# Some basic acoustics laws

How to calculate oscillation of a body into freqency:
The length of time a celestial body requires to rotate around its axis and to revolve around the sun, can be converted into sound (and color) by means of the law of the octave.

Any things that moves or oscillate has a frequency measured in a time unit, usually one second and is called Hertz after the German physicist. Frequency express the number of repetitions of a periodic phenomenon during a certain length of time. (A tuning fork that vibrates 272 times back and forth in one second, has a frequency at 272 Hertz (Hz.)).
Time is not a concept of its own, but a duration. Oscillation is a repetition of the same phenomenon, e.g. the movement of a string, the swing of a pendulum, or astronomical constellations such as a day or a year.

There is a simple relationship between the period of oscillation (the time it takes to move back and forth) and its frequency. It is an inverse proportion: the period = 1/ frequency; (or frequency = 1/ period.) It means that the reciprocal value of a period of time represents its frequency (in seconds).
Frequency range can be measured in a basic unit called octaves. In practice one can form an octave by multiply or divide the frequency with 2.

### The Octave

This interval is the very most outstanding division of sound and music and is recognized in all musical traditions through time on the globe. The division of the octave has been done differently depending of musical tradition, but all over the world in all times the octave has been recognized as the basic unit that constitute a beginning and an end. Octave derived from Latin and means the eighth. It is the 8th step in the diatonic scale consisting of 7 tones, 5 full tones and 2 semi tones. The eighth tone in the diatonic scale, which is the most common in the world, complete the octave on a pitch, that in frequency is the double of the fundamental tone.

Graphically one could say that an octave express or represent a circle. Several octaves shapes a spiral where the same fundamental is above or below. The straight out mysterious about octaves is that tones an octave apart sounds similar though the frequency is the double or the half. They contain so to speak to the same family; from the same root, unfolding in the spectrum of frequencies. They have the same Chroma. The whole time double up the frequencies in the ascending mode or halve in descending mode.

Again we see the basic, universal division of one into two as we first refer to in the paragraph about the standing waves.

Just remember the awesome sight of the pregnant egg-cell dividing itself.

The law of octaves belong not only to the realm of sound but can be observed as manifesting itself though out Nature and above in astronomy.

### Chroma

Two tones an octave apart are perceived as having a kindred relationship - as being, in a sense, "the same note". This recognition of similarities for tones an octave apart is and was applying to any major musical culture all over the globe since the beginning of the history of man.

The sense of pitch associated with a tone has two different attributes: the frequency of the tone, the ”height”, and the chroma, which is a quality shared by all notes described by a particular letter. (Bachem 1950) Thus C2 and C3 are separated in pitch by an octave, they both have the same chroma - that associated with all C’s. They are in the same family.

In one interpretation of the pattern recognition theory of pitch, the brain searches for a distribution of peaks along the basilar membrane corresponding to a harmonic series of vibrations. On this basis, the fact that C2 and C3 evoke almost the same sensation can be readily understood.

### Sound wave

The sound wave is a chain reaction where the molecules of air by elastic beats push the other molecules in the longitudinal direction quiet similar to a long train getting a push from a locomotive.
It is a longitudinal displacement of pressure and depressor in a molecular medium such as air or water initiated by an oscillator, the vibration source for the sound.
The sound waves shall not be mistaken as waves in water caused by e.g. a stone in a pond, though the picture appears to be alike.
Sound waves are longitudinal, back and forth. Water waves are transverse, the main movements are up and down.

### Calculations of sound ratios

Another feature in the realm of sound is the exponential factor, because sound as many other physical events behaves exponentially - not in straight lines.
There are two ways to calculate ratios of frequencies:
1) One can work with the ratios as they are, and when adding two sound ratios one have to multiply; to subtract you have to divide, and to divide a sound ratio you have to take the square root.
A common example: in order to divide the octave into 12 equal parts as the case is for the Equal Temperament. You have to calculate the 12th. root of 2, (21/12) . If you want to divide the whole tone, 9/8, you have to take the square root of 9/8, or (9/8)½ = 3/2*2½ .
2) The other way, which makes the calculation more straight forward, is to convert the ratios into logarithmic unities such as cent or savarts, which makes the calculation of ratios more simple by adding, subtraction and dividing.

