What is good intonation? To me it is when pitch clearly shows its harmonic function and role within a piece of music. It is when pitch is a completely intrinsic manifestation of key and tonality, tension and relaxation. It has a lot less to do with whether A equals 439 or 445, but where A is in relation to the key that you’re playing in, and what chord that specific A belongs to in any given moment. In other words, good intonation needs context.  

To understand a note’s context, one needs to examine chords. And to examine chords, one can’t overlook intervals. A dear pianist friend likes to say “feel the interval”, and I find that to be so true. A perfect fourth can feel harmonious but slightly vacuous, compared to the dissonant clash of a major seventh. To play those in tune requires more than matching the pitches to the exact sound frequencies. The difference in distance and tension in the two different intervals needs to be felt in order for them to sound “in tune”. On the other hand, sometimes an identical interval in two different contexts need to sound and be felt completely differently too. Take the major third C-E. Whether they are the bottom two notes of the C major triad or the top two notes of the A minor triad, they serve completely different purposes and should sound as such. 

Sometimes we take for granted that the ability to listen is a given, when this is a skill that needs to be deepened continuously. Have you ever had the experience, especially during a performance, where you notice that things are getting out of tune but you have no idea how to even begin to adjust because you feel so completely lost and disoriented in a rush of notes? Have you ever played a passage of double-stops or chords, not being able to tell exactly where you’re going out of tune but you keep playing it over and over anyway, even if it hasn’t seemed to help move you beyond a point of being stuck? I have been in both of these situations, and I’ve come to find that it’s because I was focusing on the wrong thing - the hand, when I should have been directing my attention to the ear. But how to work on listening? I’ve found the following exercise to be practical and helpful.

Exercise to train the ear for good intonation:

Pick a key, let’s say D major. First play the tonic in all different octaves. See if you can let go of what sounding in tune normally means to you, and include in your listening how all the different D’s ring, and try to match them in how they do. Expand your notion of listening. Listen to the quality of your tone, how much bow speed/pressure you’re using. Listen to how tight or how loose your sound is. Feel how much pressure you’re applying with your fingers on the fingerboard. (Or all the equivalents for your instrument.) Notice how all these factors could affect your pitch.

Now add the fifths. So in this case you’d play D, A, D, A, cycling through all the octaves up and down. Feel the resonance of the open fifth. How the A lies in the middle of the two Ds. Again, feel the DISTANCE between the notes, as if you’d need to traverse actual physical distances to reach the notes.

Add the thirds. (F#) Teach your ear to become sensitive to how the F# is not just a note that comes after D, but remember where the A is supposed to be from playing only fifths a minute ago, and how those resonant perfect fifths felt, and now place the F# between the D that you’ve already sounded and the A that you haven’t. In other words, you are listening for the major third of the D-F#, the minor third of the F#-A, the perfect fifth of the D-A, and on top of that, how the whole chord feels as a unit, as a whole. The more you can listen across different planes of interval relationships and retain memories of relating any given note to another - even if a note had already been played 5 notes ago or has not even been sounded - the more in tune you’ll be. (E.g., simultaneously listening and paying attention to how all the same pitches match across octaves and how adjacent intervals should sound as such.)

So now you have the D major arpeggio. I see this as the basic skeletal structure for the ear to anchor upon. You can then fill in the rest of the notes of the D major scale, using the D major chord as an anchor to place the other notes. You can even fill in the chromatic scale after that if you feel like it. Feel the tension/relaxation in all the intervals. I find that the more one does this, the concept of good intonation becomes wonderfully elastic and naturally undogmatic. An F# could sound deliciously high or mischievously low. (Obviously if we’re playing with other people, that would have to be factored in as well.) When the ear can take the lead in listening for and adjusting pitch this way, intonation serves what it’s meant to - as a means to an end in music-making. It also helps one find a much wider palette of colors and expression.

You can apply this exercise to any passage. Find the chord tones of the passage, practice listening to and playing them as described earlier to form a solid anchoring structure for the ear, then find the rest of the notes according to this skeleton.

When one finds this kind of innate connectivity between the notes, and this kind of listening that encompasses many layers, it’s common for other things to improve as well. I find that this can also increase one’s sense of security during playing. When you can find more tangible points of anchoring for the ear, good intonation no longer feels arbitrary (Teacher A said this note needs to be played higher but now teacher B is saying the opposite!!!). You feel much more rooted in the tonality, thus the music itself. And it usually translates to the body also feeling more anchored, more secure. Ideas about phrasing can sometimes become immediately crystallized after this intonation exercise because it is really at its heart an intimate exploration of intervals, tonality, and connections. After all, how could one truly shape a phrase without examining the connection between the smallest units of interval?

Traditionally we often practice scales before arpeggios. The scale provides more physical anchors for the hand to help play the arpeggio more in tune. But I think to flip it around this way is a practical and very effective way to arrive at pitch understanding and discernment on a deep level. It also helps train a much, much more elusive, and I believe a much more difficult, but often overlooked skill: the ability to listen deeply. I think it also helps hone the important skill of being able to react and interact instantaneously (even automatically) to sounds coming either from our own instrument or a musical partner’s.

Good intonation should feel and sound inevitable, and that doesn’t come from pitches matching the needle on a tuner. “Correct” pitches as determined by sound frequencies are merely markers on a map. They’re two-dimensional, and are in fact totally inaccurate representations of what musical notes are, what music is; just as a blue patch representing an ocean on a map does not at all reflect what an ocean is. It is simply a symbolic representation of what the real thing is. So it is with intonation, and with most other things in music, too.