Don Berry v. 0.9 Copyright ©1994 Don Berry |
Don Berry v. 0.9 Copyright ©1994 Don Berry |
The original intent of this chapbook was simply to provide an easy way for contemporary musicians to compose and play in the scales of the classic Greek modes. From that point on, the booklet evolved backwards.
Parts I and II are historical, metaphysical and mathematical. Part III is musical.
If you wish to read solely for practical musical purposes, you can skip Parts I and II, and go directly to Part III, which is complete in itself. It shows how to use the modern chromatic scale to approximate the classic Greek diatonic modes.
Then I thought it would be interesting to demonstrate some of the simple Pythagorean musical mathematics -- for example:
(1) the structure and musicality of the diatonic scale
(2) the metaphysical "infrastructure" of that scale and the modes derived from it
(3) the simple arithmetic that underlies harmonious sound, as discovered and analyzed by Pythagoras.
(Werner Heisenberg, architect of modern quantum theory, called this "certainly one of the most momentous discoveries in the history of mankind...")
This turned out to be Part Two.
In researching the musical mathematics, I was fascinated by the alien concepts of more general Pythagorean mathematics.
The musical mathematics are quite easy for us to comprehend, but other concepts, discarded centuries ago, sometimes seem like mathematics from the third moon of Saturn. In order to grasp them at all I had to think about number in ways that had never occurred to me.
This was so much fun, I succumbed to the temptation of including a kind of Sampler of basic Pythagorean theory. This is very far from complete, and should be considered as entertainment, not analysis.
This turned out to be Part One
The period around 500-600 B.C. was extraordinary for the number of men whose thought would profoundly affect the world from that time forward.
In India, Prince Siddhartha was becoming the Gautama Buddha. In China, it was the time of Lao-tse and Confucius. In the western world, it was the time of Pythagoras.
In our modern perspective on "history", everything before Plato and Aristotle is murky, and even semi-mythic. We tend to see everything before the rise of Periclean Athens as primitive; an arrogant and fallacious perspective. Pythagoras, some seven generations before Plato, was a philosopher/scientist in a line of teaching already thousands of years old, the Orphic tradition.
The major names we know from this ancient line are Orpheus (semi-mythological), Hermes Trismegistus of Egypt (legendary), Pythagoras, (historical personage), and Plato. The classic writers regarded Orpheus as the greatest spiritual master, Pythagoras the greatest scientist, and Plato the greatest philosopher in this line of teaching.
From our perspective we see the historical Pythgoras as an originator, but it would be more accurate to see him as the inheritor of a very ancient body of teaching, as is demonstrated in his own biography Most of his life was spent traveling, studying the accumulated wisdom of the ancient world from Egypt to India.
We can trace his path fairly accurately from Roman and Greek sources. Pythagoras left his birth island of Samos (in the third year of the 53rd Olympiad), at the age of 18, to spend the next 40 years studying with the greatest teachers of all schools in the ancient world. He spent 22 years in Egypt, and another 12 years in Babylon. He also studied in India, and with teachers in Crete and Sparta.
It was not until the age of 56 (in the 62nd Olympiad) that Pythagoras settled in the Italian city of Crotona. Crotona was one of the many Greek colonies around the northern Mediterranean, the autonomous cities of Magna Graecia.
In Crotona he established his Academy and its religious-scientific- philosophical-political movement, the secret wisdom school known as the Pythagorean Brotherhood. The Academy was to endure, in some form, for approximately 200 years after Pythagoras' death.
At about the same time Pythagoras married for the first time. His wife Theano was the daughter of Pythagoras' most famous disciple, Milo of Crotona, from whose house Pythagoras managed his school. (Men and women were admitted to the Academy on an equal basis, and Theano was a disciple at the Academy in her own right. Pythagoras' father-in-law and eminent disciple, Milo of Crotona, was the most famous wrestler of antiquity, winner of six Olympic Games.)
Pythagoras and Theano had seven children, four girls and three boys. After the murder of Pythagoras, Theano took over management of the Academy and one of the daughters, Damo, was entrusted with preserving, and keeping secret, her father's writings.
The Pythagorean Brotherhood was the archetypal Secret Society, whose inner teachings were available only to the initiates. It was a severe and authoritarian discipline. For the first five years of apprenticeship the applicants were not permitted to speak or to ask questions. Their teacher spoke to them from the other side of a curtain. When students, male or female, were initiated into the esoteric inner school, they joined an active dialogue "behind the curtain."
The body of Pythagorean teaching is known through the writings of others. Only two preserved letters are believed to have been directly written by Pythagoras. The wisdom of the initiates was never intended as public knowledge.