Savarts named after a French physicist and cents are logarithmic systems developed to make it easy to compare intervals on a linear scale instead of using fractals or frequencies ratios.
Savarts is calculated as the logarithm (base 10) of the frequency ratio and, for convenience, multiply with 1000. We shall then have an interval expressed in terms of a unit call savarts.
The logarithm of 2 being 0.3010... the interval of the octave is equal to 301 savarts.
Savarts is more exact than the widely used American system, cents, which by definition is based on the tempered scale of 1200 cents.

Cent is also a logarithmic unit, which make it easier to operate with pitch intervals or frequencies ratios, since the size of a pitch interval is proportional not to the frequency ratio, but to the logarithm of the frequency ratio.
It is a simple relationship: pitch interval (in octaves) = log2 (f2 : f1). Since there are chosen to have 1200 cents in an octave, this is equivalent to:
pitch interval (in cents) = 1200 * log2 (f2 : f). In practice, logarithms to base 10 (log10) are normally used.

### Moving string

Plucked strings exhibit transverse waves in a back and forth movements locally producing a pulse along the direction in which the wave itself travels with a speed depending of the mass of the string and its material but usually lower than airwaves. A good explanation is given by The University of New South Wales, Australia.
(The frequency of the string itself is the same as the frequency of the air waves.  The wavelength is different due to the dissimilarity in speed.)

The lengths of the vibrating part of the string is inverse proportional to the frequencies. The period of oscillation = 1 / frequency. This is a important acoustic law, that applies to any conversion of  period into frequencies. If you, e.g. divide an octave string with 2/3, the ratio of the sound will be 3/2 of an octave, a fifth.

### Oscillators

To produce sounds a vibrating body, an oscillator, is needed.
An oscillator can be any kind of a vibrating body from an atom to an astronomical object, but since we here are working with musical sounds, we are referring to oscillators such as musical instruments or the human voice box, that produce standing waves or periodic waves in a system of resonators that enhances and amplify the tone and generates harmonics. The heart and aorta formed a special resonant system when breathing is ceased. Then the heart beat seems to wait until the echo returns from the bifurcation (where the aorta forks out in the lower abdomen). Then the next heart beat sets in. In this synchronous way a resonant, standing wave of blood is established with a frequency about seven times a second. This harmonious mode requires for its sustenance a minimum amount of energy, which is an intelligent response from the body. In deep meditation a similar mode is established. It is interesting to notice that this mode of 7 Hz is closed to the Schumann resonance.

### Standing waves

Standing waves are a kind of echo, that moves back and forth, since the waves are reflected between two solid points, basically, a hamper or fixed string. For wind instruments with an open end, the impedance (the resisters from the air) works in a similar manner. There are also closed pipes, that resonates a bit different.

When a fixed string is plucked, the potential energy is released in a transverse wave, that in a split second begins to displace a division of the string in different moving parts, where some points are not moving, which are called nodes. How many nodes the string is divided into when it vibrates depends of the material, the tension, and especially how and where it is plucked or bowed ,etc, but here we try to get a general picture of the nature of standing waves in a plucked string.

When the potential energy is stopped in the fixed ends,  the kinetic energy is at its maximum and continue in a 180° phase shift the opposite way. We have two waves with the same frequency and amplitude traveling in the opposite direction. Where the two waves add together or superpose, movements is cancel out and we have moving less nodes. That occur a half wavelength apart and constitute the standing waves.

The repeating shifts between potential and kinetic energy in a moving string draws ones attention to a similar pattern we can observe in a pendulum and its simple harmonic motion.

The numbers of nodes, the no moving points, in a standing wave, is equal to the number of harmonics or partials created in the standing wave.

The same pattern can be observed with fine sand on a metal plate in vibration by a bow. The standing waves automatically divide the length and width of the plate into an integral number of half wave-lengths. It is only then, a standing wave can be sustained. That pattern is the most energy effective form nature can provide. (A similar pattern is the rhythm entrainment, where random oscillation after a while begin to oscillate in unison).

Standing waves can not exist unless they divide their medium into an integral numbers of half waves with its nodes. A standing waves having a fractional wavelength can not be sustained.
The same standing waves pattern can be preformed in a 3 dimensional box too. This pattern will look just like a highly enlarged crystal, if we assume that the aggregated particles or grains in the box fluid are analogous to the atoms in a crystal.

The key word in standing waves is order. In short, by using sound we have introduced order where previously there was none.

