Musical Acoustics

The important acoustical characteristic common to members of the musical brass instrument family is not the material of construction, but the way in which the note is sounded by vibrating the lips against the rim of a mouthpiece. The lips act as a valve, open and closing periodically to modulate the flow of air into the instrument. The resulting pressure changes in the mouthpiece set up standing waves in the air column contained by the walls of the instrument. A small fraction of the sound energy stored in the standing waves is radiated at the bell of the instrument, and this is the sound heard by a listener. The frequency and pitch of the sound depend both on the resonant frequencies of the air column and on the natural resonant frequency of the lips, which can be controlled by the muscles of the player's mouth.

The original Hammond used rotating tone wheels to generate harmonics which were added using drawbars; not true harmonics though. The principle is known as additive synthesis. The organ is usually played through a Leslie speaker which utilizes the Doppler principle to produce chorale and tremolo effects. Modern digital synthesizers such as the Nord Electro simulate both the drawbars and the Leslie speaker. In subtractive synthesis, used for example on the Korg MS20, tones with complex spectra are generated and unwanted components filtered out. The most common transient generator is the ADSR where the letters stand for Attack, Decay, Sustain and Release. The Yamaha DX7 uses Frequency Modulation (FM). Sidebands are generated in the frequency spectrum spaced at multiples of the modulation frequency from the carrier. MIDI allows you to control the sounds of one instrument from the keyboard of another.

In concert halls it’s desirable that the time between the arrival of the direct sound and the first reflection is not greater that 20ms. Highly reflecting parallel walls may cause undesirable flutter echo. The distance from the source where the intensity levels of the direct and reverberant sound are the same is known as the Room Radius; a typical value for a large hall is 5m. Closer than the room radius you hear mostly direct sound and at greater distances mostly reverberant sound. Room modes are particularly important in small rectangular rooms such as recording studios. Tangential modes are combinations of two axial components and oblique modes are combinations of all three. The vineyard design of concert hall gives an even distribution of sound. Seats should be designed to minimize changes in reverberation between the hall being empty and full.

Sound rays obey Snell’s law of reflection. When they strike a surface the fraction of sound energy absorbed is known as the absorption coefficient, which varies with frequency. The time for the reverberant sound in a room to drop by 60dB is known as the reverberation time R. This can be calculated from the formula R = 0.16 V/A where V is the volume in cubic metres and a is the total absorption in metric Sabin i.e. the sum of the surface areas times their absorption coefficients. If R is large the sound in the room will lack clarity but the sound level will be large. If R is small the sound will be clear but the sound level will be low. Typically R will be 1s for a theatre and 1.5-2s for a concert hall.

In Pythagorean tuning the 5ths are true i.e. in the frequency ratio 3/2. In 1/4 Comma Mean Tone the major 3rds are true i.e. in the frequency ratio 5/4. Both tunings give rise to the wolf which restricts the number of playable keys to 6 major and 3 minor. In Equal Temperament all keys are playable but neither the 3rds or 5ths are true. In circular temperaments the total reduction in all 5ths round the circle is one comma; all keys are playable but some usually sound better in tune than others.

The Boehm flute has a cylindrical body and a tapered headjoint, which is required to compensate for the flattening effect of increased lip cover over the embouchure hole in the upper register. Like the saxophone, the clarinet has a mouthpiece and a single reed. It has distinctive tone qualities in the low (Chalumeau) register and higher (clarinet) register. The tone quality of the oboe and bassoon, which both have double reeds, is characterized by distinctive formants.

In a woodwind the sound is generated by the resonating air column within the instrument, the material of construction being of secondary importance. To a first approximation the flute can be considered as a cylindrical tube open at both ends, the clarinet as a cylindrical tube closed at one end and the saxophone, oboe and bassoon as conical tubes closed at one end. All these instruments generate a complete harmonic series and overblow the octave, except for the clarinet which generates predominantly odd harmonics in the low register and overblows the twelfth.

This podcast is about the modern grand piano. A full size piano usually has a compass of 7 1/4 octaves, from A0 to C8. For the notes A0 to A1 there is only one string per note, for the octave above (Bb1 to Bb2) two strings per note and for the notes above these three strings. All strings up to Bb2 are overwound to increase their mass per unit length. The plane and overwound strings have separate treble and bass bridges which are designed to give the correct balance between volume and sustain. Increasing bridge impedance increases the sustain but decreases the volume. The right sustaining pedal raises dampers to allow the strings to vibrate freely. Due to sympathetic vibrations, this affects the sound on even the highest notes which have no dampers. The left una corda pedal shifts the keyboard and action so that two instead of three strings are struck in the higher registers.