It was probably resentment of this elitist discipline of the Brotherhood that led to Pythagoras' murder at 80. The most frequent story goes that the richest, most powerful citizen of Crotona, named Cylon, applied to Pythagoras for discipleship, and was refused for reasons of bad personal character -- specifically, being "of a harsh, violent, turbulent Humour."
Enraged by the rejection, Cylon assembled a small private army. Waiting until a meeting at the disciple Milo's house, Cylo's thugs set the house afire, killing Pythagoras and forty of his disciples. This was in the 4th year of the 70th Olympiad, after Pythagoras had lived in Crotona for 20 years.
Other sources claim Pythagoras' murder was a simple political assassination, owing to the enormous political influence the Brotherhood had acquired in the colonies of Magna Graecia.
The Pythagoreans believed that the underlying order of the Cosmos was mathematical in nature. Specifically, they considered mathematics to be the creative connection between the Divine Mind and the manifest universe.
Modern physicists also believe this underlying order can be described mathematically, but the Pythagorean belief went deeper than simply the possibility of description. They held that the Cosmos pre-existed in the mind of God in immaterial form, as Number.
The mathematical sciences were considered to be preparatory to the study of philosophy -- to purify the mind, "bringing it by degrees to the contemplation of eternal, incorporeal things.
Pythagoras was the first to use the term "philosophos", lover of wisdom, to describe an intellectual discipline
The most consistent principle underlying Pythagorean philosophy and mathematics is a dialectic procedure, involving the relationship of polar opposites.
This classic dialectic is not the same as either the dialectical materialism of Marx and Engels, or the philosophical Hegelian dialectic. It does, however, somewhat resemble the dialectics of some Indian philosophical schools, such as the Uttara Mimamsa.
In some ways, the very term "dialectic" has been so co-opted by political philosophies I wish we had a more neutral term -- perhaps something like "the dual logic", except that this dialectic is quite different from the Aristotelian two-valued syllogism.
In Pythagorean dialectic, for example, music is defined as "a union of many, and consent of differences. . .Its end is to unite, and aptly conjoyn. God is the reconciler of things discordant, and this is his chiefest work. . . to reconcile enmities." In short, the reconciliation of opposites - the dialectic.
In Pythagorean thought we find the dualism of opposites at every conceptual level. The logic develops as a kind of nested dialectic, dualism within dualism, all flowing from the creative spring of the One. In Pythagorean mathematics, the Principle of Oneness, of Unity, is represented by the Monad.
The first level of the mathematical dialectic is between the One and the Many. From this dialectic is generated all that is knowable and numerable. The One becoming Many has been represented by different symbolisms all around the world -- in the case of the Pythagoreans, it is seen as number. The nature of the world is number.
In regard to that which is Knowable, the polarity is between the Intelligible, knowable to the mind, and the Sensible, knowable by the senses.
In characterizing the kinds of number, the polarity is between Immaterial and Sciential.
In characterizing the Order of number, the polarity is between Odd and Even.
In characterizing the field of mathematics, the polaarity is between Multitude and Magnitude.
There is also a traditional structure of polarities which forms the basis of Pythagorean dialectic logic. Not surprisingly, this was anciently conceived as a "decad",a "ten", the most metaphysically important number in Pythagorean theory. This is the structure:
The Pythagorean Decad of contraries:
In addition to their inherent "oppositeness" the polarities in the left column partake of the nature of the Monad, the metaphysical One; completeness, perfection, eternity, the Unchanging, the permanent, etc. (The physical manifestation of the Monad is, of course, the sciential number 1.)
The right column partakes of the metaphysical nature of the Duad. The Duad is the principle by which the One becomes many, the Unchanging is allowed the change, the permanence of the Divine Mind transforms into the transient multiplicity of a cosmos of physical manifestation and experience. (The physical manifestation is the sciential number 2.)
The next few segments will describe these characteristics
in a little more detail, to show how the metaphysical character of number
reveals itself in the physical world.
In Pythagorean number theory, two fundamentally different kinds of Number are recognized:
(I) Intellectual (or immaterial) Number.
This kind of number is "preexistent in the Intellect of God, maker of the world." It was called "the principle, fountain, and root of all things. . .that which before all things exists in the Divine mind; from which and out of which all things are digested into order, and remain numberedby an indissoluble series."
(II) Sciential Number.
Sciential number is closer to number as moderns recognize the term. That is, number applied to objects, and things in the manifest world Whereas Intellectual Number exists only in the mind of God, Sciential Number "is not separate from sensible things."
