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Pythagorean by Stephen M
Phillips
“In ancient times, music was
something other than mere pleasure for the ear: it was like an algebra
of metaphysical abstractions, knowledge of which was given only to
initiates, but by the principles of which the masses were instinctively
and unconsciously influenced. This is what made music one of the most
powerful instruments of moral education, as Kong-Tsee (Confucius) had
said many centuries before Plato.” G. de Mengel, Voile d’Isis 1. Introduction
I shall explore in this article how
the ‘harmonies of heaven’ manifest here on earth in our acoustic
responses to different tunings. We are aware that Pythagoras was
supposed to have discovered the mathematical basis of music, although
the various legends surrounding his discovery of its laws probably
contain little truth. But we are, perhaps, not aware that the
Pythagorean theory of music was but one application of the principles of
his holistic philosophy, which was not only a modus vivendi
but a
system of understanding the immanence of God in nature through the study
of number. As part of a continuing programme of research into the
connection between Pythagorean number philosophy and contemporary
particle physics, I would like to set out here what I believe are some
fruits of this work.
For some this may be a journey into an
unfamiliar and abstract world, and I would fully understand if some
readers lost their way as they followed my route. However, my analysis
has to be mathematical because number is the ultimate language of logic.
That said, I shall avoid using technical terms familiar only to a
professional mathematician.
My initial discussion will centre on
the geometrical figure called the tetrahedron. This solid shape
has four corners and four equilateral triangles as its faces. It
symbolized for the Pythagoreans the number 4, which will appear
regularly in my analysis: we find it in the tetrachord and in the
two ‘perfect fourths’ into which the ancient Greeks divided their
musical scales. Most importantly, we find it in the tetractys,
the sacred symbol of Divine Wholeness at the heart of the number
philosophy of Pythagoras. We shall discover that it expresses the
mathematical character of the various musical ‘modes’ that the
Church adopted from ancient Greece. My research indicates that this is
an example of a universal principle at work, prescribing the nature of
physical and superphysical reality.
As most people’s hearing has been
blunted by life-long exposure to equal-temperament, it is difficult for
us now to recreate with ‘innocent ears’ the original musical
experience of the Greek musical modes. My article will show, however,
that they have far more significance than historians of music have
considered. They exhibit a certain mathematical completeness and a
beautiful symmetry and proportion that has its remarkable counterpart in
a certain class of numbers that some physicists believe may describe the
basic units of matter. In other words, vibration – whether it is of
violin strings or string-like subatomic particles – may be governed by
mathematically analogous laws: studying the proper (ie Pythagorean)
mathematical basis of music may therefore give one insight into the
‘holy grail’ of physics, namely, the so-called ‘theory of
everything’ that explains all the forces of nature. More important
still, it may help us to understand our spiritual origins. 2. The Pythagorean Musical
Scale
Music has four aspects: the physical
(sound/wave vibration), the aesthetic (feeling/mood), the conceptual
(theme/meaning) and the spiritual (abstract/ archetypal)[1].
This article will explore the lattermost facet, to which the quotation
above refers.
To do this, we need to realize that a
musical score really signifies a huge set of numbers — the frequencies
of all its notes — and that a musical experience is determined partly[2]
by what these numbers are and how the composer has ordered in time the
playing of the notes. For, just as geometry is the study of number in
space, so music is the study of number in time. The 12 tone octave in
use today is a development of the ancient Pythagorean scale which
consisted of 8 notes: C
D E
F G
A B
C'.
It is well-known how it came to be
conceived by the early Pythagoreans. They knew that successive octaves C
and C' are separated in pitch by a factor of 2. Assigning to the tonic C
the frequency 1, the arithmetic mean[3] of
1 and 2 is 3/2, which is the tone ratio[4]
of G, the perfect fifth, whilst their harmonic mean[5]
is 4/3, the tone ratio of F, the perfect fourth.
As 4/3x3/2 = 2 = 3/2x4/3, the interval
between the perfect fourth F and the octave C1 is a perfect fifth,
whilst the interval between the perfect fifth G and the octave is a
perfect fourth. As 4/3x9/8 = 3/2, the interval between F and G is 9/8.
This defines the Pythagorean tone interval. If note D is a tone interval from C and if E is a tone interval from D, then the interval from E to F is 256/243 because 9/8x9/8x256/243 = 4/3. This interval between the third and fourth notes is the ‘leimma’ (leftover) or semitone. The pattern of intervals and leimmas from C up to F is repeated over another interval of a perfect fourth from G up to C'. G is the geometric mean[6] of F and A. The tone ratios of the seven intervals in the Pythagorean, diatonic scale are:
where T denotes a Pythagorean tone
interval (about four cents[7] higher in pitch than the equal-tempered
interval) and L is the leimma, which is about ten cents lower in pitch
than the equal-tempered half-interval. Five tone intervals and two
leimmas therefore span the Pythagorean octave.
Twelve intervals of a fifth are
approximately equal to seven octaves, but are in reality slightly more
than seven octaves, the discrepancy being the Pythagorean comma of 312/219, or roughly 74/73. As tone intervals are in the ratio 9:8, and
leimmas are in the ratio 256:243, two leimmas are slightly less than one
tone interval. In fact, the Pythagorean, Philolaus of Tarentum, noted
that one whole note is equal to two leimmas plus a Pythagorean comma:
9/8 = 256/243x256/243x312/219. 3. The Tetrahedral Platonic Lambda
The Pythagoreans represented the
numbers 1, 2, 3 & 4 by the ‘tetractys’:
It was the cornerstone of their
philosophy not merely (as most historians of Greek science believe)
because it symbolised for the Pythagoreans the perfect number 10, or
decad, but for the following, far more profound reason: as I have
demonstrated in the context of particle physics,[8] the tetractys
expresses the physical and superphysical wholeness of systems that
actualize the 10-fold nature of Divine Unity in four stages denoted by
the four rows of the tetractys.
