Performing with superior intonation is a universal problem in all instrumental families. Musicians are bombarded by an array of difficulties when addressing intonation. Not only are there specific problems associated with particular instruments, but other problems arise such as scoring, instrument design, temperature, and physical manifestations of the performer. Intonation should be addressed early in the training of the elementary band students. Of paramount concern should be the establishment of a workable embouchure, which is allowed to respond freely with a good wind supply and support structure. In the first band classes, brass players should be introduced to good breathing techniques that will transfer readily to their instruments. Only when the tone is well supported and driven by the wind column in a singing style can intonation be addressed. There are several inherent problems regarding brass instrument intonation.

Initially, a competent performer needs to be aware of the natural intonation tendencies of the harmonic series. Secondly, one needs to understand the design of the brass instruments and the function of the valve mechanism. Thirdly, all musicians need to be aware of the different tuning problems when performing with a keyboard instrument (using equal temperament) and an instrumental ensemble (using just/natural intonation). Finally, there is the difficulty of using this knowledge in practice in order to make intonation adjustments in performance.

The Harmonic Series

Brass instruments operate under the principles of physics. Pythagoras' experiments with mathematical ratios and how musical tones relate to one another is the basis of the perception of musical scale in Western culture. Through a system of ratios, he was able to construct a series of notes that could be produced on a one-stringed instrument (a monochord). The structural component of this scale was the Pure Fifth a ratio of 3:2. Overblowing a flared end-blown tube may produce this same series of notes. Thus the law of physics governs the following sounds. The first note, designated by the number one is the fundamental. This is signified that there is only one tonal vibration within the tube. Other notes are made available by increasing the air velocity in the tube and firming one's lips (overblowing). The numbers underneath the notes indicate the overtone number and coincide with the amount of standing waves that occur in the end-blown tube. I have stopped the overtone series at the sixteenth vibrating mode, however it can go on indefinitely.

One notices that as the series progresses, the overtones become closer together. After the sixteenth overtone, pitches are one-half step apart and continue to meld into a final gliss.

Another property that is peculiar are doublings of numbers (1-2-4-8, 3-6-12, etc.) produce octaves. Thus octaves keep the same intonation tendencies. 2 According to equal temperament, all octaves coming off the fundamental are in tune. All octaves based on the third overtone are slightly sharp while the octaves of the fifth overtone are noticeably flat. The ninth overtone is only slightly sharp and is used on occasion by brass players. Items marked by an asterisk are badly out-of-tune and are not performed as a natural harmonic. Instead these notes are played on a valve combination descending from the nearest available open overtone.

Prior to the addition of valves into brass instrument design, brass players were exclusively confined to the notes within this series. Not until around 1750 were there experiments with hornist Anton Hampel in Bohemia to use the right hand to bend these overtones so as to help fill in the gaps within the central diatonic scale. While the horn's capacity to play a diatonic scale in the middle range increased, the trumpet was limited to diatonic movement in the top-most octaves. Johann Sebastian Bach frequently wrote his trumpet obbligato parts up to the 24th overtone! To understand how this figures into our concerns, we need to know the transposed fundamental of each brass instrument. The term "transposed fundamental" and not the actual concert pitch.

To explain this, let us look at the trumpet. Trumpets are pitched in Bb, but they read in the key of C, thus they sound a whole step lower than written. Thus, the musician will read and play their C, but a Concert Bb is sounded. Thus, the open fundamental of the Bb trumpet is:

This relationship of the Bb side of the horn may be confusing to some readers. The Bb side of the double horn is pitched a perfect fourth higher than the F side of the instrument. The performer still thinks of the double horn as an F pitched instrument and learns a completely different set of fingerings for the Bb side of the instrument. This can be confusing for the novice educator. Thus, the Bb side of the instrument is not considered as a true Bb pitched instrument.

