from a mathematics and acoustic point of view,
by Thomas Váczy Hightower.
The focus will be on the acoustic laws behind the musical scales and how numbers and mathematics play a part in creating the intervals in the octave. Which factors have significance for creating a musical scale? Why is the division of the octave so basically common in different musical traditions, and what make them differ? Why is the ancient Greek Pythagorean scale basically identical with the old Chinese scale? What causes the modern Western musical scale, the Equal Temperament, to be ‘disharmonious’ compared to the Eastern scales?
Music has often played an important part in shaping a culture. Some say that music is the hidden power in a culture. In ancient societies it was considered a serious public matter, a foundation for the culture. The musical scale itself and the right tuning of intervals can make all the difference as to how chaos or order. It also ensures that humans are in accord with earthly as well as celestial influence.
The more metaphysical aspects of music and sound and its influence at the level of consciousness and healing can be studied in my second part, The Musical Octave II, where I will mix different levels and categories into a larger picture.
In this thesis I will perform an analysis of four different musical traditions and their basic scales:
· the ancient Chinese
· the Indian musical tradition
· the old Greek music
· the following European musical scales.
By looking at the many tuning systems worldwide, one common factor is outstanding, the octave. The word derives from Latin and means ‘eighth’. It is the 8th step in the diatonic scale consisting of 7 tones, containing 5 full tones and 2 semitones. The eighth tone in the diatonic scale, which is the most common in the world, completes the octave on a pitch that in frequency is the double of the fundamental tone.
This universal unit, that divides the realm of sound by the factor 2, can be subdivided in three basic ways:
1) By a geometric
progression, with any number of equal intervals, such as the
common Western mode, the Equal Temperament with 12 semitones, and
A geometric progression is a sequence in which each term (after the first) is determined by multiplying the preceding term by a constant. This constant is called the common ratio of the arithmetic progression. The octave sequence is a geometric progression; so is the golden section.
2) By proportions
with low number ratios, e.g. Just Intonation with its triads
of major Thirds, or by other harmonic relations to the tonic (Modal
music), e.g. Pentatonic or Septonic (e.g. Indian music).
Systems of proportions are used in Modal music, e.g. the harmonic mean and the arithmetical mean in the division of an octave.
3) By generating
Fifths, e.g. Pythagorean Tuning or The Chinese Scale.
There are hybrids too, such as the Mean Tone Temperaments.
The reason there are so many different
ways to divide the octave and display such a range of scales can be
found in the fact that there are no formula that can fit the octave
perfectly. The different ratios expressed in numbers are prime
inter-related, so a common divisor is not possible in an octave -
unless some notes or keys are sound disharmonious.
Different musical traditions embrace this schism depending on what they consider best fit for their musical expression. The culture in which the musical scale has emerged is a profound reflection of that particular culture.
The Eastern music tradition considers the fine-tuned intervals of much more importance than the Western, which prefers first harmonious chords in any key. Consequently there are intervals which are perceived as consonant in the West, but considered dissonant in the East.
What it comes down to is habits. A musical scale is deeply ingrained. It shapes the way one hears tones in succession in a fixed pattern. There have to be at least three elements in defining a mode, just as three notes are needed to define a chord.
In the modulating cyclic systems, where
every sound is mobile, it is necessary to repeat the ‘body of
harmony’ (tonic, fifth or fourth and octave) in order to
establish the meaning or mood of the note, but in the modal system
one note alone, by changing its place, can produce the effect of a
The modal frame, being fixed and firmly established in the memory of the listener, has no need to constantly repeat chords as in harmonic music, in order to express the numerical relationship.
That shape of ingrained intervals goes more or less out of tune, when changes of key or transposition moves the frequencies up or down. It is the way enharmonic notes arise. Increasing pitch by a half tone is not the same as decreasing by a half tone. They are two different notes.
