It occurred to me after I posted my last lesson summary that we discussed one more thing: using ringing notes to check intonation. Rather than editing that post to add that point, I thought listing all the ways I know of to check notes (without electronic assistance) might be a useful exercise.
* Check against an open string. This applies to playing C, G, D, or A anywhere on the cello by playing the note, then the corresponding open string, and comparing the two notes.
* Use vibrations of open string. In this case, playing the C, G, D, or A should elicit sympathetic vibration of the corresponding open string when in tune.
* Double stop with adjacent open string. Works best with 3ds, 4ths, 5ths, and 6ths low on the fingerboard.
* Play passage against an adjacent open string drone. Works best when the drone is the tonic, 4th, or 5th in the key.
* Ringing notes. Use a fast, light bowstroke that releases the string, and listen for the resonant ringing. This works best for the "string" notes, C, G, D, and A, but can actually be heard to some extent with all the notes when they are in tune.
And finally, this is how I would "find" each of the notes in 1st position, beginning with C and working upward.
C - open
C# - play open D, then 2nd finger on D IV, then C# is leading tone with 1
Db - play open C, then Db points to C
D - play open D, then compare. Ringing note.
D# - get E in tune (see E) then D# as leading tone
Eb - tune D as above, Eb points to D
E - double stop with either open G with A, depending on the key. Tune A as below.
F - tune F on II with open A, then compare. Hear 4th with open C.
F# - leading tone to open G
G - open
G# - leading tone to A
Ab - points to open G
A - compare with open A. Ringing tone.
A# - tune B. Leading tone to B
Bb - points to A. Or tune Bb on I, then compare. Or double stop with open D.
B - double stop with open D. Or tune B on I and compare.
C - compare to open C. Ringing tone.
C# - leading tone to open D
D - open
D# - tune E, then leading tone to E
Eb - points to D. Compare double stop with open G to double stopped G-D
E - double stop with G or A, depending on the key. Compare to E harmonic on I.
F - double stop with A
F# - Tune G. Leading tone to G. Or double stop with open A.
G - compare to open G. Ringing tone.
G# - leading tone to open A.
A - open
A# - Tune B. Leading tone to B.
Bb - points to A. Compare double stop wth open D to double stopped D-A.
B - double stop with open D.
C - compare to oopen C. Ringing tone.
C# - Tune D. Leading tone to D.
Db - points to C
D - compare to open D. Ringing tone.
D# - Tune E. Leading tone to E.
Eb - Tune D. Points to D.
E - compare to harmonic under your finger.
For notes in 4th position and higher, I make use of open strings and available natural harmonics for comparison. In general, the harmonics available on each string correspond to the major triad with the string note as root. So:
C - C, E, G
G - G, B, D
D - D, F, A
A - A, C#, E
That's what I know about intonation in a nutshell. I welcome all comments and hints.