The treble clef baritone is treated as a transposing Bb instrument like the trumpet. However, it is pitched an octave lower than the trumpet, but would read exactly the same music. Thus the trumpet examples will suffice for the treble clef baritone. To illustrate:

Trombone, Euphonium and Tuba are pitched in Bb, but are treated as concert pitch instruments, sounding the same note that is read. Thus, the fundamental pitches for these instruments are:

In common practice, brasses will normally not play beyond the 12th overtone. If 5 demands necessitate this, then specialty instruments of higher pitch may be used, such as piccolo or F-trumpet. In the case of the Horn, a descant horn may be used. For tuba, an F-pitched instrument may be in order.

Brass Instrument Design

There are several valve systems that are in use. These are 1) the standard three-valve system, 2) the four-valve system, and 3) the compensating valve system. The standard three-valve system has not changed since around 1830. Buttons or levers manipulate valves of piston or rotary design to engage additional loops of tubing that lower the pitch of the open instrument. The valves are numbered 1 (that valve which is closest to the mouthpiece) through 3 (that valve which is closest to the bell of the instrument). The function of each valve is as follows:

1st valve: Lowers the pitch of the open instrument one whole step
2nd valve: Lowers the pitch of the open instrument one-half step
3rd valve: Lowers the pitch of the open instrument one and one-half steps (equals the length of the first and second valves combined).

This configuration is used on trumpets, horns, marching brasses, and less expensive low brass concert instruments. Intonation with the open instrument and with single valves can be slightly problematic, but when valves are combined in combination, major problems occur with respect to intonation.

The three-valve system is the minimum configuration that allows complete chromatics between the third and fourth overtone. To illustrate how problematic valve intonation is, let us imagine that we have a valved brass instrument 100 inches long that is pitched in the key of C. Thus, when we play the open instrument on the fundamental pitch a resultant C will be sounded. Next, let us look at the length of the individual valve loops: 1st valve: 8 inches (twice as long as 2nd valve) 2nd valve: 4 inches 3rd valve: 12 inches (equal to valves 1&2) Thus, these valves will have their own overtone series. Let us look at the following example of this hypothetical instrument pitched in C. Thus, the first 12 overtones of this open instrument will be:

Being that the third valve is equal to the first and second valves combined, this last overtone series is the same for this combination. However there is another intonation difficulty that needs to be considered which greatly affects intonation properties of all 8 the valved brasses. This problem deals with the basic design of the valve system. The first problem encountered is the inherent intonation problem when valves are used in combination with one another. Referring back to the 100 inch instrument built in C, we know that adding an additional 4 inches to the open instrument (second valve), lowers the instrument one-half step, thus lowering the pitch of this instrument to a B. To play a Bb, the open instrument will need to be lowered by depressing the first valve, adding the required 8 inches to bring the pitch down one whole step. Thus:

100" pipe = C pitch (open instrument)
104" pipe = B pitch (adding second valve)
108" pipe = Bb pitch (adding first valve)
112" pipe = slightly sharp A pitch (first and second valves or third valve)

However, if we want to go a half of a step lower, to A, a significant problem arises. If we have the first valve engaged, making a 108 inch pipe in Bb, then we must add the 4 inch loop of the second valve, making a total length of 112 inches to bring the pitch down to A. However, we can readily see a problem of mathematical proportions that affect pitch when valves are combined. 9 If 4 inches added to 100 inches lowers the pitch one-half step, that same 4 inches will not lower a 108 inch pipe to the same degree.

It is a problem of ratios; 4/100 is not equal to 4/108. Thus the first and second combination is sharp. Even the third valve loop alone will be sharp. This problem compounds itself as the valve combinations get increasingly longer. The valve combination 1&2 are slightly sharp, but as we go to combinations 2&3, 1&3, 12&3, the instrument goes excruciatingly sharp. Though problematic in itself, there are additional items that compound these problems. Using a horn model, we notice that some notes will be predictably out-of-tune while some others are so inferior in intonation as to needing to be avoided altogether. Complications arise when we start combining valves. Notice that certain pitches have their own natural intonation tendencies.

