Unidade 1

Introdução à Música: um guia prático para o iniciante


Music is a moral law. It gives a soul to the universe, wings to the mind, flight to the imagination, a charm to sadness, gaiety and life to everything. It is the essence of order,and leads to all that is good, just, and beautiful, of which it is the invisible, but nevertheless, dazzling, passionate, and eternal form.

We are a spectacular, splendid manifestation of life. We have language. We have affection. We have genes for usefulness, and usefulness is about as close to a "common goal" of nature as I can guess at. And finally, and perhaps best of all, we have music.
Lewis Thomas

I call architecture silent music.
J. W. von Goethe

Think of an equation with a thousand variables. Every piece of music that you will ever hear is like that equation. Instead of plugging in ones, twos, threes, etc., you calculate with social movements, political events, musical innovations, etc. Change any variable in either one of them and you will certainly see a change in the final result.

Music above all, and for this, choose the irregular.
Paul Verlaine

A man that has a taste of music, painting, or architecture, is like one that has another sense, when compared with such as have no relish of those arts.
Joseph Addison

Criticism perishes, human generations die out, systems change, but when one day the world goes up in flames like a scrap of paper it will be the works of art that glow the longest before they return unto God, and darkness descends. . .
An unknown woman named Caroline in a letter to philosopher Wilhelm August von Schlegel

Capítulo 1 -- Os que Falavam de Matemática

uma pequena estória de ficção científica

In the recent wave of budget cuts, it seemed that the space program was again the hardest hit. The research science lab had been one of the first facilities to shut down causing a wave of angry but futile letters to be sent to the government. It had been a grave disappointment to all at the base, but as one of the senators had said, hard times necessitated hard measures. In spite of impassioned arguments, piles of petitions, and even a stack of letters from retired astronauts who'd become national heroes, the politicians had decided to shut it all down. J. H. Tooms had walked through these same halls only last week when they were as silent as a deep cave. Things had seemed so bleak just a few days ago.

The tall grey-haired scientist stopped at a turn in the corridor and smiled to himself surveying the flurry of activity. "Perhaps there is such a thing as justice after all," Tooms muttered out loud to no one in particular. His observations were interrupted by several white-coated interns frantically racing down the crowded hallway toward the main conference room. "J. H.! It does one good to see your ugly face around here again!" called a familiar voice, that of his lab partner who had been laid off almost half a year earlier.

Tooms laughed, turned to the speaker, shook hands and then motioned toward all of the happy commotion around them. "Don't you just love to see things in their proper order?"

Phil Steevies laughed. "It's good to be back where I--where we all belong."

Their good natured banter was interrupted by an announcement over the building intercom. "Your attention. Your attention please. This morning's press conference will begin in five minutes in auditorium three. This conference will be televised. All level four and above personnel are requested to attend. Again, press conference in five minutes."

"Well, I didn't feel like having a second cup of coffee anyway this morning," Tooms half-grimaced. "I bet if we try we can get front row seats."

Phil held out his right arm in a gesture indicating 'after you' and followed J. H. down the corridor. A small flood of white-coated researchers had already begun to pour into the conference room. Stuffed with reporters, camera crews and still photographers, the former dining hall seemed an unlikely place to be the focus of a stunned world who would be straining to hear every word that came from it.

Just at the entrance, Tooms found himself pulled off to the side by his friend. "You leaked it, didn't you," Phil said quietly, feigning seriousness. "You were the one who blabbed, right?"

Tooms made a valiant attempt to look innocent. "Blabbed what?"

"Come on, I may be getting old, but I'm not stupid. So how are you going to pass your lie detector test when Big Man Hinson lowers the boom?"

Lowering his voice, Tooms whispered, "Phil, I wish I HAD done it. A discovery like this is just too big to keep. Whoever 'blabbed' did the world a big service. Besides, I think it's time that we were given a little of the credit we're due. This thing will blow over faster than we would like, so let's enjoy the fame while we have it."

