Brass Instruments, Tuning and Harmonics
Created | Updated Feb 9, 2008
The brass family is one of the oldest families of musical instruments. Confusingly, not all members of the family of 'brass instruments' are made of brass, and not all instruments made of brass are in the family. An instrument is a 'brass instrument' if the sound is produced by using the player's lips as a sound generator. The family includes instruments such as trumpets, trombones, tubas (which are in fact normally made of brass) and also a few odd ones such as the didgeridoo and the alpen horn.
Overview: Wind instruments
Wind instruments are characterized by the fact that the tone is produced by oscillation of an air column. The pitch is determined by the length of that column.
There are three different families of wind instruments1: 'flutes'; 'reeds' and 'brass'. They differ from each other in the way the tone is created:
- Flutes: (in the very broadest sense) create a tone by guiding a stream of air over a sharp edge. This can be done by blowing into the instrument (recorders, ocarinas, ...) or by blowing over the instrument (transverse flutes, panflutes, ...).
- Reeds: have a little piece of solid material (wood or reed, hence the name) that vibrates in the air flow, thus producing an air vibration within the instrument. There are double reed instruments (oboe, bassoon, cor anglais) which have two vibrating reeds tied together, and single reed instruments (clarinets, saxophones) which have only one vibrating reed attached to a mouthpiece.
- Brass Instruments: create their tone by the vibrating of the player's lips at the mouthpiece.
Brass Instruments - a closer look
The first brass instruments (which were not even made of brass, but of horns or conch shells) had a fixed length. The player had only a very limited choice of tones to play, all of which are harmonics. To understand the concept of harmonic notes a little physics and a little mathematics are needed. All the following calculations are simplified as much as possible, but the principle should shine through the formulas. Most of them are also valid for reeds and flutes.
Are tube length and frequency range somehow correlated?
Yes, they are. The longer the tube, the deeper the sound. Sound travels at a speed of approximately2 1,000 ft/sec. If a tone has a frequency of 440 cycles/sec, or 440Hz, the wavelength of this tone is calculated as follows: 1,000 ft/sec divided by 440 cycles/sec = 2.272727... ft.
Tubes do not create a sound, they just reinforce it. A tube of a certain length will not reinforce all notes of every possible frequency, but just notes of particular frequencies. There has to be an exact ratio between the tube length and the sound wave length. Brass instruments are open tubes, that is, they are open at each end, so the largest sound wave that will fit in them is one whose wavelength is exactly twice the length of the tube. To put it the other way around, the tube length must be half the length of the wave we want to produce.
Wavelength and frequency are related in that the wavelength times the frequency gives the speed of the wave. With a speed of sound of 1,000 ft/sec and a frequency of 440Hz (the 'A' produced by a tuning fork), the tube must be 1.136 ft in length. This calculation neglects some factors such as the width of the tube and its exact shape, but these don't have a huge effect on the result. So instruments can be made to play in different keys by making their tubes different lengths. A typical trumpet plays in B flat with a tube length of about 4.5 feet. You can also buy a C trumpet which is pitched slightly higher and has a correspondingly shorter tube.
How is it possible to produce different pitches without changing the tube length?
Take a fictional open tube with a length of 5 ft. Its basic tone would have a frequency of (1,000 ft/sec divided by 5 ft) / 2 = 100Hz. If you trigger3 the air in that tube with a frequency of exactly 100Hz, the resulting wavelength is exactly twice the length of the tube, so the wave fits exactly or 'stands' in the tube. If the frequency of the sound source is gradually raised, the resulting wave length will be shorter, no longer fitting exactly. The waves within the tube are no longer 'standing', but floating back and forth, interfering with each other and thus extinguishing themselves. When the sound source reaches a frequency of 200Hz, the wavelength will be 5ft. One of those waves will fit exactly ('standing') into the tube, which means that the first harmonic4 of that special tube has been found. Raise the pitch to 300Hz and get a wavelength of 3.333... ft, which stands in the tube three times, and so forth.
