Joined: 16 December 2005
Joined: 19 March 2005
Joined: 16 December 2005
Composer Profile - Leo Tolstoy
Joined: 16 December 2005
Joined: 16 December 2005
"We shall therefore borrow all our Rules for the Finishing our Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat." Leon Battista Alberti (1407-1472)
Pythagoras (6th century BC) observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer.
Number (in this case "amount of weight") seemed to govern musical tone...
See if you can hear the sound in your imagination before it comes, by judging from the proportions of the string lengths. The first one's easy.
Then mouse-over the strings (-if you dont hear anything you'll have to click, it depends on your set-up.)
Again, number (in this case "amount of space") seemed to govern musical tone. Or does musical tone govern number? He also discovered that if the length of the two strings are in relation to each other 2:3, the difference in pitch is called a fifth:< name=2 src=fifth.au =true =audio/basic autostart="false" enable="true">
...and if the length of the strings are in relation to each other 3:4, then the difference is called a fourth.< name=3 src=fourth.au =true =audio/basic autostart="false" enable="true">
Thusthe musical notation of the Greeks, which we have inherited can be expressed mathematically as 1:2:3:4
All this above can be summarised in the following.< name=4 src=all.au =true =audio/basic autostart="false" enable="true">
(Another consonance which the Greeks recognised was the octave plus a fifth, where 9:18 = 1:2, an octave, and 18:27 = 2:3, a fifth;)< name=5 src=tetrad.au =true =audio/basic autostart="false" enable="true">
This triangular figure of numbers in the shape of the Greek letter Lamda is the Tetrad of the Pythagorians.
As was discussed by Plato in his dissertation on the Composition of the Soul, it is a set of numbers whose relationships with each other seemed to summarize all the inter-dependent harmonies within the universe of space and time.
Thus to have established the relationship between music and space/number fired the imagination of the Pythagorians and was taken up especially by the School of Plato and the subsequent Neo-Platonists. Pythagoras himself wrote nothing which has survived, and so it is the Platonists we have to thank for recording and developing what had hitherto been passed down through two hundred and fifty years of oral tradition.
Pythagoras taught that each of the seven planets produced by its orbit a particular note according to its distance from the still centre which was the Earth. The distance in each case was like the subdivisions of the string refered to above. This is what was called Musica Mundana, which is usually translated as Music of the Spheres. The sound produced is so exquisite and rarified that our ordinary ears are unable to hear it. It is the Cosmic Music which, according to Philo of Alexandria, Moses had heard when he recieved the Tablets on Mount Sinai, and which St Augustine believed men hear on the point of death, revealing to them the highest reality of the Cosmos. (Carlo Bertelli, Piero della Francesca, p. 60.) This music is present everywhere and governs all temporal cycles, such as the seasons, biological cycles, and all the rhythms of nature. Together with its underlying mathematical laws of proportion it is the sound of the harmony of the created being of the universe, the harmony of what Plato called the "one visible living being, containing within itself all living beings of the same natural order".
For the Pythagorians different musical modes have different effects on the person who hears them; Pythagoras once cured a youth of his drunkenness by prescribing a melody in the Hypophrygian mode in spondaic rhythm. Apparently the Phrygian mode would have had the opposite effect and would have overexcited him. At the healing centers of Asclepieion at Pergamum and Epidauros in Greece, patients underwent therapy accompanied by music. The Roman statesman, philosopher and mathematician, Boethius (480-524 A.D.) explained that the soul and the body are subject to the same laws of proportion that govern music and the cosmos itself. We are happiest when we conform to these laws because "we love similarity, but hate and resent dissimilarity". (De Institutione Musica, 1,1. from Umberto Eco, Art and Beauty in the Middle Ages. p. 31).
Joined: 16 December 2005
Music of the Spheres
There is geometry in the humming of the strings
... there is music in the spacing of the spheres.
The History of Philosophy (c.1660) by Thomas Stanley.
|Pythagorean Number Symbolism|
|Music of the Spheres|
From Egypt we move across the Mediterranean Sea to the Greek island of Samos, the birthplace of Pythagoras, whose ideas dominate most of the material in this course. We'll introduce Pythagoras and his secret society of the Pythagoreans.
We'll look at the Pythagoreans' ideas about numbers, as a prelude to our next unit on number symbolism. Finally, we'll introduce a new idea that will be recurring theme throughout this course, the musical ratios, which will reappear in discussions of the architecture of the Renaissance.
