the creation of musical scales

5/21/2018 The Creation of Musical Scales

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The Creation of Musical Scales, part I

from a mathematics and acoustic point of view,

by Thomas Vczy 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 isthe 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 publicmatter, 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 thathumans 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 OctaveII,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 theirbasic 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,

theoctave. The word derives from Latin and means eighth. It is the 8th step in thediatonic 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, completesthe 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 thecommon Western mode, the Equal Temperament with 12 semitones, and other

numbers.
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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) Byproportions 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 orSeptonic (e.g. Indian music).

Systems of proportions are used in Modal music, e.g. the harmonic mean and thearithmetical 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 habits of hearing

The reason there are so many different ways to divide the octave and display such arange 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 hasemerged is a profound reflection of that particular culture.

The Eastern music tradition considers the fine-tuned intervals of much moreimportance than the Western, which prefers first harmonious chords in any key.

Consequently there are intervals which are perceived as consonant in the West, butconsidered 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 threeelements 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 establishthe meaning or mood of the note, but in the modal system one note alone, by changingits place, can produce the effect of a chord.

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 thenumerical relationship.

That shape of ingrained intervals goes more or less out of tune, when changes of keyor transposition moves the frequencies up or down. It is the way enharmonic notesarise. Increasing pitch by a half tone is not the same as decreasing by a half tone. They

are two different notes.


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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 commaor the smaller Syntonic comma (the comma of Didymos).

The notion of harmony is different too. In the West the perception of harmony isvertical - 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 thememory to harmonious chords.

Laws of acoustic

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 soundan 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 elasticbeats, push the other molecules in a longitudinal direction - quite like a long traingetting 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 rangeof 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 circularmotion.Please note that longitudinal pressure waves will reappear in the description below oflogarithmic, standing pressure waves.

Moving string

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 speeddepending on the mass of the string and its material but usually lower than

airwaves. (A good explanation is given byThe University of New SouthWales,Australia.)
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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 acousticlaw 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.

Oscillators

To produce sounds, a vibrating body, an oscillator, is needed.

An oscillator can be any kind of a vibrating body from an atom toan astronomical object, but since we are here working withmusical 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 aortaforks out in the lower abdomen). Then the next heart beat sets in. In this synchronousway 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 amountof energy, which is an intelligent response from the body. In deep meditation a similarmode is established. It is interesting to notice that this mode of 7 Hz is close to the

Schumann resonance.

Standing waves

Standing waves are a kind of echo that moves back and forth, since the waves arereflected 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 similarmanner. 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, thatin a split second begins to initiate a divisionof the string into different moving parts,

where some points are not moving. Thesestationary points are called nodes. How

many nodes the string is divided into when itvibrates 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 standingwaves in a plucked string.
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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 havestationary nodes that occur a half wavelength apart and constitute the standing waves.

The repeating shifts between potential and kinetic energy in a moving string drawsones 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 thenumber of harmonicsor 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 besustained. That pattern is the most energy-effective form nature can provide. (A

similar pattern is the rhythm entrainment, where random oscillations after a whilebegin to oscillate in unison).Standing waves cannot exist unless they divide their medium into an integral number

of half waves with their nodes. A standing wave having a fractionalwavelengthcannot 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 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, produces harmonics. Formusical 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, the2nd and 3rd by a perfect fifth (3:2), the 3rd and 4th by a perfect fourth (4:3), and 4thand 5th by a major third (5:4), and the 5th and the 6th by a minor third (6:5), and so

on.

A composed (periodic) tone contains a multiple of various frequencies in wholenumbers, (integers2,3,4,5,625) 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 thesound because the ear and the brain perform a Fourier analysis of the sound. (Some

wind instruments, for example, produce only odd harmonics).
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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 curvein this case any toneis composed of a set of sinecurves that

contain the fundamental sine frequency + another sine curve with double thefrequency + a sine curve with triple the frequency, and so on.

A simpleHarmonic motionistypified by the motion of apendulum, which issinusoidal in time and demonstrates a single resonant frequency.

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 whenspeed 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 havelengths 2L, L, 2L/3, L/2. We could write this as 2L/n, where n is the number of theharmonic.

The fundamental or first mode has frequency f1= k/wavelength = k/2L,

The second harmonic has frequency f2= k/wavelength = 2k/2L = 2f1The 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 nthharmonic has frequency fn = k/wavelength = nk/2L = nf1.
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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 themodes (and the sounds they produce) are called the harmonics of the string. Thefrequencies 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 newharmonics increase with the factor of 2.

How Nature performed such a mathematical division, an arithmetic progression, is beyond myapprehension, but it is surely a mighty prominent and well-proven law. Intuitively I feel that the

number 2, or its inversion , is the mega number. Remember the integer numbers of 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.

The Harmonic Scale

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
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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 establishedby 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 7/4lower than the

usual Bb, and forming with upper C', the maximum tone 8/7.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 atexact octave intervals from earlier harmonics in the series.

We have to understand the way the harmonic series display itself in a chain ofoctaves, 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, wethus obtain a scale of eight tones formed of the following intervals:

Notes C D E F# G A- Bb B C'

Ratios 1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1

Savarts 51 46 41 38 35 32 30 28

Notice that each interval gets smaller as the pitch rises.

Calculations of sound ratios

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 of frequencies:

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; tosubtract you have to divide; and to divide a sound ratio you have to take the squareroot.


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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)= 3/2*2.

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 makeit 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 thefrequency ratio. This makes the calculation of ratios simpler, by a plain process of

adding, subtracting or dividing.

Savarts, named after a French physicist, and cents are both logarithmicsystems

developed to make it easy to compare intervals on a linear scale instead of usingfractals or frequency ratios (f2/f1).

A Savart is calculated as the logarithm (base 10) of the frequency ratio and, forconvenience, multiplied by 1000. We then have an interval expressed in terms of asavart unit.The interval of an octave in savarts is the logarithm of 2, which is 0.3010... expressed

as 301 savarts.Savarts have an advantage over the widely used American system, cents, since savartsis designed to fit any frequencies ratios (f2/f1), while cent by definition is based on

one scale, the 12 semitones in the Equal Temperament.