Why are these correlated pitches called 'harmonics'?
If two of the harmonic pitches are played simultaneously, say on two different trumpets, they tend to sound good together, whereas two non-harmonic pitches tend to sound bad. Example: a 100Hz tone and a 200Hz tone have an interval of exactly one octave and sound absolutely harmonic. Lots of untrained ears will not even notice that they hear two different tones. A 200Hz and a 300Hz tone form a perfect fifth, which is also one of the pure intervals. As the ratio between the frequencies gets more complex, the tones will start to sound less well together. A ratio of 3:4 is known as a 'perfect fourth' and it still sounds pretty good, but not as good as the 2:3 we've already seen. By the time you get to a ratio of 15:16, the notes are no longer considered to be 'harmonic' together. This interval is called a 'semitone'. Playing two notes a semitone apart is a good way of giving someone a headache.
These notes form an array of tones like a ladder or 'scale', the so-called natural harmonics scale. This scale starts with a really big step (one octave), followed by a smaller one (a fifth), again a smaller one (a fourth), again a smaller one (a major third), and so forth. This means that no matter what frequency you start with, the note an octave higher will always have twice that frequency, the note a perfect fifth higher will always have 3/2 times the frequency, .... The following table shows frequency ratios and the corresponding intervals:
| Frequency ratio | Corresponding Interval |
|---|---|
| 1:1 | Unison (i.e. no interval) |
| 2:1 | Octave |
| 3:2 | Perfect Fifth |
| 4:3 | Perfect Fourth |
| 5:4 | Major Third |
| 6:5 | Minor Third |
| ... | ... |
A brass instrument with a fixed length pipe pitched in the key of C will be able to play the following tones:
| Harmonic | Tone |
|---|---|
| 0 | C |
| 1 | C |
| 2 | G |
| 3 | C |
| 4 | E |
| 5 | G |
| 6 | Bb (slightly flat5) |
| 7 | C |
| 8 | D |
| 9 | E |
| 10 | F# (very flat6) |
| 11 | G |
| 12 | A (very flat) |
| 13 | Bb (slightly flat) |
| 14 | B |
| 15 | C |
This first tone in this table, denoted by 0, is the fundamental tone, whose wavelength is twice the length of the pipe. This note is called the pedal tone, but it is quite a difficult tone to reach on a typical instrument such as a trumpet, and the normal lowest note of the instrument is considered to be the one denoted as '1' in the table. It can be seen that even with such a limited instrument, a scale can be played, although the some of the notes will be out of tune. This must be compensated for by changes in the tension in the player's lips.
This is the challenge that presented itself to brass instrument players up to about 1800 - the natural brass instrument (with the exception of the trombone) had a fixed length, so the notes produced fell into the natural harmonics series given in the table above.
How can all the other notes which are not in the harmonic scale be played?
Before the invention of valves or slides (as used with the slide trombone) the answer was quick and simple: They can't. This was an enormous drawback both for composers and musicians. The key to the other notes was the length of the tube. Instead of changing to another instrument (with different tube length and different natural scale) in the middle of a tune, which would have required acrobatic skills, valves which opened some extra tubes were invented. The modern standard trumpet, first built around the year 1815, has three valves, which allow the player to lower the pitch of the whole natural scale by anything from one to six semitones7. From the 1st harmonic up to the top of the range, the biggest gap in the scale is between the 1st and 2nd harmonics, a gap of seven semitones. By using the valves, any note can be lowered by up to six semitones, so this gap is filled in. It is therefore possible to play a complete chromatic scale over the full range of the instrument.