This division of number into two kinds reflects the general view of the ancient philosophers that the Knowable was, in general, of two kinds: The Intelligibles, knowable by the mind, and the Sensibles, knowable through the senses.
Sciential Number was divided into two Orders, Odd and Even. The Pythagorean definition of these terms was, again, more complex than our modern usage.
"Even number is that which at once admits division into the greatest and the least; into the greatest Magnitudes (for halves are the greatest parts) the least in Multitude (for Two is the least number) according to the natural opposition of these two kinds. Odd is that which cannot suffer this, but is cut into Two unequals."
Odd numbers were "of Masculine Virtue," "full and perfect" and "proper to the Celestial Gods, to whom they sacrificed always of that Number."
Even numbers were "indigent and imperfect and Female, and proper to the subterraneous Deities, to whom they sacrificed Even things."
Pythagoreans considered the female Even numbers to be infinite and unbounded, and the male Odd numbers to be finite and bounded.
The reasoning was that Even numbers can be divided into equal parts, "which is infinite, and by its proper Nature generates infinity in those things in which it exists. But it is limited by the Odd; for that being applied to the Even, hinders its dissection into Two equal parts."
The generative power of the Monad is shown by the fact that when the Monad is added to the Odd, it becomes Even; when added to the Even, it becomes Odd. Thus all transformation is generated by the principle of Unity interacting with the principle of multiplicity.
The Pythagoreans divided the study of mathematics "into four parts, attributing one to Multitude, another to Magnitude, and subdividing each of these into two. For multitude either subsists by itself, or is considered with respect to another; Magnitude either stands still or is moved.
Arithmetick contemplates Multitude in itself: Musick with respect to another: Geometry unmoveable magnitude, Sphaerick, moveable." ('Sphaerick apparently referred to Astronomy.')
[... Multitude/Magnitude is approximately the same dialectic as our modern differentiation between Digital and Analog, i.e. the discontinuous vs the continuous. Contemporary mathematician Rudy Rucker has even suggested that the development of left-brain/right-brain capacities in humans evolved as a way of dealing with this fundamental polarity...]
Another ancient philosophical school of music, known as the Aristoxenian, was based on the senses, again reflecting the ancient distinction between Intelligibles and Sensibles.
"Pythagoras dijudicated it (music) by reason,
Aristoxenus by sense."
[[ Summary: ]] Principle of the unchanging and principle of the changing -- the one and the many -- the indivisible and the divisible. From the interaction of these two is derived the entire number system. This is the engine that drives the subsequent linear-sequential arithmetic from 3 to infinity. The characteristic of that system is that multiplication is more powerful than addition, exponentiation is more powerful than multiplication (and inverselywith subtraction, division, and extraction of roots, of course.)However, at the generative level of the Monad and Duad, these laws of arithmetic have not yet come into being.
The Monad and Duad obey different principles, and generate different arithmetic powers.
The Monad:
1+1=2
1x1=1 (addition is more powerful than multiplication)
1 squared=1 (multiplication and exponentiation are equal)
1-1=0
1/1=1 (subtraction is more powerful than division)
sqr root of 1=1 (square root and division are equal)
In short, only addition and subtraction have any effect. All other operations leave the monad unchanged.
And for the Duad:
2+2=4
2x2=4
2 squared=4
2-2=0
2/2=1
sqr root of 2=irrational, (i.e. a number which cannot be finitely defined. )
The discovery of irrational numbers was perhaps the most important metaphysical/mathematical discovery of the classical world, and it is usually attributed to the Pythagoreans.
The irrationals may be considered to belong to a different dimension from the natural numbers. They cannot be precisely defined as multitude (though that is the essential nature of number), but are easily defined as magnitude (i.e.in geometric space.)
(The discovery is made when the Pythagorean theorem is applied to the diagonal of a square whose sides = 1. The result is 2, which can be calculated to infinity without ever reaching an end or a reptition. It simply cannot be represented as a finite ratio of two whole numbers. (Thus the name irrational...)
This discovery of numbers which were not numbers was so shocking some stories said it was kept secret for many years. Others say it was so important that Pythagoras sacrificed a yoke of oxen to the Goddess ingratitude. (Neither story is true -- Pythagoras absolutely opposed any killing of animals.)
However, the importance given to the nature of irrational and transcendental number is shown by Plato's dictum:"Whoever does not know the diagonal of a square is incommensurable with one of its sides is not a man, but a beast.
In a sense the irrationals are the visible resolution of the dialectic of multitude/magnitude, -- the reconciliation of the discontinuous and the continuous -- and reveal the singularity of the Divine Mind of which they are the manifestation.