### Harmonics

Any vibrating body that is set in a standing, resonant motion,  produce harmonics. For musical sounds the harmonics series is usually expressed as a arithmetical proportion:

1,1:2, 1:3, 1:4, 1:5, 1:6...1:n.
The first and 2nd harmonics are separated by an octave, frequency ratio 2:1, the 2nd and 3rd by a perfect fifth (3:2), the 3rd and 4th by a perfect fourth (4:3), and 4th and 5th by a major third (5:4), and the 5th and the 6th by a minor third (6:5), and so on.

A simple Harmonic motion is typified by the motion of a pendulum and is sinusoidal in time and demonstrate a single resonant frequency.
The formula for The Harmonic Series is the sum,
å 1/n = 1 + ½ + 1/3 + ¼ + 1/5 +1/6 +…diverges to infinity, when n goes from 1 to infinity.

Since the Harmonic Series is so important in the construction of musical scales, another common way to express the harmonics is a simple multiplication by whole numbers: the fundamental f, then 2f, 3f, 4f, 5f....nf harmonic.

How Nature performed such a mathematical division, an arithmetic progression is beyond my apprehension, but it is surely a mighty prominent and well proofed law. Intuitively I feel that the number 2 or its inversion ½ is the mega number. Remember the integer numbers of ½ waves (nodes) in the standing wave.

The harmonic series is special because any combination of its vibrations produces a periodic or repeated vibration at the fundamental frequency.

### Harmonics of the string

Imagine an idealized stretched string with fixed ends vibrating the first 4 modes of the standing waves. This can be expressed as the relationship between wavelength, speed and frequency, a basic formula where the wavelength is inversely proportional to the frequency when speed is a constant (k) since it is the same string:

Let's work out the relationships among the frequencies of these modes. For a wave, the frequency is the ratio of the speed of the wave to the length of the wave: f = k/wavelength. Compared to the string length L, you can see that these waves have lengths 2L, L, 2L/3, L/2. We could write this as 2L/n, where n is the number of the harmonic.
The fundamental or first mode has frequency f1 = k/wavelength = k/2L,
The second harmonic has frequency f2 = k/wavelength = 2k/2L = 2f1
The third harmonic has frequency f3 = k/wavelength = 3k/2L = 3f1,
The fourth harmonic has frequency f4 = v/wavelength = 4k/2L = 4f1, and, to generalize, The nth harmonic has frequency fn = k/wavelength = nk/2L = nf1.

All waves in a string travel with the same speed, so these waves with different wavelengths have different frequencies as shown. The mode with the lowest frequency (f1) is called the fundamental. Note that the nth mode has frequency n times that of the fundamental. All of the modes (and the sounds they produce) are called the harmonics of the string. The frequencies f, 2f, 3f, 4f etc are called the harmonic series.

The diagram below displays the harmonics in a span of 5 octaves, where the fundamental is C with the frequency of 32 Hz. As the octaves progress the numberes of new harmonics increase with the factor of 2. ### Composite sound

A musical sound or a tone is a composite sound containing a multiple amount of overtones or harmonics. In musical practice the tone is not only depending of its pitch and amplitude (loudness), but also of its specific numbers of harmonics, (formats) which "color" the tones so each instrument or voice have their characteristic sound.
This has nothing to do with musical compositions aiming to "paint" colors, or the blue notes in jazz music. Overtones originate from the German Obertone which refer to the various numbers of partials or harmonics that are produced by the strongest and lowest fundamental tone and fused into a compound or complex tone.
Herman von Helmholtz  formulated in his book "On the Sensation of Tone" from 1877, the theory about the consonant and dissonant intervals based on the numbers of beats there are generated, when two tones or a chord are played.
It was first about 100 years later Promp could prove a more consistent theory, the Consonant Theory, which now is generally accepted

## Harmonics

In order to understand the composed tone one has to turn to a French mathematician from the 19th. Century, Jean-Baptiste Fouier, who in 1822 proved that any complex periodic curve or in this case any tone is composed of a set of sinus curves, that contains the fundamental sinus frequency, plus another sinus curve with the double frequency, plus a sinus curve with a triple frequency, and so on.
A composed (periodic) tone contains a number of various multiple frequencies in whole numbers, integers, (2,3,4,5,6…25…) of the fundamental frequency.
They are named harmonics. Each voice or musical instrument (Some wind instruments, e.g. produce only odd harmonics) produces its own characteristic set of harmonics, also called formats, that makes the ear able to identify the sound because the ear and the brain also perform an Fouier analyze of the sound.