A few minutes later the agency's director of public affairs Hinson Boyd was facing the glare of several dozen camera lights and a forest of microphones. The buzz of conversation in the room quieted quickly as Boyd began. Boyd was well known to the press for his matter-of-fact working style as well as a dry sense of humor. "Good morning everyone. I would like to start by publicly thanking our research staff for compiling information so quickly. Many of them have been working around the clock for several days to analyze the data we will be presenting," Boyd paused and added, "which is indeed unique in our history."

"Yes. THE RUMORS THAT YOU HAVE HEARD ARE TRUE. On one of our routine space explorations, we discovered the remains of a spacecraft that has come from another world, another civilization. There really IS intelligent life out there. Someplace."

For a few seconds the only sound that could be heard was the dull whoosh of the air conditioning system. In that moment of stunned silence it seemed as if everyone in the room forgot to breathe. Then at least twenty-five reporters began shouting out questions simultaneously, bringing the atmosphere back to life in a roar of confusion.

"Gentlemen, please. Gentlemen. We'll get to your questions in a moment. Now to continue, we have found the remains of an alien spacecraft. It has come an unknown distance from a planet and a star that has not yet been discovered by our observations. Let me start by saying that we are in no danger of contamination from this civilization. We are in no danger of attack. This spacecraft was an unmanned, exploratory spacecraft launched what would seem to be thousands of years ago. It has been drifting powerless for most of that time. From the damage present, we believe that several meteors and other space debris have collided with it, knocking it off of its original course any number of times. At this point, we have no way of knowing where or when the craft was launched. We are safe from them, and I'm sorry to have to add, they are safe from us."

The faces of many of the news crews reflected disappointment. The director paused to let that sink in. He then quickly added, "All is not lost, however. They have enclosed a package of data meant to be examined by anyone who may have picked up the spacecraft. We are working around the clock on it. At this point I'll take a few questions."

A dark-haired reporter in the front row quickly barked, "What kind of data did you find? What can we tell about them at this point?"

"Some of the information is conveyed by symbols and pictures obviously meant for an occasion like this. There are some drawings that would seem to represent the beings. Those will be reproduced and distributed later this morning.

"We can guess that they communicate by sound, like us. The spacecraft contained a series of recordings that would seem to be a greeting of some sort along with samplings of their language. Their language, however, is unlike ANYTHING we have ever encountered. Our preliminary analysis is that they speak in and understand mathematics."

"Mathematics??" several reporters simultaneously gasped.

"It would seem so. Their data contains a series of vibrations at complex but related wavelengths. The first sounds we have analyzed contain a series of vibrations 'spoken', for lack of a better word, by voices that are alternately metallic, woody, or stringy.

"My guess is that it would seem to be some kind of a greeting by a community of different speakers, or perhaps a discussion of a topic that we have never encountered before."

"Does this mean that they have an intelligence beyond our ability to measure?" another voice shot.

Boyd chuckled. "No, actually it doesn't. It means that--that they're different, that's for sure. As to intelligence, we can't tell from that. They have unmanned spacecraft. So do we. Don't forget that some of our pets have much more highly developed senses of smell and hearing than we do and we don't suggest that they're smarter than we are."

"What type of mathematics do they have in their sound?"

"I'm afraid that at this moment, it's a little too early to tell. We have duplicated the data and spread it through several of our departments. It will be examined in a number of ways, in hopes that we can unlock whatever secrets are encoded in it.

"When do you anticipate the completion of your analysis?"

"We're going to be cautious about this. Six months looks to be the earliest date that we can come up with a preliminary conclusion, although with intelligence as completely alien as this, the process could take years if not lifetimes. During that time we plan to have regular press conferences, daily if needed. Don't worry--we know you all have to write stories and we'll give you at least enough to keep your jobs!" A few scattered chuckles came from the audience.

"When, if ever, is a copy of the data going to be distributed to the other governments, or to the public?" a young reporter asked sharply.