Centis also a logarithmic unit, which by definition is based on the tempered scale of1200 cents/octave. A semitone is therefore 100 cents. This definition is a bit more

complicated than the plain savart, since the exact relationship of frequencies to centsis expressed by this formula: 1200 * (f2/f1) / log 2= 3986 * log10(f2/f1).E.g. the interval of the perfect fifth calculated in cents is: the log103/2 = 0.1761.. The

fifth in cents is 3986 * 0.1761 = 702 cent.

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 theoctave has been made differently depending on musical tradition, but alr the world inall 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 thediatonic 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 obviousmystery about octaves is that tones an octave apart sound similar, though the

frequency is the double or the half.


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They pertain, so to speak, to the same family; from the same root, unfolding in the

spectrum of frequencies. They have the sameChroma.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 inthe 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 ofsound, but can be observed as manifesting itself throughout Nature around, and in

astronomy above.

The Fourth

The very harmonious Fourth is a kind of a puzzle, with its prime interval in the ratioof 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 areseparated 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 , 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 oppositemovementlowering by 4:3makes it possible to fit the generated intervals into one

octave.

Music and mathematics

Music and Numbers are often said to be as brother and sister, different but related. Inaddition, we have to take into consideration numerical representation, which plays animportant role in Eastern music but is ignored in the Western tradition.

Composite sound

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 andamplitude (loudness), but also on its specific numbers of harmonics (formats), whichcolor 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 bluenotes 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.
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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 consistenttheory, the Consonant Theory, which now is generally accepted.

Beats

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 thedifferent waves, which alter the amplitude so an intensified rhythmic beating, floatingtone is heard as a third tone.

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 thanothers, and those are picked up by theear as unclean, muddy or unpleasant,

and are labeled dissonant.

The intervals which make fewest beatsare called consonant, such as an octave,

the perfect fifth, the perfect fourththe

three prime intervals, or The body ofHarmony as described by Aristotle; the

basis for the musical scale.

A general rule about sound ratios is thatthe simpler the ratios between sounds

are, the more their relations are

harmonious, while the morecomplicated the ratios are, the moredissonant 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 ofharmonia 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 Spheresaroused deep emotions in me. It alludes tothe seven planets known at that time, and has puzzled generations since it wasdeclared. Johannes Kepler dedicated most of his life to attempting to solve that notion.

The auditory system
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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 hearingfunctions.

When the frequency ratios are narrowed down to suchsmall intervals that our auditory system is not capable

of differentiating, the harmonics become fusedbecause of the critical band, a relatively new

discovery, (around 1970-80 by Plomp a.o.) which

refers to the overlapping amplitude envelopes on thebasilar membrane in the Organ of Corti in the

Cochlea.

Trained ears are able to detect the harmonics up to the

6th or 7th harmonics.Schematicgraph of the Cochlea

When the interval between two tones decreases, their amplitude envelopes overlap toan increasing extent. A rough, harsh tone will be heard, which anyone can hear whentwo notes with less than minor 3rd separationare played simultaneously. This is very

shortly the key to understand the theory of dissonance and consonants, which is thefoundation 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 earperforms apartial frequency analysisof 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 ofthe 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. Thenthe 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.

Breakthrough in the science of hearing

Helmholtz beat theories was commonly accepted for about 100 years, before the

Noble Prize winning Hungarian scientist Bksy 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 basilarmembrane and frequency of maximum response. A high frequency pure tonegenerates a wave that travels only a short distance along the basilar membrane before


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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 anddying away. Signals from nerve fibers connected to this region

of the basilar membrane evoke a low-frequency sensation inthe brain.

Other theories than the above place theory have been brought

forward, among them the temporal theories, i.e. emphasizingthe use of the timing information in nerve signals.

Helmholtz dismissed

The modern Consonance Theory of Plomp extended the

discoveries of Bksy with some new important findings, thatgave whole new meanings to the concept of hearing. The beat theory of Helmholtzwas finally dismissedin favor of the well experimented and proven Consonance

Theory, in which the ears Discrimination Frequency and its Critical Bandwidth playsan important part.

The Critical Band

As the interval between two tones decreases, their amplitude envelopes on the basilarmembrane 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, theamplitude 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 theiramplitude envelopes, we say that their frequencies lie within onecritical band.Thisconcept 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.

Logarithmic intervals and frequency distributions

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 mostimportant biological discoveries of the 20th century. The published work wasStructure of Fauna and Flora with Regard to Body Size of Organisms (Lomonosov-
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University, Moscow).

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 speciesfind 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 pressureof 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 certainsizes would be advantageous for the survival of a species, and what these advantages

actually were.

The logarithmic frequency distributions by Dr. Hartmut Mulier

Cislenko's work inspired the German scientist Dr. Hartmut Mller to search for otherscale-invariant distributions in physics. The phenomenon of scaling is well known to

high-energy physics.

Mller found similar frequency distributions along the logarithmic line of sizes,

orbits, masses, and revolution periods of planets, moons and asteroids. Being amathematician 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 anumber with measuring unita 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 numbersequence 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 notchange, 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!

Any scale is a logarithm


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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 willaccelerate growth so as to overcome these sections as quickly as possible.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 scalesare 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 6units 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 itselfthe cause for the standing pressure wave.

Leonard Euler (1748 ) had already shown, that irrational and transcendental numberscan be uniquely represented as continued fractions in which all elements (numerators

and denominators) will be natural numbers.

Prime 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 naturalnumbers; 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 noformula 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 409.


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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, itdetermines the frequency of distribution of matter on all physically calibrated

logarithmic lines - the line of ratios of size, that of masses, of frequencies, oftemperatures, velocities, etc.

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 thecalculation 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 intechnological applications.

I wish to thank Dr. Willy de Maeyer for his help in the subject of this deeper scientific nature of

scales. More similar kinds of mind-puzzling statements in sound and music can be found on my

pageThe Sound of Silence.The Creation of Musical Scalesfrom a mathematic and acoustic point of view, Part II,

by Thomas Vczy Hightower

My first search was to look at musical practice in ancient times, not only in Europe

but all over the world. There were several other musical scales besides the diatonicscale, where the semitones were located in other places than from me-fa and si-do. In

the Gregorian modals, for instance, the different placement of the semitones createsthe specific modes.