C c g c1 e1 g1 bb2 c2 d2 e2 f#2 g2 # b # * (#) b * #

If we are using the first and second valve combination, all the pitches will be slightly sharp. Those that have naturally flat tendencies will be canceled out, making these notes in-tune. However, those notes that have naturally sharp tendencies will become even sharper. Thus the resulting intonation of the overtone series using the first and second valve combination will be:

Thus, to simplify this process, a good rule of thumb is that the longer the valve combination that is used, the sharper the instrument becomes. This is a universal problem shared by all valved brasses.

In 1847, a British instrument designer by the name of Blaikley devised a four-valved system that would help alleviate the extreme sharpness of the first/third and first/second/third valve combinations. His added fourth valve loop was equal in length to the first and third valves combined. However the accurate tuning of this loop is slightly low to the third overtone of the harmonic series. Tuning to the exact pitch of this overtone will make this and all combinations using this valve sharp. Thus, the player can readily substitute the fourth valve for the sharp first/third combination making a satisfactory pitch.

Likewise, the valve combination first/fourth will substitute for the poor first/second/third combination. Even so, engaging additional valves with the fourth valve makes the aforementioned ratio problems come into play. In the present day, this system is commonly found on low brass instruments, flugelhorn and piccolo trumpet. The superior intonation afforded by a four-valved instrument outweighs the slight difference in cost.

In the late Nineteenth Century, another approach to improving brass instrument design resulted in the compensating valve system. Again, this was a British influence. The idea behind this design was to incorporate additional knuckles of tubing that were engaged only when valves were used in combination. Thus, if the first and second valves were used, an additional windway connecting these two valves was opened that would lower the pitch sufficiently. Likewise, additional knuckles were made available to the 23, 13, 123 valve combinations. Though this idea is sound, an unexpected problem was the increase in resistance afforded by the additional valve loops, making the instrument much more stuffy on long valve combinations. This compensating system is also in use today, used in three and four-valved formats.

Intonation Adjustments During Performance

The universal intonation adjustment on brass instruments is that of "lipping" the pitch. A slight opening or closing of the lip aperture will cause the pitch to bend. The concept of lipping should be introduced early to the elementary bandsman. It will be easier for the younger bandsmen to bend the pitch down. Instruct these players to keep the embouchure firm and slightly pull the jaw down and leave the lips behind. This will assure that your students will keep the firm set of natural corners, firm cheeks, and a pointed chin.

Using your hand as a pitch indicator, have the students follow the pitch as you slightly lower your hand. When this is successful, have them return to the original pitch. As this becomes more polished, one may start ascending from the starting pitch. Instruct your students that any adjustment should be made within the mouthpiece. Thus, only the vibrating edge of the lip does the focusing. The motion is similar to that of a goldfish breathing ("woo-woo"), but very subtle.

An added benefit to these lipping exercises is that you will notice if some students are buzzing off the resonant center of the pitch. For advanced students, I have them find the outer parameters of the pitch; bending up and down until they slip off the note. Next I have them play the central pitch again. As they go high and low, I ask them to locate the position where the note sounds the loudest. Most can do this readily. Almost all find that this resonating center is lower than their normal placement. This will have the students play much more relaxed improving efficiency and intonation.

In addition to lipping, each of the brass instruments uses their own unique ways of adjusting pitch during performance. Adjustments can be done with special fingerings or certain mechanical adjustments.

Trumpet Intonation

Trumpet players need to learn to adjust intonation using the third valve and first valve slides. All trumpet and cornet players in the band should have a third valve tuning ring or kick trigger. This is an area that is grossly neglected by music educators, especially those teaching beginning band. It is essential that beginning instruments have this third valve tuning adjustment. For those students with used instruments, replacement universal rings can be purchased. This tuning accessory should be a requirement for all trumpet players.

