Brass instruments are among the oldest of all instruments. In antiquity, instruments such as the Scandinavian lur and the Roman buccina admirably fulfilled their ceremonial and musical functions. As each epoch unfolded, instruments were modified to serve the musical requirements of the new era. The line of development from ancient to modern is a process of refinement of a basic idea: the sounding of a flared tube through the vibration of the lips. While the outward appearance of the instruments has changed, their internal operation is unaltered from a millennium ago.

In years past brass players often conceived of the instrument as something like an old-fashioned phonograph horn or megaphone which amplified a buzzing sound made by the lips. Now, due to the research of acousticians such as Arthur H. Benade,¹ such conceptions are known to be false. Actually, the lips do not make an audible sound, nor does a horn function like an amplifier to radiate sound into the surrounding environment. The bell flare of a brass instrument is designed to contain acoustical energy within the instrument in order to set up standing waves at specific frequencies.

The player's embouchure may be seen as a flow-control valve acting on the steady air flow coming from the lungs. Puffs of air are emitted into the mouthpiece, setting in motion a sound wave which eventually reaches the instrument's expanding bell. As the bell flare widens, the wave encounters a drop in impedance (resistance) which, perhaps surprisingly, causes it to reflect back toward the mouthpiece (Fig. 1.1). It is then reflected at the mouthpiece where it is modified by the motion of the lips, encouraging a specific frequency. The vibratory motion of the lips is itself modified by the reflecting wave so that its pattern of vibration corresponds to the instrument's timbre and the desired pitch. As the wave bounces back and forth while interacting with the instrument and the vibrating lip, the standing wave characteristic of brass instrument sound is gradually formed. In reality, the process takes only a few hundredths of a second.

Although some acoustical energy leaks through the "barrier" in the expanding bell flare, most is reflected in middle and low frequencies. As frequencies rise, the reflective threshold moves ever closer to the mouth of the bell and less energy is reflected. This is the reason why high notes are more difficult to play than pitches in the middle register.

Figure 1.1. Approximate point of wave reflection in a horn bell.

THE HARMONIC SERIES

Notes of the harmonic series are familiar to all brass players since a certain amount of practice time is usually devoted to studies based on them. Prior to the invention of valves, these were the only notes available to the natural trumpet and horn, although a technique of handstopping (used after 1750) allowed hornists to fill in the gaps between partials, What is not often recognized is the importance of the bell flare in deriving a usable harmonic series. If one attempts to play the overtone series on an appropriate length of cylindrical pipe, such as a garden hose, the following series will result:⁴

Another important aspect of the harmonic series is that overtones of the series also sound in greater or lesser degree when a note is played. This is what defines the characteristic tone quality of an instrument. Also, notes with less sharply defined resonance peaks (making these notes more difficult to produce) are made more stable by the participation of other harmonically related peaks when the instrument is played at medium and loud dynamic levels.

Since the 7th, 11th, 13th, 14th, and 15th partials of the harmonic series are not in tune within the equal temperament system in use today, they are substituted by valve notes. In the interest of finger dexterity, the 1-2 combination, which also lowers the fundamental a tone-and-a-half, is normally used in place of the third valve alone. A basic problem of the valve system is inadequate tube length when the valves are used in combination, causing sharpness. Various approaches are used to correct this deficiency.⁷

Piston valves offer a light, quick action but have slightly less direct and accurate windways than the other two types. A shorter finger stroke may be used on rotary valves, but their action is not quite as immediate as piston valves. An advantage of the rotary type is that the diameter of the windway is maintained with somewhat greater consistency, providing less resistance. Vienna valves (now found only on Vienna horns) cause the least disturbance to the air column enabling the Vienna horn to play and sound more like the natural instrument. With the valves closed, the air column goes straight through the valve section, avoiding the sharper angles and misshapen windways inherent in rotary and piston designs. Although their action is not quite as fast as other valve types, Vienna valves contribute greater fluency and smoothness to slurred passages.⁸

In using any type of valve, it is important to recognize that there are only two positions, open and closed. Therefore, valves should always be depressed as quickly as possible to avoid an audible discontinuity between notes. In slow passages, students often have a tendency to move their valves sluggishly. This produces an unattractive sound, particularly on slurs. Sometimes placing the finger tips slightly above the valve caps or levers encourages a quicker motion.