This unitive state, which they called
the Monad, corresponds in arithmetic to the number 1 (represented by the
mathematical point), although for the Pythagoreans it was not a number
per se but the metaphysical source of all numbers. The Monad is the
undivided One described by mystics; the tetractys is its 10-fold
differentiation to become the Many within the One.
Plato learnt much of his mathematics
from the few books on Pythagorean philosophy that existed during his
time.[9] In his Timaeus, Plato describes how, having blended the three
ingredients of the World Soul — Sameness, Difference and Existence —
into a kind of malleable substance, the Demiurge took a strip of it and
divided its length into portions measured according to the simple
proportions of the first three powers of 2 and 3.
This became known as his ‘Lambda,’
so-called because of its resemblance to the Greek letter L
(Figure 1).
These numbers line two sides of a tetractys of ten numbers, from whose
relative values the physicists and musicians of ancient Greece worked
out the frequencies of the notes of the octaves of the now defunct
Pythagorean musical scale.
The Table of Lambda (Figure 2), handed
down to us by the Pythagorean mathematician and mystic, Nichomachus of
Gerasa (fl. c. 100 CE), extends the numbers in the Lambda. The
horizontal rows of integers in Figure 2 are geometric progressions by 2
and the rows descending diagonally are geometric progressions by 3.
Consecutive numbers in vertical columns are to each other in the ratio
3/2, defining the perfect 5th, whilst consecutive, diagonally paired
numbers (shown by arrows) are in the ratio 4/3, defining the musical
fourth.
The tetractys of integers at the
beginning of the Table of Lambda serves as the basis of musical harmony:
Using the octave of 6:12, their
arithmetic mean is 9, which, in relation to 6, is in the ratio 3:2. This
is the perfect fifth. The harmonic mean of 6 and 12 is 8. This
proportion, 8:6 or 4:3, is the perfect fourth. But 12:9 is also 4:3,
that is, a perfect fourth, whilst 12:8 is also 3:2, which is a perfect
fifth. The ratio 9:8 defines a whole tone. The central integer, 6, which
is absent from the set of integers (1, 2, 3, 4, 8, 9, 27) in Plato’s
Lambda, is pivotal to discussion of the mathematical proportions of the
musical notes for the following reason.
If the integer n is chosen as the
fundamental frequency, then the first overtone, or octave, has a
frequency of 2n. The harmonic mean of these integers is (4/3)n, which is
the perfect fourth. The arithmetic mean of n and 2n is 3n/2, which is
the perfect fifth. Because n has to be divisible by both 2 and 3, that
is, by 6, both means are integers if n = 6N, where N is an integer, ie
the smallest value of n defining integer values of both the perfect
fourth and fifth is therefore 6. If the tonic has a tone ratio of 6N,
then the perfect fourth has a tone ratio of 8N and the perfect fifth has
a tone ratio of 9N, thereby defining a musical interval of 9/8.
If the Lambda and its underlying
tetractys are regarded purely as a Pythagorean construction, then it is
incomplete. This is because the numbers 1, 2, 3 and 4 were the basis of
Pythagorean number mysticism (as exemplified par excellence by the
tetractys) and its application to the study of natural phenomena such as
the sounds made by vibrating strings, whereas the number 4 is missing as
a generative factor from the Lambda, which uses only 1 (the monad), 2
(the dyad) and 3 (the triad) to generate its numbers.
Figure 3 shows how the Pythagorean
wholeness of the Lambda is restored naturally by realising that it is
but two edges of a tetrahedron having 1 at its apex and a third edge
with the first three powers of 4 arranged along it.
The resulting set of 20 numbers has
the following musical virtue: the Table of Lambda generates the tone
ratios of octaves along one side and perfect fifths along another side.
But the numbers starting with 6 and generating the perfects fourths have
to be added by hand, so to speak, following ad hoc rules of
multiplication by 2 and 3 that were not part of Plato’s cosmological
theory and whose justification is merely that they conform to the same
pattern of multiplication. In Figure 3 we see moreover, that whereas the
pairing of numbers separated by octaves or intervals of the perfect
fifth follows the natural geometry of the array of numbers set by the
extended boundary of the Lambda, the pairing of successive perfect
fourths does not respect the same symmetry because it occurs in diagonal
fashion across the array. Worse still, the other possible diagonal
pairing of numbers whose tone ratios differ by a factor of 3 plays a
relatively weak role in generating twelfths of the diatonic scale. The
traditional construction of the tone ratios of the diatonic scale
clearly lacks symmetry. This is because the classical scheme is
mathematically incomplete.
In Figure 4, on the other hand, the
fourth face of the tetrahedron is a tetractys of numbers whose pairings
parallel to its three sides create octaves, perfect fifths and perfect
fourths with, respectively, the tone ratios, 2/1, 3/2 and 4/3. The
hexagonal arrangement of these pairings means that, when extended in the
manner of the Lambda tetractys forming the first face of the
tetrahedron, every number becomes surrounded by six others that are
octaves, perfect fourths or perfect fifths. Parallel lines of successive
octaves are at 120° to parallel lines of successive perfect fourths,
which in turn are at 120° to parallel lines of successive perfect
fifths.
The integers 6, 8, 12 & 24 at the
centres of the four faces of the tetrahedron are in the ratio 1:4/3:2:4,
ie if 6 corresponds to the tonic with tone ratio 1, then the number 24
at the centre of the fourth face defines the tone ratio 4 of the second
octave. Through these centres, the tetrahedron defines the 15 notes
spanning two complete octaves. We shall return later to the musical
significance of this.
Figure 5 displays the lattice of tone
ratios, starting with 1, the fundamental, that are created by dividing
the numbers in Figure 4 and their higher octaves by any one of them. The
same infinite lattice of tone ratios results, whichever number is
chosen, because the perfect symmetry generated by each number being an
arithmetic, harmonic or geometric mean of its neighbour favours no
particular number.
Overtones are shown in circles, lines
sloping towards the right connect octaves (x2), horizontal lines connect
perfect fourths (x4/3) and lines sloping towards the left connect
perfect fifths (x3/2). The dashed line connects successive notes of each
octave. It zigzags between octaves, thirds, fourths and sevenths, ie
between the extremities of the scale and its midpoint.