Pentatonic Music

In pentatonic folk music semitones do not exist. By practical experience, people have

found out that the five-note scale allowed the possibility of playing in any key withoutsignificant disharmony. Theorists would say that the scale was composed of

ascending and descending fifths, only in two steps in each direction. A pentatonic

scale can be played by only using the black keys on the piano.

EASTERN MUSIC
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After a study of ancient main cultural music, mainly Chinese and Indian, I realized

how universal the concept of the octave was in every musical culture.

According to Helmholtz, the Arabic and Persian scales, and the Japanese and the

Pacific scales are also within an octave. However, the division of the octave differsfrom culture to culture.

Arab music divides the octave into sixteen unequal intervals. The Persians dividedtheir octave into 24 steps, so they must have used quarter tones. From excavated

Egyptian flutes, a seven note scale C, D, E, F#, G, A, B, has been discovered, which isidentical with the Syntolydian scale of ancient Greece. Japanese music used mainly a

pentatonic scale.

Chinese music

Music was the cornerstone of the Chinese civilization, the longest living culture in

history. It was considered to embody within its tones elements of the celestial order.

The audible sound, including music, was but one form of manifestation of a muchmore fundamental form of Super-physical Sound. The fundamental Primal Sound was

synonymous with that which the Hindus call OM. The Chinese believed that this

Primal Sound, Kung or Huang Chung (directly translated yellow bell) was, thoughinaudible, present everywhere as a Divine Vibration.Furthermore, it was also divided into 12 lesser Sounds or Tones. These twelve Cosmic

Tones were emanations of, and an aspect of, the Primal Sound, but were closer invibration to the tangible, physical world. Each of the 12 Tones was associated withone of the12 zodiacal regionsof the heavens.

Audible sound was conceived as being a physical level manifestation of the 12 tones.Sound on Earth was a kind of sub-tone of the celestial vibration. It was believed tocontain a little part of the celestial tones' divine power.

As above, so below, as the Egyptian Hermes Thot said.In the Lords Prayer, asimilar wish is spoken.

For the ancient Chinese, the alignmentwith the divine prime tone was the Emperor's

most important task. The alignment of earth with heaven, and man with the Supreme,

was literally the purpose of life. The entire order and affairs of the State were

dependent upon the right tuning of the fundamental tone, the yellow bell, or Kung.

As an ancient text warns: If the Kung is disturbed, then there is disorganization; theprince is arrogant.

If the Kung was out of tune, because the celestial realm has changed, disorder andinharmonious behavior in society became obvious. Every instrument (including

measuring instruments) was tuned and utilized in accordance with the holy tone.

The instrument that could give to man the fundamental tone for a musical scale inperfect harmony with the universe was the key to earthly paradise, and essential to the

security and evolution of society.It became the Chinese Holy Grail.

One legend tells of the amazing journey of Ling Lun, a minister of the second
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legendary Chinese Emperor, Huang Ti. Ling Lun was sent like an ancient Knight of

King Arthur to search for a special and unique set of bamboo pipes. These pipes were

so perfect that they could render the precise standard pitches to which all other

instruments throughout the land could be tuned.

That sacred tone, which relates to the Western modern pitch of F, was considered as

the fundamental cosmic tone. The Chinese were aware of the slow changing cosmicinfluence, and consequently the Kung has to change accordingly. The Emperor had

the task of tuning the Kung so it was in alignment with the cosmic tone.

Tuning the Sacret Kung

Cousto has in his book The Cosmic Octave an interesting observation on this matter.

He relates the Kung to the frequency of the Platonic Year.The duration of the

Platonic Year, (The Pythagorean Great Year) is about 25,920 years and represents the

amount of time the axis of the Earth takes to complete a full rotation.

The vernal equinox is the point at which the equator (of Earth) intersects the ecliptic(or zodiac), which is the position of the sun at the beginning of spring - March 21st.

The vernal equinox takes an average of 2,160 years to travel through one sign of the

zodiac. This period of time is known as an age. It is not possible to state exactly whenone age is ending and a new beginning, because the signs overlap to a certain degree.

The journey of the vernal equinox through each of the 12 signs of the Zodiac equalsone great year of approximately 25,920 years. (Presently we are on the cusp of

Aquarius as the age of Pisces is ending).

This number of years is close to the high number ofgenerating fifths when we comeinto a cycle of 25,524 notes.

Cousto calculates the note of the Platonic year to be F in the Western Equal

Temperament pitch, which is found in the 48th octave with a frequency of 344.12Hz., or in the 47th octave to be 172.06 Hz. Note that the corresponding a' has a

frequency of 433.564 Hz. (Modern Western concert pitch is 440 Hz.)

Calculation: 31 556 925.97(the tropical year in seconds) * 25,920 (Platonic year).Since the length (of a vibrating string, or the period of time) is in reverse

proportionality to the frequency, the length of the Platonic year in seconds shall be the

denominator. The frequency is very low, so we will raise the frequency to the range ofhearing by multiplying with the necessary amount of octaves, e.g. 48 octaves so wearrive to 344,12 Hz. (47 octaves will be the half, 172,06 Hz.)

If we want to reach the spectrum of light, we multiply with 89 octaves which leads usto a frequency of 1/31 556 925.97 * 1/25 920 * 2 89 = 7,56 * 10 14 Hz. corresponding

to a wavelength of 0.396 micrometer, which we perceive as violet near the ultraviolet. This is the color of the Platonic Year. The complementary color to violet is

yellow. Their fundamental tone was called the yellow bell.

It is a wonder for me how the ancient Chinese could be aware of their sacredfundamental tone, Kung, being in accordance with the Platonic Year, and choose the

great rhythm of the Earth.
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Creation of a scale

It might be a surprise that the diatonic scale was the foundation for the ancientChinese and the Indian music, though the musical theory and practices differ from theWestern.

For the old Chinese, their musical scale was developed by the circle of perfect fifths

up to 60 degrees or keys, the 60 L, though they usually only used the first 5 fifths intheir pentatonic music, because they knew that these represent the limit of consonance

in modal music. In addition, the ancient Chinese saw a symbolic representation in the

pentatonic scale, rooted in their belief in music as being the representation of therelationship between heaven and earth (the five elements).