Hinson Boyd accepted the statement as the thinly disguised challenge to the government and space agency policy that it really was. Leaning forward to the battery of microphones he paused for a few seconds. "A week to ten days. As it was our government's money that supported the space program, and we discovered and retrieved the spacecraft, we feel we deserve first crack at it. However, something like this is far too important to keep to ourselves. Within ten days we will be gradually releasing some of the data. You yourselves may be among the first non-scientists to hear some of the alien sounds if you stick with us.

"Thank you for your questions. I'm going to conclude my part of the conference and turn the podium over to the astronauts who actually made the discovery. There is another conference scheduled for this time tomorrow in which we will update you on anything new. . ." Boyd quickly left the stage.

The room was lit up by dozens of flash pictures taken as the director left the little room and three now very well known astronauts filed up to the microphones.

J. H. leaned over to Steevies. "Did you hear how pleased he seemed when that twit asked him that last question? You don't suppose it was the Big Man himself who leaked the discovery?"

Phil stared off into space for a few seconds. "You know, you've got me wondering. Keep your eyes open. I've always hated these press conferences. I'd rather be back at work cracking this particular nut."


Six months very quickly passed. Most of the data and information surrounding the spacecraft had been given to the press and other governments. With the blessings of the President the space agency held onto a little, most of it being associated with the alien spacecraft itself. Of anything revealed to the public, the pictorial information along with some strange symbols created the biggest stir. It showed basically what the aliens looked like. Countless imaginative directors had consulted special effects technicians. The first movies depicting the aliens were due out in a few weeks.

When the first official analysis began hitting the public, it had an almost overwhelming effect--on those that could comprehend it.

The verdict was -- mathematics. Pure mathematics in sound. This alien civilization apparently spoke in, and understood mathematics. Because of this, the unknown aliens were dubbed "the Speakers of Mathematics".

What elegance they had achieved! Many of the programs, especially the first one, seemed like discussions. One voice made a statement, another voice repeated that statement, adding its own meaning, and then another voice picked up the discussion. They all seemed to be talking simultaneously, but instead of clashing or creating chaos, there was nothing but pure harmony! Somehow, this alien people had discovered a logic and order in what would seem to be near chaos! It was a true community comprised of individuals contributing to a greater whole.

The scientists marvelled at the order in the language, that was organized around the powers of two. Any analysis from any direction found it saturated with twos, fours, eights, sixteens, etc. Whatever civilization spoke in such language must have been absolutely remarkable. There was debate that the language may have been written for computers instead of the spacecraft's builders, but little credence was given to that theory.

One of the most striking patters of twos was in the length of each of the little blocks of sound, each being closely related by the formula. There were twos, fours, eights, sixteens. There were even "words", as they came to be called, that seemed to be lengths of one-half, one-fourth, one-eighth!

The overall structure of the sounds seemed to be organized in large scale patterns that again repeated in patterns of fours and eights and sixteens. One of the data analysts went so far as to suggest that this number was related to a deity or deities--they must worship twin gods. Another argued with her, saying that it was possibly related to the fact that they had two arms and legs as shown on a primitive drawing found on the spacecraft. Another suggested that perhaps it had something to do with the symbolism of the two sexes that their species had.

Even the very sounds of the individual voices had mathematical patterns! The mixtures of wavelengths used to create each of these sounds, some woody or metallic or stringy, had been carefully crafted in regular mathematical patterns.

One of the things that made these mathematical programs so difficult to understand was that almost all of them consisted of mixtures of these sounds--as if a whole group was each talking simultaneously, each one saying something different. When they did sound together and the vibration frequencies were compared a very common ratio of numbers that came up were 4, 5, and 6. Over and over, one sound vibrating at a 4x times per second, another vibrating at 5x times per second, and another at 6x times per second.

Taking the length of each of the programs into account, the scientists found another startling discovery--something happened at a precise point in many of them. A little past halfway in the program of sounds, many of them would reach a climax, or begin a pattern again. The formula they discovered was expressed as

             B/A = A/(A+B)

In the little set of recordings, there were a number of programs, most of them based on these mathematics. A few of them seemed to contain more chaos than others, but they still came back to that mathematical order.