Figure 6 displays the remarkable,
mathematical harmony of the Pythagorean scale. Each tone ratio is the
arithmetic, harmonic or geometric mean of its nearest neighbours
(indicated by single or double-headed arrows).
As in Figure 4, successive notes of
the scale for each octave are joined by lines that zigzag through the
corners and centres of sets of three joined hexagons, successive sets
overlapping corner to centre:
This hexagonal pattern depicts the
notes of the Pythagorean scale in a more symmetric way than the Table of
Lambda in Figure 2. But why should this beautiful proportion exist in a
musical scale that was discarded when musicians found that they could
not tune their instruments to it? Could it be that a musical scale based
upon such mathematical harmony actually has another relevance in Nature?
Indeed. My research [10] has proved that this mathematical pattern is not
confined to music but applies also in the subatomic world to the
vibrations of what physicists call “superstrings.” 4. The Seven Greek Musical Modes
Evolution is perceived today as
proceeding inwardly from the material to the spiritual world or,
according to Darwinian biology, as traversing the planet from one
species to another by accident and without purpose. According, however,
to ancient Greek thinking, man had descended from the Gods. This
fundamental difference in how they saw their place in the universe was
reflected in the way they regarded the musical scale. Instead of
considering, as musicians do now, the notes of the major scale: do, re,
mi, fa, sol, la, ti, do, as ascending in pitch, the Greeks thought in
terms of a descending scale, with its eight notes divided into two sets
of four notes, or ‘tetrachords.’
The most important note in a scale was
the
meoh (mesê, or middle note) which was the highest note of the lower
tetrachord. [11] It set the quality of music played according to a
particular scale because it was often the most played of the notes. The
Pythagoreans compared its position to that of the sun amidst the
planets: it held the melody together as its fulcrum. Altering the
sequence of whole tones and semitones within a tetrachord generated
different scales.
Seven distinct scales or modes came to
be recognised, although this is a simplification, as variations of these
were known: Bishop Ambrose of Milan codified and simplified them in the
4thC into four ‘authentic modes.’ To these Pope Gregory used the
writings of the medieval musical theorist, Boethius, to add four
‘plagal’ modes. Their ecclesiastical names, ascending from C, are:
Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian (the
eighth was the Phrygian with a different tonic and dominant).
They were named not after their first
or last notes (being sequences of descending notes, tonics within these
scales were not definable) but after the various peoples of ancient
Greece who were reputed to have preferred one mode to another, depending
on their character and temperament. Ancient Greeks were very familiar
with the distinct qualities of melodies played according to some of
these modes and the psychological effects they had on those listening to
them.
The following story illustrates this:
according to his biographer Iamblichus,[12] Pythagoras devised melodies to
heal the soul by soothing its passions and cleansing it of negative
conditions such as sorrow, rage, pity and anger. He would play this
music to his disciples as they prepared for bed in order to remove all
the emotional disturbances they had acquired during the course of the
day. When they woke up the next morning, Pythagoras freed them from the
heaviness and torpor of their sleep by certain kinds of singing and
playing of the lyre.
Iamblichus tells the story of
how a young man, angered by seeing his girl-friend leave the house of
his rival, had eaten and drunk all night, his jealousy inflamed by a
piper playing Phrygian music, which made him unable to resist his
impulses. Pythagoras persuaded the piper to play slow and heavy spondaic
music probably of the Dorian mode, which had the effect of restraining
and calming the man,[13] who returned home in a sober and orderly fashion.
Whether this tale is true or not (and
one must bear in mind that Iamblichus was wont to glorify his hero by
passing off as factual all the legends surrounding him), it serves to
illustrate the belief in ancient Greece that music based upon their
musical modes could transform the behaviour of the hearer in positive or
negative directions; the music of some modes was thought to be
character-building, whilst that of others led it astray. Plus ca change,
plus c’est la meme chose!
Plato had strong views about the
rightness and wrongness of various modes. In his Republic, which sets
out his ideas about an ideal society, he advocated only the Dorian and
Phrygian modes. In referring to the Lydian, Plato regarded it as a
"mixed" mode, seeing a distinction between the “tight”
Lydian (by which he refers perhaps, jokingly, to the upper tetrachord)
as “wailing modes” suitable for women, whilst the “slack” Lydian
(perhaps the lower tetrachord or Hypolydian) and the Ionian he thought
encouraged drunken-ness, softness, and idleness. He probably meant his
words to be a joke for his readers because certain strings were tight
for some modes and slack for others, whilst he also meant the word
‘tight’ to indicate that the mode reached into higher notes that put
a strain on the male voice and altered its quality, whereas in the slack
Lydian mode the melody stayed more in the lower part of the octave.
Plato makes Socrates remark: “Just
leave that mode which would appropriately imitate the sounds and accents
of a man who is courageous in warlike deeds and every violent work, and
who in failure or when going to face wounds or death or falling into
some other disaster, in the face of all these things stands up firmly
and patiently against chance. And, again, leave another mode for a man
who performs a peaceful deed, one that is not violent but voluntary,
either persuading someone of something and making a request –whether a
god by prayer or a human being by instruction and exhortation– or, on
the contrary, holding himself in check for someone else who makes a
request or instructs him or persuades him to change, and as a result
acting intelligently, not behaving arrogantly, but in all these things
acting moderately and in measure and being content with the
consequences.
“These two modes — a violent one
and a voluntary one, which will produce the finest imitation of the
sounds of unfortunate and fortunate, moderate and courageous men —
leave these.”[14] The modes referred to here are the Phrygian and the
Dorian. Aristotle[15] declared Socrates to have been wrong in including
the Phrygian mode for an ideal state, for the exciting, emotional
quality of its melodies made it apt for Bacchic celebrations.
Plato might have been expected to
deprecate this aspect but, surprisingly, he ignored it. Perhaps the
Phrygian mode had other qualities that he esteemed sufficiently not to
reject it. More likely, however, there was a variety of Phrygian modes,
not all of which were associated with orgiastic cults, and one of them
met Plato’s approval.
