Brass Instruments - Tuning and Harmonics Content from the guide to life, the universe and everything

Brass Instruments - Tuning and Harmonics

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A tuba player in an orchestra.

The brass family is one of the oldest families of musical instruments. Confusingly, not all members of the brass family 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 generated by the player's lips. 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.

Wind Instruments - an Overview

Wind instruments are characterised 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, such as with a recorder or ocarina, or by blowing over the instrument such as with the standard orchestral flute or with 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 column 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, as stated previously, 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 animal horns or conch shells) had a fixed length. The player had only a very limited choice of tones to play, all of them 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 be evident 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 440Hz3, the wavelength of this tone is calculated as follows: 1,000ft/sec divided by 440 cycles/sec = 2.2727ft.

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 1,000ft/sec and a frequency of 440Hz, the tube must be 1.136ft 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. 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 4.5ft. 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 5ft. Its basic tone would have a frequency of (1,000ft/sec / 5ft) / 2 = 100Hz. If you trigger4 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 in the tube. If the frequency of the sound source is gradually raised, the resulting wave length will be shorter and will no longer fit 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 into the tube, which means that the first harmonic5 of that special tube has been found. Raise the pitch to 300Hz and get a wavelength of 3.33ft, 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 good 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, and so on. The following table shows frequency ratios and the corresponding intervals:

Frequency ratioCorresponding Interval
1:1Unison (i.e. no interval)
2:1Octave
3:2Perfect Fifth
4:3Perfect Fourth
5:4Major Third
6:5Minor Third

A brass instrument with a fixed length pipe pitched in the key of C will be able to play the following tones:

HarmonicTone
0C
1C
2G
3C
4E
5G
6Bb (slightly flat6)
7C
8D
9E
10F# (very flat7)
11G
12A (very flat)
13Bb (slightly flat)
14B
15C

This first tone in this table, denoted by 0, is the fundamental tone, the wavelength of which 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 until the 19th Century: 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 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 for both 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 trumpet, which was first built in 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 semitones8. From the first harmonic up to the top of the range, the biggest gap in the scale is between the first and second 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 standard B-flat trumpet:

Tone9Fingering
Bbno valve
B1+2+3
C1+3
Db2+3
D1+2 (or 3)
Eb1
E2
Fno valve
Gb2+3
G 1+2 (or 3)
Ab1
A2
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 Bb-instrument at the touch of a valve. They have, of course, the standard set of three or sometimes four valves. The French horn is the only brass instrument where the valves are operated with the left hand10. 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 likely 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 and 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 more obvious, however: the flugelhorn has a wider bore, which makes the instrument significantly larger in size. The tube length, however, is still the same (approximately 4½ft) as the trumpet's. To be exact, the flugelhorn should not be seen as a wider trumpet but as a soprano tuba — 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.

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, Bb, C and Eb). 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 generally changed by a slide at the bottom of the instrument, though valved trombones are sometimes seen. Trombones come in different voices. There is even a soprano trombone which has the tonal range of a trumpet, and is sometimes considered to be a slide trumpet. The most common variety 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 player's 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 From Other Materials

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 conch shell performance beyond the limit of what was imaginable before.

1The pipe organ, the biggest instrument in musical history, is a combination of flutes and reeds, as there can be pipes of both families. Accordions and mouth organs might also be called wind instruments. Their pitch, however, is determined only by the length and mass of little metal tongues, not by the length of a resonating air column.2This is not absolutely correct, but it's close enough and easy to calculate.3The frequency of the A above middle C, the note produced by a tuning fork and used as a standard for most instruments.4Triggering involves attaching a 100Hz sound source (like the player's lips) to one end of the tube.5There are two different counting methods: One begins with the fundamental tone as 'first' harmonic, the next one as 'second' harmonic and so forth. The other one refers to the fundamental tone as the 'fundamental', the next one as 'first' harmonic, etc. This entry uses the latter method.6The slightly flat notes are about a quarter of a semitone flat.7The very flat notes are about a half of a semitone flat.8Two semitones with the first valve, one semitone with the second and three semitones with the third one, plus all possible combinations of the three, in which case the semitone downshifts are just added. The fourth valve, which is quite common on tubas, allows a downshift of 5 semitones, thus adding some extra combinations.9Please note that, as an example, Eb stands for D# as well. The same fingering is used on all common valved brass instruments (trumpet, flugelhorn, baritone, tuba, sousaphone, etc). 10The availability of lefthanded instruments which are built 'the other way around' is neglected in this entry.

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