Music and math relationship graph

Music and mathematics - Wikipedia

music and math relationship graph

Music is richly mathematical, and an understanding of one subject can earliest nontrivial results in graph theory, change or method ringing. It is perhaps even more surprising that music, with all its passion and such graphs provide a new perspective on the relationships between. Mathematics and music play very different roles in This essay examines the relationship between mathematics and music from Figure 1).

When all three notes are played together, they form the "C major chord", which is a sweetly harmonious, happy sound, like a barbershop quartet. It is the basis for music as diverse as Row, row, row your boat, and the symphonies in C Major of Mozart and Beethoven and Schubert.

Math and Music Pt1/5

Why do these three notes - C, E, and G - sound so sweet together? Let's have a look. We see that certain of the X's line up almost perfectly.

Indeed, 4 time periods for Middle C, and 5 time periods for Middle E, and 6 time periods for Middle G, are all practically equal. So, about once every 0. So, even though certain of the X's do not line up for these three notes, quite a number of them do.

This combination of variety and consistency is just what is required to produce the C Major chord - one of the most pleasing sounds known to humanity, and the basis of multitudes of tunes from Mendelssohn to Metallica.

So, humanity's quest for beautiful music amounts to finding creative and interesting ways to line up the air pocket X's of different musical notes. Who decides which notes will have their X's line up well, and which notes will not? The answer is, mathematics decides!

Remember that High C has a frequency which is twice as large as Middle C. On the other hand, High C is twelve semi-tones above Middle C. What are their frequencies? How do they fit in?

Mathematics & Music

Early musicians - as far back as the Greek mathematician and musician Pythagoras of the sixth century B. This would make the X's line up perfectly, so the notes would fit together exactly right. However, this system required different tunings depending on which notes you were planning to play, or in which key your music was written.

Over the last couple of hundred years, a more universal system has been used instead. This system - called equal tempering, a version of well tempering - spaces all twelve notes of the octave equally.

In this way, a single tuning can be used regardless of the key being chosen, or the music being played.

music and math relationship graph

Equal tempering is a system for breaking up each octave into twelve equal semi-tones. Since each octave represents multiplying the frequency by a factor of 2, each semi-tone represents multiplying the frequency by the twelfth root of 2 - a number which produces two when twelve copies are all multiplied together.

This number is about 1. So what does this mean? Then Middle D, two semi-tones above Middle C, has a frequency which is 1. Continuing in this way, we eventually reach Middle G, seven semi-tones above Middle C. The frequency of Middle G is 1.

This explains why every three time periods for Middle G, correspond to two time periods for Middle C. Similarly, the frequency of Middle E is 1. This ratio is very close to 1.

music and math relationship graph

Hence, like G, the note E also fits in well with C. Another good example is the note Middle F, whose frequency is 1. This is very close to 1. And, indeed, C and F also fit well together their interval is called a "fourth". On the other hand, Middle F-Sharp has a frequency which is 1. That is why the notes F-Sharp and C do not fit well together. So, to figure out which notes fit well together, we don't need to guess, or use trial and error, or study musical theory for years.

We just need to remember the equal-tempering principle, and multiply copies of 1. Unlocking the key Experienced musicians are always discussing what key to play in.

There is no cooler moment for a jazz musician than, when asked if they can play I got rhythm, to be able to reply, "Sure, dude. This change might make the song more convenient to play on a particular musical instrument, or more comfortable for a vocalist to accompany.

The magical mathematics of music |

If done properly, the key change should have essentially no effect on the way the song "sounds" - it should be just as recognisable, just as lovely, and just as catchy, in the new key as in the old. Indeed, it should be exactly the same song, just performed at a higher or lower pitch. How can this be?

How can a song be the same, but different? How can we change the pitch without changing the tune? From our understanding of frequencies, the answer is clear.

Music and mathematics

To make the notes higher saywe make their frequencies faster, i. But to make the song sound the same, we leave the relationships between the X's just as they were before. We compress each row of X's by exactly the same factor. Equivalently, we raise each note by exactly the same number of semi-tones.

With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings. In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such as musical keyboards.

music and math relationship graph

Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world. Equally tempered scales have been used and instruments built using various other numbers of equal intervals. The 19 equal temperamentfirst proposed and used by Guillaume Costeley in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal semitone equal temperament at the cost of a flatter fifth.

  • The magical mathematics of music

The overall effect is one of greater consonance. Twenty-four equal temperamentwith twenty-four equally spaced tones, is widespread in the pedagogy and notation of Arabic music.

However, in theory and practice, the intonation of Arabic music conforms to rational ratiosas opposed to the irrational ratios of equally tempered systems.

These neutral seconds, however, vary slightly in their ratios dependent on maqamas well as geography. Indeed, Arabic music historian Habib Hassan Touma has written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture.

This is great intuitive preparation for the concepts of discrete vs continuous generally. For example, you can investigate the mathematical relationship between the circle of fifths and the circle of half-steps. Fractal self-similarity is probably one of the defining pleasures of good music in general. Some speculation My experiences in both music and math have convinced me that music is a severely underutilized resource for math teaching.

There are many ways to learn besides manipulating symbols on a page or computer screen. In his book AnathemNeal Stephenson imagines monks solving proofs and running cellular automata by chanting melodies that evolve by systematic rules. When I was trying to learn how wrap my head around binary numbers, I eventually just wrote a song that counts in binary from one to sixty-four and back down.