One of the most widely used modes in
the 5thC BCE and probably earlier, the Dorian was always well regarded.
It was a versatile mode, often employed for choral song but not confined
to it, and compatible with more than one mood. On the whole, however, it
was regarded as dignified and manly. Aristotle said that “everyone
agrees that it is the steadiest and the one that most has a manly
character.”[16] It is mentioned in The Hymns of
Orpheus, a great poetic
work of ancient Greece:
’Tis thine all Nature’s music to
inspire,
Plato wanted to eliminate from his
ideal society the Lydian mode as emotional, fit only for tragedy, as was
deemed the Mixolydian. Indeed, Sophocles introduced it to his plays for
this very reason. However, in his book Music in Ancient Greece and
Rome,
Landels remarked: “One suspects that Plato is being a bit puritanical
here, as the Mixolydian is described elsewhere as combining (hence the
prefix Mixo-) the emotional quality of the Lydian with the nobility of
the Dorian, and therefore being suitable for tragedy.”[18]
The Hypodorian and Hypophrygian modes
were not identified under these names before the late 5th or early 4thC
BCE. The music scholar M.L. West conjectured[19] that the Hypolydian was
an invention of Eratocles, who enumerated the seven species of the
octave in one genus, devising the name for the sake of parallelism, so
that Lydian, Phrygian and Dorian each had a corresponding Hypo- species
starting on the note a fourth higher in the scale. There were several
rival schemes of classification, with general agreement only on the
sequence: Dorian, Phrygian and Lydian. As will be discussed shortly, the
confusion caused Aristoxenus to base modal scales on keys rather than
consider them as octave species, using the seven names that Eratocles
had applied to his octave species. This purely musical emphasis
prevailed, with the consequence that the mathematical nature of the
unfolding of the modes ceased to be of interest to anyone. Music had
passed out of the hands of mathematicians and philosophers looking for
divine design (and perhaps divine inspiration) in music and had become
the practical business of musicians more concerned with how to tune
their instruments properly so as to play music that just entertained,
rather than melodies that could heal and elevate the soul.
In antiquity there seems to have been
general agreement on the sequence: Dorian, Phrygian, Lydian, but only
partial agreement about others. This caused Aristoxenus to disregard the
Pythagorean basis of the musical scale by adhering to the simple rule
that a fourth is 2.5 tones and that all intervals must be measured in
tones and fractions of a tone. For some years a pupil of Aristotle,
Aristoxenus no doubt shared his teacher’s rejection of Pythagorean
principles in general and the Pythagorean basis of music in particular.
He worked out a system of melodic scales based not on modalities but on
keys — thirteen of them in fact, arranged at regular semitone
intervals over a whole octave. He did not refer to the seven species of
scale by their modal names, although he adopted existing nomenclature to
name them. Aristoxenus added an eighth scale above the Mixolydian to
complete the octave, calling it the Hypermixolydian. Later, his system
was reformed to that of a 15-key system.
Both were criticised eventually
(although to no avail, as the latter had already become firmly
established) by the musical theorist Ptolemy, who (rightly) condemned
the completion of the octave with the Hypermixolydian as mere
duplication. 5. The Tonal Structure of the Modes
Historically speaking, musical scales
were always divided into eight notes because the ancient Greeks regarded
them as composed of two tetrachords, which were together known as the
diapason, or full range, a term preserved in the English organ stop of
the same name. Because the musical scale is based entirely on octaves
and fifths, that is, two notes, it is called the ‘diatonic scale.’
As we have seen, it comprises five tone intervals of 9/8 and two smaller
intervals, or leimmas, of 256/243. Its tone interval structure is:
C
D
E
F
G
A
B
C1
tone – tone – leimma – tone –
tone – tone – leimma.
Seven (and only seven) musical scales[20] can be generated by successive translations of a diatonic
interval (T) or leimma (L):
(Numbers 1–6 indicate the number of translations). There are no
more than seven distinct sequences because the scale created by shifting
Mode 7 by an interval creates the interval pattern: T
T L
T T
T L, which is Mode 1.
It does not matter where in the infinite sequence of successive
intervals of the ascending Pythagorean scale: … T L T T T L T T L T T
T L T T L T T T … a starting point is chosen for a
sequence of seven successive intervals and then shifted upwards in pitch
by an interval or leimma, for this translation will always generate one
of the patterns of intervals of the seven scales.
Figure 7 demonstrates this by
representing the seven intervals of the Pythagorean scale by arcs of a
circle. The smaller arc denotes the leimma (L) and the larger one
denotes the tone interval (L). Wherever one starts — indeed, whether
the selection is made in a clockwise direction (increasing pitch) or an
anticlockwise direction (decreasing pitch) — there are seven, and only
seven, different patterns of Ls and Ts that can be chosen sequentially
before one arrives back at the interval from where one started. Whether
one starts with a sequence of descending or ascending notes, successive
transpositions generates the same set of seven patterns of intervals.
When they start with the same note, these patterns become the seven
Greek musical modes. Just as a Pythagorean musical scale is a cycle of
seven intervals, each successive octave being a repetition of this
cycle, so, too, the seven modes constitute a cycle comprising 13
intervals (9 tone intervals T and 4 leimmas L) and 14 of the 15 notes
spanning two complete octaves. The difference in pitch between their
lowest and highest notes is 243/64 = (3/2)5/2, ie an octave below the
fifth perfect fifth, which has a tone ratio of (3/2)5.
Table 1. Tone ratios of the 56 notes
in the seven Greek Modes
Grey cells contain tone ratios not
belonging to any octave of the diatonic scale. The table shows that Mode
1 is unique in that it encompasses all the notes of the Pythagorean
scale (no grey cells).
There are 16 (=42) such tone ratios.