To the surprise of many, especially those who discovered and analyzed the data, recordings of the sounds were suddenly in great demand by the public. To the even greater surprise of the scientists, they were greatly enjoyed by nearly all who heard them. Those aliens, whoever they were, must have been privy to some great knowledge.


Two years flew by as quickly as if they had been two weeks. The pace of new discoveries and analysis had slowed down greatly. What was coming out of the investigations consisted mostly of theories. As part of the ongoing project the heavens were constantly scanned for a second spacecraft, a second sign of some other life out there, but with universe this big, it was unlikely.

Even though the operation had begun to phase down somewhat, the space agency had more than enough funding to keep it going. Just the idea of someone else out there, someone new and different was enough to spark the imagination of the whole planet.

At the end of the two years, J. H. Tooms had decided to retire from the project and spend time travelling with his wife. Their children had grown and many of the ties that had kept him to his life patterns were slowly becoming weaker. Someone else, someone a little younger, a fresher perspective could step in to wring out the last secrets of the 'speakers of mathematics'.

After a very well attended retirement party and too many embarrassing speeches, Tooms had one final request. He wanted to say goodbye to the alien object in his own personal way. It could have gotten security chief Knear in trouble, but he figured that Tooms had surely earned one last look at the alien spacecraft that had put some excitement in everyone's life recently.

Through the big skylights, the glow of the night sky provided most of the illumination in the heavily secured chamber. Tooms and his wife spent a few minutes in silence looking at a little plaque with some drawings that had come from somewhere up there in the stars.

Whoever they were, these speakers of mathematics were a true wonder. There was the discouraging possibility that perhaps in the thousands of years that this little exploratory satellite had been travelling, this civilization had died out, never knowing if its communique had been received. It was like putting a message in a bottle and throwing it out into a sea of infinite size. There was no way of knowing if they were alive or extinct. Silently, Tooms wished them well.

The couple took one final close look at the plates which included drawings of the strange aliens. Funny, how a species could be so different, having only five fingers on each hand. There was even a little diagram of what must have been their solar system--and the spacecraft coming from the third of nine planets.

Tooms reached out to touch the little inscription and its foreign marking symbols that meant something to these speakers of mathematics. He hoped someday his people would understand more. A lot more.

The door to the chamber closed behind them, and the little spacecraft sat, a multitude of secrets waiting to be unlocked. The only thing breaking the silence of the little chamber was the sound of one of the "programs" playing from a speaker off in the corner. This program in particular had intrigued the scientists by its complexity and order that almost seemed to contradict each other. The name of its programmer, Johann Sebastian Bach, would be one of the enigmas forever locked in its mystery.

Light from the twin moons reflected like jewels off of the bronzed plaque, with the etchings that spelled

United States of America

Perhaps someday the etchings might be gently persuaded to give up at least a few of their secrets. . .

There's a saying that whoever discovered water, it certainly wasn't a fish--meaning that we often need to see things from a new perspective in order to understand them fully. When we are immersed in something--like the music that connects with us through radios, elevators, shopping malls, the movies--we often don't think about it or realize how marvelous it really is. I hope you've been taken out of the "water" for a few moments.

Science fiction is often thinly disguised fact with a "what if" approach. I hope you have found "The Speakers of Mathematics" entertaining, and even more, I hope you have found it informative. All of the things that the aliens discussed are actual descriptions of our own music--the sounds that we listen to and take for granted on a daily basis. The Voyager spacecraft that have been sent out to explore the outer planets and then fly off into deep space have human relics on them, mostly in the form of . . music. It's not inconceivable that the conclusions reached by the characters above would be quite logical if one were to take a different approach to music. For the record, the Classical Greeks considered music to be yet one more branch of mathematics.

Let's take another look at some of the ideas discussed in the above story.


(If you haven't studied mathematics in depth lately, the phrase "powers of two" refers to a series of numbers generated by taking the number 1, multiplying the result by two, multiplying that by two, multiplying that result by two, et cetera. The numbers generated are a series of perfectly symmetrical numbers; 1, 2, 4, 8, 16, 32, 64, 128, and so on.)

