Wave, Sound, and Music

Sound

A compressional wave in air can be set up by the back-and-forth motion of a speaker as shown in Figure 09. Here, the air molecules are alternately pressed together and pulled apart by the action of the speaker. The result is a propagating wave in which the pressure (and density) of the air varies with distance in a regular way - the pattern is, in fact, exactly the same as the displacement pattern of a transverse wave on a string (see Figure 01 and 02). Compressional waves in air are called sound waves, which are always longitudinal waves with the vibration parallel to the direction of propagation. Most of the previously mentioned concept about waves can be applied to the sound wave without modification except the formula for the wave velocity in Eq. (3) where the tension is replaced by the "bulk modulus" (change in pressure / change in volume) and the linear density is just the density of the air. It turns out that the velocity of sound at STP is about 330 m/s.

Figure 09 Sound Wave
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Pitch (frequency) -- The term "frequency" is referred to the objective measurable rate of vibration of an object, while "pitch" is the subjective sense of that "frequency" to the human ears. We can hear frequencies ranging from about 20 Hz to 20,000 Hz. The upper range, in particular, decreases substantially with age. Pitch is perceived according to an exponential relationship. For example, a frequency of 200 Hz is perceived as an octave above a frequency of 100 Hz. The frequencies of each octave above a given tone "f₀" is calculated by the following formula:

f = f₀ x 2ⁿ, ---------- (8)

where n = 0, 1, 2, 3, ... describes the octave relationship such as
2⁰ (unison), 2¹ (one octave), 2² (two octaves), 2³ (three octaves), ...

Within the range of an octave, there is a series of frequencies called consonant intervals, which is known to produce the most pleasing sounds to the ear. They are usually combinations of notes related by ratios of small integers, such as the fifth (3/2) or third (5/4). Many musical instruments are tuned according to these intervals. Unfortunately, this kind of tuning depends on the scale - the tuning for C Major is not the same as for D Major. The "equal-temperament" scale solves the problem by dividing the octave into twelve equal intervals, each has a size of 2^1/12 x f, where f is the fundamental or harmonic frequency. It was developed for keyboard instruments, such as the piano, so that they could be played equally well (or equally bad) in any key. It is a compromised tuning scheme.

Figure 10 Musical Scale
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Table 01 depicts the consonant intervals (sometimes referred to as "harmonic tuning" or "Just Scale") in rational number with the corresponding decimal and the "equal temperament" (E-T) scale for comparison. The difference is shown in the last column. It is evident that, the frequencies in "equal temperament" are close but not quiet the same as the consonant frequency. Since the ear can easily detect a difference of less than 1 Hz for sustained notes, differences in scale of 0.001 can be quite significant. The syllable for the solfege and numerical systems of sight-singing is presented in the second column. A list of the frequencies in "equal temperament" scale is shown in Figure 10 from C4 (middle C at 261.63 Hz) to C8. The E-T unit in cents (¢) is defined as 100 cents equal to one equal tempered interval. The number of cents between two frequencies f₁, f₂ is computed by the formula:

¢ = (1200/ln(2)) x ln(f₂/f₁) or f₂/f₁ = 2^¢/1200 ---------- (9).

For example, the difference of frequency in cents between minor third and the E-T interval (with n=3) is (see Table 01):

¢ = (1200/ln(2)) x ln(1.2/1.189207) = 15.6
or the minor third can be expressed in cents by: ¢ = (1200/ln(2)) x ln(1.2/1.0) = 315.6