Expressed graphically, the frequencies’ ratios behave exponentially - in a non-linear curve - (which is displayed e.g. by the logarithmic spacing of the frets on the neck of the guitar), so a discrepancy is produced by moving the set frame up or down. This discrepancy is expressed in the different ‘commas’, such as the Pythagorean comma or the smaller Syntonic comma (the comma of Didymos).
The notion of harmony is different too. In the West the perception of harmony is ‘vertical’ - meaning as chords played at once. The Eastern tradition of harmony is ‘horizontal’. Each tone is carefully played, and by attention over time adds up in the memory to harmonious chords.
Before we deal with the creations of
musical scales, we have to dwell on the underlying foundation of
scales, namely the physical laws of sounds.
Acoustics is a branch of physic that is complicated and extensive, so I have only chosen - in a brief form - those parts we need to look at in order to understand the invention of musical scales.
Sound is vibrations, but three
conditions have to be in place, if a sound is to be heard:
1) The vibrating source for the sound – an oscillator.
2) A medium in which the sound can travel, such as air, water or soil.
3) A receiver for the sound, such as a functional ear or a microphone.
The sound wave is a chain reaction where
the molecules of the medium, by elastic beats, push the other
molecules in a longitudinal direction - quite like 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. Any sound is initiated by an oscillator, which can be a huge range of devices and instruments, each one having its own definite characteristic sound.
The sound waves should not be imagined
as waves in water caused by e.g. a stone in a pond, though the
picture appears to be alike.
Sound waves are longitudinal: pressure waves - back and forth.
Water waves are transverse: the main movements are up and down in a circular motion.
Please note that longitudinal pressure waves will reappear in the description below of logarithmic, standing pressure waves.
Plucked strings exhibit transverse waves in a back and forth movement, locally producing a pulse along the direction in which the wave itself travels, with a speed depending on 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 length of the vibrating part of the string is in inverse proportion to the frequencies. The period of oscillation = 1 / frequency. This is an important acoustic law that applies to any conversion of period into frequencies. If, for example, you divide an octave string by 2/3, the ratio of the sound will be 3/2 of an octave, a fifth.
produce sounds, a vibrating body, an oscillator,
An oscillator can be any kind of a vibrating body from an atom to an astronomical object, but since we are here 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 amplifies the tone and generates harmonics.
The heart and aorta form a special resonant system when breathing is ceased. Then the heartbeat 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 of 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 close to the Schumann resonance.
Standing waves are a kind of echo that moves back and forth, since the waves are reflected between two solid points, basically, a 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 resonate a bit differently.
When a fixed string is plucked, the potential energy is released in a transverse wave, that in a split second begins to initiate a division of the string into different moving parts, where some points are not moving. These stationary points are called nodes. How many nodes the string is divided into when it vibrates depends on 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 at the fixed ends, the kinetic energy is at its maximum and continues in a 180° phase shift the opposite way. We thus have two waves with the same frequency and amplitude traveling in opposite directions. Where the two waves add together or superpose, movement is canceled out and we have stationary 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 one’s attention to a similar pattern we can observe in a pendulum and its simple harmonic motion.
The numbers of nodes, the non-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 set 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 that 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 oscillations after a while begin to oscillate in
Standing waves cannot exist unless they divide their medium into an integral number of half waves with their nodes. A standing wave having a fractional wavelength cannot be sustained.
The same standing waves pattern
can be formed in a three-dimensional box. 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
The key word in standing waves is order. In short, by using sound we have introduced order where previously there was none.
Any vibrating body that is set in a
standing, resonant motion, produces harmonics. For musical sounds the
harmonic series is usually expressed as an arithmetical proportion:
1,1:2, 1:3, 1:4, 1:5, 1:6...1:n.
The first and second 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.
composed (periodic) tone contains a multiple of various frequencies
in whole numbers, (integers – 2,3,4,5,6…25…) of
the fundamental frequency.
They are named harmonics. Each voice or musical instrument produces its own characteristic set of harmonics, also called formats, that enable the ear to identify the sound because the ear and the brain perform a Fourier analysis of the sound. (Some wind instruments, for example, produce only odd harmonics).