The 49 notes of the 7 modes other than octaves C1 comprise (49-16=33)
diatonic tone ratios, of which seven are 1s, leaving (33-7=26) such tone
ratios other than 1. 26 is the sum of the number of combinations of ten
objects H, I, J, etc arranged in the four rows of a tetractys and 33 is
the total number of their permutations:
The presence of these two numbers[21] in
the context of the seven Greek musical modes is no coincidence but
reveals the fundamental role of the tetractys in expressing the
mathematics of music. Indeed, those tone ratios other than 1 that occur
more than once in the seven modes themselves form a tetractys:
The six tones above the tonic of Mode
1 occupy the first three rows of this tetractys and the four
non-diatonic tone ratios above the tonics of four other modes that occur
more than once occupy its fourth[22] row.
The even integers 2 and 4 and the odd
integers 1, 3 and 5 in Table 2 measure the deviation of the six
non-Pythagorean modes from the perfection of the Pythagorean scale
because they are the numbers of tone ratios not belonging to the
diatonic scale (Figure 8). Notice how, starting from Mode 7, the most
imperfect mode with five non-diatonic notes, there is an oscillation
between modes with even and odd numbers of such notes as they develop
into the perfect Mode 1 with no non-diatonic notes.
Although, according to Table 1, both
Modes 4 and 5 contain the same number of diatonic notes, it can be
argued that the latter is less perfect than the former because it does
not contain a perfect fourth. It is for this reason that Mode 5, rather
than Mode 4, is shown in Figure 8 as the penultimate stage in the cycle
of completion of the perfect Pythagorean scale.
Figure 9 shows how the Pythagorean
character of the modes increases as they converge to the perfect
Pythagorean scale. The ancient Greeks regarded all scales as descending
and as composed of two tetrachords of four notes, their mesê being the
highest note of the second tetrachord. Figure 9 indicates that the first
tetrachord of Modes 7, 6, 5 & 1 comprise, respectively, 1, 2, 3
& 4 notes of the diatonic scale (shown encompassed by the grey
triangle). The geometric symbol of Divine Wholeness — the tetractys,
which symbolises these integers, therefore expresses the progression of
musical modes to their perfection in the fourth, the Pythagorean scale.
This is another example of the
‘Tetrad Principle’ referred to earlier, whereby an infinite sequence
of a class of mathematical objects finds its completion and perfection
in its fourth member, which means that it always embodies a number
characterising the cosmos in some way. Here, the Tetrad Principle
determines the Pythagorean scale — the original basis of music itself
— as the fourth of the musical modes. Another example of this
principle at work prescribing the arithmetic properties of the seven
modes is that the (7?8 = 56) notes of the seven octave scales (7 = 4th
odd integer, 8 = 4th even integer) contain (42 = 16) non-diatonic note
and (56 - 16 = 40) diatonic notes, where
When their tonics (and therefore octaves) coincide, the 56 notes become (1+7?6+1=44) notes, showing how 4, the Pythagorean tetrad, aptly expresses the very number of notes in the seven modes when they start and end on the same notes! Some of the notes between their tonic and octave are, of course, the same.
Of these,
16 are non-diatonic, leaving (44-16=28) that are diatonic, where
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7 is the second perfect number.[23]
Also, 28 = 4?7, where 7 = 4th odd integer. The modes with the same tonic
have 43 notes other than their common octave.
This is the number of dots in a
heptagon divided into tetractyses. In Figure 10 the dot at its centre
shared by the seven tetractyses symbolises the tonic common to all seven
modes and the six dots per tetractys denote the six other tones of each
mode.
Finally, as illustration of the Tetrad
Principle, notice that the seven modes descend from the second octave
with tone ratio 4, which is both the fourth harmonic and the (42 = 16)th
note. This provides new insight into the tetrahedral generalisation of
the Platonic Lambda tetractys, for this is but one face of a tetrahedron
(Figure 3) whose fourth face generates the octaves, their perfect
fourths and perfect fifths in a symmetrical way. The musical proportions
at the centres of its faces are 6, 8, 12 and 24. 6 is the musical
proportion of the tonic of the diatonic scale, whose mathematical
structure is determined by the equation of proportion between its tonic,
perfect fourth, perfect fifth and octave that is known to every student
of music: 6:8::9:12.
Relative to the tonic, the musical
proportion 24 represents the second octave with tone ratio 4. In other
words, this tetrahedron defines through its first and last faces the two
octaves with 15 notes spanned by the seven Greek modes. In accordance
with the Tetrad Principle, it is the fourth face of the tetrahedron
whose central number defines something of fundamental musical
significance. In fact, 15 is the total number of ways of grouping four
objects either singly, in pairs, in groups of three or as a group of
four:
4C1 + 4C2 + 4C3 +
4C4 = 4 + 4 + 6 +
1 = 15. [24] It is therefore the number of
corners, edges, triangles and tetrahedra in this Platonic solid. The
tetrahedron of musical proportions can be said not only to generate the
notes of the diatonic scale but also to define in potential the seven
Greek modes themselves.
It is sometimes a shock to musicians
to learn that the names of the Church modes that they became familiar
with during their studies are incorrect. Confusion in nomenclature
persists to this day. It arose from the Church’s taking over ancient
Greek names on the assumption that they referred to ascending scales,
whereas in fact the Greeks always regarded scales as descending. We do
not know what the interval structures of the various scales were,
although there is agreement on the sequence: Dorian, Phrygian and
Lydian.
As was shown earlier, the same set of
patterns of intervals results whether an ascending or descending
Pythagorean scale is considered. However, when individual scales are
named, the particular assignment of these names will depend on whether
the notes are regarded as ascending or descending. This is illustrated
in Figure 11 (below). The interval patterns of the seven Church modes
(written in brackets) are indicated in an upward, staggered sequence.
Suppose they are assigned the ‘ancient Greek’ names shown (the
actual ones used is irrelevant for the present purpose). If, keeping the
ordering of interval patterns the same, they had been correctly
considered as descending in pitch, the names would have been re-ordered
to that shown in the upper half, although the relative ordering would
have been unchanged. But what, say, the Church called the ‘Lydian’ (TTTLTTL)
should really have been what it called the ‘Locrian’ (LTTLTTT)
because the intervals should have been regarded as descending, not
ascending. To obtain the correct name for a given Church mode, one has
to reverse its interval pattern and assign to the new pattern whatever
is the name agreed by musical scholars. The names given in Figure 11
agree with the 1998 edition of Encyclopaedia Britannica.