The Chinese were well aware centuries ago of the existence of our modern Equal

Temperament. They dismissed such a tempered scale not only for its badly false

notes, but mainly because the tuning was not in alignment with the cosmic tone.According to the book by David Taime, The Secret Power of Music, 3 was the

symbolic numeral of heaven and 2 that of the earth; sounds in the ratio of 3:2 willharmonize heaven and earth. As a way to apply that important concept, the Chinesetook the foundation note, Huang Chung, and from it produced a second note in the

ratio of 3:2.

A more in-depth explanation made by Alain Danilou in his Music and the Power ofSound:

Music, being the representation of the relationship between heaven and earth, must

quite naturally have this confirmation of a center or tonic (gong) surrounded by fournotes assimilated to the four directions of space, the four perceptible elements, the

four seasons, and so on. "

The pentatonic scale thus presentsa structure that allows it to be an adequaterepresentation of the static influence of heaven on earth. But a static representation of

a world in motion could not be an instrument of action upon that world. It is necessary

to evolve from the motionless to the moving, from the angular to the circular, from thesquare to the circle. To express the movements of the universe, the sounds will haveto submit to the cyclic laws that, in their own field, are represented by the cycle of

fifths.

The spiral of fifths

As we have already seen, the fifth is thethird sound of the series of harmonics, the

first being the fundamental and the secondits octave. According to the formula of theTao-te ching, One has produced two, two

has produced three, three has produced allthe numbers, we can understand why the

third sound, the fifth, must necessarily


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produce all the other sounds by its cyclic repetitions.

Observe the feminine & masculine notes respectively pink & blue.

The first to be produced will be the four principal sounds, which form comparativelysimple ratios with the tonic.

For the sake of convenience we will use Western notes: SeeChinese & Western

Music.I, C

II, G = 3/2III, D = 9/8 = (3/2)2* (lower an octave)

IV, A+ ( a comma sharp) = 27/16 = (3/2)3* (lower an octave)

V, E+ (a comma sharp) = 81/64 = (3/2)4* (lower 2 octaves).These five primart sounds represent the elementary structure of the perceptible world,

the pentatonic scale. These sounds are used in music, as you can play the five black

keys on the piano. Howevwer, the next two fifths have to be added as two auxiliry

sounds:VI, B+, (a comma sharp) = 243/128 = (3/2)5* (lower 2 octaves)

VII, L+F? (sharpen a major half tone) = 729/512 = (3/2)6* 1/8 (lower 3 octaves).

The seven-notes Chinese scale

C D E+ (F)L+F#1/1 G A+ B+

1/1 9/8 81/64 4/3 (729/512) 3/2 27/16 243/128

Let us note here that the most striking difference between the system of fifths andthat of harmonic relations to a tonic, resides in the perfect fourth, which is an essential

interval in the scale of proportions, but in the scale of fifths it is an augmented fourthas its sixth fifth, (3/2)

6..

The two auxiliary sounds243/128 and 739/512should not be used

as fundamentals, though they are needed for transpositions, because they belong to thescale of invisibleworlds, and therefore we can neither perceive their accuracy nor

build systems upon them without going out of tune.

Instead of starting from C, we could have begun one fifth below, that is to say, from

F, and we would have obtained this essential note without changing anything in ourscale, except that, since we begin with a masculineinterval instead of

a feminineinterval, the character of the whole system is modified.

The five successive fifths, whether in an ascending or a descending series, representthe limit of consonance in modal music too. Beyond this limit, no interval can appear

harmonious, nor can it be accurately recognized. A rule originating from the same

principle was also known in medieval Europe, where the tritone was prohibited asdiabolical, that is, as connected with forces that are supernatural and therefore

uncontrollable.Folk music in its pentatonic form had understood this too by only using the span of

two fifths up and down.
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After these seven notes, the next five notes generated by the series of fifths are:

VIII, bDb lowered a minor half tone, IX, bAb lowered a minor half tone, X, bEb a

minor half tone lower, XI, bBb a minor half tone lower, XII, F+ a comma sharp.

We now have twelve sounds, which divide the octave chromatically into twelve half

tones.

The twelfth fifth (note 13) in a 7 octave span brings us back to the fundamental, butwith a slight difference.

It is higher than the fundamental by one comma, the Pythagorean comma (312 / 219 =531,441/524,288, (5.88 savarts or 23.5 cents). It is, therefore, in our notation, C+, one

comma sharp.

In this way, successive series of twelve fifths will be placed one above the other atone-comma intervals, up to the 52nd fifth (note 53) which fill the octave.

The Chinese continued the cycle of fifths up to 25,524 notes, with a basic interval of

0.0021174 savarts. This cycle is very near to that of the precession of theequinoxes,or the Pythagorean Great Year, which is of 25,920 solar years. Why the

Chinese continued so many octaves in the cycle of fifths could have something to do

with their reference tone,Kung.

In practice, for reasons that are symbolic as well as musical, after the 52nd fifth (53rdnote) the Chinese follow the series only for the next seven degrees, which placethemselves above those of the initial seven-note scale, and they stop the series at the

60th note. The reason given is that 12 (the number of each cycle) * 5 (the number of

the elements) = 60.

The scale of 60 L

The Chinese scale, being invariable, constitutes in effect a single mode. Every change

in expression will therefore depend upon modulation, a change of tonic.Firstly, the choice of gender: fifths whose numbers in the series are even are feminine.

The odd numbered fifths are masculine.

The choice of tonic is dependent on complicated rules and rituals, whose mainpurpose is to be in accordance with celestial as well as earthly influx orcircumstances. Accordingly, the Chinese have to choose the right key for the hour of

the day and the month, even during a performance.It is an extensive scheme, but to get an idea we can say that it corresponds to politicalmatters, seasons, hour of the day, elements, color, geographic direction, planets and

moon.This scale of fifths, perfect for transposition because of its extreme accuracy, also

allows the study of astrological correspondences and of terrestrial influx in theirToneZodiac.

