Andrew Meronek

I find it to be a vast gaping hole in the repertoire of a composer and in performers of pitch-adjustable instruments when they are unaware of pitch drifting that happens due to counterpoint.

First, a bit of background. In the vast history of music, tempering happened. Tempering is the ‘evening out’ or ‘fudging’ of small musical intervals so that certain kinds of harmonies are easier to play on a keyboard, because tempering reduces the required number of keys to bang on in one octave. Because our music notation is based on the keyboard layout, what we read is also tempered. A side-effect of extensive use of tempering is that when we go to untempered instruments (such as a violin or the human voice, particularly when without vibrato) our ears naturally want to keep things untempered, because tempering is a deliberate distance away from where our ears hear a harmony as being the most stable or in tune, and we naturally gravitate toward these points of stability, reintroducing these small intervals that are tempered out in our music notation.

First, look at and listen to this very common pitch sequence:

The arrows to the left of note heads are Sagittal accidentals. I include them to indicate precisely where a tone would need to be tuned in order for the chord to be in tune. In this sequence, I also kept common tones between sequential chords to be exactly the same pitch. The first accidentals are just to get the major thirds in tune, as you probably surmised when seeing them on the thirds of the first two chords. But notice how, during the IV-ii progression, what was a third in the IV chord becomes a fifth of the ii chord. It forces the D of the ii chord to be lowered in order to be in tune. And then the D, staying lowered, is a common tone in the V chord, and the subsequently lowered G is a common tone with the final I chord.

The very last chord is the same pitch level as the first chord, for direct comparison to make it easier to hear how the sequence drifts. I think that the pitch drift here is striking, and because this is a fairly common chord sequence, undoubtedly affects many musical performances.

Why does this happen?

We can think of forming a major scale by taking the notes from the three primary triads: the I, IV, and V chords. In C major, this gives us CEG, FAC, and GBD; sorted this covers CDEFGAB. But if we base this scale on in-tune triads, the language of just intonation shows a clearer picture of what happens. Just intonation defines pitches as integer ratios, and the major triad is 1, 5:4, and 3:2. Relative to C major, the ratios of the primary triads work out to be:

C major: 1, 5:4, 3:2
F major: 4:3, 5:3, 1
G major: 3:2, 15:8, 9:8

Then, we can sort the terms and form a major scale again: 1, 9:8, 5:4, 4:3, 3:2, 5:3, 15:8.

The minor triad is similar to the major triad, but in just intonation is defined as 1, 6:5, and 3:2. Two minor triads emerge from this scale: the iii and vi chords. The ii chord is NOT in this major scale. In order to create a ii chord, we must replace one or two pitches. We can either keep the 9:8 and replace the 4:3 and 5:3 with 27:20 and 27:16 respectively, thus deviating significantly from the original IV chord, or we can keep the 4:3 and the 5:3 and replace 9:8 with 10:9, where the altered pitch is from the V chord. It is, in fact, a mathematical impossibility to have both a ii chord and a IV chord in this major scale without adding at least one note.

Now, look back at how I notated my chord sequence. In Sagittal notation, the half-down arrow in front of each of the major thirds, in front of the D of the ii chord, and elsewhere, notates exactly the different in pitch between 9:8 and 10:9. That accidental is in front of the major thirds of the I and IV chords because in Sagittal, the base major scale is actually a Pythagorean major scale, and this difference in pitch also corrects the Pythagorean scale pitches to the in-tune major thirds that we want to hear. That accidental is, in fact, extremely important in music theory, and it has it’s own name: the Syntonic Comma. For discussing this pitch drifting example, noting that Sagittal uses a Pythagorean major scale is not particularly important, except for clarity. However, noting the small difference in pitch forced by using the ii and IV chords in sequence is vital.

So, how can performers deal with this?

The best solution is to have composers write music that doesn’t do this. But, we have hundreds of years of great music already created which has these kinds of pitch drift problems. So, the next-best solution is to recognize that written common tones do not necessarily have to be exactly the same. Consider this solution:

The most significant difference here is that I added a natural sign in front of the D in the bass of the V chord. This is to indicate that the lowered D from earlier in the measure is now un-lowered. It changes the 10:9 back into a 9:8, which is the correct pitch for the V chord. This also means that in this example, the bass would have to make a small adjustment in pitch upward between those two Ds.

Melodic considerations also apply. I find that the voice which has the melodic prominence (usually, but not always, the top voice) should maintain the most diatonicity in pitch, meaning that this voice should try to avoid making these small pitch adjustments if possible. If, in this example, the bass was actually the prominent melodic voice, this solution may not be ideal, and we might want to find somewhere else to introduce the upward shift in common tones in this progression. Putting it in-between the IV and ii may be tough because having two voices singing the shift makes it more noticeable, and it’s simply harder for two performers to make this kind of small pitch adjustment in agreement than it is for one performer to handle it. Maybe in-between the I and IV chords, which only have one common tone: the C. Or maybe between the final V and I chords, with their common tone of G. Alternatively, if this is a lyricised chorale, maybe the word that the basses sing on their second D deserves to sound a bit brighter (which will be accomplished when singing it slightly sharper) to enhance the word painting. It is also possible, although difficult, to simply sing the ii chord out of tune, but this kind of out-of-tune-ness will starkly stand out of the progression in a bad way, especially when the other chords are resonantly in tune, and is probably an even worse solution than simply allowing the pitch drifting to happen:

As a general rule, progressions that include the ii chord with either the IV chord or the vi chord will need this kind of pitch adjustment somewhere to deal with drifting. And this is not the only kind of pitch drifting that can happen in music; far from it. This is merely the most common.

You might say that performers have been dealing with our limited western music notation just fine, and already have ways of dealing with pitch drift. But, methods of dealing with it rely on a performer’s memory of tonal center. For the very best performers, they recognize when pitch drift starts to happen and adjust, but even performers with the best ears who do not consider that pitch drift is forced by counterpoint like I outlined here will encounter problems with figuring out how to both prevent pitch drift and sing consonant harmony. These tools of just intonation provide the solutions. Musicians who deal with music which has pitch drift problems who learn these tools gain a competitive advantage.