As proof of the incorrectness of the
names of the Church modes, compare the inconsistent statements of
Aristotle (384-322 BCE) and Odo, Abbot of Cluny (d. 942) regarding the
Mixolydian mode:
Aristotle (Politics): “The musical
modes differ essentially from one another, and those who hear them are
differently affected by each. Some of them make men sad and grave, like
the so-called Mixolydian; others enfeeble the mind, like the relaxed
modes; another, again, produces a moderate or settled temper, which
appears to be the peculiar effect of the Dorian; and the Phrygian
inspires enthusiasm.”
Cluny: “Both in its original seat,
and when transposed, it breathes majesty, boldness, and joy, such as
might fitly become the celestial hierarchy in their ‘Gloria in
excelsis’ when the ‘Christ the Lord’ was born in Bethlehem.”
The Dorian mode favoured by Plato for
teaching music to the youth of his ideal state is the Pythagorean scale.
It alone is mathematically complete and perfect. 6. Pitch Differences between Modes
& Equal-tempered Scale
The Pythagorean scale is divided into
seven intervals, five of which are whole tones differing in pitch by 9/8
and two of which are leimmas, differing in pitch by 256/243. In the
modern equal-tempered scale the scale is divided into twelve semitones
separated by equal intervals, so that two semitones are together equal
to one tone. This means that the difference in pitch between successive,
equally tempered semitones = 21/12 and that the difference in pitch of a
tone interval = 21/12x21/12 = 21/6. The tone ratios of the notes in this
scale corresponding to those in the diatonic scale are: C D E F G A B C-
1
21/6 21/3
25/12 27/12
23/4 211/12
2
The difference in pitch between two
notes separated by a semitone is defined as 100 cents. This means that
an octave spans 1200 cents because it spans 12 semitones. Two notes with
pitches f1 and f2 separated by n semitones will have pitches in the
ratio f2/f1 = 2n/12, that is, they differ by 100n cents. Therefore, n =
12log2(f2/f1), that is, their pitch interval in cents =
1200log2(f2/f1).
Noting that log2Xxlog102 = log10X,[25] where X is any positive number,
their pitch interval = (1200/log102)log10(f2/f1)
≈ 3986.3log10(f2/f1).
This formula may be used to calculate
the pitch differences in cents between the notes of the seven modes and
their corresponding notes in the equal-tempered scale.
Table 2 above shows in cents the
difference: modal note pitch - equal-tempered note pitch. Table 3 below
shows the notes corresponding to these values.
As a comparison, the Pythagorean tone interval 9/8 is 204 cents,
slightly larger than the equal-tempered tone interval of 200, and the
leimma is about 90 cents, lower than the semitone of 100 cents. 21 of
the 42 notes between the tonic and the octave are higher in pitch than
their counterparts in the equal-tempered scale and 21 are lower in pitch
(Figure 12).
The Dorian mode is closest in pitch to
the equal-tempered scale, or rather it ought to be said that this
artificial scale is nearest the mathematically perfect Pythagorean scale
compared with the other modes. With variations of pitch between 2 and 10
cents, the two are virtually indistinguishable to most ears, at least
for single notes — chords can be another matter. Over the frequency
range most common in music (500 Hz to 4,000Hz), the ear can just detect
an interval of less than one-thirtieth of a semitone, ie about three
cents.[26] But it is harder to detect differences of pitch in real life
than in the laboratory, and sensitivity falls below 500 Hz, dropping to
about 30 cents at a frequency of 62 Hz. On the other hand, differences
of pitch are easier to detect in musical sounds than in pure tones. All
the notes of Mode 1 except the perfect fourth are slightly higher in
pitch than the modern scale. But all 16 non-diatonic notes differ
sufficiently to be differentiated. 15 of them are lower in pitch than
their equal-tempered counterparts. Mode 4 is the only one whose notes
are all lower in the equal-tempered scale, the fourth being more than a
semitone lower (112 compared with 100). Mode 7 is the only mode whose
notes in the modern scale are all higher than their classical
counterparts, five of the seven notes being more than a semitone higher.
Melodies played in modes other than Mode 1 (the Dorian) on instruments
tuned to the modern scale will sound differently to their playing
according to their pure pitches. Plato would have approved the modern
major scale at least musically (if not mathematically) speaking because
it most approximates the mode favoured by him as the ideal one! It must
not, however, be forgotten that it is only an approximation, albeit a
good one. 7. Mirror Symmetry of Modes
The ancient Greeks thought in terms of
descending musical scales. As we have seen, the issue of whether the
modes should be generated from ascending or descending notes does not
arise. It is only we humans who, finding it easier to think in terms of
increasing, rather than decreasing, numbers, wonder whether the modern
view of the musical scale as ascending might be incompatible with its
mathematical basis. Another reason for this is as follows: writing the
Pythagorean whole tone interval 9/8 as T and the Pythagorean leimma
256/243 as L and taking in turn successive intervals (not notes) between
the tonic and the octave of the Pythagorean scale (the true Dorian
mode), the seven modes ascending in pitch from left to right are:
Notice that: 1. the order of tone
intervals in Mode 3 is the mirror image or reverse of that of Mode 1
(Dorian), 2. Mode 4 is the reverse of Mode 7, 3. Mode 5 is the reverse
of Mode 6, and 4. Mode 2 is unique among the seven modes in that the
orderings of its ascending and descending tone intervals are the same
— it is its own mirror image.
What does this reflection symmetry
between pairs of modes (or within itself, in the case of Mode 2) imply?
As only the relative ordering of tone intervals and leimmas defines the
differences between the seven modes, not the absolute values of the
pitches of their notes, it means that, if we started at any note of any
mode and selected successive sets of descending notes with tone
intervals of either 1/T or 1/L, we would get precisely the same sets of
combinations of descending tone intervals as that shown above for the
ascending notes. It makes no difference whether we regard the scales as
descending (as the ancient Greeks did) or as ascending (as musicians do
now). This is because the seven descending modes are, in terms of the
patterns of their tone intervals, the very same as the seven ascending
ones, the remarkable mirror symmetry displayed by their sets of tone
intervals generating seven, and only seven, different ways of ordering
them.
