Title: Loudness' We start with pure tones or a pure frequency'
1Loudness. We start with pure tones or a pure
frequency. The intensity, a physical quantity, is
related to to the amplitude square and for sound
waves, pressure waves, to the variation of
pressure. In fact the precise formula is
I 0.00234 . (?p)2 W/m2 ,
and the pressure expressed in Newtons/m2 The
faintest sound we can perceive at 1000 Hz is
10-12 W/m2 which means a pressure variation of
2.10-5 Newtons/m2 which is 2.10-10 normal
atmospheric pressure! A typical musical
instrument has a typical output, assuming a
spherical wave emission (not really true)
between 0.01 Watt for a clarinet to 6.4 W for a
trombon playing fortissimo. This is the output in
all directions. Given a tone of a given
frequency there are two important concepts
the threshold of hearing or lowest audible
intensity and the upper limit of hearing beyond
which one feels pain.
2Instruments by family
Pianoforte family lute, koto,viguela, cembalo,
harpsichord, harp, pianoforte, modern electronic
versions of some of these Violin family violin,
viola, cello, counterbass Woodwind
familyclarinet, oboe, basson, flute, recorder
(in many forms) harmonica Brass
instrumentsclarinet, oboe, basson, horn,
saxophon, shofar Percussion triangle, celesta,
drums, cymbal, bells, Organ on a class if its
own Electronic many of the above plus ondes
Marthenot Voice a reed instrument classified as
its own class
3instrument family
4Plucked instruments
5Plucking mechanism for harpsichord
6Radiated spectrum of harpsichord
7Harp mechanism
8Guitar radiation zones
9Guitar structure
10Guitar vibration modes
11Vibrations when bracing is introduced
12Modes of the complete system
13waveforms
14Frequencies of different elements
15Time and frequency analysis
16More on time
17Piano generalities
- f 1/2LvF/µ
- This contains the three Mersenne laws
- Frequency is inversely proportional to length.
- Frequency is inversely proportional to mass
density per unit length (µ) - The frequency is proportional to the tension.
- These considerations determine how the modern
piano is built - 243 strings varying in length from 2m to 5 cm.
- 8 are wrapped in a single wire envelope, 5 pairs
also wrapped , - 7 made of 3 wrapped strings and 68 of a r
unwrapped ones. - This makes a total of 88 notes or a bit more of
seven octaves.
18Piano history
19Piano history 2
20Piano history 3
21Piano history 4
22Piano action
23Science helps striking
24Details of hammer striking
25Subtleties in hammer structure
26Piano touch and sound dependence
- The piano is the upmost example of an instrument
related to a given music period. The relation of
the player to the instrument is strong. - However, is the touch relevant? Pianists pay
attention to the way the notes are pressed but in
fact as we will soon see amplitude and frequency
depend only on the velocity of the hammer. The
loudness and the timbre are intimately linked
because the strength of the hit determines the
excitation of the harmonics. As usual there is
more to it since there are nonlinear effects due
to the deformation and inelasticity of the felt.
In short, hitting harder does increase the power
but also increases the proportion of higher
harmonics and therefore - The timbre is more brilliant.
27Piano spectra as a function of hammer hardness
28Diagram of a piano
29Piano resonance cavity modes
30Piano inharmonicity
31String and hammer inharmonicity
String inharmonicity and hammer effects
The hammer gets deformed striking the chord and
it is not anymore a point contact, this
introduces inharmonicity. The string reflects
waves to the hammer and these in turn combine to
produce an effect. The string should be as tight
for power reasons see below and as thin as
possible to avoid inharmonicities. In fact
with B0.004
The inharmonicity at the 17th partial is 1
which is quite audible.
B p2ESK2/TL2
Where E is Young modulus, T the tension, S the
area, K the radius of giration
32Decay time
33inharmonicity
34 Position of the notes
35Parallel and perpendicular vibration decay
36Tuning and inharmonicity
37Radiated power and position of source
38Piano radiation pattern