Taming the Beast—Revolutionize Your Piccolo Intonation!

Taming the Beast—Revolutionize Your Piccolo Intonation!

Taming the Beast—Revolutionize Your Piccolo Intonation!
By Stephanie Mortimore
Piccolo, New York Metropolitan Opera Orchestra

Playing in tune is probably the most challenging part of playing the piccolo. If you struggle with tuning, don’t despair: every piccoloist wrestles with this issue occasionally, if not often. Learning some basic physics and simple math related to the principles of sound help you understand why and put you in a better position to address the problem. First, let’s get a few basic definitions out of the way.

Making Cents of it All

A cent is a unit of measure that stands for one hundredth of an equal tempered semi-tone. For example, there are 100 cents between A and A#, 100 cents between A# and B, and so on. In all, there are 1200 cents in each octave.


I’m Losing My Temper

Musical temperament is, quite simply, a system of tuning. Equal temperament describes a tuning system in which the twelve tones of the chromatic scale are divided into twelve equal intervals. Pianos are usually tuned using a variation on equal temperament. This system is used not because it sounds best but rather, it is a compromise which sacrifices true harmony for the convenience of allowing music to be played in any key and have it sound the same. Pianos are tuned using a variation on equal temperament called well temperament in which the piano tuner, starting from a base of equal temperament, stretches the octaves, making the lower octaves increasingly flatter, and the upper octaves increasingly sharper. It is important to remember this because, when played with piano, the high register of the piccolo will have to be slightly sharpened.

By contrast, orchestras generally play using just intonation. This system is based on the physics of sound waves and, by extension, the harmonic series, so it results in a purity and stability of harmony that is perceived as consonance by the human ear. When using just intonation, players adjust of all of the notes of the equal temperament scale by playing a few cents higher or lower depending on the note. By changing the notes in this way, the two notes in any interval become related by whole number frequency ratios. This technical definition may sound complicated, but in reality, playing with just intonation is something that seasoned orchestral musicians do as second nature. And it’s something you can train yourself to do using the simple tools described in the “Practical Application” section. Take a look at the chart below to see how equal tempered scales (both major and minor) must be altered, note by note, to become a scale in just intonation.

Major Scale

Scale Degree Root Major 2nd Major 3rd Perfect 4th Perfect 5th Major 6th Major 7th Octave
Change in Cents 0 +4 -14 -2 +2 -16 -12 0

Minor Scale

Scale Degree Root Major 2nd   minor 3rd Perfect 4th Perfect 5th minor 6th Major 7th Octave
Change in Cents 0 +4 +16 -2 +2 +14 -12 0

It Hertz My Ears

Sound is made up of waves which, if you could see them, would look very much like the waves you see when you visit the ocean. You have probably noticed that ocean waves vary a great deal —from tiny waves quickly lapping the shore to massive ones slowly rolling in. It is exactly the same with sound waves. The more waves there are in a given period of time—independent of size—the higher the frequency.

In sound, frequency (which is measured in hertz) refers to the number of waves per second; in music, this corresponds to pitch. The higher the frequency, the higher the pitch. For example, A440, or 440 Hz, refers to the A just above middle C. The A one octave above that is A880, or 880 Hz. If you keep going up by octaves, the hertz doubles with every leap.

This upward ascent brings us, eventually, into the highest realm of the piccolo range. While the next A in succession (A1760) is the top A on the flute, the highest A on the piccolo is twice that (A3520). Just to remind you, that’s 3,520 sound waves per second! Unfortunately for our poor ears, this is where piccolo players are required to hang out all the time.


Why is This Instrument So Hard to Play in Tune?