(On the other hand, Plato's teacher Socrates once said, "I never knew a mathematician who could reason...")
This version of the traditional story is from Nicomachus, and is preserved in the 1687 Edition of The History of Philosophy by Thomas Stanley.
Pythagoras wanted to invent, or discover, some "solid and infallible...instrumental help for the ear."
The sense of sight had been made precise by the compass and rule of Geometry, the sense of touch was empowered by the invention of precise scales and systems of measurement. Pythagoras was now searching for some equivalent instrument of precision for the study of music.
One day he passed a blacksmith's shop in Crotona, and noticed that the sound of several hammers striking on the anvils were in perfect harmony in all combinations but one.
Specifically, he heard the principle intervals of the diatonic scale:
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Pythagoras made on-the-spot experiments to determine that the concordant sounds were not according to the force of the blow, nor the design of the hammers, nor the kind of metal being forged. He determined that the difference was due to the weights of the hammers.
He weighed the hammers precisely, and found the proportions between them to be 6:8:9:12.
He then constructed his experimental system by fastening beams to opposite walls of his house and connecting them by musical strings of "the same substance, length, swiftness and twist." From the end of each string he hung a weight corresponding to the 6:8:9:12 proportion of the blacksmith's hammers.
By striking these strings in pairs, he discovered the following proportions invariably produced the harmonious intervals:
(In Pythagorean theory, the whole note is defined as the difference between Diatessaron and Diapente )
These relationships hold true for all parameters of the stretched string. That is, the ratio 2:1 produces an octave interval whether derived from length of string, tension of string, frequency of vibration, or wavelength, etc.
These parameters are all physical manifestations of the simple ratio 2:1.
The full diatonic scale is created by the combination of the Diapente andthe Diatessaron. (From do - sol is Diapente, from sol-do is Diatessaron.)
The most harmonious relationships are created by the simplest ratios; that is to say, the ratios which depart least from the One.
The traditional story as told is not completely true; the tonality of hammers could not be derived accurately from their weight alone. However, weighting the strings in those proportions would in fact yield the harmonic ratios.
Pythagoras' experimental device was accurate. He later invented the monochord, a single string with moveable bridges, with which all the pitch relationships can be accurately determined.
I've gone into this anecdote in detail, partly to correct a common misapprehension about Pythagoras' mysticism.
Like all mystics, Pythagoras was eminently practical. Mysticism is based on direct experience, as is experiment. The principle difference is that the results of one are easily communicable, and the results of the other are not.
However, history, as we know, is written by the winners, and seven generations after Pythagoras the domain of science/philosophy was occupied by Aristotle's forces.
For his own reasons, Aristotle wished to portray the Orphic school (and his own teacher Plato) as pure theoreticians, idealists, impracticable visionaries. Most subsequent writers on Pythagoras simply parroted Aristotle's generally hostile, and sometimes dumb-headed, opinions.
This has been almost uniform in the past couple of centuries. The astronomer Carl Sagan presently holds the Aristotelian posture with regard to Pythagoras.
From Nicomachus' account, much abbreviated above, we can also see the emergence of a fully fledged experimental method, relying on precise measurement, which was also a part of the Pythagorean school.
With this combination of mathematical representation and experimental verification, the Pythagorean school had, by 500 B.C. prefigured the entire development of the physical sciences in the Western World.
The fundamental division of the octave
The primary importance of these intervals is reflected in modern music in several ways. The chordal structure of folk music, for example, frequently consists entirely of I-IV-V. The only chords are those based on Diatessaron and Diapente intervals.
In European classical music, cadences (ending sequences) are of two kinds:
The "authentic" cadence moves from V -> I (the Diapente.). The "plagal" cadence moves from IV -> I (the Diatessaron.)
These are the chord progressions that give us the sensation of "coming home." In metaphysical metaphor, "returning to the One." The return to the One (musically, the tonic) gives us the sense of completion and satisfaction.
The diatonic scale and the Pythagorean ratios were developed in a musical system that was conceived melodically. The above examples show how these melodic principles carried forward into chordal progressions.
The intervals that we recognize as "Harmonic" are described by the ratios of small integers. That is, 2:1, 3:2, 4:3, etc.
The harmony is ear-evaluated with extremely high precision. The diatonic scale, as known to the ancients, was created by the pleasurable sensation of harmony.
The diatonic scale based on these ear-pleasurable relationships had existed for several thousand years before Pythagoras.
Pythagoras was simply the first to recognize the mathematical relationships that underlie that pleasure.