In much of our music there is a very strong feel for beat patters of two or four. Musical sound attracts us--and also reflects us--in a wide variety of ways. Appealing to our physical senses can be an extremely powerful means of expression.

Where does the link with the physical element come in? When we walk, we walk RIGHT, LEFT, RIGHT, LEFT--in a cycle of two. (The pattern of four is just a slight variation on the pattern of two.) If we try to keep a rhythm with our two hands, the same pattern comes up. Our breathing pattern is in two cycles. Our heart beats in two cycles. Pediatricians and researchers have found that the human fetus is capable of hearing noises in the womb before birth. Perhaps our needs for rhythmic regularity and symmetry come from our mother's heartbeat and are ingrained in us long before we are born. It shouldn't be too surprising that our music, something that comes from deep inside of us, would take on that comforting pattern which becomes known to us so very early in our lives.

Let's take this binary sense of regularity one step further. A very large percentage of music, popular and otherwise, consists of these patterns of four repeated as wholesale units. What numbers typically come up when you count these repetitions? Fours, eights, sixteens, etc.

Any music that is based on physical motions, such as dance music, must necessarily be obsessed with that regularity. Disco and marching music take this element about as far as it can go. A strong argument can be made that a huge percentage of our music derives itself from dance roots--even classical symphonies and chamber music.

Listen to a typical song on the radio. If you count the measure numbers, you will often find a section (such as a verse or chorus) will consist of four or eight lines of poetry, each taking two beat patterns (or in musical terminology, two measures). Even instrumental solos will often very rigidly stick to those patterns based on the powers of two.

For the record, not every piece of music ever written rigidly adheres to the powers of two. Quite often you will find a composition that will suddenly jump out of this symmetry and throw in a five or a seven. This is a good device to catch your attention and perhaps signal that something unusual is coming. It will then often settle back down into regularity.

This feeling of regularity is the principle part that makes us want to move to a piece of music. A common name for this is the "beat".

Even our musical notation revolves around the powers of two, reflecting the sound that it is capturing. The names of the notes (in the English language) are a dead giveaway to this relationship. They are called whole note, half note, quarter note, eighth note, sixteenth note, etc. and have a duration directly related to their names (for example: a sixteenth note lasts exactly half as long as an eighth note in a given composition).

Western music wasn't always like this. One of the few times in history when our music didn't have that multileveled regularity was during the Middle Ages. This music was called "Gregorian chant", also known as "plainchant". Gregorian chant was meant (and felt) on a very spiritual plane with little physical inspiration desired. Things that appealed to the physical aspects in this era were even considered sinful. Not coincidentally, death--the liberation of the spirit from the human body--was looked upon as an end to suffering during these times.

Our music affects us deeply on a physical level.


This category is both the simplest and the most complicated to explain.

One of the reasons that a composition has a particular emotional effect on us is because of the sound created by the notes sounding simultaneously. A less technical name for this is "harmony". The harmonies that we say sound the best are, as cited in the story, closely related mathematically. The ratio of 4-5-6 that was mentioned is the ratio found between the three notes of a major chord, probably the most common harmony and certainly the most pleasing to our ears. If we work to the simpler ratios (3-4-5 or 2-3-4) they still sound pleasing. When you generate notes with very complex frequency ratios (9-10-11 or 3-7-11), you will get something that doesn't sound nearly as pleasant.

It is those mixtures of air vibrations that help to cause a piece of music to sound happy, sad, funky, mellow, defiant, or produce nearly any other feeling that you can think of.

In addition to relationships found between notes, each note that we hear is not just one vibration, but a mixture of air vibrations. Perhaps not coincidentally, the components of these notes are related in the same way as described above. Regular patterns sound normal, giving us a feeling of relaxation and familiarity. Irregular patterns sound strange to our ears, conveying a feeling of tension.

Our music affects us on an emotional level.


If you take a line and divide it in the middle, you will have found what is called its "mean". Classical Greeks found another type of dividing point, which they called the "Golden Mean". At about 61% of the way through a line there is a point where you can divide it so that the ratio of the smallest to the largest is the same as the largest to the whole of the line.

