Words like elegant and beautiful are used constantly by mathematicians to describe paths of reasoning and proofs. Certainly many tasks in the life of a musician fall into this category. Arranging a melody on an instrument and finding fingerings that correspond to certain sequences of notes is definitely a type of math problem. Playing the same melody on different instruments is math, as is playing a stringed instrument and changing the tuning. And when you find the best key to play a certain melody on a guitar, for example, there is a sensation that is known to math insiders as elegance.
Mathematicians praise each other for the elegance of a proof, referring to the esthetic beauty of it. When you write a new piece of music, when you find the best fingering on a stringed instrument for a sequence of notes, or when you arrange a piece of music for an ensemble, you can experience nearly identical sensations of elegance. As you learn about music and about chord theory, you learn to recognize chord changes, and you experience a mathematics of musical structure also. Playing harmonies, playing the same song in different keys, taking solos on unfamiliar songs-- these things all involve recognizing the structure of a piece of music.
Good musicians can often listen to a song, observe the musical structure, and play along with it, without really knowing it or rehearsing it, because they recongnize patterns and familiar shapes. This type of thinking is very much like the way you think when you study mathematics. Both music and math have concepts, and special symbols. What is a musical key? What is a number? The definitions of things in both disciplines are somewhat circular and vague, unless you understand what they are.
You cannot define a number, but you know what they are much of the time and you can use them. It's no different with a musical notion like a minor key. Once you know what it means you can spot one, though you cannot really define it rigorously. There are many things in music that are obviously math-related, and many musical notions can be explained in numbers. But it is important to note that numbers are not some way to describe music-- instead think of music as a way to listen to numbers, to bring them into the real world of our senses.
The ancient Greeks figured out that the integers correspond to musical notes. Any vibrating object makes overtones or harmonics, which are a series of notes that emerge from a single vibrating object.
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The fundamental musical concept is probably that of the octave. A musical note is a vibration of something, and if you double the number of vibrations, you get a note an octave higher; likewise if you halve the number of vibrations, it is an octave lower. Two notes are called an interval ; three or more notes is a chord. The octave is an interval common to all music in the world.
Many people cannot even distinguish between notes an octave apart, and hear them as the same. In western music, they are given the same letter names. If you blow across a coke bottle and it produces the note F, and you drink enough so that the air remaining in the bottle is twice as much, the note will be also an F, but an octave lower.
If you shorten a string exactly in half, it makes a note an octave higher; if you double its length, it makes a note an octave lower. You can think of the concept of octave and the number 2 as being very closely associated; in essence, the octave is a way to listen to the number 2. We call it a 5th, because it is the 5th scale note of the Western do-re-mi scale, but it represents the integer 3.
Incidentally, the 5th is the only interval other than the octave that is common to all musics in the world. Strings of a violin are tuned a 5th apart. Men and women often sing a 5th apart, and most primitive harmony singing involves octaves and fifths. In fact, they say that when you are learning to tune a stringed instrument, you can only trust your ear to hear octaves and fifths, and you should not rely on your ability to compare other musical intervals properly.
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The next note in the harmonic series corresponding to the number 4 is 2 times 2 and thus a second octave. The number 5 produces a new note, called the musical 3rd. The 3rd is the other note in the fundamental chord, called the major triad, which is made up of 1st, 3rd and 5th notes of the Western scale.
The number 6 produces a note an octave higher than the 5th, and it is also a very harmonious note. The number 7 produces the first dissonant note in the harmonic series, which has some numerological and religious significance.
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Also of spiritual and numerological interest-- the next dissonant overtones are the 11th and the 13th. If you build a musical system out of these integer notes, it is what is now called the Pythagorean scale, as used by the ancient Greeks. If you bore holes in a flute according to integer divisions, you will produce a musical scale.
Oddly enough, if you try to build complex music from these notes, and play in other keys and using chords, dissonances show up, and some intervals and especially chords sound very out of tune. Our Western musical scale paralleled the evolution of the keyboard, and finally reached its modern form at the time of J. Bach, who was one of its champions.
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After a few intermediate compromise temperings , as systems of tuning are called, the so called even-tempered or well-tempered system was developed. Even-tempering makes all the notes of the scale equally and slightly out of tune, and divides the error equally among the scale notes to allow complex chords and key changes and things typical of western music.
Our ears actually prefer the Pythagorean intervals, and part of learning to be a musician is learning to accept the slightly sour tuning of well-tempered music. Tests that have been done on singers and players of instruments that can vary the pitch such as violin and flute show that the players and singers tend to sing the Pythagorean or "sweeter" notes whenever they can.
More primitive ethnic musics from around the world generally do not use the well-tempered scale, and musicians run into intonation problems trying to play even Blues and Celtic music on modern instruments. Old bagpipes had the holes drilled in places that sound "sour" to modern ears. The modern musical scale divides the octave into 12 equal steps, called half-tones. The frets of a guitar are actually placed according to the 12th root of 2, and 12 frets go halfway up the neck, to the octave, which is halfway between the ends of the strings.
On fretted instruments we are actually playing irrational numbers! And any of you who have trouble tuning your guitars might get a clue as to why they are so hard to tune. Our ears don't like the irrational numbers, but we need them to make complex chordal music.
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The student of music must learn to accept the slight dissonances of the Western scale in order to tune the instrument and to play the music. Studying mathematics can also assist you in daily life as a musician. I cannot tell you how many times I have actually needed to solve an equation or refer to one of my math textbooks, but the answer is a very small integer. I think the only time I ever needed to do that was to compute how many combinations of a guitar capo that allows you to selectively capo any combination of strings at a given fret, rather than just clamp across all the strings as capos have traditionally done.
I am not talking so much about solving the little algebra problems of life like changing money when you tour in foreign countries.
That's kind of tricky, though it is junior-high school math involved. Being a former math student makes it easier I think for me to use and understand my computer, which is an essential tool for a working musician today. Print out a list of everybody who has signed up in the last 2 years who either lives in northern Mass, coastal NH or Southern Maine, but only if they are media, and sort them by zip code. It's a math problem. When you are setting up a sound system for a band, you might have a 16 channel mixer, with a monitor send and an effects send.
How can you plug in your wires to send a mix to the main amps to send to the audience, send another mix to the monitors for the band to hear, and maybe a 3rd mix to a radio feed or a tape recorder. The wiring of sound systems and the routing of signals is a type of mathematics. The noise in a signal is determined by a theory called gain structure, where it passes through from 5 to 15 different devices and wires of different lengths through pre-amps, delays, choruses, reverbs, mixers, tuners; learning to understand and optimize your use of these things is definitely a math problem.
Troubleshooting a sound system 30 minutes before the gig is a math problem. One of the speakers is not working. I usually make my stories smaller but this one needed seven.
I use reader's theaters to encourage my students to practice fluency. It is a fun way to do something that they ne. Reading , Native Americans , Drama.