The Twenty-First Century
by Don Robertson

© 2005 Rising World Entertainment

Part Five: The Spiral of Octaves

<-previous                                                              

The Basics: Consonance and Dissonance

The reality of consonance and dissonance is very important for an understanding of the effects of music. The traditional teachings about the subject, however, were often tossed aside by writers, musicologists, teachers, and composers during the twentieth century in order to accommodate the negative music that dominated so much of the century. It is time now to bring those teachings back. I believe that this is the most important musical concept that we will embrace in the 21st Century.
     I found that the music theory website Theory on the Web has a very nice graphic representation on the subject. Let's see what they have to say:

The Octave as a Sine Wave

     Georges Gurdjieff, the important Russian spiritual teacher said that the law of octaves was based upon the discontinuity of all vibrations in nature, as they do not develop uniformly, but with periodic accelerations and retardations. The original impulse becomes alternately stronger and weaker. This could be a phenomenon similar to the AC cycle. This cycle of alternating electrical vibrations can be viewed by means of an oscilloscope or with a computer program such as Sound Forge. Called the duty cycle, it appears as such. 

    All sounds are made up of these cycles. The cycles take place in the time domain. By using a brilliant mathematical formula known as the Fourier transform, these cycles can be translated into the frequency domain, where we find that the timber of the sound depends upon the strength of overtones, the higher harmonics of the fundamental tone.  
     The duty cycle can be equated to a circle broken at its diameter with the bottom half twisted 180 degrees or to the continuous movement of a spiral. Alternating current (AC) obeys the laws of the duty cycle: energy alternates with no energy. Home electricity in the United States alternates sixty times per second, thus it is called 60 cycles, or 60 Hertz. An electric light bulb does not continually burn, but flashes sixty times in one second, faster than our eyes can follow, so we perceive the appearance only of continuous light. This would not be the case when light is powered by a battery, as batteries produce direct current (DC).  
     Musical tones can be represented using the mathematical symbols of trigonometry, signs and cosigns, and the imaginary numbers of calculus. Computer systems that generate musical tones, such as synthesizers, do so using these mathematical formulas. All of this is based on the alternation of the duty cycle.  
    I have something new that I would like to present, something that to my knowledge has not been proposed before: the octave as a sign wave:

    In the great keyboard of nature, encompassing all phenomena, octaves accumulate in a giant spiral, each phenomena appearing in a different frequency range, in different octaves. Matter vibrates at a lower frequencies, sound occupies another region of this "cosmic keyboard," color appears in another, higher, octave, and so on. It is all vibratory: "maya," as it is called in India. The phenomenon created by the vibrations in any octave is always different, but the octave patterns themselves are the same...the same law controls everything. That is why Newlands called his original research "The Law of Octaves."
     The duty cycle of the sine wave has two phases: Phase I is the positive phase of the duty cycle, Phase II is the negative. Notice that there are two places where the cycle touches "0" (the line): at the beginning of the duty cycle ("C"), and as it crosses over the line from the positive phase to the negative at F#, the interval that cuts the octave into two sections, and as we see from the "Theory on the Web" chart, the most dissonant of intervals, called the tritone.
     During the twentieth-century, many composers, just as I did for several years, made an icon out of the tritone (also called the augmented forth or diminished fifth), but in times past, this wasn't the case. From the 13th Century to the 19th it was called diabolus in musica, or the devil in music, and composers were very careful of how they handled tritones in their music. Taken by themselves, the tritone creates a very disturbing sound. To my knowledge, it was probably only during the late 19th century that tritones were first attempted in a bare configuration by composers to denote the presence of evil and suspense.
     The cycle begins on the left on moves to the right as it makes it's course moving away from C towards Db, D, and E. To understand the consonance or dissonance of the pitches, we implement two tone generators, one maintaining a constant "C" pitch, the other increasing in pitch as it follows the path of the duty cycle forward. The dissonance begins right away, as the second tone generator moves up slightly from C on to Db, the discordant minor 2nd interval, then it gradually decreases: the two pitches grate against each other at first, but the sound becomes less dissonant as the interval of D (a major second) is reached, then Eb (the minor third interval), then E (the major third interval). Finally we arrive at the perfect interval of the forth. The cycle re-approaches the line, and when it touches it, we hear the tritone that splits the octave in half. Next we move to the perfect fifth. Even though F and G are as close to the line as Db and B, they are perfect concords, unlike Db and B, both dissonant intervals. This is a very interesting situation. 
      Continuing the movement from the perfect fifth concord, next we arrive at Ab (minor sixth), A (major sixth) as dissonance increases to its maximum just before touching the line at the octave, the most perfect consonance.
     [Notice that the three notes in the first half of the duty cycle are DEF (The first three notes of a minor scale), and in the other half GAB (the first three notes of a major scale).]
     When you look at a scale in this manner, some of the tuning ratios appear very flexible. For example, there is an area around E that is at the top of the curve. E can be a little sharp, a little flat...it is still the third. That is why there are different thirds used in some scales of other cultures, ranging from the low third of Darbari Kanada in North Indian classical music, to the high third of some middle-eastern music. The same goes for sixths. The musicians in India know this, that is why they identify variations of tone with shrutis, which are not a 22-note scale per say as some western musicians would have us believe, but ways to account for slight variations of pitch. And this is why the slight out-of-tuning condition of the equal-tempered scale doesn't kill us (as some would have us believe).