Note	Syl- lable	Consonant Interval	Ratio	Decimal	E-T Interval (n)/(¢)	E-T Scale, 2^n/12	Difference/(in ¢)
C	Do / 1	octave (fundamental)	1/1	1.000000	n = 0 / 000	1.000000	0.000000 / 00.0
C#		minor second	25/24	1.041667	n = 1 / 100	1.059463	-0.017796 / -29.3
D	Re / 2	major second	9/8	1.125000	n = 2 / 200	1.122462	+0.002538 / +3.91
D#		minor third	6/5	1.200000	n = 3 / 300	1.189207	+0.010793 / +15.6
E	Mi / 3	major third	5/4	1.250000	n = 4 / 400	1.259921	-0.009921 / -13.7
F	Fa / 4	fourth	4/3	1.333333	n = 5 / 500	1.334840	-0.001507 / -1.96
F#		diminished fifth	45/32	1.406250	n = 6 / 600	1.414214	-0.007964 / -9.78
G	So / 5	fifth	3/2	1.500000	n = 7 / 700	1.498307	+0.001693 / +1.96
G#		minor sixth	8/5	1.600000	n = 8 / 800	1.587401	+0.012599 / +13.7
A	La / 6	major sixth	5/3	1.666666	n = 9 / 900	1.681793	-0.015127 / -15.6
A#		minor seventh	9/5	1.800000	n = 10 / 1000	1.781797	+0.018203 / +17.6
B	Ti / 7	major seventh	15/8	1.875000	n = 11 / 1100	1.887749	-0.012749 / -11.7
C	Do / 1	octave (1st harmonic)	2/1	2.000000	n = 12 / 1200	2.000000	0.000000 / 00.0

Table 01 Musical Scale

Note: There are 11 semitones or half-steps (between both white and black keys), and 7 major notes (tones, the white keys) within an octave on a piano for example (see Figure 10).

Loudness (amplitude) -- The greater the energy of vibration, the louder the sound is emitted, and correspondingly the amplitude A in Eq.(4b) has a higher value. A musical sound has an overall loudness, but each note itself has a changing loudness that is called its envelope. Each musical instrument has its own characteristic envelope, and this is partly how we recognize the instrument. Periodic variations in loudness are called tremolo. Sound levels are measured using a unit called decibel. A decibel is not an absolute value, but a comparison expressed by the following formula:

dB = 10 x log₁₀(I₂/I₁), ---------- (10)

where I₂ and I₁ are the intensity of two sounds.

The reference point for the sound level can be taken arbitrarily. Usually it is referred to the pressure level that is just below the audible sound. This scale is divided into 130 dB up to a level at the threshold of pain. Table 02 shows the intensity level (in watts/m² as well as in dB) corresponds to various loudness.

Loudness	Intensity (watts/m²)	Intensity Level (dB)
Threshold of hearing	10^-12	0
Rustle of leaves	10^-11	10
Whisper	10^-10	20
Watch ticking at 1 m	10^-9	30
Radio (low)	10^-8	40
Quiet conversation	10^-7	50
Quiet motor at 1 m	10^-6	60
Busy street traffic	10^-5	70
Door slamming	10^-4	80
Heavy truck, 50 ft	10^-3	90
Power mower	10^-2	100
Pneumatic drill	10^-1	110
Near aeroplane engine	1	120
Physical damage	10	130

Table 02 Sound Intensity Levels

Timbre (spectrum) -- A musical sound is a composite of many harmonics. While the fundamental gives the sound its pitch, the harmonics give the sound its characteristic color or timbre. The sounds of a clarinet, violin, and piano are different even if they are all playing the same pitch. The difference is caused by the complex mixture of harmonics from each instrument as shown in Figure 11a. Timbre typically changes over the life of a note sounded by an instrument. New harmonics are added to

the sound as it gets louder. A piano tone constantly changes in timbre as it decays after it is sounded. Timbre is altered by the player for expressive reasons. Subjectively, strong upper harmonics are responsible for making an instrument's sound "bright" or "piercing". If the lower harmonics become dominant, then the sound would be "darker" or "duller". Figure 11b shows explicitly the generation of the fundamental and harmonic frequencies when a note is played on a piano key, e.g., the middle C (C4) key. Mathematically, such combination of frequencies is unavoidable because sound wave is governed by a linear differential equation such as Eq.(4a), which is similar to the Schrodinger equation in quantum mechanics, where the amplitude is related to the probability of finding the system in a particular quantum state (instead of the intensity of each frequency in the note).