Brass Instruments

The Physics of Musical Instruments

Class Notes – October 31

Pipes driven with air pulses; Brass instruments

The Physics of Musical Instruments – Fall 2017 – October 31 – 1 Topics

• Self-sustaining vibrators driven by puffs of air • Effect of the mouthpiece • Recipe of the air puffs • Privileged frequencies – the bugle • Slides and valves • Recipes of real brass instruments • Trumpet demonstration

The Physics of Musical Instruments – Fall 2017 – October 31 – 2 Elements of a brass instrument

• Player’s lips • Mouthpiece • Mouth pipe • Long tube Pulses of air from the player’s lips combine with • Bell reflected pressure wave from the impedance changes at the bell end get synchronized into a sustained vibration with rich harmonic content. This is a NONLINEAR, hence, complicated, process.

The Physics of Musical Instruments – Fall 2017 – October 31 – 3 A simple model

• The “water trumpet” has all the components

This is a terrific model. The key thing is that the pulse of water enters when the water (analogous to the air pressure) is high.

The model doesn’t show how or why the water wave gets reflected at the bell end. In an actual trumpet, this reflection is produced by the changing acoustic impedance in and at the end

of the bell. The Physics of Musical Instruments – Fall 2017 – October 31 – 4 Adding air at high pressure is critical • Without new air the vibration dies out • Air added at the peak strengthens the wave • Air added at the trough damps the wave

The Physics of Musical Instruments – Fall 2017 – October 31 – 5 So how do the lips do it?

• If the lip vibration frequency is higher than the air column fundamental frequency, the lips can keep up with the air column resonance and add air to build up the resonance

• If the lip vibration frequency is lower than the air column fundamental frequency, it can’t keep up with the air column resonance and adds air so as to damp out the resonance

• We will demonstrate this with the bugle (later)

The Physics of Musical Instruments – Fall 2017 – October 31 – 6 Topics

The Physics of Musical Instruments – Fall 2017 – October 31 – 7 Mouthpieces

• A brass mouthpiece has a cup for the lips and then a transition into the tubing of the instrument • Trumpet, trombone, tube – A cusp in the mouthpiece • French horn – A smooth taper (much more difficult to excite vibrations compared to the others)

The Physics of Musical Instruments – Fall 2017 – October 31 – 8 Air column resonances without a mouthpiece • The acoustic Trumpet tubing impedance at the entry to the instrument displays the modes of the air cavity as maxima in the impedance (it’s a closed pipe)

The Physics of Musical Instruments – Fall 2017 – October 31 – 9 Air column resonances with a mouthpiece

Trumpet tubing • The mouthpiece shifts all the resonances to slightly lower frequency (added volume) and modifies the strength of the various modes

The Physics of Musical Instruments – Fall 2017 – October 31 – 10 Topics

The Physics of Musical Instruments – Fall 2017 – October 31 – 11 Air puffs have their own recipe

• Examples from the text – Top – 1/20 of a cycle

– Middle – 1/4 of a cycle

– Bottom – 1/8 of a cycle

The Physics of Musical Instruments – Fall 2017 – October 31 – 12 Putting it together

• The pipe itself has many modes • The mouthpiece shifts the modes • The air puffs have their own recipe that may or may not match the pipe modes • AND – the modes that do get excited all have to synchronize with the same set of air puffs • The math for this is WAY BEYOND our scope – it is “oscillator theory” – nonlinear differential equations • The result is a complex waveform that contains those modes that are capable of synching with the recipe components of the air puffs

The Physics of Musical Instruments – Fall 2017 – October 31 – 13 Topics

The Physics of Musical Instruments – Fall 2017 – October 31 – 14 Privileged frequencies

• A cylindrical pipe closed at one end has only odd harmonics • But the system is nonlinear • Other frequencies can enter the game and be part of the resulting waveform

The Physics of Musical Instruments – Fall 2017 – October 31 – 15 What are privileged frequencies? • Privileged frequencies are sub-multiples of the actual pipe modes, and there are dozens of them • The key point is that when a bunch of privileged frequencies from different resonances match, those frequencies become possible notes for the instrument • The “pedal” tone of a brass instrument is such a case – it is very hard to play and virtually never used, but it appears as if it were the fundamental of the higher modes

The Physics of Musical Instruments – Fall 2017 – October 31 – 16 Bugle pitches

• Ignore the tuning in this drawing and look just at the series of notes for Position 1 • The pedal tone is not playable • The lowest bugle note is middle C

The Physics of Musical Instruments – Fall 2017 – October 31 – 17 Topics

The Physics of Musical Instruments – Fall 2017 – October 31 – 18 To get a scale, we must change the length of the pipe

• One needs enough length variation to span the first fifth, from middle C to G, and that requires seven half steps. • It can be done with a slide -- trombone • It can also be done with three valves – trumpet, tuba, french horn

The Physics of Musical Instruments – Fall 2017 – October 31 – 19 Trumpet valve

• The plumbing inside a trumpet valve simply adds a short length of tubing when the valve is depressed • There are many compromises to get a decent scale, as discussed in the text

The Physics of Musical Instruments – Fall 2017 – October 31 – 20 Fingering Chart

The Physics of Musical Instruments – Fall 2017 – October 31 – 21 Topics

The Physics of Musical Instruments – Fall 2017 – October 31 – 22 We need to add the bell

• Fall off at low frequencies is the added effect of the mouthpiece • Fall off at high frequencies is the effect of the bell

The Physics of Musical Instruments – Fall 2017 – October 31 – 23 A comparison

The Physics of Musical Instruments – Fall 2017 – October 31 – 24 Topics

The Physics of Musical Instruments – Fall 2017 – October 31 – 25