Each mode turns into another one (its
mirror reflection) whilst Mode 2 turns into itself. The difference
between the descending and ascending patterns is one of reflection of
the order of tone intervals. The descending Dorian: is like the ascending Mode 4: the ascending Mode 4: is like the
descending Mode 7: and the ascending Mode 5: is like the
descending Mode 6: One could say that the members of each
pair are polar opposites, moving in opposite directions of pitch but
similar in the ordering of their two types of intervals.
The seven Greek modes therefore
consist of three such chiral pairs, one the mirror reflection of the
other, and one (Mode 2) that is invariant with respect to reversing its
tone intervals. This can be represented by a hexagon with polar opposite
pairs at diametrically opposite corners and the mirror-symmetric Mode 2
at its centre:
As well as turning into one another by
mirror reflection of their tone intervals, the seven modes are generated
by successive translations of a Pythagorean interval or leimma shifted
upwards or downwards in pitch by an interval, for this translation will
always generate one of the patterns of intervals of the seven musical
modes (see Figure 11). This is because the endless sequence of intervals
of the Pythagorean musical scale has the very important property of
being its own mirror image. The infinite sequence of Pythagorean
intervals is composed of repeated cycles of seven sets of seven
intervals because, when Mode 7 is translated by one interval, it becomes
Mode 1, further, successive translations merely repeating the cycle.
The cyclic nature of the interval
structure of the seven modes is best illustrated by a circle with seven
points denoting the modes equally spaced along its circumference:
The polar opposite Modes 1 and 3 turn
into each other by one reflection (denoted by the double-headed, dotted
line arrow) or two successive, interval translations (denoted by solid
line arrows). Modes 7 and 4 change into each other by one reflection or
four successive translations. Modes 6 and 5 turn into each other by one
reflection or six translations. An even number of translations creates a
reflection.
Figure 13 shows how the T/L interval
structure of the seven modes with the same tonic and octave can be
mapped by representing their 44 notes as points on three great circles
that are 60° apart and intersect at the South Pole (tonic) and North
Pole (octave) of a sphere. The longer arcs denote Pythagorean tones (T)
with an interval of 9/8 and shorter arcs signify leimmas (L) with an
interval of 256/243. As the reflection of Mode 1, Mode 3 spans a
semicircle (solid line) opposite that spanned by the former scale. Its
second note is diametrically opposite the penultimate note of Mode 1,
its third note is diametrically opposite the sixth note, and so on.
Similarly, the notes and intervals of the mirror image Modes 4 and 7 are
diametrically opposite one another on another great circle (dotted
line), as are the notes of the mirror image Modes 5 and 6 lying on the
great circle made up of dashes and dots. The notes of Mode 2 are
situated at points that are mirror images of one another along the
vertical axis of the sphere connecting its poles
Each of four modes (2, 3, 6 & 7)
has four intervals that are different compared with those of Mode 1; two
modes (4 & 5) have two intervals out of place with respect to this
mode. We found earlier that Modes 4 and 5 each has one non-diatonic
note. These two modes therefore most resemble Mode 1, which is what the
ancient Greeks knew as the Dorian mode. Accordingly, the
psycho-spiritual quality of music based upon them may be expected to be
closer to that of the Dorian than the other four modes. Conclusion
For many people, the fact that the
tetrahedral generalisation of the Platonic Lambda generates the tone
ratios of the Pythagorean musical scale will be merely a mathematical
curiosity without even fundamental significance, let alone, spiritual
meaning. This is because their denial of the metaphysical, or
archetypal, origin of concepts makes them unable to recognise universal
patterns as such linking the particulars.
For a Pythagorean, the connection
between music and the tetractys (as we have seen, so much more than a
mere symbol of the integers 1, 2, 3 & 4) has a more profound
implication. It demonstrates the universal applicability of the sacred
tetractys not only as an abstract representation of the ten-fold nature
of Divinity but also as a pattern that actually prescribes our musical
experiences, although only indirectly and faintly now through the
equal-tempered scale. For this potent symbol transcends fleeting,
cultural paradigms and man-made theologies. It expresses the ten-fold
nature of Divine Immanence in the phenomenal world, as particle
physicists working on the theory of superstrings have unconsciously
revealed in their discovery that space-time must be ten-dimensional.
Just as mathematics is the language of
the Divine Mind in which, as Galileo said, the book of Nature is
written, expressing all possible states of the physical world, so music
is the natural language for expressing all the states of the soul. Its
mathematical composition must therefore reflect the similar, perfect
harmony of the soul. Because the seven notes of the Pythagorean scale
and the seven Greek musical modes are the exact, musical counterpart of
a universal (that is, cosmic) principle that leads, among other things,
to a seven-fold spectrum of consciousness in all life, melodies played
according to them have qualities that must resonate in the awareness of
the hearer, in whom these states exist. There must be in fact a
correspondence between the psychological qualities of music played
according to each mode and these seven, primary modes of
consciousness.[27] This would help to explain why simply varying the
positions of their half-intervals creates Church modes with such
different characters and moods.
The highest function of music is not
to entertain but to transform and refine all the subtleties of
awareness— to invoke a consciousness of and ease accessibility to all
the levels of one’s being and its infinite variation. Because the
modern, equal-tempered scale does not reproduce the exact, tonal
frequencies of the Pythagorean scale, contemporary and classical music
cannot have the power of, say, the music used during the rites of
ancient Mystery religions like Orphism to resonate with different levels
of human consciousness and to transport the hearer to the celestial
heavens. Only such pure sounds could be recognised and responded to at
some subtle level of the psyche through the working of a principle of
sympathetic resonance. This is not to say that equal-tempered music
cannot induce epiphany — it can and does, but perhaps only when
composed with the artistic genius of a Mozart or a Bach and for
fundamentally different reasons.