We notice that the Chinese scale is very similar to the Pythagorean tuning, which was

also produced by generating a perfect fifth (3: 2). When the Chinese derived theirscale goes back to 3000 BC, when European stone-age man was still beating wooden
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logs. The prevalent opinion in the West about our music superiority should hereby be

moderated.

The Indian music system

The ancient Indians had a less formalized approach to their music than the Chinese.Generally speaking they emphasized the personal inner contemplation more than the

outward organized rituals. One can say that they sought inner alignment with thedivine supreme by means of the sounds AUM or OM, which were (are) the earthly

sound of the prime creator, Brahman.

For the Hindus, as the Chinese, the spoken or chanted words were the carrier of some

of the creative energy, and composed by the prime Creator. Pronounced correctly, it

was believed that special words were able to alter humans thoughts and feelings and

literally change and form physical matter.

Raga is the basic form in classical Indian music. There is a whole system of Ragas,which differ respectively between North and South India. Originally there were only 7Ragas. These may have been the remnant of an ancient reference to the seven Cosmic

Tones: the seven principal notes, or savaras, connected with the seven main planets,

and two secondary notes corresponding to the nodes of the moon. This brings the totalnumber of notes in the scale to nine principal notes, which is related to the nine

groups of consonants of the Sanskrit alphabet.

The Raga system grants musicians freedom of expression within the limitations of a

certain inviolable mode. Since music was so important a force in altering phenomenaupon Earth, they considered it would be unwise, dangerous, and perhaps even suicidal

in the long run to allow musicians to perform whatever they wished.

The Indian solution was then to apply a system of rules which, while effectively

determining what typeof music was performed and even its spiritual atmosphere and

the period of the day, did not indicate the notesthemselves. This was a convincinglysuccessful solution to the problem that the music of ancient civilizations always cameup against.

The Chinese had a more rigid system. They created variations by use of instruments,

and especially in the expression of the single note. The dimensions of tone color, ortimbre, were highly developed in the East. The ear had to learn to distinguish subtle

nuances. The same note, produced on a different string, has a different timbre. The

same string, when pulled by different fingers, has a different timbre, etc. Furthermore,and very important, the whole spiritual being of the musician himself was

crucial.That applies also to Indian music.

As in the Western diatonic scale, the Indian scale was based on 7 main notes: SA, RE,

GA, MA, PA, DHA and NI. If we go back to the most ancient texts on music, thescales were divided intotwo tetrachords, similar to the ancient Greeks, and later put

together with a whole tone (9/8) between, MaPa, so a full octave was completed.


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The Indian notes relate broadly to the Western ratios, though the tuning is very

harmonious and creates a world of difference. We have to emphasize that the use of

harmony as we know it was, but is no longer, musically practised.

Here is a crucial point. The Indian music is modal. There is a strong relationship to

the tonic. When a third is played it always relates to the third degree; whereas in

Western harmonious tradition the third has a relative position, because it can be theroot, the fifth or third of a chord.

Eastern listeners often make remarks such as: Beethoven symphonies are interesting,but why have all those chords been introduced, spoiling the charm of the melodies?

The modal music of India is 'horizontal' as the Western is 'vertical'. The vertical,

harmonious system, in which the group of related sounds is given simultaneously,might be more direct though also less clear. The accurate discrimination of the

different elements that constitute a chord is not usually possible.

The modal, horizontal system, on the other hand, allows the exact perception andimmediate classification of every note, and therefore permits a much more accurate,

powerful and detailed outlining of what the music expresses.

One can say that the attention span in the Eastern musical language has to be muchlonger since, in time, the different and distinct sounds adding up in the listeners mindcreate the chords or the whole musical idea. Only then, by remembering with attention

all the elements that constitute the musical image, can the full meaning finally beunderstood.The Indian musical system operates with a combination of fixed and mutable pitch, so

the key can be recognized along with variable notes. The 2nd, 3rd, 4th, 6th and 7th

notes are variable, but the 1st (Sa or Do) and the perfect 5th (Pa or Sol) are immutable

and of a fixed pitch. The drone is accordingly often Do-Sol (Sa-Pa), which becomesthe ultimate open chord containing all other notes within it as a series of subtleharmonics.

This drone (a constant note or tonic), whether actually played on an instrument likethe tampura or simply heard within oneself as the Om sound, is the constant referencewithout which no Indian musician would play.

One must not be confused by the vast use of micro intervals, sliding or bending the

notes, prominent in Indian music. The musicians can freely use these microtones asprivate points, often moving freely between two notes as a kind of infinitely

exploitable space, eventually returning home to the tonic of the Raga. The musician

has a freedom to play tones as his inspiration demands so long as he obeys the sacredrules of types and its mood.

The 22 Shrutis (degrees)

Musical intervals can be defined in two ways, either by numbers (string lengths,

frequencies) or by theirpsychologicalcorrespondences, such as feelings and images

they necessarily evoke in our minds. There is no sound without a meaning, so theIndians consider the emotionsthat different intervals evoke as exact as sound ratios.


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The feeling of the shrutis depends exclusively on their position in relation to the tonic,

and indicates the key for the ragas.

The 22 different keys or degrees encompass what the Indians consider the mostcommon feelings and reflections of the human mind. They were aware of the division

of the octave into 53 equal parts, the Pythagorean Comma, and its harmonic

equivalent, the comma diesis, (the syntonic comma, the difference between the majorand the minor tones).

However, they chose the 22nd division of the octave for reasons based on the limit ofhuman ability to differentiate the keys, as well as for psychological and metaphysical

reasons. The symbolic correspondences of the numbers 22 and 7, (7 strings and main

notes), could also play a part since the relationship between the circle and thediameter is expressed as the approximate value of Pi, 22/7.

The modal or Harmonic division of the octave

Indian music is essentially modal, which means that the intervals on which the

musical structure is built are calculated in relation to a permanent tonic. That does notmean that the relations between notes other than the tonic are not considered, but thateach note will be established first according to its relation to the fixed tonic and not,

as in the case of cycle of fifths, by any permutations of the basic note.

The modal structure can therefore be compared to theproportionaldivision of thestring (straight line) rather than to the periodic movement of the spiral of fifths.