The third valve slide should move freely. If not then the slide may need to be cleaned, especially if the instrument is older and not well maintained. For step-up instruments, not only should there be a third finger tuning adjustment, but a first-valve tuning adjustment such as a saddle or a spring operated lever is necessary.

First, the instrument must be cleaned thoroughly. A professional chemical cleaning is the best solution. Barring that, run warm water through the instrument and all valve loops. Drain the water from the instrument. With a paper towel or lint free cloth, take out all slides and wipe them clean. Keep inserting and wiping off the slides until there is no grime left. This can be a time-consuming process, but is necessary. If the slides have corrosion, drop valve oil on the slide and lightly (I cannot emphasize this enough) buff off the corrosion with 0000 steel wool. Rinse and wipe thoroughly afterward.

Once the slide is free from interior grime, heavy grease, or corrosion, one may lubricate this slide with either valve oil alone or mix in a few drops of STP oil treatment. Others have used oil and liquid graphite, but this can stain clothing. Spacefiller® is another recommended product. For this tuning slide, lightweight oil is essential. Stay away from tuning slide grease, Vaseline, or anhydrous lanolin. While this is fine for the tuning slide and second valve slide it is not appropriate for those slides that are frequently used in intonation adjustments.

Assuming that the instruments are in good working order, students should be instructed to use this third finger adjustment immediately. Directors should remember that these elementary bandsmen come to you as a clean slate. It makes no difference to them if you require them to engage the trigger mechanism. This will save you countless hours of headache and will be the single most effective thing you can do to improve the performance of your band.

As the third valve is used in the longest combinations, the kick trigger needs to be placed accordingly. For the second and third combination, the slide needs to be moved out nearly one-third inch. The first and third combination requires the slide to be moved at least one inch. For all three valves combined, at least one and one-half inches are required. If the trumpet player has a first valve tuning adjustment, then this should come into play to alleviate excessive third trigger movement. If time permits, it would be advisable to have these trumpet players work with a chromatic tuner to see where these slides need to go. For a temporary aid, the director can mark the designated adjustments on the slide.

Horn Intonation

Intonation adjustments for hornists can be made several ways. The right hand can make most intonation adjustments by slightly flattening out the cup of the hand (raising pitch) or adding a little more cup to the hand (lowering pitch). A good analogy would be to scoop up water in the palm of the hand in order to drink. If the hand is sufficiently far enough into the bell, only a minimal amount of movement is necessary. Some hornists may use the tips of their right hand fingers to lower the pitch. When tuning the double horn, I recommend to my students to tune the third valve loops slightly low to the first and second combination. This will give the hornist a satisfactory third finger a, e, and a1 that is well in-tune. In addition, the second and third combination will not be nearly as sharp, allowing the hornist to keep the throat of the bell more open to help tonal consistency. Because the horn performs commonly in its upper tessitura, there is little use of the 13, and 123 combination. However, in some solo and orchestral work these pitches are occasionally demanded.

Single F horns are at a distinct disadvantage here as little can be done to exact proper intonation. If one has a double horn, these problematic pitches, low d (13) and c-sharp (123) are best performed on the Bb side using the 12 (or 3) and 23 combinations respectively. Also, if other problematic notes arise, the performer has the discretion to use the opposite side of the instrument.

Trombone Intonation

In this discussion, trombone has been absent. However, there are definite considerations that must be taken with this instrument. The trombone is the only brass instrument that can be played perfectly in tune. Sadly to say, it has been my experience that trombonists are often the most blatant offenders of poor intonation. This comes from inadequate ear training in their formative years. Often young trombonists are instructed to place the slide in an arbitrary spot for position assignment. Trombone positions are not universal among individual players. All trombone positions are approximate. It is up to the director to go slowly and make sure all trombonists are matching pitch on beginning band unison exercises. While this is time consuming, it is well worth your while, as it will help the intonation of the intermediate and advanced ensembles in the future.