                A         B

The formula expressed by this is B/A = A/(A+B)

The percentage can be found by taking the square root of five, subtracting one, and dividing the result by two, giving a result of approximately 61.8 per cent. In other words, take any line, and at about 61.8% of the way from one end to the other you will find its Golden Mean. The ratio of the two parts is then called the "Golden Proportion."

What is so important about the Golden Proportion? Historically, it was one of the first numbers found that couldn't be expressed as a simple fraction (or a ratio, giving us the term "irrational").

In addition, it's a proportion that we find instinctively pleasing. No one's really sure why--it just seems to be born in us. When you look at a good photograph or painting, the chances are that the dominant lines of the scene do NOT divide the picture into halves or fourths, but approximately into thirds. Many of the design proportions of our books, desks, houses, etc. are divided closely approximating the ratio of the Golden Proportion. Take a look at a magazine. The placement of the pictures on the page, the page layout, the placement of the pictures, even the shape of the pictures will likely reflect the golden proportion. Maybe even the shape of the magazine itself!

To complicate the issue, the Golden Proportion is a ratio found in nature in the structure of certain plants and animals. The outward spirals of shells or seeds in a sunflower are designed along the pattern of the Golden Proportion. While it can be argued that much of our sense of perfection can trace itself to our Greek cultural heritage, that argument collapses when Nature comes into the picture.

Where does music fit into this whole scheme of things? By now you might have guessed that the order in music, being as much second nature as it is, would instinctively reflect the Golden Proportion somehow.

Take musical compositions, time them, and find the place that represents the golden mean (about 60% of the way through). A surprising amount of the time you will find some unusual things happening at or very close to that location. Often, you will find the composition's climax, or its epiphany, or perhaps it will come to a stop and replay the music heard at the beginning. Not every piece of music is based on this ratio--but many of them are. Pure coincidence would seem an unlikely explanation for the amount of times this mathematical patterns occurs.

Our music touches us on a deeply ingrained intellectual level with its proportional perfection.


I had a difficult time trying to squeeze this idea into "The Speakers of Mathematics" and decided to skip it for the sake of brevity and clarity in the story line. However, the element of melody--the rise and fall of pitches--and how it affects the listener is vital to the understanding of music itself. The way it affects us may be the most miraculous point of them all.

Even in purely instrumental music the leading musical line will have many characteristics of our most natural sound, speech. By varying the sense of pitch or tension in our voices, we are capable of expressing a whole dimension of emotions. Melodic lines go up and down and, aided by rhythm and harmony, can convey almost anything wished by a composer. Keep in mind I'm still talking about nonvocal music--the addition of the voice and lyrics opens up a whole new world of musical expression. We can easily imagine the first primitive humans wailing a cry of sadness from a loss or tragedy; perhaps the boisterous chanting and cheering of a successful hunting or warring party; perhaps a shriek of terror coming from the victim of a surprise attack by a predatory animal. All of these complex emotions and more can be--and were--easily be conveyed without understandable words.

These first stirrings of what evolved into music rest with us on a level more basic, more a part of us than speech. The descendants of the above three examples, although much more refined, are still with us in the wailing musical lines of jazz, the fight song of a football team, or the shrieking high violin sounds that help to terrify us during a horror movie.

It's quite probable that you've heard a few conversations spoken in a language you didn't understand. Yet, you could easily tell from the melody and rhythm of the speech if the speakers were angry, happy, or somewhere in between. Animals pick up on this element quite easily. If you were to go up to a sleeping cat and angrily scream, "YOU'RE SUCH A NICE FRIENDLY KITTY! I'M GOING TO GIVE YOU SOME FOOD", you'll probably scare the cat out of the room. On the other hand, if you say to your dog in your smoothest, most friendly, most reassuring voice, "I'm going to beat you to within an inch of your flea bearing life", he'll probably wag his tail and look very contented. The actual language in these two cases is unimportant. The inflection, the pitch, the tension--the essence of a musical melody--does all the communication necessary. Because music conveys this so easily, it is often called the "universal language". Emotions conveyed in music have little trouble breaking cultural or language barriers.