Our main link between Egypt and Greece seems to be Thales c 640-550 BC, father of Greek mathematics, astronomy, and Philosophy, and was one of the Seven Sages of Greece. A rich merchant, his duties as a merchant took him to Egypt, and so became one of the main sources of Egyptian mathematical information in Greece. It was Thales advised his student to visit Egypt, and that student was Pythagoras.
Raphael's School of Athens
Pythagoras is shown in this famous painting, done by Raphael in 1510-11, which also shows most of the Greek philosophers.
Socrates sprawls on the steps at their feet, the hemlock cup nearby.
His student Plato the idealist is on the left, pointing upwards to divine inspiration. He holds his Timaeus, a book we'll talk about soon.
Plato's student Aristotle, the man of good sense, stands next to him. He is holding his Ethics in one hand and holding out the other in a gesture of moderation, the golden mean.
Euclid is shown with compass, lower right. He is the Greek mathematician whose Elements we'll mention often.
|Slide 3-2: Pythagoras in Raphael's School of Athens
Janson, H. W. History of Art. Fifth Edition. NY: Abrams, 1995. p.497
Finally, we see Pythagoras (582?-500? BC), Greek philosopher and mathematician, in the lower-left corner.
Pythagoras was born in Ionia on the island of Smos, and eventually settled in Crotone, a Dorian Greek colony in southern Italy, in 529 B.C.E. There he lectured in philosophy and mathematics.
He started an academy which gradually formed into a society or brotherhood called the Order of the Pythagoreans.
Disciplines of the Pythagoreans included:
|silence||music||incenses||physical and moral purifications|
|rigid cleanliness||a mild ascetisicm||utter loyalty||common possessions|
(whatever that means)
|pure linen clothes|
We see here the roots of later monastic orders.
For badges and symbols, the Pythagoreans had the Sacred Tetractys and the Star Pentagram, both of which we'll talk about later.
There were three degrees of membership:
1. novices or "Politics"
2. Nomothets, or first degree of initiation
The Pythagoreans relied on oral teaching, perhaps due to their pledge of secrecy, but their ideas were eventually committed to writing. Pythagoras' philosophy is known only through the work of his disciples, and it's impossible to know how much of the "Pythagorean" discoveries were made by Pythagoras himself. It was the tradition of later Pythagoreans to ascribe everything to the Master himself.
Pythagorean Number Symbolism
The Pythagoreans adored numbers. Aristotle, in his Metaphysica, sums up the Pythagorean's attitude towards numbers.
"The (Pythagoreans were) ... the first to take up mathematics ... (and) thought its principles were the principles of all things. Since, of these principles, numbers ... are the first, ... in numbers they seemed to see many resemblances to things that exist ... more than [just] air, fire and earth and water, (but things such as) justice, soul, reason, opportunity ..."
The Pythagoreans knew just the positive whole numbers. Zero, negative numbers, and irrational numbers didn't exist in their system. Here are some Pythagorean ideas about numbers.
Masculine and Feminine Numbers
Odd numbers were considered masculine; even numbers feminine because they are weaker than the odd. When divided they have, unlike the odd, nothing in the center. Further, the odds are the master, because odd + even always give odd. And two evens can never produce an odd, while two odds produce an even.
Since the birth of a son was considered more fortunate than birth of a daughter, odd numbers became associated with good luck. "The gods delight in odd numbers," wrote Virgil.
1 Monad. Point. The source of all numbers. Good, desirable, essential, indivisible.
2 Dyad. Line. Diversity, a loss of unity, the number of excess and defect. The first feminine number. Duality.
3 Triad. Plane. By virtue of the triad, unity and diversity of which it is composed are restored to harmony. The first odd, masculine number.
4 Tetrad. Solid. The first feminine square. Justice, steadfast and square. The number of the square, the elements, the seasons, ages of man, lunar phases, virtues.5 Pentad. The masculine marriage number, uniting the first female number and the first male number by addition.
6 The first feminine marriage number, uniting 2 and 3 by multiplication.
The first perfect number (One equal to the sum of its aliquot parts, IE, exact divisors or factors, except itself. Thus, (1 + 2 + 3 = 6).
The area of a 3-4-5 triangle
7 Heptad. The maiden goddess Athene, the virgin number, because 7 alone has neither factors or product. Also, a circle cannot be divided into seven parts by any known construction).