The following table shows the fingerings for a chromatic scale, played on a Bb-trumpet, which is the standard today:
| Tone8 | Fingering |
|---|---|
| Bb | no valve |
| B | 1+2+3 |
| C | 1+3 |
| Db | 2+3 |
| D | 1+2 (or 3) |
| Eb | 1 |
| E | 2 |
| F | no valve |
| Gb | 2+3 |
| G | 1+2 (or 3) |
| Ab | 1 |
| A | 2 |
| Bb | no valve |
Brass Instruments
French Horn
The French horn is related to the signalling horns that were (and still are) used for hunting. It has inherited the typical round form, but it has been equipped with a sophisticated set of valves. Today's standard French horns are double horns, which means that they can be turned from a F-instrument into a Eb-instrument at the touch of a valve. They have, of course, the standard set of three, sometimes four valves. It's the only brass instrument where the valves are operated with the left hand9. The right hand is stuffed into the bell of the horn, which contributes to the soft, mellow sound. While the hand does affect the sound, rather like a mute, it's also used to bend the pitch slightly.
The French Horn as originally designed and built had no valves. It came with several extra pieces of tubing (called 'crooks' because they looked a bit like the top of a shepherd's crook). To achieve chromatic notes that were not in the original harmonic set, the player had to remove a piece of tubing (a crook) and replace it with another one of slightly different length ... preferably in between playing two notes!
This was the arrangement in Mozart's time. He was aware of it, and wrote deliberately challenging parts for French Horn. The best-known is (probably) the third movement of the fourth Horn Concerto, which is quite difficult even with a valved horn. On the manuscript copy of the horn part, Mozart added comments for the player, including:
I bet you can't play this bit!
Trumpet
Trumpets are the highest pitched brass instruments. Their tube has a standard length of 4½ft. The sound of a trumpet can be martial or festive, but also very subtle when played with a mute. There are two basic types of trumpets, one with rotary valves, the other one with piston valves.
Cornet
The cornet is very often mistaken for a trumpet. The differences are in fact very small. The cornet has a wider bore with a slightly different shape, which causes a slightly softer tone. One of the best known cornet players was 'Satchmo' Louis Armstrong, who was also an accomplished trumpet player.
Flugelhorn
Even the flugelhorn is sometimes mistaken for a trumpet. The differences between the flugelhorn and the trumpet are obvious, however: The flugelhorn has a wider bore, which makes the instrument significantly larger in size. The tube length, however, is still the same (approx. 4½ft.) as the trumpet's. To be exact, the flugelhorn should not be seen as a wider trumpet, but as a soprano tuba, as its conical bore is typical for the tuba family.
Tenor horn and Baritone
These two horns (the baritone is a baritone horn) have a fairly straight bore, similar to the bore found in a cornet. They have poor lower ranges and volumes, and so are frequently used as backing.
The Euphonium
The Euphonium is essentially a small tuba, with a wide conical bore and an excellent lower range.
Tuba
The tuba is the largest brass instrument, playing the lowest notes. It comes in different shapes (tuba, helical sousaphone) and tunings (F-tuba, Bb-tuba, C-tuba, Eb tuba). Deeper tubas are commonly noted as BBb and EEb, and there have been some even deeper tubas which were largely experimental.
Trombone
The trombone is the only chromatic brass instrument which doesn't usually have valves. The length of the tube is changed by a slide at the bottom of the instrument. Sometimes a valve trombone is seen, too, which has a regular set of valves instead of a slide. Trombones come in different voices. There is even a soprano trombone which has the tonal range of a trumpet (experts say that it's in fact a slide trumpet). The most common one is the tenor trombone which has the tonal range of a baritone horn. Some trombones have additional valves ('triggers') which are operated with the player's thumb, enabling tenor trombones to be played as bass trombones, and making lower registers easier on the arms.
'Wooden' Brass Instruments
The Australian didgeridoo is just a long, more or less straight wooden pipe without any moving parts. The tone is created by the players lips, along with some singing and grunting into the instrument. This causes the specific, magic sound of the 'didge'.
The European 'alphorn' is a really long straight wooden tube with a curved-up muzzle. Typical alphorns have an average length of 15ft.
Brass Instruments made of animal horns or Conch shells
Sometimes animal horns or conch shells are used as wind instruments. As their length and hence their pitch is a quirk of nature, they hardly fit into a standard musical context. Some players, especially Steve Turre, have pushed the conch shell performance beyond the limit of what was imaginable before.