Again, the answer comes down to physics. When two concurrently played notes are close to—but not quite—a perfectly tuned unison, the sound waves interfere with one another and produce beats that can be heard as a distinct buzzing. As the notes approach each other in frequency, thus getting closer to a true unison, the buzzing slows. As they get further apart, the buzzing speeds up.
Now, using some simple math, let’s apply this knowledge to some theoretical orchestral situations. Let’s say you and a colleague are playing the flute and both of you are asked to play A440. Easy enough, you might say. But let’s assume you are having a bad day and, instead of playing perfectly in tune, you play the note 10 cents sharp. I won’t bore you with the more complicated math of cents-to-hertz conversion, so you’ll have to trust me when I tell you that when played 10 cents sharp, A440 becomes A443 (rounded to the nearest whole number)—a difference of 3Hz. You will produce 3 beats per second—not ideal, but not such a big problem.

Let’s compare that with a slightly different scenario. You and your colleague are now asked to switch to piccolo and to play, in unison, A3520, the highest A on the instrument. And let’s assume that your day still hasn’t improved and you play this note 10 cents sharp too. Your sharp note would actually be A3540—a difference of 20Hz. You will now produce 20 beats per second. Bzzzzzzzzz!! This can start to be really painful for everyone within earshot.

The unfairness of the situation becomes even more clear when you start looking around the orchestra. All of those other musicians (who at this point are glaring at you) don’t have anywhere near the same challenges as you, the poor piccoloist. Take the cellists for example. Pretend two cellists are attempting to play A220 (the A just below middle C) in unison. For them to be to be producing 20 beats per second, one of the cellists would have to be playing 150 cents sharp (or flat). That’s one-and-a-half semi-tones apart—the difference between an A and a really flat B. That’s one bad cellist.

We could go through the rest of the instruments of the orchestra in this same way but, while that might make us feel better, it should be clear by now why the piccolo is the most difficult of any instrument to play in tune. So what to do? Unfortunately, even though it’s not your fault, you still have to fix the problem.

In a well-meaning but misguided attempt, many players try to use the indicator on their tuner which shows them, visually, how many cents sharp or flat they are. The problem with this approach is threefold. Not only do most tuners register improperly for the high notes of the piccolo, but almost all of them use equal temperament. Because orchestras play using just intonation, tuning visually can be destructive to your ability to learn to play in tune with your colleagues. And most importantly, why would you want to train your eyes to do a job that your ears should be doing? Playing in tune has nothing to do with having perfect pitch. It is a learned skill. In order to learn the skill properly you have to train the right muscles. Think of it this way: You weren’t born knowing how to ride a bike. When you wanted to learn did you watch a video or read a book about it? No, you probably just went out and got on a bike. Similarly, when you want to learn how to play in tune, you have to train your ears, not your eyes. Fortunately, there is another way to make a difference in your intonation.

Make a Difference


While the properties of sound production can be traced back to pure physics, hearing is a more complicated matter—biology also enters into play. When sound waves enter the ear they are translated into neural impulses which can be perceived by the brain. When two notes of different frequencies are heard simultaneously, events inside the ear or brain—there is some debate on this matter—often cause the listener to perceive a third tone. This “ghost note” is known as a difference tone, so called because of the mathematical relationship it has to the two primary notes; its frequency is the difference between the frequencies of the other two notes (the frequency of the higher note (f2) minus the frequency of the lower note (f1) is equal to the frequency of the difference tone(D). )

f2—f1=D

When the two notes are close together (less than about 30Hz apart) they will produce beats as described above. When the two notes are further apart (more than about 30Hz), they will begin to produce a difference tone which is audible as not just a buzzing, but as a separate note. Difference tones then actually combine with the primary notes to form the illusion of three note chords. As a result, the difference tone becomes a powerful tool for improving your intonation. By paying close attention to and tuning difference tones instead of the primary notes you will develop the skills you need to play in tune with yourself and with your orchestral colleagues. To use the bike analogy again, difference tones are your training wheels.

This training will have two major positive effects. First, by sensitizing yourself to difference tones in your individual practice, you will develop muscle memory around how to adjust your piccolo so that you can play in tune with yourself (meaning you use just intonation to play arpeggiated intervals). Second, by using these skills when you play in the orchestra, you will hear the harmonic relationships which exist between your notes and your colleagues’ notes. When you hear difference tones which are not harmonically related and, therefore, not aesthetically pleasing to the ear, you will know how to adjust your instrument accordingly.