### Beats

When two tones (or chords) are played simultaneously, another important acoustic phenomenon is taking place called beats. It can clearly be heard when the frequency of the two tones are close to each other as a periodical beat caused by the interference of the different waves, which alter the amplitude so an intensified rhythmic beating, floating tone is heard as a third tone. There are other interference patterns than beat frequencies, but this will do in this contest Some intervals or chords produce more beats in the higher harmonics than others and those are pick up by the ear as unclean, muddy or unpleasant and are labeled dissonant.

The intervals which make fewest beats are called consonant, such as an octave,  the perfect fifth, the perfect fourth, the three prime intervals,  “The body of Harmony” as described by Aristotle; the basis for the musical scale.

A general rule about sound ratios is, that the simpler the ratios between sounds are, the more their relations are harmonious, while the more complicated the ratios are, the more dissonant are the sounds.

Pythagoras was the first in the West who formulated the law of musical pitches depending on numerical proportions. From this he based his underlying principle of "harmonia" as a numerical system bound together by interlocking ratios of small numbers. This discovery probably led him to the idea of the Harmony of the Spheres.
His vision of "The Music of the Spheres" aroused deep emotions in me. It alludes to the seven planets known of that time, and has puzzled generations since it was declared. Johannes Kepler dedicated most of his life in order to solve that notion.

### Breakthrough in the science of hearing.

Helmholtz beat theories was commonly accepted for about 100 years, before the Noble Prize winning Hungarian scientist Békésy in 1960 made a new breakthrough by his discovery of the role the basilar membrane plays in the hearing of pitch. He derived by anatomical studies a relationship between distance along the basilar membrane and frequency of maximum response. A high frequency pure tone generates a wave that travels only a short distance along the basilar membrane before reaching its peak amplitude; the hair cells at the position of the peak are fired, and the brain receives signals from the corresponding nerve fibers. These fibers evoke a "high frequency" sensation.
A low frequency tone generates a wave that travels most of the way to the helicotrema before rising to its peak amplitude and dying away. Signals from nerve fibers connected to this region of the basilar membrane evoke a "low-frequency" sensation in the brain.

Other theories than the above "place theory" have been brought forward, among them the "temporal theories", i.e. emphasizing the use of the timing information in nerve signals.

### Helmholtz dismissed

The modern Consonance Theory of Plomp extended the discoveries of Békésy with some new important findings, that gave whole new meanings to the concept of hearing. The beat theory of Helmholtz was finally dismissed in favor of the well experimented and proven Consonance Theory, in which the ears Discrimination Frequency and its Critical Bandwidth plays an important part.

### The Critical Band.

As the interval between two tones decreases, their amplitude envelopes on the basilar membrane overlap to an increasing extent. A significant number of hair cells will now be responding to both signals. When the separation is reduced, e.g. to a tone, the amplitude envelopes overlap almost completely, implying a strong interaction between the two sounds, which is heard as a harsh, rough sound: a dissonance.

When two pure tones are so close in frequency that there is a large overlap in their amplitude envelopes, we say that their frequencies lie within one critical band. This concept has been of great importance in the development of modern theories of hearing and one must add gives a much better explanation for the ears determination of consonant or dissonant intervals.

For each 'reference' tone (X axis in the diagram), there is a frequency difference value (called Discrimination Frequency, 'Discrim. Freq.' label in the diagram), under which the second tone is not distinguished from the reference tone; for example, the semitone difference (the blue curve in the diagram) in the dodecaphonic system (12 semi tones Equal Temperament) is, for each reference tone, under the 'Discrimination Frequency' curve and can therefore not be detected.

In addition, for each 'reference' tone, there is a second important frequency difference value, called 'Critical Band'; if the frequency difference is below 25% of the Critical Band the two tones are judged 'clearly dissonant'; if it is between 25% and 50% of the Critical Band (the sky blue curve in the diagram) the two tones are judged 'partially dissonant'; if it is more than 50% of the Critical Band, they are judged 'consonant'; for example the tone difference (the violet curve in the diagram) in the dodecaphonic system is 'consonant' for reference tone  frequencies > about 500 Hz, where as the minor third difference (the yellow curve in the diagram) in the dodecaphonic system is 'consonant' for reference tone frequencies > 300 Hz.
(With permission by Francesco Caratozzolo, Electronic Engineer (Ph.D. in Bio-Engineering)).

Main page for the acoustic laws behind the musical scales.

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