The trombone is perhaps the most painful instrument to learn how to play. It is physically painful and awkward to told. Secondly, with respect to the slide mechanism, there is no concrete spot for most slide positions. This, in addition to the awkwardness of holding the instrument are the main reasons the trombone section exhibits the highest drop-out rate for beginning bands.

Also there are directors that neglect to tell these players that the instrument has a tuning slide. It is essential that the instrument be in tune in the first position. Tune to a concert Bb and adjust the main tuning slide accordingly. On step-up models some trombone slides have a spring mechanism, similar to a shock absorber at the very end of the slide. For these instruments, have the person tune the first position where the slide barely touches the spring mechanism. This will give the player some latitude in raising the first position fifth harmonic (d1).

Another aid for the trombonist is the alternate position. The sixth position f and the short fourth position d1 are common substitutions and should be introduced to your beginning bandsmen. It is imperative that you spend adequate time with your trombonists in unison playing with one another, with other bandsmen, and with a tuning device to help develop a secure sense of pitch.

Baritone and Marching Brass Intonation

The three-valve baritone has a very limited intonation adjustment. There are no mechanisms for mechanical adjustments on these instruments. Thus the fifth harmonic and its octave will be very flat and the third harmonic and its octaves will be sharp. The later intonation problem will be compounded with the combination of valves. This knowledge is of special concern for those programs using specialty marching brasses (mellophones, marching baritones, marching tubas and sousaphones). These instruments all share like problems of inferior intonation.

There are only a few solutions that can be recommended for these instruments. The tuning slide may be able to be adjusted for a few problematic notes in performance, but this is not a viable solution. Some concert baritone manufacturers offer a spring-loaded lever that will kick the tuning slide out for problematic notes. However this is not an option for the majority of these instruments. I recommend that the first valve slide be extended a short amount and that the third slide be extended to the point that a reasonably in-tune 13 combination may be accomplished. However, this will make the notoriously low 5th harmonic much lower on the first valve. For this reason, I recommend that the bandsmen substitute the sharp lower octave fingerings for the flat fifth harmonic. In effect, this will help cancel out the competing intonation tendencies. These limited solutions are the only viable alternatives to help with intonation on these instruments.

Euphonium and Tuba Intonation

It is essential that your low brass students have access to a four-valve instrument. This will improve the intonation of the extended valve combinations. For euphonium, the compensating four-valve system is the most recommended system for the most successful negotiation of good pitch. The downside is that some instruments are stuffier than others are. If you were considering purchasing one for your band inventory, it would be wise for a competent euphonium player to select a suitable instrument. If a compensating euphonium cannot fit the budget, a standard four-valve system can suffice. However, there are the inherent tuning problems regarding the 12, 23 valve combinations, as well as any combination involving the fourth valve.

When purchasing a four-valve instrument, it is advisable to look for an instrument with the fourth valve located independently on the side of the instrument. This valve will be operated by the right index finger. Models that have all four valves in a cluster alignment inhibit finger dexterity in advanced players.

Tubas purchased for the band and orchestra programs should have four valves. The choice between piston and rotary valves is arbitrary, as there are advantages to each. Though compensating models of tubas are manufactured, it is not necessarily the best choice. For piston action instruments the American design, that which places the plane of valve travel perpendicular to the corpus of the horn, is the most desirable. The traveling distance of the piston valve is greatly reduced and the fourth finger will have more dexterity in performance.

I recommend choosing a tuba (piston or rotary) that offers the opportunity for the tubist's left hand to grasp the extended first and third valve loops in order to adjust intonation. Many European tubas have this feature and now are being copied by some domestic manufacturers. Tubas that have valve tuning slide loops that drape down to the bottom should be avoided, unless they are compensating models. Recommended models are those with tuning loops that rise upwards allowing for ease of adjustment by the player. It is also necessary to have these slides readily mobile. This can be accomplished in the same manner as recommended for the trumpet's third valve slide.