A sceptic might argue that the pitch
differences between the notes of the diatonic and equal-tempered scales
are not noticeable to the human ear, at least when played in quick
succession as a musical composition, so that there could not be any
difference in the psycho-spiritual effects of music employing these
scales. In fact, the differences between the notes themselves are
detectable, though only just. Nevertheless, this argument might seem
incontestable as a scientific argument if, as it assumes, consciousness
were merely the product of a brain whose neurological activity is
affected by nerve signals issuing from the organ of Corti in the
fluid-filled cochlea of the inner ear when it is set vibrating by sound
waves in the air. However, many traditions of esoteric knowledge
contradict the unproved, working assumption of science that the brain
creates consciousness. The perception-altering effects of sound may
depend in part on some supersensory part of a human being and so they
would bypass his ears. Certainly, like Beethoven, one does not need
functioning ears to ‘hear’ and to be moved by music. As Plato’s
World Soul in the microcosm, the human soul — as well as its physical
organ of hearing — would be attuned naturally to the Pythagorean scale
by the principle of correspondence that only like can affect like.
This would explain why, when a study[28]
of professional violinists was carried out in the 1930s, the average
value of the interval that they played was much nearer to that of the
Pythagorean scale than to the equal-tempered scale. When his instrument
allows him to create his own sounds, the musician seems unconsciously to
play notes most approximating those of the Pythagorean scale! Music
played according to the latter may therefore affect the psyche in
subtle, little-known ways that music played on the equal-tempered scale
(or, for that matter, on any other scale) cannot do in principle. What
may play a role here is the fact that there is a greater number of exact
resonances between various notes of the modes in just intonation as
compared to equal temperament, in which such harmonic possibilities are
weakened by the inexact tuning.
Perhaps a human being is like a
musical instrument. It has been tuned by God to the ‘music of the
spheres’ — the ideal, mathematically perfect harmony of the
Pythagorean scale. But it is played out of tune in an imperfect world
that is ignorant of spiritual principles and the true power and purpose
of music. Our function in life is to play this instrument so that all
its potential melodies of consciousness become alive. Then at last, like
Pythagoras, we shall hear the divine harmonies within us and the
cosmically attuned instrument shall become its player. __________________
[NOTE; The number of each footnote is a link back to the corresponding point in the text] [1]
As dimensions of musical experience, they correspond to science, art,
philosophy and religion. [2]
Only partly, because the psychological qualities of musical sounds
depend also on the timbre of the sounds made by a musical instrument, a
property that is not reducible to single frequencies and numbers but
which depends on the distribution and relative intensity of their
partial tones. [3]
The arithmetic mean of two numbers A and B is (A+B)/2. [4]
A tone ratio is the ratio of the frequencies of the note and the tonic,
whatever the latter may be. [5]
H is the harmonic mean of two numbers A and B if (B-H/(H-A) = B/A, ie H
= 2AB/(A+B). [6]
X is the geometric mean of two numbers A and B if (B-X)/(X-A) = X/A, ie
B/X = X/A, so that X2 = AB. [7]
100 cents define a half-interval, or semitone, the whole octave of 12
half-intervals being 1200 cents. [8]
See
Articles 1-14 on my website at http://www.smphillips.8m.com. [9]
Its arcane teachings were written down for the first time by Philolaus
of Tarentum, whose book, Fragments, Plato obtained from Philolaus’
parents. [10]
See my Articles 13 & 15 at http://www.scimed.org or at
http://www.smphillips.8m.com. [11]
The mesê originally referred to the middle string of the seven-stringed
lyre. It came later to denote the fourth lowest note of a musical scale. [12]
Life of Pythagoras, translated by Thomas Taylor, London, 1818, ch. 15. [13]
Ibid., ch. 25. [14]
The Pythagorean Plato, Ernest G. McClain (Nicolas-Hays, Inc., 1984). [15]
Pol.1342a32-b12, cf. 1340b4, Procl. Chrestomathy, ap. Phot. Bibl. 320b. [16]
Pol. 1342b12, cf. 1340b4 [17]
The Hymns of Orpheus, Hymn XXXIII, translated by Thomas Taylor (The
Philosophical Research Society, Inc., Los Angeles, California, 1981). [18]
Music in Ancient Greece and Rome, John G. Landels (Routledge, 1999),
p.101. [19]
Ancient Greek Music, M.L. West (Clarendon Press, Oxford, 1994), pp.
228-229. [20]
The labels ‘Mode 1’, ‘Mode 2’, etc do not refer to those used
for the Church modes. [21]
For more details about the significance of the numbers 26 and 33, see
Article 12, pp. 11-17 at http://www.scimednet.org or at
http://www.smphillips.8m.com. [22]
Notice how the Pythagorean tetrad (4) expresses these properties. It is
an example of my Tetrad Principle, whereby the number 4 mathematically
prescribes or expresses numbers of cosmic significance (see Article 1 on
my website at http://www.smphillips.8m.com). [23]
A perfect number is one that is the sum of its factors. 1, 2, 4, 7 &
14 are the factors of 28. [24]
The binomial coefficient nCr is the number of combinations of n objects
taken r at a time. nCr = n!/r!(n-r)!, where n! =
1_2_3_…_n. [25]
Let log2X = Y and log10X = Z. Then X = 2Y =
10Z. Taking the
logarithm to base 10 of each side of this equation, Ylog102 = Z, as
log1010 = 1. Therefore, Y = Z/log102 = log10X/log102 =
log2X. Therefore, log2Xxlog102 = log10X. [26]
The Physics of Music, 7th ed., Alexander Wood (Chapman and Hall, 1975),
p.83. [27]
This is explored in Article 14 in my website at
http://www.smphillips.8m.com. [28]
Paul Greene, Journ. Acous. Soc. Amer., Vol. 9, p. 43 (1937). This is
discussed in The Physics of Music, Alexander Wood (Chapman and Hall,
1975), pp. 193-194. |
and vice versa
and vice versa,
and vice versa.