In order to understand the composed tone, one has to turn to a French mathematician from the 19th Century, Jean-Baptiste Fourier, who in 1822 proved that any complex periodic curve – in this case any tone – is composed of a set of sine curves that contain the fundamental sine frequency + another sine curve with double the frequency + a sine curve with triple the frequency, and so on.
A simple Harmonic
motionis typified by the motion of a pendulum,
which is sinusoidal in time and demonstrates a single resonant
The formula for The Harmonic Series is the sum, ∑ 1/n = 1 + 1/2 + 1/3 + 1/4 + 1/5 +1/6 +…diverges to infinity, when n goes from 1 to infinity. Another common way to express the harmonics is, the fundamental f, then 2f, 3f, 4f, 5f....nf harmonic.
To have an idear of the harmonics in the string you have to 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 n'th 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 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.
Since the harmonic series plays such an important part in music, it should be obvious to use the harmonic series as the notes in a scale. This is also valid since the harmonic series contains all the possible intervals used in music, although the order in which those intervals appear does not properly constitute a musical scale. The main difficulty is that all its intervals differ from one another and become smaller as the scale rises.
The problem with modulation is obvious
since each interval is not alike. Further, the need for a fixed
structure to establish a musical scale, a body of harmony established
by the three prime intervals, cannot be fulfilled by using the
harmonic series as a musical scale.
Nevertheless, the series of the first sixteen harmonics can be considered to form a mode that is interesting in comparison with the musical scales used throughout the history of music.
If we take C as a starting point, we
first notice the appearance of the octave, C', 2/1, then the fifth,
G, 3/2, then the third, E, 5/4, then the harmonic Bb 7/4 –
lower than the usual Bb, and forming with upper C', the maximum tone
After this appears the major second, D, 9/8, which forms with E a minor second, 10/9.
Then come the harmonics F#, 11/8, A-, 13/8, and finally, the seventh, B, 15/8.
The remaining eight of the first sixteen harmonics add no new notes, as they are at exact octave intervals from earlier harmonics in the series.
We have to understand the way the harmonic series display itself in a chain of octaves, where each new octave contains twice as many harmonics as the last octave.
By looking at the ratios, the denominator indicates the octave, the numerator states the number of harmonics in that octave. Considering only the first sixteen harmonics, we thus obtain a scale of eight tones formed of the following intervals:
Notice that each interval gets smaller as the pitch rises.
Another feature in the realm of sound is the exponential factor, because sound, like many other physical events, behaves exponentially - not in straight lines. Harmonics are not linear either.
There are two ways to calculate ratios
1) One can work with the ratios as they are, often pretty long numbers, and the calculation is a bit twisted, since in adding two sound ratios one has to multiply; to subtract you have to divide; and to divide a sound ratio you have to take the square root.
A common example is the Equal Temperament, where the octave has to be divided into 12 equal parts.
One semitone is 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 straightforward, is to convert the ratios into logarithmic unities such as cents or savarts. Logarithmic calculations make it easier to operate with pitch intervals or frequency ratios, since the size of a pitch interval is proportional not to the frequency ratio, but to the logarithm of the frequency ratio. This makes the calculation of ratios simpler, by a plain process of adding, subtracting or dividing.
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 made differently depending on musical tradition, but
alr the world in all times the octave has been recognized as the
basic unit that constitutes a beginning and an end.
‘Octave’ derives from Latin and means the ‘eighth’. It is the 8th step in the diatonic scale consisting of 7 tones, 5 full tones and 2 semitones. The eighth tone in the diatonic scale, which is the most common in the world, completes the octave on a pitch that in frequency is the double of the fundamental tone.
Graphically, one could say that an
octave expresses or represents a circle. Several octaves shape a
spiral, where the same fundamental is above or below. The obvious
mystery about octaves is that tones an octave apart sound similar,
though the frequency is the double or the half.