In communication mediums two items that are conveyed are ideas and emotions. Written language is very precise in the way that it communicates specific ideas. Music is particularly lousy in that category. There are very few specific ideas, if any, that music can communicate. However, it's been suggested that the purpose of those words and sentences is to "program" the reader with a set of emotions so that he'll be entertained by that emotional program. The words, sentences, paragraphs, and chapters merely keep the reader occupied while the author sneaked his "program" into the reader's subconsciousness. If that's the case, Language gets to the finish line of the race and finds Music impatiently waiting for him to catch up.

Our music is linked to our sense of the aesthetic. It touches our sense of expression and communication on a deep and profound level, and where its creator wishes, touches us on a spiritual level.

A question often comes up from this discussion: "Does everyone who writes a piece of music always think of it in terms of mathematical proportions? I would be lying if I claimed that even the greatest masters of music always thought in these terms. Many musicians have never thought of powers of two or the Golden Proportion, they can't even read or write musical notation, but that doesn't stop them from a deep understanding of it. Again, music is instinctive--it comes from deep within us, deeper even than speech. Music is a language all its own, with its own ways of achieving tension and relaxation. Many musicians, even the well-trained ones, are capable of just hearing a piece of music and being able to tell if it "feels right" without counting measure numbers or composing with a calculator in hand.

Music is its very own language, only able to be understood in terms of music itself. Many times someone has asked a composer "What did you mean by that piece of music?" The stock reply is "If I had meant for it to be explained in words, I would have written a novel instead."

If a lot of this discussion strikes you as a radically different way of looking at music, then I have succeeded in at least one of the goals of this text. I hope you begin to understand the complexity of this field that we so often take for granted. When we turn on the radio, when we watch a movie and are affected by the music in the soundtrack, when we go to a concert, we are often deeply affected by what we hear. A very mysterious and sophisticated process is taking place. That we don't have to think about the operation is due to the fact that music does its job so well. To a casual listener, music actually operates on a subliminal level.

Unit one on musical notation presents a system that in certain ways makes more sense than our own decimal mathematical system and will familiarize you with a few of the symbols of the written language. Lately the computer has been redefining the role of music and its notation, but instead of making changes, it may be linking us back to the ancient times when music was considered mathematics.

Music can be studied as a science. Unit two on Acoustics explores the physical production of sound. It can even be discussed as a branch of psychology because of the many ways it touches our memory, our intellect, our emotion.

Music can be studied as history--but with something extra added. That something is the fact that the documents, the thoughts, the feelings are as alive as they ever were. They were not only alive in their own era--but remain living and breathing to this day. Unit three discusses a few common musical forms, ideas, and the important composers of the past with several purposes in mind. One of them is to touch upon the great ideas that will link us with our cultural heritage. Another very important one is to learn and understand more about our own era and ourselves through our own artistic eyes. In a reference to a line a little earlier, we must be like the first fish that crawled out of the water to take a look at different surroundings in order to better understand our own, what it means to us, and to our times, which is the subject of unit four.


  1. What is music? Explain (a)in a precise "scientific" way, and (b)what it means to you personally.
  2. Describe/explain some of the mathematical patterns found in ordinary music.
  3. "Music is like a cultural sponge--it absorbs everyt hing in the society around it." Either defend or refute this statement, using musical examples to support your case (feel free to refer to your own favorite music).
  4. Think of some examples in movies, television, or commercials where music was used to "program" you or enhance your emotions (such as make the product sound more exciting, heighten your emotions in a dramatic situation, etc.)
  5. Imagine you have a job "programming" various types of music for certain moods, tasks, times of day, etc. Choose a vacation day, a weekend day, a work day, or a school day for your framework. Prepare a program of music (use specific compositions) that would run for the cycle of a full day from dawn to dusk and into the night. Include music to accompany meals, various activities, entertainment, etc. Explain the emotions your music should convey at the appropriate times.