Practical Application

Let’s give it a try. For this part, you will need a tuner that plays a reference tone chromatically at least up to B6 (written as the B just above the staff for the piccolo), and that will play loud enough for you to hear well while playing but without causing pain. The louder your tuner is, the more audible and identifiable the difference tones will be.

First, let’s listen for beats. Put your tuner on A6 (this note is written as the A directly above the staff—A1760). Now, on your piccolo, play the same A. Try to move above and below the tuning note by allowing your hands to turn the piccolo, first in, then out. Listen for the beats. Notice how the buzzing speeds up as you get further away from the tuning note. Notice too how it slows down and eventually stops as you approach and arrive at a perfect unison.

Now let’s listen for difference tones. Put your tuner on A6 again. On your piccolo, play the C# which is a Major 3rd above the sounding A (C# 2200). Listen closely—in addition to the two primary notes you should hear a ghostly but very distinct third tone. If you carefully tune this note, you will find that it is, in fact, A440—the difference tone (2,200Hz  — 1,760Hz = 440 Hz).

Useful Intervals

All of the twelve possible intervals within a chromatic scale produce difference tones, but six of them are particularly useful for tuning difference tones. This is because they are all members of the same harmonic series—that of the tonic. Each of these six intervals, when played in tune, will produce perfectly tuned difference tones which are also members of the same harmonic series.

The following chart lists the intervals which are particularly useful for tuning:

Interval Difference Tone
minor 3rd (m3) minor 6th (m6)
Major 3rd (M3) Root    (R)
Perfect 4th (P4) Perfect 4th (P4)
Perfect 5th (P5) Root    (R)
minor 6th (m6) minor 3rd (m3)
Major 6th (M6) Perfect 4th (P4

Now work your way through all of the intervals in the chart. Again play A6 on your tuner. As you ay each of the intervals on your piccolo, listen for the corresponding difference tone.

The wonderful thing about just intonation is that is works in any key. If you do the same exercises with different tonic notes and play the same intervals on your piccolo, you will hear difference tones at the same scale degrees.

This exercise can also be applied while you are working on excerpts or solo works. Identify the tonic of the passage you are working on, set your tuner to that note, and play the passage over the top of the tuning note, all-the-while paying careful attention to the difference tones that are produced.

How Do I Remember All of This?


This may seem like a lot to absorb, but don’t be too concerned if you can’t remember which difference tone is produced by which interval. Just use your ears and try to tune the difference tone; use the chart as a reference when you need some help hearing where the third note should be.

That said, the generation of musical notes is, at its most basic, a combination of math and physics so it’s not surprising then that certain patterns emerge which will make memorization easier. These are some of the patterns I have discovered:

  • If you look at the six intervals which produce useful difference tones, you will notice that there are two minor intervals (m3, m6), two major intervals (M3, M6), and two perfect intervals (P4, P5).
  • Look first at Major 3rds and Perfect 5ths. These are the intervals which are probably most useful in tuning as they form the basis for the Major triad. Notice that both of these intervals will produce a difference tone of the Root.
  • The minor intervals are opposites of each other; if you play a minor 3rd above the root you get difference tone of a minor 6th. Conversely, if you play a minor 6th above the root you get difference tone of a minor 3rd.
  • All of the difference tones sound within the octave immediately below the root with the exception of those occurring with Major and minor 3rds. These sound within the octave which is two below the root.


In addition to the above patterns, if you memorize the order of the intervals from smallest to largest you can use the following mnemonic device to help aid your memory.

Interval Difference Tone

Mnemonic Device

m3 m6 six
M3

R

rudiments
P4 P4 for
P5

R

revealing
m6 m3 tuning
M6 P4 fortitude

These six rudiments really can reveal your tuning fortitude. Just a few minutes each day with these principals will change the way you think of tuning and make all the difference in your piccolo intonation and, hopefully, in your enjoyment of the instrument.

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