They pertain, so to speak, to the same family; from the same root, unfolding in the spectrum of frequencies. They have the same Chroma. They always double up the frequencies in the ascending mode or halve them in the descending mode.
Again we see the basic, universal division of one into two, as we first mentioned in the paragraph about the standing waves. Just remember the awesome sight of the pregnant egg-cell dividing itself. The law of octaves belongs not only to the realm of sound, but can be observed as manifesting itself throughout Nature around, and in astronomy above.
The very harmonious Fourth is a kind of
a puzzle, with its prime interval in the ratio of 4 : 3. It is not
represented in the first 16 harmonics in the series, though the 3rd
and 4th harmonics are separated by a Fourth. It has taken me some
time to figure it out.
In order to understand the importance of the Fourth, we have to look at the previous prime interval, the Fifth, with the ratio of 3 : 2. The 2nd and 3rd harmonics are separated by a Fifth. These two intervals together constitute an octave. They are complementary intervals.
Furthermore, by going down by a Fourth into the octave below, one reaches the Fifth in the sub-octave, which has half the frequency. In other words: a descending Fifth, 2:3, divided by ½, equals 4:3, a Fourth.
In the musical language the Fifth is
called the Dominant and the Fourth the Sub-dominant, which plays a
very dominant role in music all over the world.
In all the musical scales that are obtained by the generating interval 3:2, the opposite movement – lowering by 4:3 – makes it possible to fit the generated intervals into one octave.
Music and Numbers are often said to be as brother and sister, different but related. In addition, we have to take into consideration numerical representation, which plays an important role in Eastern music but is ignored in the Western tradition.
A musical sound or tone is a composite
sound containing a multiple of overtones or harmonics. In musical
practice the tone is not only dependent on its pitch and amplitude
(loudness), but also on its specific numbers of harmonics (formats),
which ‘color’ the tones so that each instrument or voice
has its characteristic sound.
This has nothing to do with musical compositions aiming to ‘paint’ colors, or the blue notes in jazz music. ‘Overtone’ originates from the German Obertone, which refers to the various numbers of partials or harmonics that are produced by the strongest and lowest fundamental note, and fused into a compound or complex tone.
In his book On the Sensation of Tone
from 1877, Herman von Helmholtz formulated the theory about the
consonant and dissonant intervals based on the numbers of beats
generated when two tones or a chord are played.
It was first about 100 years later that Promp was able to prove a more consistent theory, the Consonant Theory, which now is generally accepted.
When two tones (or chords) are played
simultaneously, another important acoustic phenomenon takes place,
called ‘beats’. When the frequencies of two tones are
close to each other, a periodical beat can clearly be heard, caused
by the interference of the different waves, which alter the amplitude
so an intensified rhythmic beating, floating tone is heard as a third
There are other interference patterns besides beat frequencies, but this will do in this instance.
Some intervals or chords produce more beats in the higher harmonics than others, and those are picked 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, or ‘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 to formulate 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 at that time, and has puzzled generations since it was declared. Johannes Kepler dedicated most of his life to attempting to solve that notion.
The receiving part, the human ear, is equally important. The recent discoveries (The Consonant Theory) of the function of the basal membrane in Cochlea as a Fourier analyzer, and the role the critical band plays in the perception of rough or smooth sound, dissonance and consonance, gives a consistent theory for some of the hearing functions.
the frequency ratios are narrowed down to such small intervals that
our auditory system is not capable of differentiating, the harmonics
become fused because of the critical band, a relatively new
discovery, (around 1970-80 by Plomp a.o.) which refers to the
overlapping amplitude envelopes on the basilar membrane in the Organ
of Corti in the Cochlea.
Trained ears are able to detect the harmonics up to the 6th or 7th harmonics.
←Schematic graph of the Cochlea
When the interval between two tones decreases, their amplitude envelopes overlap to an increasing extent. A rough, harsh tone will be heard, which anyone can hear when two notes with less than minor 3rd separation are played simultaneously. This is very shortly the key to understand the theory of dissonance and consonants, which is the foundation in the origin of scales.