8 The first cube.
9 The first masculine square.
Incorruptible - however often multiplied, reproduces itself.
10 Decad. Number of fingers or toes.
Contains all the numbers, because after 10 the numbers merely repeat themselves.
The sum of the archetypal numbers (1 + 2 + 3 + 4 = 10)
27 The first masculine cube.
28 Astrologically significant as the lunar cycle.
It's the second perfect number (1 + 2 + 4 + 7 + 14 = 28).
It's also the sum of the first 7 numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 = 28)!
35 Sum of the first feminine and masculine cubes (8+27)
36 Product of the first square numbers (4 x 9)
Sum of the first three cubes (1 + 8 + 27)
Sum of the first 8 numbers (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)
The Pythagoreans represented numbers by patterns of dots, probably a result of arranging pebbles into patterns. The resulting figures have given us the present word figures.
Thus 9 pebbles can be arranged into 3 rows with 3 pebbles per row, forming a square.
Similarly, 10 pebbles can be arranged into four rows, containing 1, 2, 3, and 4 pebbles per row, forming a triangle.
From these they derived relationships between numbers. For example, noting that a square number can be subdivided by a diagonal line into two triangular numbers, we can say that a square number is always the sum of two triangular numbers.
Thus the square number 25 is the sum of the triangular number 10 and the triangular number 15.
One particular triangular number that they especially liked was the number ten. It was called a Tetractys, meaning a set of four things, a word attributed to the Greek Mathematician and astronomer Theon (c. 100 CE). The Pythagoreans identified ten such sets.
Ten Sets of Four Things
|Living Things||seed||growth in length||in breadth||in thickness|
|Ages of a Person||infancy||youth||adulthood||old age|
|Parts of living things||body||three parts of the soul|
Gnomon means carpenter's square in Greek. Its the name given to the upright stick on a sundial. For the Pythagoreans, the gnomons were the odd integers, the masculine numbers. Starting with the monad, a square number could be obtained by adding an L-shaped border, called a gnomon.
Thus, the sum of the monad and any consecutive number of gnomons is a square number.
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
and so on.
While speaking of groups of four, we owe another one to the Pythagoreans, the division of mathematics into four groups,
giving the famous Quadrivium of knowledge, the four subjects needed for a bachelor's degree in the Middle Ages.
Music of the Spheres
Jubal and Pythagoras
|Slide 3-4: Theorica Musica|
F. Gaffurio, Milan, 1492
Lawlor, Robert. Sacred Geometry. NY: Thames & Hudson, 1982. p.7
So the Pythagoreans in their love of numbers built up this elaborate number lore, but it may be that the numbers that impressed them most were those found in the musical ratios.
Lets start with this frontispiece from a 1492 book on music theory.
The upper left frame shows Lubal or Jubal, from the Old Testament, "father of all who play the lyre and the pipe" and 6 guys whacking on an anvil with hammers numbered 4, 6, 8, 9, 12, 16.
The frames in the upper right and lower left show Pithagoras hitting bells, plucking strings under different tensions, tapping glasses filled to different lengths with water, all marked 4, 6, 8, 9, 12, 16. In each frame he sounds the ones marked 8 and 16, an interval of 1:2 called the octave, or diapason.
In the lower right, he and Philolaos, another Pythagorean, blow pipes of lengths 8 and 16, again giving the octave, but Pythagoras holds pipes 9 and 12, giving the ratio 3:4, called the fourth or diatesseron while Philolaos holds 4 and 6, giving the ratio 2:3, called the fifth or diapente.
|8 : 16 or 1 : 2||Octave||diapason|
|4 : 6 or 2 : 3||Fifth||diapente|
|9 : 12 or 3 : 4||Fourth||diatesseron|
These were the only intervals considered harmonious by the Greeks. The Pythagoreans supposedly found them by experimenting with a single string with a moveable bridge, and found these pleasant intervals could be expressed as the ratio of whole numbers.
Pythagoras in the School of Athens
|Slide 3-3: Closeup of Tablet
Janson, H. W. History of Art. Fifth Edition. NY: Abrams, 1995. p.497
Raphael's School of Athens shows Pythagoras is explaining the musical ratios to a pupil.
Notice the tablet. It shows:
The words diatessaron, diapente, diapason.
The roman numerals for 6, 8, 9, and 12, showing the ratio of the intervals, same as in the music book frontispiece.
