy".When not playing with the piano, keep in mind that the placement for any given note is dependent on the context. For example, if I play an F# in a D major chord (D-F#-A), it will be slightly different from an F# in an F# diminished chord (F#-A-C) leading to a G major chord. Specifically, the F# in the D major chord will be placed a bit lower than the F# in a diminished chord. Or if I am thinking melodically, the F# in a D-E-F#-G passage tends to sound best when placed a bit higher.
Certain intervals permit more variance than others. 2nds, 3rds, 6ths, and 7ths are more negotiable than the perfect intervals such as PUs, P4s, P5s, and P8s.
So don't think of intonation in terms of 12 chromatic notes. What we really have are 12 regions. If this all sounds a bit complex, ultimately the ear needs to be the final judge. And there is a certain amount of subjectivity and acceptable variance to intonation. If you are a student, you may need the assistance from your teacher to tune each note to the drones.
Another facet of intonation is listening to how your cello reacts to your intonation. Your cello will communicate with you! When playing notes that are the same as open strings, listen for the sympathetic vibrations or ringing of the open string and the harmonics. Your cello will give you feedback if you just listen!
One last important componant of developing a centered intonation is to sing. In fact, as an exercise, i recommend practicing singing before your next note before you play it. You see, you need to hear in your mind (audiation) what your next note is before trying to play it. The mind is an amazing thing. I find I have a much better chance of hitting the note if I imagine it in my mind. And the best way to develop this skill is to sing. So practice singing the pieces you are working on!
Once you are getting the hang of diatonic intonation (staying within the traditional scales), you can branch out into quarters tones (notes between our half steps). For this I will pass you along to Russell Rolen who more experienced than I on playing quarter tones on the cello.
In my intonation exercises I have linked below, for each drone note I begin with the perfect intervals because they are often the easiest to tune. From there I move to the 3rds and 6ths followed by dissonant intervals concluding with the tritone.