All the notes obtained in the harmonic system are distinct from those of the cyclic

system, which is based on different data. Though the notes are theoretically distinctand their sequence follows completely different rules, in practice they lead to a similar

division of the octave into fifty-three intervals.

The scale of proportions is made of a succession of syntonic commas, 81/80, whichdivide the octave into 53 intervals. Among those, 22 notes were chosen for their

specific emotional expressions:

Note degree Interval Value in cents Interval Name Expressive qualities

1

1/1

0

unison

marvelous, heroic, furious

2 256/243 90.22504 Pythagorean limma comic

3 16/15 111.7313 minor diatonic semitone love

4 10/9 182.4038 minor whole tone comic, love

5 9/8 203.9100 major whole tone compassion

6 32/27 294.1351 Pythagorean minor third comic, love

7 6/5 315.6414 minor third love

8 5/4 386.3139 major third marvelous, heroic, furious

9 81/64 407.8201 Pythagorean major third comic

10 4/3 498.0452 perfect fourth marvelous, heroic, furious

11 27/20 519.5515 acute fourth comic

12 45/32 590.2239 tritone love


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13 729/512 611.7302 Pythagorean tritone comic, love

14 3/2 701.9553 perfect fifth love

15 128/81 792.1803 Pythagorean minor sixth comic, love

16 8/5 813.6866 minor sixth comic

17 5/3 884.3591 major sixth compassion

18 27/16 905.8654 Pythagorean major sixth compassion

19 16/9 996.0905 Pythagorean minor seventh comic

20 9/5 1017.596 just minor seventh comic, love

21 15/8 1088.269 classic major seventh marvelous, heroic, furious

22 243/128 1109.775 Pythagorean major seventh comic, love

The Ancient Egyptians

The ancient Egyptians had similar beliefs to the Chinese and Hindus. In their Book ofthe Dead and other sources, it is stated that God, or his lesser servant gods, createdeverything, by combining visualization with utterance. First the god would visualizethe thing that was to be formed; then he would pronounce its name: and it would be.

From as late as the reign of Alexander II, a text dating from about 310 BC still has the

God of Creation, Ra, declaring: Numerous are the forms from that which proceeded

from my mouth. The god Ra was also called Amen -Ra, with the prefix Amen. TheEgyptian priesthood understood well the word Amen, or AMN, and it was equated

with the Hindu OM.

Egyptian music, as does Greek, most probably had its roots in Indian music, or at leastin that universal system of modal music whose tradition has been fully kept only by

the Indians.The pyramid can easily be a symbolic representation of Earth with its four perceptible

elements, and all its characteristics that are regulated by the number fourthe four

seasons, four directions of space, etc.; especially the projection of the single into themultiple.

WESTERN MUSIC

Pythagoras

The Greek philosopher Pythagoras (570 - 490 BC) spent 22 years in Egypt, mainly

with the high priest in Memphis, where he became initiated into their secretknowledge of Gods. When the Persians conquered Egypt, he was kept in captivity in

Babylon for sixteen years before he could return to Greece and begin his teaching.

I began to study the theory of the Pythagoreans and their esoteric schools. Very littleis known of them. Pythagoras demanded silence about the esoteric work. This historic

school was founded in the Greek colony Kroton, in southern Italy, about 2,500 years

ago.


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I realized after reading dozens of books about the matter what an outstanding role that

school played in the establishment of western civilization. He created an entirely new

concept. Any person - man or woman - who had a sincere wish for knowledge could

enter the school stepwise, with a number of initiations. The tradition of a priesthoodsmonopoly of knowledge of God was broken.

Pythagoras' study of the moving string and his discovery of the harmonic progressionof simple whole numbers was the first real scientific work and creation of modern

science. But his vision went far beyond present science in his deep understanding ofthe integration of the triad: Ascience, Bwork on being, Clove and study of

God. Something modern science could learn from!

Nicomachus of Gerasa

Nicomachus the Pythagorean (second century B.C.) was the first who wrote about

Pythagoras legendary encounter with the harmonious blacksmith and the weights ofthe 4 different hammers being 12, 9, 8 and 6, that determined the variation in the

pitches Pythagoras heard.This story illustrates how the numerical proportions of the notes were discovered. Hismethodical measuring of the hammers and how the sound was produced and related

(collecting data), then making experiments with strings, their tension and lengths(repeating the findings and, with mathematics, formulating them into a law), was thefirst example of the scientific method.

We will not dwell on the question of the force of the impact or the tension of the

strings, which later was discovered as the square root of the force, but just stick to theproportion of weights and the pitches he heard, which led him to his discovery.

Pythagoras' experiments led to the combination of two tetrachords, (two fourths),

separated with a whole tone, 9/8, which constitute an octave. He changed thetraditional unit in Greek music, the tetrachord, into the octave by an octachord.

In the time of Pythagoras the tradition was strongly based on the seven strings of the

lyre, the heptachord. The Greeks considered the number 7 sacred and given by thegod Hermes, who handed down the art of lyre playing to Orpheus. The seven-stringlyre was also related to the seven planets, amongst other things the ancients

venerated.The lyre often, but not always, consisted of seven strings comprising two tetrachords,each one spanning the most elementary concord, the fourth, both joined together on

the note mese.

According to legend, a son of Apollo, Linos, invented the four-stringed lyre with threeintervals, a semitone, whole tone and a whole tone comprising a fourth; the fourth,the first and most elementary consonance as Nicomathuscalls it, and from which all

the musical scales of ancient Greek music eventually developed.

Trepander of Antissa on Lesbos, born about 710 B.C., assumed a mythological statusfor his musical genius. His most lasting contribution was perhaps his transformation

of the four-stringed lyre to the instrument which became institutionalized by tradition


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to the heptachord.

Trepander did before Pythagoras extend the heptachord from its minor seventh limits

to a full octave, but without having to add the forbidden eighth string.

He removed the Bb string, the trite of the conjunct tetrachord, and added the octavestring, E1, yielding a scale of E F G A C D E1.

This arrangement left a gap of a minor third between A and C, and seemed to haveenhanced the Dorian character of Trepander's composition.