The Enneagram

     While teaching in Russia during the first world war, Gurdjieff described something about octaves that had not been publicly known before. He stated that there were two places in the octave, represented by the major scale, where retardation of the vibrations takes place. These places are between the notes E and F and between the notes B and C. 
     These are the two places on the keyboard where there are no sharp or flat note (black) keys between the white keys. Octaves, according to Gurdjieff, develop according to whether or not these places where retardation of vibrations take place are “filled.” In the right development of these octaves, these places are filled by the energy of octaves that run in parallel to the given octave. This process of filling in Gurdjieff calls a shock. Processes, institutions, laws and other things all progress through time according to the law of octaves, and therefore these two shocks in the octave are a part of the development  process. Gurdjieff stated that if development continues without the shocks being filled in, then the line of development constantly changes. 
     The way he described it, if octaves – as a principal of nature – develop unimpeded, the development looks as if it is proceeding along a straight line, but every time one of the two points in the octave is reached where a shock occurs, even though development of the process that the octave represents appears to be progressing forward along a straight line, the development of the process actually jogs a little, causing what would be compared to a jog in the line. The development continues and after the progress through the series of octaves has gone on for a particular amount of time, what was considered to be development along a straight line has actually turned into a development in the opposite direction! Further continuation will bring the development back to the same direction that it started in. This is actually a full duty cycle, or one full revolution on a spiral. 
     The law of octaves, with this intrinsic circular motion generating a spiral, governs all activity and shows how organizations, religions, and governments -- if allowed to develop without conscious knowledge concerning the application of the ‘filling in’ of these shocks -- always develop into something that is opposite to what was originally intended.  

The Octave with Two Added Shocks

     Gurdjieff explained how an octave of seven notes was transformed into one of nine, the seven notes and the two shocks, by applying the following fascinating bit of arithmetic.

     He divided the number ‘1’ into seven equal fractions:

1/7, 2/7, 3/7, 4/7, 5/7, 6/7, 7/7

     The decimal equivalents of these seven fractions are arrived at using the time-honored tradition of dividing the numerators by the denominator:

1/7 = .142857
2/7 = .285714
3/7 = .428571
4/7 = .571428
5/7 = .714285
6/7 = .857142

     Of course 7/7 equals 1

     Studying the numbers that result from our division is a fascinating process. We find that we have a reoccurring sequence of numbers that is present in every case! This is the sequence 7142857142857….that continues on and on ad infinitum.

              .142857142857142857 etc….  
          7/1.00000000000000000

     (Numerical patterns such as these are very telling. They describe a situation that is naturally occurring and should be taken seriously by the researcher).

     The nature of this pattern is very interesting. It starts with the number ‘7’, which is the number of the octave, the number that we are dealing with here. Next in line is double that number, ‘14’. Following that we have another doubling: ‘28’, and after that another doubling: ‘56’…whoops, not 56, but 57, the ‘7’ being the start of another sequence! It takes some thought about this to fully appreciate the beauty of this sequence and this change from 56 to 57 to start another series. This just really breathes “cycle,” does it not?
  There are other interesting things about this series of numbers. If we add each line in the set of six decimals above, we always get the number ‘28’ as a result. This is because each of the six decimal sequences is ordered in such a way that when you add downwards, only the six numbers of the sequence appear, there are no duplications. We have six pieces of the whole (7/7), each with a sequence of six numbers, the same sequence in each piece, ordered in the very unique fashion. This is something to contemplate.  

The Periodicity of Nine

     The notes of the octave combined with the two shocks show us what is called the periodicity of nine. Creating a circle with nine positions and connecting the numbers from our sequence above, we have the figure that Gurdjieff calls the enneagram.

     The numbers that are not in the sequence are 3, 6, and 9. When a they are connected, the result is a triangle. The numbers 3 and 6 are the shocks.

     By applying the color spectrum and musical notes to the enneagram, we have the following:

    When asked why one shock correctly appeared between the E and the F, but the other, instead of appearing between B and C, appeared between G and A, Gurdjieff stated that this demonstrated the nature of the shock that occurred when passing from one octave to the next. He also said that every completed whole, every organism, is an enneagram, but that each of these enneagrams does not have an inner triangle (3,6,9). 142857 is the working of the corporeal scheme, the triangle is the presence of higher elements in the organism. This inner triangle is possessed by plants, for example, from which come our “drugs”: hemp, poppy, hops, tea, coffee, tobacco.