There are a lot of more acoustic laws and theories of fusion of pure tone components and other acoustic phenomena such as masking, except to state that the inner ear performs a partial frequency analysis of a complex musical tone, a Fourier analysis, sending to the brain a distinct signal recording the presence of each of the first seven or eight harmonic components; in addition the brain receives signals from the part of the basilar membrane activated by the unresolved upper harmonics.
Several experiments by different
scientists suggest that the brain determines the pitch of a complex
tone by searching for a harmonic pattern among the components
separately resolved in the inner ear. If the deviation from a true
harmonic series is too large, the brain gives up the attempt to find
a single matching set of harmonics. Then the components are heard
separately, rather than as a fused tone.
This explains the ‘missing fundamentals’ in the harmonic spectrum of a bassoon playing E3, because the ear does not ‘hear’ the fundamental tone, but the harmonic.
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.
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.
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 ear’s determination of consonant or dissonant intervals.
This portion is a bit of off the key with the musical scales. However, when (in 2007) I read about Cislenko's logarithmic intervals in the book Tools of Awareness, I felt immediately that here is new, first-class research about the basic concept of a scale. You have to go above the level of sound and reach up to the level of sizes of bodies.
In 1980, the Russian biologist Cislenko
published what is probably one of the most important biological
discoveries of the 20th century. The published work was Structure of
Fauna and Flora with Regard to Body Size of Organisms
His work documents that segments of increased species representation were repeated on the logarithmic line of body sizes in equal intervals (approx 0.5 units of the decadal logarithm).
The phenomenon is not explicable from a biological point of view. Why should mature individuals of amphibians, reptile, fish, bird and mammals of different species find it similarly advantageous to have a body size in the range of 8 - 12 cm, 33 - 55 cm or 1,5 - 2,4 m?
Cislenko assumed that competition in the
plant and animal kingdoms occurs not only for food, water or other
resources, but also for the best body sizes.
Each species tries to occupy the advantageous intervals on the
logarithmic scale where mutual pressure of competition also gave rise
to crash zones.
However, Cislenko, was not able to explain, why both the crash zones and the overpopulated intervals on the logarithmic line are always of the same length and occur in equal distance from each other. He was unable to figure out why only certain sizes would be advantageous for the survival of a species, and what these advantages actually were.
Cislenko's work inspired the German
scientist Dr. Hartmut Müller to search for other scale-invariant
distributions in physics. The phenomenon of scaling is well known to
Müller found similar frequency distributions along the logarithmic line of sizes, orbits, masses, and revolution periods of planets, moons and asteroids. Being a mathematician and physicist he did not fail to recognize the cause for this phenomenon in the existence of a standing pressure wave in the logarithmic space of the scales/measures.
Scale is what physics can measure. The
result of a physical measurement is always a number with measuring
unit – a physical quantity.
Imagine that we have measured 12cm, 33cm and 90cm. Choosing 1 cm as the standard measure (etalon), we will get the number sequence 12 - 33 - 90 (without measurement unit, or as the physicist says: with unit 1). The distances between these numbers on the number line are 33 - 12 = 21 and 90 - 33 = 57.
If we were to choose another measuring unit, such as the etalon with 49,5cm, the number sequence would be 0,24 - 0,67 - 1,82. The distances between the numbers have changed into 0,67 - 0,24 = 0,42 and 1,82 - 0,667 = 1,16.
However, on the logarithmic line, the
distance will not change, no matter what
measuring unit we choose. It will always remain constant.
In our example, this distance amounts to one unit of the natural logarithm (with radix e = 2,71828...): ln 33 - ln 12 ≈ ln 90 - ln 33 ≈ ln 0,67 - ln 0,24 ≈ ln 1,82 - ln 0,67 ≈ 1. Physical values of measurement, therefore, own the remarkable feature of logarithmic invariance (scaling).