Harmonia

Only Pythagoras escaped censure for adding an eighth string to the ancient and

venerated lyre because of his position as a great master and religious prophet. His

purpose was to teach man the unifying principle and immutable laws of harmonia by

appealing to his highest powers - the rational intellect and not to his untrustworthyand corruptible senses. Pythagoras altered the heptachord solely to engage man's

intellect in proper fitting together - harmonia- of the mathematical proportions.

Plutarch (44-120 B.C.) states that for Pythagoras and his disciples, the word harmonia

meant octave in the sense of an attunement which manifests within its limits boththe proper fitting together of the concordant intervals, fourth and fifth, and thedifference between them, the whole tone.

Moreover, Pythagoras proved that whatever can be said of one octave can be said of

all octaves. For every octave, no matter what pitch range it encompasses, repeats itselfwithout variation throughout the entire pitch range in music. For that reason,

Pythagoras considered it sufficient to limit the study of music to the octave.

This means that within the framework of any octave, no matter what its particularpitch range, there is a mathematically ordained place for the fourth, the fifth, and for

the whole tone. It is a mathematical matter to show that all of the ratios involved inthe structure of the octave are comprehended by the single construct: 12-9-8-6.

For the Pythagoreans, this construct came to constitute the essential paradigm - of

unity from multiplicity.

The arithmetic and harmonic mean

We see that 12:6 expresses the octave, 2:1; 9 is the arithmetic mean, which is equal tothe half of the sum of the extremes, (12 + 6)/2 = 9.

Further, 8 is the harmonic mean of 12:6, being superior and inferior to the extremes

by the same fraction.Expressing this operation algebraically, the harmonic mean is 2ac/a+c, or in this

series, 2*12*6/12+6 = 8.


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Among the peculiar properties of the harmonic proportion is the fact that the ratio of

the greatest term to the middle is greater than the middle to the smallest term: 12:8

>8:6. It is this property that

made the harmonic proportionappear contrary to the arithmetic

proportion.

In terms of musical theory, these

two proportions are basic fordivision of the octave since the fifth, 3/2, is the arithmeticmean of an octave and the

fourth, 4/3, is the harmonicmean of an octave.

The principle of dividing the string by an arithmetical proportion is done by theformula: a:b is divided by 2a:(a+b) and (a+b):2b.

The ancient Greeks presumably did such division in their studies of the singing string

of the monochord.

The semitone

We have already seen that in the diatonic genus each tetrachord was divided into two

full tones and one semitone. A full tone derives from a fifth minus a fourth, 3/2 - 4/3 =9/8. The semitone will be 4/3 - (9/8 + 9/8), or 4/3 - 81/64 = 256/243.

This semitone is called leimma, and is somewhat smaller than the half tone computedby dividing (for musical ratios dividing means the square root) the whole tone in half:

(9/8)= 3/2*2.

The square root of 2 was for the Pythagoreans a shocking fact, because their concept

of rational numbers was shattered. (For me it represents the beauty of real science,because it revealed the flaws in the Pythagorean paradigm of numbers). Their own

mathematic proved with the Pythagoreans doctrine of the right-angle triangle (thesum of the squares of the two smaller sides of a right-angled triangle is equal to the

square of the hypotenuse) that in music, as in geometry, there are fractions, m/n, that

are incommensurablessuch as the square root of 2, which cannot be expressed withwhole numbers or fractions, the body of rational numbers, but with irrational numbersnot yet developed.

This discovery was held as a secret among the Pythagoreans and led to the separationof algebra and geometry for centuries, until Descartes in the 17th century united themagain.

For music it meant that there was no center of an octave, no halving of the whole tone,no perfect union of opposites, no rationality to the cosmos. The semitone could be the door to other dimensions!

My task here is to give some clues to the meta-physical functions of semitones, which

seem to involve the potential to shift to a different world or enter another dimension.The key to attaining a different spiritual world exists in the search for the exact right
http://vaczy.dk/pix/oktavdeling.png


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tone that resonates with that particular door to other dimensions and worlds. The

human being contains more dimensions than just threespatial dimensions.

Philolaus

We have to bear in mind that Pythagoras himself left no written record of his work; itwas and is against esoteric principles. Neither did those few students who survived the

pogrom of Pythagoras. It is one in the nextgeneration of Pythagoreans, Philolaus(ca.480- ? B.C.), who broke the precept of writing down the masters teaching.

However, Philolaus' records are lost, so it is Nicomachus fragments of his writing, in

his Manual of Harmonics, that is actually the only source posterity has.

According to Nicomachus / Philolaus, the whole tone, 9/8, was divided differently

from the Pythagoreans method, by representing the whole tone with 27, the cube of 3,

a number highly esteemed by the Pythagoreans. Philolaus divided the whole tone in

two parts, calling the lesser part of 13 units a diesis, andthe greater part of 14 units,apotome. Philolaus had, in effect, anticipated Plato's calculations in the Timaeus!

Timaeus by Plato

Plato (427-347 B.C.) gave in his work Timaeus a new meaning to the Pythagorean

harmonic universe byin a purely mathematical methodenclosing it within themathematically fixed limits of four octaves and a major sixth. It was determined by

the numbers forming two geometrical progressions, of which the last term is the

twenty-seventh multiple of the first term:

27 = 1+2+3+4+8+9The two geometric progressions in which the ratios between the terms is 2:1 and 3:1

are, respectively:1-2-4-8 and 1-3-9-27.

Combining these two progressions, Plato produced the seven-termed series: 1-2-3-4-8-9-27. The numbers in this series contain the octave, the octave and a fifth, the

double octave, the triple octave, the fifth, the fourth and the whole tone. The entire

compass from one to twenty-seventh multiple comprises, therefore, four octaves and amajor sixth. In numerical terms it contains four octaves, 16:1 * 3:2 (a fifth) * 9:8 (awhole-tone) equals 27:1.

Plato then proceeded first to locate in each of the octaves the harmonic mean, thefourth, then the arithmetic mean, the fifth. By inserting the harmonic and thearithmetic means respectively between each of the terms in the two geometric

progressions, Plato formulated mathematically everything Pythagoras had formulatedby collecting acoustic data.

Plato did, however, independently of the Pythagoreans, compute the semitone in the

fourth, which consists of two whole tones plus something, which is less than the halfof a whole tone, namely 256:243, the leimma.