The word for the tone, EPOGLOWN, at the top.
Under the tablet is a triangular number 10 called the sacred tetractys, that we mentioned earlier.
The Harmonic Scale
|Slide 3-5: Gafurio Lecturing |
F. Gafurio, De Harmonia musicorum instrumentorum, 1518, Wittkower, Rudolf. Architectural Principles in the Age of Humanism. NY: Random, 1965. 43a.
This diagram from a book written in 1518 shows the famous Renaissance musical theorist Franchino Gafurio with three organ pipes and 3 strings marked 3 , 4, 6. This indicates the octave, 3 : 6 divided by the harmonic mean 4, into the fourth, 3 : 4, and the fifth, 4 : 6 or 2 : 3.
The banner reads, "Harmonia est discordia concors" or Harmony is discordant concord, propounding the thesis that harmony results from two unequal intervals drawn from dissimilar proportions. The diagram shows compasses, suggesting a link between geometry and music.
So after experimenting with plucked strings the Pythagoreans discovered that the intervals that pleased people's ears were
|octave||1 : 2|
|fifth||2 : 3|
|fourth||3 : 4|
and we can add the two Greek composite consonances, not mentioned before . . .
|octave plus fifth||1 : 2 : 3|
|double octave||1 : 2 : 4|
Now bear in mind that we're dealing with people that were so nuts about numbers that they made up little stories about them and arranged pebbles to make little pictures of them. Then they discovered that all the musical intervals they felt was beautiful, these five sets of ratios, were all contained in the simple numbers
1, 2, 3, 4
and that these were the very numbers in their beloved sacred tetractys that added up to the number of fingers. They must have felt they had discovered some basic laws of the universe.
Quoting Aristotle again ... "[the Pythagoreans] saw that the ... ratios of musical scales were expressible in numbers [and that] .. all things seemed to be modeled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number."
Music of the Spheres
|Slide 3-6: Kepler's Model of the Universe |
Lawlor, Robert. Sacred Geometry. NY: Thames & Hudson, 1982. p. 106
"... and the whole heaven to be a musical scale and a number... "
It seemed clear to the Pythagoreans that the distances between the planets would have the same ratios as produced harmonious sounds in a plucked string. To them, the solar system consisted of ten spheres revolving in circles about a central fire, each sphere giving off a sound the way a projectile makes a sound as it swished through the air; the closer spheres gave lower tones while the farther moved faster and gave higher pitched sounds. All combined into a beautiful harmony, the music of the spheres.
This idea was picked up by Plato, who in his Republic says of the cosmos; ". . . Upon each of its circles stood a siren who was carried round with its movements, uttering the concords of a single scale," and who, in his Timaeus, describes the circles of heaven subdivided according to the musical ratios.
Kepler, 20 centuries later, wrote in his Harmonice Munde (1619) says that he wishes "to erect the magnificent edifice of the harmonic system of the musical scale . . . as God, the Creator Himself, has expressed it in harmonizing the heavenly motions."
And later, "I grant you that no sounds are given forth, but I affirm . . . that the movements of the planets are modulated according to harmonic proportions."
Systems of Proportions based on the Musical Ratios
Slide 17-1: Villa Capra Rotunda
What does this have to do with art or architecture? The idea that the same ratios that are pleasing to the ear would also be pleasing to the eye appears in the writings of Plato, Plotinus, St. Augustine, and St. Aquinas. But the most direct statement comes from the renaissance architect Leone Battista Alberti (1404-1472), "[I am] convinced of the truth of Pythagoras' saying, that Nature is sure to act consistently . . . I conclude that the same numbers by means of which the agreement of sounds affect our ears with delight are the very same which please our eyes and our minds."
Alberti then gives a list of ratios permissible, which include those found by Pythagoras. We'll encounter Alberti again for he is a central figure in the development of perspective in painting.
We'll also discuss another architect who used musical ratios, Andrea Palladio (1518-1580), who designed the Villa Capra Rotunda shown here.
|Slide 3-7: Correspondence School in Crotone
W. S. Anglin. Mathematical Intelligence V19, No. 1, 1997
I always wanted to make a pilgrimage to Crotone, site of the Pythagorean cult, but this is all that's there to mark their presence. Pythagoras and his followers died when their meetinghouse was torched. We'll have more on the Pythagoreans later, in particular their fondness for the star pentagram.In this unit we've had some Pythagorean number lore and soon we'll add to it by talking about number symbolism in general, especially numbers in astrology and the Old Testament. Somewhere I had read that one answer to the question, Why study history? was To keep Pythagoras alive! I've forgotten where I read that, but anyway, it makes a nice goal for this course.
