Double stops are a way of playing harmony lines with yourself. As a result, the use of double stops makes for a richer, fuller solo, than by playing single notes. Just listen to any good fiddle player like Bobby Hicks or Johnny Gimble, and you will hear for yourself how much artistry is conveyed by double stop useage.
Where do double stops come from? They are a pair of notes from the scale you are using to play the tune you are performing. My discussion of double stops will be mostly confined to Western Major scales typically used in playing Western music. You will have to evolve your own specific set of double stops for other scales and modes.
The first note of the double stop is the melody note you wish to play (say a "C" note in the key of "C"). Your double stop harmony note would be any other note of the "C" Scale that is higher in pitch than the melody note. The most common harmony note in western music is the third interval. So, if you want to harmonize a "C" note, count up, using the melody note as "one", three notes in the scale to get the note "E" ("C"=1, "D"=2, "E"=3). On the mandolin you could play these two notes in the following places (I'm sure there are others, this is just to give you an idea of what I'm talking about):
E |-------0------- A |-------3----7-- D |--2--------10-- G |--5------------ E E E C C C
Notice that the two notes stay in the same relationship with each other. The lower-pitched note ("C") is always three frets higher on the fingerboard than the higher-pitched note ("E"). This relationship between the notes is echoed in the middle two notes of a "Chop" C chord as played on the 7th fret of the mandolin:
E |-----8-- A |-----7-- = "E" Note D |----10-- = "C" Note G |----12--
"C" Chop chord played at the seventh fret
Notice how the two notes tie together the playing of melody notes and chords. If want to harmonize the second note of the "C" Scale, "D", you must count up three notes of the "C" Scale, starting with "D" = 1 to the "F" note. These two notes played together would look like:
E |-------1------- A |-------5----8-- D |--3--------12-- G |--7------------ F F F D D D
Notice there are now FOUR frets between the two notes (this interval is technically a minor third interval-not a Major third one, so the higher pitched note has to be lowered in pitch one fret, thus widening out the space between the two notes. These notes relate to the middle two notes of a "D" minor chop chord (rarely) played at the ninth fret:
E |----10-- A |-----9-- = "F" Note D |----12-- = "D" Note G |----14--
"D" minor Chop chord played at the ninth fret
What's happening is we are slowly building a harmonized scale in the key of "C". Remember that a major scale is harmonized by having the first note of the scale be a Major chord ("C" Major in the key of "C"), the second chord is a minor chord ("D" minor), the third chord is a minor chord as well ("E" minor). The fourth chord is a Major chord ("F" Major), the fifth chord is a dominant seventh chord (G7--for all intents and purposes similar to a Major chord, and played as such usually in Bluegrass). The sixth chord is a minor ("A" minor). The seventh degree of the scale is called by some a diminished chord; others name it as a minor flat-fifth chord ("B" minor b5). The eighth degree of the scale brings us back to another Major chord based on the key note of the scale ("C" Major).
So, a "C" Major scale harmonized in thirds would look like:
E |--------------------------------------0---- A |-----------------0----2----3----5-----3---- D |--2----3----5----3----5----7----9---------- G |--5----7----9------------------------------ E F G A B C D E C D E F G A B C
Notice carefully that the first four double-stops have the same fingering relationships as the second four. If we move the entire series of double stops up one fret so that we are harmonizing the "C Sharp" scale, we have a nice moveable harmonized scale that can be played anywhere on the mandolin fingerboard-your choice:
E |--------------------------------------1---- A |-----------------1----3----4----6-----4---- D |--3----4----6----4----6----8---10---------- G |--6----8----10----------------------------- E# F# G# A# B# C# D# E# C# D# E# F# G# A# B# C#
Just remember to start the scale with the lowest-pitched note of the double stop pair on the first note of the scale and keep the fingering relationships the same for the rest of the scale. That way you don't have to bother counting from one note to the next or wonder what the third chord of an Eb scale is. It is all patterns. Let's play a harmonized scale in thirds in the dreaded key of Bb:
E |------------------------------------------- A |----------------------0----1----3-----5---- D |--0----1----3----5----3----5----7-----8---- G |--3----5----7----8------------------------- D Eb F G A Bb C D Bb C D Eb F G A Bb
Notice the fingering requirements had to change a little to accommodate the fact we are starting off on a fingered note and an open note, but the relationship between the first four double stops of the scale and the last four stay the same. If we want to play a Bb scale that adheres to the fingering shown in the "C#" scale, we would have to start the scale off with the Bb played on the thirteenth fret of the "G" string, or the Bb on the eighth fret of the "D" string:
E |-----------------3----5----6----8----6----10- A |--5----6----8----6----8---10---12------or-13- D |--8---10---12-------------------------------- G |--------------------------------------------- D Eb F G A Bb C NA D Bb C D Eb F G A Bb Bb
So you can see the thirds double stops only require learning a few fingering relationships, and then extrapolating them over the length of the neck. Practice playing these harmonized scales melodically the same way you manipulate the single-note versions of the scales. Tremolo them. Play them against drone notes. Play the notes individually, one after the other. You will find them very satisfying. If you want to play a more Bluegrass or country sounding scale, omit the fourth and seventh degrees of the scale (making it a pentatonic scale). Try variations on this. I think you will recognize these double stops from the playing of Bill Monroe, John Reischmann, Ronnie McCourey, Jethro Burns, and most of the other greats playing out there.