Natural Intonation vs. Equal Temperament Intonation

There is a dichotomy with respect to intonation regarding all instruments. Younger students notice that tuning in ensembles is distinctly different than when performing with a fixed pitched keyboard instrument. This because equal temperament is a stopgap measure that allows keyboards to play in any key. However, this adjustment bends the natural intonation found in physics. In my earlier discussion of Pythagorean tuning, a diatonic scale was devised by a system of mathematical ratios using the pure fifth 3:2. However, once the cycle is completed the return to the home pitch is not exact. The return to the tonic is flat, thus the meaning of the term "Pythagorean comma". However, our ears hear such tuning as proper, allowing for summation and difference tones to be produced in performance.

In later developments other tuning systems such as Just Intonation (use of the pure fifth and the natural third) developed, however this was not fully satisfactory to keyboard instruments. It was not until the system of equal temperament could keyboards perform in all 24 keys. In equal temperament, the octave is divided into 12 equal semitones. Each semitone is divided into pitch increments of 100 called cents. Thus, there are 100 cents between successive half steps. Within the octave, some chromatic tuners will identify pitch tendencies by displaying the cents flat or sharp. Pitch variance of plus or minus five cents from a central pitch is indistinguishable by the ear. This dichotomy between equal temperament and pure intonation, hereafter called natural intonation, explains the difficulty a wind or string player has in playing with a keyboard instrument. As performers, we naturally want to hear natural intonation.

Steve Colley, developer of Tune-Up Systems, states in his intonation workbook that "pure intonation is based on a tonal "context" developed from a single pitch, the tonic that acts as the "anchor" for the key. All other pitches are measured and judged to be correct by the lack of beats between the tonic and interval." When using natural intonation, instrumentalists want to eliminate the beats in a composite sound. According to Christopher Leuba, "beats are the periodic swelling of a sound followed by its dying away, as a result of two frequencies first coinciding with each other and going out of phase." One can readily hear this when matching pitch on unisons and octaves.

This same phenomenon occurs with other intervals. The inclination of a competent musician is to eliminate such beats. In doing so, other phantom overtones appear that make the composite sound rich and ringing. In natural intonation, fifths need to be stretch and fourths need to be slightly lowered. In addition, major thirds and major sixths need to be placed lower. Conversely minor thirds and minor sixths need to be raised. The leading tone of the scale creates controversy. Performers may perceive that this leading tone needs to be significantly raised, but in fact it is best if this is lowered. More problematic is the interval of the minor seventh. When used as the seventh of a dominant chord, this note needs to be lowered more than a quarter tone even though this same note is only slightly sharp when occurring as the fourth scale degree from the tonic. A comparison of pitch tendencies between equal temperament and natural intonation within a given scale is as follows. Notice the deviation in the "cents."

Major Scale Step Intonation

Equal Temperament                          Natural Intonation          Deviation from ET I 

I                              0                                   0                                      0 
II                          200 cents                       204 cents                           4% higher 
III                         400 cents                       386 cents                          14% lower 
IV                        500 cents                       498 cents                           2% lower 
V                         700 cents                       702 cents                           2% higher 
VI                        900 cents                       884 cents                          16% lower 
vii  (7th of V7)      1000 cents                       971 cents                          29% lower 
VII (leading tone)  1100 cents                     1088 cents                          12% lower 
VIII                      1200 cents                     1200 cents                           0 

Equal Temperament            Natural Intonation         Deviation from ET 
I                     0                                   0                               0 
II                 200 cents                      204 cents                      4% higher 
iii                300 cents                      316 cents                    16% higher 
IV               500 cents                      498 cents                      2% lower 
V                700 cents                      702 cents                      2% higher 
vi                800 cents                      814 cents                    14% higher 
VII (L.T.)    1100 cents                     1088 cents                    12% lower 
VIII            1200 cents                     1200 cents                     0