Joined: 16 December 2005
Nietzsche's musical output - concentrated in his early years - provided him with a medium in which to experiment and refine ideas that would later reappear in more finished forms in his mature works. When specific instances of musical experimentation are examined, a pattern of musical 'prototypes' emerges in which Nietzsche utilizes musical composition to extrapolate the effects of altering accepted concepts.
Joined: 16 December 2005
It is not well known that Friedrich Nietzsche wrote music, and among those who do know it, he has a reputation as having been a terrible composer. This is a little unfair. It is true that he never developed a personal style, and his few attempts at longer forms are mostly unfinished and awkward. The bulk of his pieces, however, were written before the age of twenty; a handful of longer attempts were made in the early 1870s, and Nietzsche composed virtually nothing new after 1874, discouraged by sharp criticism from his friends Richard Wagner and Hans von Bulow. With friends like these, his music didn't need enemies.
By common consent, Nietzsche's most successful works were his songs, about 16 in number, most of which he wrote as a teenager, between the years 1861 and 1864. Some of these are quite lovely, with a stirring sense of melodicism that hews closely to the texts. In addition, the Musikalische Nachlass published for Nietzsche in 1976 contains about a dozen completed piano works, most of them brief character pieces of the albumbltter variety, and many more unfinished; fragments of a mass and the beginning of a Christmas Oratorio (it will be remembered that Nietzsche came from a family of Lutheran clergymen); a piece for violin and piano; several choral songs; and an 1872 tone poem, written for two pianos and never orchestrated, based on Byron's supernatural poem Manfred.
Most of this music was written before Nietzsche was converted to the Wagnerian cause, which happened around 1868; his early hero was Robert Schumann, whose melodicism leaves its mark on the piano pieces especially. And yet, while there are Schubertian touches in the lieder, Nietzsche's style also shows some influence of the Wagner-Liszt "Music of the Future" school, and often resembles Liszt's in its poignantly leaping melodies and tonal ambiguities. Remarkable in this regard is the Manfred-Meditation, Nietzsche's most developed large composition, which he reportedly wrote because he found Schumann's Manfred music unsatisfactory and felt he could do better. The middle development section of this piece is undeniably over-repetitious, but I leave it to those less generous to argue whether this work's often ambiguous harmonies reveal an attachment to the Weimar-centered avant-garde of the time, or are merely the result of incompetence.
The Hymnus an das Leben (Hymn to Life) represents one of Nietzsche's final musical thoughts. The original song, under the title Gebet an das Leben, was written for voice and piano in 1882 to a poem by Lou Salom, using music from a piano piece Nietzsche had earlier written in 1874, Hymnus an die Freundschaft. Part of the impetus for setting the poem was no doubt to draw closer to Salom, whom Nietzsche had met in April of 1882 in Rome. He proposed to her twice, through their mutual friend Paul Re, and was twice rejected. In August or September he set her poem to music, but in October Lou and Re left together, causing Nietzsche considerable jealousy. Much later, in 1897, Salom would go on to become the mistress of the poet Rainer Maria Rilke, and still later a close disciple of Freud.
Nietzsche had an amenuensis, Peter Gast (real name: Heinrich Kselitz) who had first come to hear the great philosopher lecture in 1875, and to whom Nietzsche, plagued by eye trouble, subsequently dictated several of his books. It was Gast who, in 1887, arranged Gebet an das Leben for chorus and orchestra, the version heard tonight. Where Nietzsche had set only the first verse of Salom's poem, Gast repeated the musical setting to include a second. The poem apostrophizes both the joy and bitterness of life in a way reminiscent of what has become one of Nietzsche's most widely quoted phrases, "Whatever doesn't kill me makes me stronger":Whether you gave me suffering or pleasure,
Despite the rather dark stoicism of the poem's sentiments, Nietzsche's setting is triumphal and full of light, if also marked by abrupt changes of key. Nietzsche never managed an imaginative use of musical texture, but the chromatic harmonies do show that he was attuned to the more musically progressive currents of his day, and the stirring melody has the air of a nationalistic German anthem.
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