[Illustration:  Musical Notation]

Then there are four kinds of artificial harmonics, only three of which are used:  harmonics on the major third (1); harmonics on the perfect fourth (2); harmonics on the perfect fifth (3); and harmonics—­never used—­on the octave: 

[Illustration:  Musical Notation]

Where does the harmonic sound in each case?  Two octaves and a third higher (1); two octaves higher (2); one octave and a fifth higher (3) respectively, than the pressed-down note.  If the harmonic on the octave (4) were played, it would sound just an octave higher than the pressed-down note.

“Now say we wished to combine different double harmonics.  The whole principle is made clear if we take, let us say, the first double-stop in the scale of C major in thirds as an example: 

[Illustration:  Musical Notation]

“Beginning with the lower of these two notes, the C, we find that it cannot not be taken as a natural harmonic

[Illustration:  Musical Notation]

because natural harmonics on the open strings run as follows:  G, B, D on the G string; D, F{~MUSIC SHARP SIGN~}, A on the D string; A, C{~MUSIC SHARP SIGN~}, E on the A string; and E, G{~MUSIC SHARP SIGN~}, B on the E string.  There are three ways of taking the C before mentioned as an artificial harmonic.  The E may be taken in the following manner: 

Nat. harmonic                     Artificial harmonic
[Illustration:  Musical Notation]    [Illustration:  Musical Notation]

Now we have to combine the C and E as well as we are able.  Rejecting the following combinations as impossible—­any violinist will see why—­

[Illustration:  Musical Notation]

we have a choice of the two possible combinations remaining, with the fingering indicated: 

[Illustration:  Musical Notation]

“With regard to the actual execution of these harmonics, I advise all students to try and play them with every bit as much expressive feeling as ordinary notes.  My experience has been that pupils do not pay nearly enough attention to the intonation of harmonics.  In other words, they try to produce the harmonics immediately, instead of first making sure that both fingers are on the right spot before they loosen one finger on the string.  For instance in the following:  [Illustration:  Musical Notation] first play [Illustration:  Musical Notation] and then [Illustration:  Musical Notation] then loosen the fourth finger, and play [Illustration:  Musical Notation]

“The same principle holds good when playing double harmonics.  Nine tenths of the ‘squeaking’ heard when harmonics are played is due to the fact that the finger-placing is not properly prepared, and that the fingers are not on the right spot.

“Never, when playing a harmonic with an up-bow [Symbol:  up-bow], at the point, smash down the bow on the string; but have it already on the string before playing the harmonic.  The process is reversed when playing a down-bow [Symbol:  down-bow] harmonic.  When beginning a harmonic at the frog, have the harmonic ready, then let the bow drop gently on the string.