The Six Percent Rule

Ed Dumas

I have always had trouble convincing students that all wind musical instruments are quite flawed in terms of musical intonation. I am also completely convinced that I am not alone in that struggle! So, I took to using the Six Percent Rule to help senior students understand WHY they needed to pay more attention to their intonation and I think it has helped, at least to some degree.

First, a little sidebar about intonation. There is a difference between tuning and intonation. Tuning is what all musicians do together so that they can agree on at least one foundational pitch. Most groups today settle on A = 440, which means that the A above middle C will be exactly 440 Hz or 440 vibrations per second. Intonation means that if you place an A at 440 Hz, then the A#/Bb just above it needs to be precisely 466.16 Hz. If you place the Bb at 475 Hz or 450 Hz, your intonation is off, even though your tuning note might be correct with others.

It is the placement of intonation that has worried me the most over the years, as many students have seemed reluctant to understand that the 6% rule means that it is practically impossible to build a wind instrument with perfect intonation. I have usually found that once students understand WHY their instruments are inaccurate, they are more likely to go along with the notion that they must learn to adjust for intonation problems on all wind instruments.

So What Is The Six Percent Rule?

I mentioned that if A = 440 Hz, then A#/Bb just above must be 466.16 Hz. Exact pitch frequencies can be found on numerous tables all over the internet and are readily available. The concept I want students to understand though is “IF A = 440 Hz.” Though we often aim for 440 Hz, there are plenty of times when conductors choose to alter that tuning frequency. I have done this myself if we were playing outside when it is very cold (such as at a Remembrance Ceremony), choosing to drop the tuning pitch slightly (A = 436 Hz) rather than try to fight the coldness all the time. The reverse is true when it is very hot and the conductor chooses to tune slightly higher (A = 444 Hz) to reduce the amount of fight with the tuner and instruments.

As an aside, you should keep in mind that string instruments such as string basses and violins move oppositely due to temperature changes. For them, cold temperatures will RAISE the pitch due to increased tension on cold & shortened strings. Similarly, raised temperatures will LOWER the pitch due to reducing tension on lengthened strings. This may affect how far you can compensate a band for temperature.

Now, let us assume that the room temperature is normal, and the musicians have chosen to tune to A = 440 Hz. Given those parameters, the A#/Bb immediately above then should be 466.16 Hz. The difference between them is 5.95 percent, since 440 X 1.0595 = 446.18 Hz. The difference of .02 Hz is a rounding error, and for the sake of students, I round this all to become “The Six Percent Rule” when in reality it is “The 5.95% Rule.” The former just sounds easier to remember!

So, depending on where we place A, the next pitch up (A#/Bb) should be about 5.95% higher. Each successive pitch higher will use the same 5.95% increase in Hz to create the next sound. This is the relative tuning that students need to develop on their wind instruments that plague so many students and bands.

Now, I should mention that I honestly believe the best place for students to develop a strong relative pitch ability is in a choral program. In a band class, beginning band students will inadvertently learn a faulty absolute sense of pitch by thinking that to get that Bb, I just “add these fingers or keys and the instrument does the rest.” But in a choir, placing a valve down or adding a key is just not a thing. There is little chance here that they will develop that faulty dependency on an instrument to make it right for them. I sincerely wish that I had a choral program available to me when I was in secondary school, and I STRONGLY encourage every wind instrument player to sing in a choral program if there is one available to you!

Additive vs Subtractive Instruments

Now consider the guitar for a moment, which is best thought of as six musical instruments in one, since it has six strings. If you play the A string open, which means no fingers on frets, that string will sound an A. To make it sound one pitch higher (A#/Bb), you then place one finger on the first fret. This removes 5.95% of the length of the vibrating string, thereby making it sound one semitone higher.

For each successive pitch on that string, you move to the next fret up along the length of the string.

Notice that the second fret that you have now pressed to get the B sound is closer to the first fret than the first fret was to the nut at the top. The reason for this is that the effective length of the string has been made shorter when you played the Bb, which means that the next 5.95 percent reduction will be shorter than the last one. This pattern continues up the string with the frets getting closer and closer while the 5.95%s keep getting lesser due to a shorter previous string.

Hence, the guitar is a “Subtractive Instrument” meaning that to change the pitch on any given string, one must remove some active length of the string by pressing down on a fret. There is no possible way of making a string longer by adding to it to achieve a lower sound than the “open” position of no frets. A different string needs to be used to achieve this lower sound.

Now consider that all wind instruments are additive, rather than subtractive. For winds, the length of the tube is fixed and cannot be shortened, but can be added to by placing down a finger or valve. On a trumpet, a player will produce the highest possible sound on that given harmonic when not pressing any valves. By adding a second valve, the player adds in the length of the tube that accompanies this valve. Let’s call it length “1” which represents one tube length on the second valve.

By adding one length of tube, the pitch now drops one semitone. To drop another semitone, the student would have to add in another length of tube. This is achieved by raising the second valve and adding the first valve since the first valve tube is double the length of the tubing of the second valve. This first valve, then, will drop the pitch by two semitones. The third valve is three times the length of the second valve, which means adding this valve alone will drop the pitch three semitones.

More semitone drops can be achieved by combining the valves. Normally a drop of three semitones would be achieved by using the first and second valves (2+1). Four semitones would be achieved with second and third valves (1+3), five semitones would be one and three (2+3), and finally, a drop of six semitones would be all three valves (2+1+3).