Octaves and the Overtone Series

     The cosmic keyboard is a repeating pattern of octaves each vibrating at different, albeit proportional, frequencies. The overtone series is quite another thing. Thus we have a Pythagorean comma. The sequence of octaves and the overtone series are two phenomenon that occur in nature. They are separate, but work together.
    A keyboard consists of octaves of pitches that range from low to high. We could extend our piano's 88 keys to included sub-sonic and super-sonic sounds, but that would be little use if we could not hear the music that we created, but this does illustrate how octaves are vibratory frequencies that extend in both directions beyond the realm of sound.
   

The Overtone Series through Time

    It is very interesting to observe that over a period of hundreds of years, as the understanding and use of harmony developed in Europe, higher partials in the overtone series were gradually incorporated harmonically and melodically into the music.
     During the time of early organum around the 9th-10th centuries, harmonizing with parallel fifths and fourths predominated. Later, the third and sixth was added, yielding the first use of three-note polyphony in European church music (C,E,G).
     During the Renaissance period, the fourth note, the seventh (C,E,G,Bb) was added, but as a passing note. During the Baroque, it began to stand on its own. Classic era composers began using other than the standard, dominant, seventh chord, mostly as passing chords (not standing alone) and the ninth chord began to come into use (C,E,G,Bb,D). Romantic composers like Schubert and especially Franz Liszt began using the ninth chord more boldly. Gabriel Fauré used ninth-chord block harmonies in his famous mid-1870s song Après un Reve. Cesar Franck introduced standalone ninth chords in his violin sonata and in one of his last works, Ghiselle, a beautiful opera that has unfortunately for music lovers remained unperformed for the past century. Richard Wagner eleventh (C,E,G,B,D,F#), and thirteenth (C,E,G,B,D,F#,A) chords in his dramatic works, along other unusual harmonies, and this opened the door to higher partials incorporated into melodic and harmonic elements in music. 
     It was the 19th century French composers, all of them studying Wagner and many of their works unknown today, who began experimenting with music that used higher-partial chords with the 11th and 13th more consistently. Debussy and Scriabin demonstrated the use of chords that employed higher partials, and it was Scriabin who was able to demonstrate the use of higher partials coordinated with the higher astral planes in some of his later works such as his 8th and 10th piano sonatas. Later in the century, jazz musicians will begin using these chords in their music, especially after Charlie Parker studies Debussy's music and incorporates upper partials in his saxophone solos. 
     The overtone series represents the laws of life itself, the inner dimension. The repeating octaves are the occurrence on the outer planes, the phenomena. The series outlines the octave and the laws of harmony that govern creation.

Tuning

     A heated subject is that of how scales should be tuned. That which is known as just intonation is the scale defined according to the overtone series, all the notes related to the scale base note (C) by rational numbers. This is nature at it's best, all the intervals are perfectly in tune.
    Pianos today are tuned by a system called equal temperament, where certain intervals were slightly detuned, making the tuning of intervals in all the keys the same. That's why a modern piano can't be tuned 'by ear', unless the tuner has a well trained ear and knows how many 'beats' (beats occur when pitches are out-of-tune with each other) per second is needed for the correct out-of-tuneness that is necessary for equal temperament. (I wrote a software system for my Synclavier called "BACH". When you changed keys, the just intonation changed with the modulation!).
     This century will unfold many different kinds of new music based on different tunings. Some of these tunings will conform to natural principles such as the overtone series and the golden mean and will prove to have a sympathetic relationship to our sympathetic nervous systems and be beneficial and healing, while others will not. 
     There have been many temperaments that have come forth over the years, such as the Pythagorean scale, the mean-tone scale, and the Werckmeister scale. What will be important to all discussions of tuning will be the work of researchers working with the science of healing, through sound, and many discoveries will be made in that area. What I propose is for my readers to take a look at what is on the web:

Tuning on the Web

Barbara Hero
An Introduction to Historical Tuning
La Monte Young's Well Tuned Piano
Just Intonation Explained
Phi Music
Jorgensen's Tuning
Jorgensen's Tuning: Amazon Link
Chrysalis Foundation
Lucy Tuning
Ernest McClain
PrimaSounds
Ray Tomes

The Last Word  

Music in the 21st Century will be based on the overtone series and the consonant intervals. In addition to new music, we will bring back to our concert halls the great music we have neglected or forgotten about from the past and the great spiritual traditions in music used by the ancient civilizations of our planet that still preserve their music. When healing returns to music, then healing will be given back to the world, and healing will be needed. As earthquakes, wars, disasters, global warming and pandemics loam before us, we realize more and more that we are creating the future. This is why the overtone series is returning to music.

All the great composers understood it: study the masterpieces.

"In the great ancient cultures of India, China, and Greece, the understanding of the transformational power of music and its role in society was much more developed than it is today. The master musicians of the Vedic civilization were also enlightened masters or rishis whose consciousness was completely in tune with all the laws of nature that govern the entire creation. They truly exemplified the meaning of the Sanskrit aphorism "Aham brahmasmi": I am the totality. Not only did they intellectually understand the underlying musicality of the structure of the cosmos, they lived and breathed it, and were thereby able to cognize and produce music of a particularly celestial nature."

From Paul Stokstad

<-previous