So, in reality, any scale is a logarithm!
It is very interesting that natural systems are not evenly distributed along the logarithmic line of the scales. There are ‘attractive’ sections which are occupied by a great number of completely different natural systems; and there are ‘repulsive’ sections that most natural systems will avoid.
Growing crystals, organisms or
populations that reach the limits of such sections on the logarithmic
line will either grow no more or will begin to disintegrate, or else
will accelerate growth so as to overcome these sections as quickly as
The Institute for Space-Energy-Research I.M. was able to prove the same phenomenon also in demographics (stochastic of world-wide urban populations), economy (stochastic of national product, imports and exports world-wide) and business economy (stochastic of sales volume of large industrial and middle-class enterprises, stochastic of world-wide stock exchange values).
The borders of ‘attractive’ and ‘repulsive’ segments on the logarithmic line of scales are easy to find because they recur regularly with a distance of 3 natural logarithmic units. This distance also defines the wavelength of the standing pressure wave: it is 6 units of the natural logarithm.
In fact, the world of scales is
nothing else but the logarithmic line of numbers known to mathematics
at least since the time of Napier (1600). What is new, however, is
the fundamental recognition that the number line has a harmonic
structure, which is itself the cause for the standing pressure
Leonard Euler (1748 ) had already shown, that irrational and transcendental numbers can be uniquely represented as continued fractions in which all elements (numerators and denominators) will be natural numbers.
In 1928, Khintchine succeeded in providing the general proof about prime numbers. In the theory of numbers this means that all numbers can be constructed from natural numbers; the universal principle of construction being the continued fraction. All natural numbers 1, 2, 3, 4, 5, ... in turn are constructed from prime numbers, these being natural numbers which cannot be further divided without remainder, such as 1, 2, 3, 5, 7, 11, 13, 19, 23, 29, 31, ... (traditionally 1 is not classed as a prime number although it fulfills all criteria).
The distribution of prime numbers on the
number line is so irregular that so far no formula has been found
that would perfectly describe their distribution.
Dr. Muller found that the distribution of numbers is indeed very irregular - but only on the linear number line.
On the logarithmic number line, large gaps of prime numbers recur at regular intervals. Gauss (1795) had already noticed this.
The reason for this phenomenon is the
existence of a standing density wave on the logarithmic number line.
The node points of this density wave are acting as number attractors.
This is where prime numbers will 'accumulate' and form composite
numbers, i.e. non-primes, such as the 7 non-primes from 401 to
Hence a ‘prime number gap’ will occur in this place. Precisely where non-primes (i.e. prime clusters) arise on the logarithmic number line, there it is that matter concentrates on the logarithmic line of measures. This is not magic; it is simply a consequence of the fact that scales are logarithms, i.e. ‘just’ numbers.
So the logarithmic line of scales is
nothing else but the logarithmic number line. And because the
standing pressure wave is a property of the logarithmic number line,
it determines the frequency of distribution of matter on all
physically calibrated logarithmic lines - the line of ratios of size,
that of masses, of frequencies, of temperatures, velocities,
Finding a node point on the logarithmic line is relatively easy, since the wavelength of the standing density wave on the logarithmic number line is known, and the calculation of all nodal points is done by a simple formula.
The distance between adjacent node points is 3 units of the natural logarithm.
The frequency ranges around 5 Hz, 101
Hz, 2032 Hz, 40,8 kHz, 820 kHz, 16,5 MHz, 330,6 MHz, etc. are
predestined for energy transmission in finite media. This is also
where the carrier frequencies for information transmission in
logarithmic space are located.
Frequencies that occur near a note point are very common in nature, as well as in technological applications.
This is a brief resume of those factors in hearing that are closely related to perceiving an interval or chord. We will skip the many other acoustic phenomena, because here we will only try to give an account of the reasons why musical scales are created as they are.
Thomas Váczy Hightower, © 2002. (edited 2009)
To continue in part II of The Creation of Musical Scales,
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