A vibrating string creates overtones in exact multiples of the tonic.
That is,a string vibrating at 100Hz also generates overtones at 200, 300, 400, 500Hz,etc. (The timbre of the tone is largely affected by how much energy is in eachof these overtones.)
The notes of the diatonic scale are the values of the overtones, divided so as to fit within a single octave. This is the purely physical manifestation of the Pythagorean ratios.
The pure diatonic scale derives from the One (the tonic, the Monad) by simple ratio and the internal (mental) comparison of overtones. All the diatonic intervals are literally contained in the overtones of the keynote.
The brain derives the sensory experience by "ratio-izing" the simple proportions.
Hence the saying, "Music is the arithmetic of the mind, unaware of its counting."
The sense of hearing is the only human sensory system capable of accurately estimating proportion. It is very much more accurate than the sense of sight in this respect.
Consider the violinist tuning her instrument-- she is easily capable of distinguishing one part in several thousand by ear.
A "mode" is a unique melodic scale with its own characteristic musical and emotional possibilities. It corresponds to the idea of raga in Indian music.
The following sketch is a simplified explanation of the Greek modes, intended for musicians, not theoreticians.
The Pythagorean musician recognized seven modes, each based on a different note of the 7-string lyre. In European music we have retained only two modes, which we call the Major and Minor scales.
In modern notation the tuning of the ancient 7-string lyre was G A B C D E F, which contains four whole steps and two half steps (B-C and E-F) This is the relationship of intervals, not absolute pitches. The idea of an absolute pitch emerged many centuries later in response to orchestral music and the factory production of keyed instruments.
We now call this asymmetrical division of the octave the diatonic scale -- do re mi fa sol la ti do; (The half steps in this scale are between mi and fa and between la and ti.)
This uneven division of the octave is immemorially old, ear-created, and is the source of all European melody. (The composer Lou Harrison has found this diatonic tuning of a lyre in a cuneiform tablet as early as 2500-3000 BC in Mesopotamia, pre-dating Pythagoras by a couple of millenia and a civilization or so.)
Thousands of years later, the European equally tempered scale was introduced, inserting half steps between all notes, and detuning almost every interval from its Pythagorean value, with the purpose of creating a 12 note octave with exactly equal intervals between every note. Something was gained by this mechanical division, (... and something was also lost. So it goes.)
The basic diatonic scale is approximately preserved in the white notes of the piano (though beginning with the keynote C instead of G) In its original tuning, the true diatonic octave is described by the small-integer Pythagorean harmonic ratios outlined in Part II above.
What gives each mode its unique musical and emotional character is the order of whole steps and half steps in the diatonic scale. That is, the character of each mode is determined by where the semi-tones fall.
Each Greek mode begins on a different keynote of the diatonic scale.
The Myxolydian mode with the original keynote of G evolved into the European Major Scale: (G A B C D E F G.)
In this mode the order of whole and half steps is:
Whole, whole, half, whole, whole, half,whole
If the same notes are played beginning with the next keynote (A), the mode is the Aeolian: A B C D E F G A. Order:
Whole, half, whole, whole, half, whole, whole
And so on for all the modes.
These are the traditional keynotes (and scales) of each mode:
MYXOLYDIAN -- G A B C D E F G
AEOLIAN -- A B C D E F G A
LOCRIAN -- B C D E F G A B
IONIAN -- C D E F G A B C
DORIAN -- D E F G A B C D
PHRYGIAN -- E F G A B C D E
LYDIAN -- F G A B C D E F
(The Locrian mode is almost useless musically because it lacks a perfect fifth -- the Pythagorean Diapente interval. (Refer to Part II.) Without the Diapente, the octave's internal logic collapses, and thus its capacity to generate intelligent musical order. I have found playing and composing in Locrian mode unpleasant at best and scary at worst. Be careful of it. All the other modes are musically whole.)
The traditional American mountain dulcimer is fretted to produce a diatonic scale, and is the only contemporary instrument I know on which modal playing is perfectly straightforward. On keyboards, modal playing is posssible by using only the white notes. Other instruments require some kind of translation between the ancient modal scale and the modern chromatic scale of equal temperament.
The following chart transposes every mode and every possible keynote into a chromatic scale for the convenience of modern players and instruments.
The chart reads from left to right. To transpose, run your finger down from the desired mode. Each note you cross can be used as a new keynote. Reading left to right from that keynote gives you the chromatic approximation of that mode.
I adapted this chart from Bob Force and Al d'Ossche's book In Search of the Wild Dulcimer.
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