According to Flora Levin in her commentary on Nicromachus' The Manual ofHarmonics, Plato went further than Pythagoras by completing all the degrees in a

diatonic scale:
http://vaczy.dk/htm/octave3.htm#dimensionshttp://vaczy.dk/htm/octave3.htm#dimensionshttp://vaczy.dk/htm/octave3.htm#dimensions


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1 9/8 81/64 4/3 3/2 27/16 243/128 2

E F# G# A B C# D# E'

Plato's calculations led to the inescapable fact of no center to the octave, no halving of

the whole tone with rational numbers, no rationality of the cosmos.Nicomachus did his part in covering up the secret by misrepresenting Plato andputting off some of the shattering discoveries of irrational numbers to some future

time.

The semitones in the different modes

Pythagoras had practiced music long before he transformed the heptachord into anoctachord that led him to discover the mathematical laws determining the basic

structure of an octave. He had fully understood the therapeutic value of music in

healing the body and soul. Most of all, he knew the set of conditions for melody. Herecognized strongly that every tetrachord on which melody was based embodies the

natural or physical musical progression of whole tone-whole tone-semitone.He maintained the fundamental structure of both tetrachords in his scale, and for

musical reasons he understood that this distribution of intervals had to be maintained

for all melodic purposes with their configurations and inversions.This was the foundation of the ancient Greek music, which further developed into TheGreater Perfect System.

The confusion of systemsThe Greek music has an inherent confusion of musical systems: a mix of the cyclicsystem of perfect fifths (Pythagorean tuning), and the modal system (tetrachords). Wecan only get a very faint idea of what ancient Greek music really was about because

European theorists through time have made errors and misunderstandings.

In reality, the Arabs and the Turks happened to receive directly the inheritance ofGreece. In many cases the works of Greek philosophers and mathematicians reached

Europe through the Arabs. Most serious studies on Greek music were written by Arab

scholars such as al-Frbi in the tenth century and Avicenna a little later, whileWesterners - Boethius in particular - had already made the most terrible mistakes.It is the Arabs who maintained a musical practice in conformity with the ancient

theory, so to get an idea of ancient Greek music, we should turn to Arab music.

The Pythagorean Tuning

The musical scale, said to be created by Pythagoras, was a diatonic musical scale withthe frequency rate as:

1 9:8 81:64 4:3 3:2 27:16 243:128 2.
http://vaczy.dk/htm/greater_perfect_system.htmhttp://vaczy.dk/htm/greater_perfect_system.htmhttp://vaczy.dk/htm/greater_perfect_system.htmhttp://vaczy.dk/htm/greater_perfect_system.htmhttp://vaczy.dk/htm/greater_perfect_system.htmhttp://vaczy.dk/htm/greater_perfect_system.htm


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This scale is identical to the Chinese cyclic scale of fifths,if we take F as the tonic.It has 5 major tones (9/8) and 2 semitones, leimma (256/243), in the mi-fa and si-do

interval.

The third, 81/64, is a syntonic comma sharper than the harmonic third, 5/4.Here is the seven-notes Chinese scale:

C D E+ (F)L+F#1/1 G A+ B+

1/1 9/8 81/64 4/3 (729/512) 3/2 27/16 243/1

Let us note here that the most striking difference between the system of fifths andthat of harmonic relations to a tonic resides in the perfect fourth, which is an essential

interval in the scale of proportions.The scale of fifths has an augmented fourth as its sixth fifth, (3/2).

The Pythagorean scale was based on the three prime intervals: the octave, the perfect5th and the perfect 4th. Everything obeys a secret music of which the Tetractys isthe numerical symbol (Lebaisquais).

By generating 12 perfect fifths in the span of 7 octaves, 12 tones were produced. In

order to place the tones within one octave, the descending perfect 4th (thesubdominant) was used, and a 12-note chromatic scale was made.

He discovered what later was called the Pythagorean comma,the discrepancybetween 12 fifths and 7 octaves gives (3:2)12> (2:1)7.Calculated

through, it is: 129.74634 : 128 = 1.014. Or in cents: 23.5. Donot mistake Pythagoras'

Comma for the syntonic comma, equal to 22 cents, which is derived from thedifference between the major tone and the minor tone in the Just Diatonic Scale, ordiscrepancy between the Pythagorean third and the third in the harmonic series whichis 5:4.

As far back as 2,500 years ago the Pythagorean figured out that it was impossible to

derive a scale in which the intervals could fit precisely into an octave. The ancientGreeks explained this imperfectionthe commaas an example of the condition of

mortal humans in an imperfect world.

This fundamental problem with the 3 prime ratios: 2:1, 3:2, 4:3which can beformulated in mathematical terms as interrelated prime numbers having

no commondivisor except unityhas been compromised in a number of differenttemperaments of the diatonic scale up to our time.

In ancient Greek music several other modes were used based on the tetrachords with a

span of the perfect fourth. Later, two tetrachords were put together with a full tone in

between so an octave was established. A number of different modes were used inpractical music performance. The different placement of the two half tones made thedifferent modes.

An account of ancient Greek contributions to musical tuning would not be completewithout mentioning the later Greek scientist Ptolemy (2nd C. A.D.). He proposed analternative musical tuning system, which included the interval of the major third based


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on that between the 4th and 5th harmonics, 5 / 4. This system of tuning was ignored

during the entire Medieval period and only re-surfaced with the development of

polyphonic harmony.

Gregorian church music

From those ancient Greek modes the Christian Gregorian church derived its music,

though their names were a complete mix-up of the original Greek names for theirmodes. What is important in this context is the placement of the two semitones in the

octave. They were placed differently in order to create different modes that produced

a special tonality or mood. The interaction between tones and semitones made eachcharacteristic mode.The Gregorian church music from the late Middle Ages developed an amazing beauty

and spirituality. We owe the monksandHildegard von Bingen- a debt of gratitude

for their partsinging to worship the refinement of the sou

the creation of musical scales

Documents

creation of musical

musical octaveii

different musical traditions

modern western musical

indian musical tradition

theirbasic scales

eastern scales

thediatonic scale