DESIGN CONSIDERATIONS

Every brass instrument consists of four basic parts: the mouthpiece with its tapered backbore, a conical leadpipe, a section of cylindrical tubing containing the valves, and the gradually expanding flare of the bell. The diameter of the bore, the shape of the tapered sections, the thickness and type of material, and overall mass are variables that cause instruments of the same type to play and sound differently.

Bore size is determined by the diameter of the tubing of the instrument's
cylindrical section, although the bell throat and leadpipe usually conform to the main bore. Instruments of smaller bore generally respond with less effort and have a lighter tone. While their timbre is exceptionally pure, they can be overblown at high dynamic levels. Large-bore instruments typically have a darker tone and retain a more even timbre from soft to loud.

How the bell is shaped is of primary importance in determining the quality of
an instrument and the character of its timbre. The size of the bell, how sharply it is flared, and especially the diameter and taper of the bell throat strongly influence tone, intonation, and response. The rate of expansion of the bell section from the valves onward also has a significant effect on timbre. (Bell tapers are discussed in relation to the horn in Chapter 4). How the bell is made is a factor governing the overall quality of the instrument. The finest bells are formed from sheet brass which is beaten on a mandrel and spun on a lathe by hand. This requires the skill of an expert craftsman and is reflected in the instrument's price.

All brass instruments consist of a tube, at one end of which is a mouthpiece shaped so that the player can make an airtight seal when the lips are placed against it. The acoustical properties of brass instruments depend on the interactions of the player (in particular tile oral cavities and lips), the air column inside the instrument, and the ambient air at the other end of the instrument. The column of air inside the tube is set into vibration when it is excited by the player buzzing his/her lips placed against the mouthpiece. A sustained sound on a brass instrument requires standing waves i.e. sound waves traveling from one end to the other and reflected from each end like water waves in a bath. Although the player opens his/her lips by blowing air through them, because he/she is buzzing his/her lips they are effectively closed for enough of the time to reflect most of the sound waves traveling towards them through the instrument. Whether the other end of the instrument terminates abruptly (as in a bugle) or terminates in a flaring bell (as in a trumpet), sound waves are reflected by the bell mouth or by the flare. The sound inside an instrument is much more intense than the sound produced by the instrument in the surrounding air. The bell of an instrument has to be carefully designed so that it reflects enough sound to allow standing waves to build up, yet allows enough sound to escape to be audible at an appropriate intensity to be useful in music. For this reason, brass instrument bells are of a limited range of patterns - one shaped like a gramophone horn, for example, would not work. 

The standing waves lose some of their energy to the ambient air as audible sound, some in friction with the walls of the instrument, and also a small part to the player's lips, which are coerced to vibrate at a frequency to some extent dictated by the instrument. At the same time, a player adds energy to the vibrating air column at just the right
frequency by blowing through the buzzing lips to replace the sound energy being dissipated. 

The air inside a brass instrument. which is effectively closed at one end by the lips and open al the other, can sustain standing waves at certain quite well-defined frequencies, known as the frequencies of the 'modes of vibration' or the air column. If the frequency of the wave is very slightly higher or slightly lower than one of these frequencies, standing waves are still possible, but will be weaker. These mode frequencies form a series which is more extensive for a narrow tube such as in a french horn or a natural trumpet than for a wide tube such as in a bugle or an ophicleide.  For a perfectly conical tube, the frequencies would correspond numerically to a harmonic series, which is defined as a series of numbers (here, frequencies) which are exact integer multiples of the lowest member (the fundamental). For a perfectly cylindrical tube, the frequencies would correspond to the odd-numbered members of a harmonic series. Real brass instruments are neither perfectly cylindrical nor perfectly conical, and the modes of vibration depend on the internal shape of the instrument. Tubes are musically most useful if several of the frequencies of several of the modes of vibration approximate to members of a harmonic series. In the case of instruments with large cylindrical portions of tubing such as trumpets and trombones, the mouthpiece and bell need to be carefully designed to make this possible. Even so, the lowest one or two members of the series of modes of vibration of trumpets and trombones diverge considerably from the harmonic series. The art of the brass instrument maker is to give the modes the most advantageous frequencies, strengths and tolerances.