Now, here is where the math gets complicated for students. If we assume that instrument makers create the valve tubes at PRECISELY lengths of 2, 1, and 3, then the larger valve combinations will be dramatically short in their percent drop. That is because the 5.95% extra length also needs to take into account the other lengths already added in. So, a first valve drop of two pitches should be the length of the horn, plus two times 5.95% of that horn length AND 5.95% MORE of the second valve length. A drop of three semitones should have an added drop of 5.95% of the entire horn that was used to create the pitch just above it, which means adding a percent of a percent.

Here is a chart of the basic formulas used to calculate the length of tubing to add in.

Lengths (L) of Additional Tube (Valve Combinations)

1L (2) 2L (1) 3L (1,2) 4L (2,3) 5L (1,3)

6L (1,2,3)

Formula

L = 1 additional valve tube length L
2L+Lx5.95%
3L + (2L + L x 5.95%) x 5.95%

4L + [3L + (2L + L x 5.95%) x 5.95%] x 5.95%

5L + {4L + [3L + (2L + L x 5.95%) x 5.95%] x 5.95%} x 5.95%

6L + {{5L + 4L + [3L + (2L + L x 5.95%) x 5.95%] x 5.95%} x 5.95%} x 5.95%

Now I don’t expect you or anyone else to memorize these formulas. Quite frankly it hurt my brain just working this out, and I am not even convinced that it is 100% correct! I even ran out of available brackets on my keyboard and had to use the { } type of brackets twice.

But the point is that the math is very complex, and students must not be complacent with just putting down 1,2,3 for a trumpet C# and be satisfied with that! The Six Percent Rule dictates that they must make some further changes to their slide lengths by using the third slide trigger when playing that note. The same is true of using 1,3 for a low D on trumpet. After hearing this from me, few students would risk hearing another math lesson on the Six Percent Rule, but would rather just opt to begin using the trigger to put those notes in tune!

Now when talking to brass players about adding some trigger length to make the 1,3 and 1,2,3 combinations in tune, students usually question how much length to add using the trigger. This is a good question, and students will find that the amount of addition varies between the various instrument makers. That is because the various manufacturers will use slightly different lengths for all the valve slides to achieve a combination they feel would work the best on their instrument. Therefore, the way to tell is to check it against a tuner.

If you are checking trigger slide lengths against a tuner, please be aware of the challenges with tuners and the bio-feedback response. If you are not sure about this effect, please review the article titled “Digital Tuners In A Concert Band” for a good explanation. Then, remind your students not to “stare” at the tuner when checking a note.

Now, in the same way that brass instruments are additive in their lengths, woodwind instruments are also additive, but not often thought of that way. The shortest instrument available for a clarinet would be the length from the tip of the reed to the first open key which is Bb using thumb & one. The next lowest sound will add 5.95% of that length by removing the thumb and allowing that key to close.

The next lower pitch after that will add 5.95% of the entire instrument on the pitch above, which includes one calculation of the 5.95% already. Hence the next pitch-down addition will be slightly longer than the previous addition. In that way, students can see that each successive addition of length means that the additions are growing longer and the holes or keys more separated.

For woodwinds, students can then get a sense of why the fingers spread as the instrument gets longer, as well as the tone holes (clarinet), get larger. Once the tone holes become too large or too far apart, keys must be used to allow fingers to complete the task. Hence larger instruments like bass clarinets do not have open holes in the same way as a soprano clarinet.

There are several more functions that I like to mention to students when discussing why their instruments all have poor intonation. The first is the size of the bore of the instrument, and how much this bore opens from the mouthpiece to the bell. The second is the use of harmonics in different registers, and how the length additions will change how all the pitches in the various harmonics will sound. Third is a consideration that the way the instrument is built will determine if the harmonics are even in tune with each other, before the use of lengthening tubes using keys or valves.

By now students are getting some dizzying understanding that the math that is involved in making a wind instrument is extremely complex. When one understands that manufacturers make some accommodations to try to help with impossible intonation, this then becomes an engineering-level challenge. For example, manufacturers will adapt the lengths of the three valve slides on a trumpet to help with some pitches, knowing that they are also then putting other pitches farther off correct. The notes that use the third valve without the trigger (2+3) tend to be pushed farther off correct if the trigger valve was made a little longer to help compensate for 1+2+3. Notes that use the 1st valve or 2nd valve could even be pushed off from correct if those slides were adjusted to help move the 1,2,3 combination to more in tune.

Simply putting down some fingers and hoping it will be in tune now is usually seen as incredibly naïve. This makes room for a discussion with students about which notes on their instrument need correction. The throat tones on the clarinet come to mind, which behave more like subtractive notes since the clarinet played “open” is a G and not a Bb. The very short notes on a flute like high Db/C# are also renowned for being fairly off-pitch. The same is true of open Db/C# on sax which can be very out of tune.

Finally, a comment about the trombone. I mentioned that due to the math challenges regarding the Six Percent Rule, it is practically impossible to build a wind instrument that plays with perfect intonation. The trombone is the closest that could be achieved for this, but it requires trombone players to understand that the distances between the various positions lengthen as they go down the slide. The distance between positions 1 and 2 will not be the same as the distance between say 4 and 5. As long as trombone players understand that the positions lengthen and that they, like everyone else, must use their ears to place a pitch, it is a fantastic instrument for intonation!

The other important concept for trombone players to understand is that the various positions will change slightly depending on which harmonic they are playing on. Hence, they may end up with a slightly shorter or longer position on some harmonics.

The discussion with students about The Six Percent Rule leads naturally into a discussion about which notes on their particular instrument need to have special attention paid to it. There are some excellent band technique books available with some good descriptions of Pitch Tendencies of the various wind instruments. Once students get a glimpse of how complex this whole topic is, they are then usually much more open to some higher-level intonation studies on their instruments, or to at least check their intonation against a tuner to find a better placement!