When a sustained sound is produced on a brass instrument, the air inside the instrument vibrates not only at the frequency of vibration of the player's lips, but also at exact integer multiples of this frequency.  These are the spectral components of the sound, sometimes called 'overtones'; the lowest component (whose frequency is that of the lip vibration) is the fundamental. The frequencies of the spectral components of the sound when a sustained single note is being played without vibrate form a harmonic series. The sound which escapes from the bell of the instrument also contains these spectral components, and it is the relative strengths of these components that determine the timbre of a sustained sound on the instrument. However, different notes played on the same instrument will have different spectra: a high note may have a significant amount of acoustic energy in only two or three components whereas a low note may have a rich spectrum with significant amounts of energy at fifteen or more frequencies. It is always easier to distinguish two brass instruments by comparing low notes than high. Loud notes not only have energy at each spectral component, but also a more extensive spectrum.  Because of this, a recording of a loud note can be recognized as such even if reproduced at low volume. 

The series of fundamental frequencies of the notes which can be sounded by a player form only an approximation to a harmonic series, though they are sometimes loosely called 'the harmonics'.  If the frequencies of the modes of vibration of the air column formed a harmonic series, then the 'note center frequencies' available to the player would also form a harmonic series. However, this is an ideal case and the behavior of real instruments is more complicated. In order for the instrument to 'speak' and produce a 'well-focused' sound, several of the harmonics of the note being played need to resonate with modes of vibration of the tube. In most cases, the fundamental of the note is very close to one of the mode frequencies; in addition, to produce the tone quality expected of brass instruments, its spectral components (harmonics) also resonate with higher modes of vibration of the air column inside the tube. The interaction of the harmonic components of the sound with the air column, termed a 'cooperative regime', is a strong effect. On the one hand, a co-operative regime can allow a sustained sound even if the fundamental does not match a mode of vibration of the air column - this is how a trombonist can sound a pedal note or a tuba player can sound 'factitious' notes not in the usual series (see Table 2). On the other hand, if the modes of vibration have a poor match with the harmonics of a note a player is sounding, the note will be 'stuffy' in quality, difficult to produce, and possibly out of tune. Since the air column can sustain standing waves at frequencies very slightly higher or lower than the mode frequencies, the player has some latitude to' lip' a note up or down in pitch and to use pitch vibrate.

In the case of wide-bore signaling instruments such as bugles, there are only a small number of modes of vibration of the air column which are of sufficient strength to contribute to the generation of sustained sound; therefore the 'co-operative regimes' are less extensive than those which allow in-tune production of the lower notes of narrow-bore instruments such as the french horn and the trombone; as a result many instruments of the bugle family are not well in tune. In the case of instruments with tone-holes such as cornetts and serpents, the situation is complex: the series of notes which can be produced with a given fingering are not generally a close approximation to a harmonic series. On these instruments, for example, a note and a note an octave higher usually have different fingerings.

So far, we have discussed only sustained sounds. In order to sound a note, the brass player has to set higher lips in vibration, sending a pulse of sound towards the bell. By the time the initial sound is reflected back and can interact with the lips to establish a stable sustained sound, the lips will have gone through at least one cycle (many cycles for high notes). A large part of the skill of the brass player consists of the ability to buzz the lips at the right initial frequency; it is to acquire this ability that many teachers recommend practice on the mouthpiece alone. The length of unsupported time is longer for a given note on, say, a natural trumpet in 7ft D than a piccolo trumpet in 2~4ft Bb. With a longer tube length, the nearest playable notes above and below the desired note are closer in pitch than with a short tube length. These are the reasons why the trend in instrument design since the invention of the valve has been to make shorter instruments.

The sound characteristics of instruments depend to a large extent on their behavior in the initial build-up of a note. If tape recordings have these' starting transients' cut out, it is almost impossible to identify the instrument being played, sometimes even to tell if it is wind or string.  Another characteristic of an instrument can be the presence of formants.  These are regions of the spectrum where the components are consistently strong regardless of the exact fundamental frequency of the note being sounded. Formants are the mechanism whereby vowels can be recognized in speech and song; they make an important contribution to wood wind character, and are less important for brass but still significant.

Opinions differ as to the importance of the material of a brass instrument. The vibration of the walls has little effect on the sound spectrum produced by a brass instrument, and the character of what the listener hears is principally determined by the shape of the bore profile of the instrument and of the oral cavities of the player. Factors such as material and wall thickness may in some cases have effects that can be sensed by the player, who is in physical contact with the instrument and who perhaps hears sound radiated from the body of the instrument. The bore profile, however, is the principal determinant of the character of the instrument - for example, whether it is a french horn, a flugelhorn or a saxhorn.