Making math with music

Sujata C

On the face of it, music and mathematics make strange bedfellows. But there is evidence of mathematical inquiry into music as far back in time as ancient Babylonia during the fourth millennium.

Order, sequence, patterns of numbers have been part of our maths textbooks right from primary classes. An arrangement of numbers in a particular pattern is a sequence. In lower classes children are made familiar with patterns in numbers like 1, 3, 5, 7, 9, 11, 13… odd natural numbers or 2, 4, 6, 8, 10… even numbers, or squares of natural numbers 1, 4, 9, 16, 25…

Go a step higher and you come face to face with the beauty that lies hidden in numbers. Squaring natural numbers gives you further natural numbers in a pattern.

12 = 1
112 = 121
1112 = 12321
11112 = 1234321
111112 = 123454321
1111112 = 12345654321
11111112 = 1234567654321
111111112 = 123456787654321
1111111112 = 12345678987654321

If you notice, the answers throw up a palindrome, i.e., they read the same backwards as they read forward. When you take sequences to the next level they are called arithmetic progression. A sequence of numbers a1, a2, a3… an is called an arithmetic progression if there exists a constant number d as the common difference of the arithmetic progression.

Eg: a2 = a1+d, a3=a2+d, a4=a3+d

We get into deeper waters when we look at the algebraic and geometric expressions of arithmetic progressions. This concept is 3000 years old and Archimedes and Aryabhatta made significant contributions to it.

Mathematical uniformity makes music
The idea of mathematical uniformity making music is credited to Pythagoras. He discovered that uniformity in numbers directly impacts harmony, but it is possible that the Indians were aware of this idea much before him. The talam in Carnatic and the tala in Hindustani music is nothing but the rhythm and the pitch, both expressions of number. We keep count with the talam and finger count. Students of advanced Carnatic music will tell you that there is plenty of algebra in Carnatic music. It is the talam that gives structure to a composition and is similar to the measure of western music but is more complex.

The Pythagoras math-based tuning system was based on the principle of perfect fifths – intervals between the first and the fifth note of the proposed scale would be tuned to a ratio of 2:3. Alternately, the ratio between the first and eighth notes (which would be an octave) would be 1:2. These ratios are nothing but the ratio of the wavelength of one note to another.

Once this basic concept was crystallised, further mathematical exploration on wavelengths established that in a combination of notes when the peaks and troughs coincide there is music and when they do not, it makes a noise.

The-golden-ratio Fibonacci sequence and the Golden Ratio
As the children progress to higher classes they are introduced to more complex patterns like Euclid’s Golden Ratio and the Fibonacci sequence. What’s interesting is that the Golden ratio was derived in the 5th century BC, and the Fibonacci sequence centuries later in the Middle Ages, when Leonardo Fibonacci studied the birth of rabbits and established the link between the two concepts.

Golden Ratio is the ratio of distances in simple geometric figures such as pentagon, pentagram, decagon, etc. It is denoted by the symbol phi and is equal to 1.61803399.

The Fibonacci sequence is a seemingly random series of numbers, where every number is the sum of the two preceding it. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…

Dividing each number in the series by the one that precedes it produces a ratio which stabilises around 1.61803399 – the Golden Ratio.

Over a period of time mathematicians discovered that the Golden Ratio and the Fibonacci numbers have several geometric representations in nature like the spiral on a snail’s shell or a new fern leaf, the spectacular patterns in the head of a sunflower or the symmetry in a pine cone. (Carry all these items to your next math class and get your students hooked onto nature’s maths, as detailed further in the article on Biology and Maths, in this issue of Teacher Plus).

logarithmic-spiral Explaining the concept further, mathematician Mark Freitag says, when we draw a set of whirling rectangles we are able to draw a logarithmic spiral. This spiral converges at the intersection of the two blue lines, and the ratio of the lengths of these two lines is in the Golden Ratio. Notice the stark resemblance of this logarithmic spiral to a snail’s shell!

Why composers love Fibonacci
Many music composers have created some chart busting music using the Fibonacci sequence as a basis for their music, beginning with Mozart himself.

Mozart wrote some of the most beautiful piano concertos, within which they say Mozart implemented the Fibonacci sequence. It is said that one can find the Golden ratio in Beethoven’s fifth Symphony. The new Hungarian art music of Bela Bartok uses the Fibonacci sequence because of its occurrence in nature. More popular is Robert Shcumann’s Traumerei, familiar to us as the soundtrack of the Titan watches commercial.

The structure of a piano is based on the Fibonacci numbers. If you observe the keys of a piano, you notice that within a scale of 13 keys, 8 of them are white and 5 are black, these five are split into groups of 3 and 2.

The Mozart Effect
The Mozart Effect was an idea that was very popular in the 90s and the hype created around it was used to sell music lessons and music products with the theme – Mozart makes you smarter. There is huge body of research done on the Mozart effect on children.

Research conducted by Rauscher, Shaw, Levine and Ky at the University of California Irvine established a casual relationship between math and music (only classical chamber music and not popular rock or heavy metal).

Another study by the same team was published in the prestigious science journal Nature in 1993; where they tested college students on spatial tasks and found that they performed better after listening to 10 minutes of Mozart’s Sonata for two pianos in D major than after listening to a spoken word relaxation tape or ten minutes of silence. The study has not gained universal acceptance and many researchers have been conducting research to debunk the Mozart effect theory.

Are you listening or learning?
What do psychologists say about this intriguing idea of music being good for learning maths? What one does with music seems to be the differentiating factor, they say. Listening to music may relax you (if you have a liking for it, yes, there are lots of people who do not particularly like music) and hence improve your performance. But learning music has a remarkably different impact. What we need to take note (pardon the pun) of is that the brain is actively involved in processing information and the similarity between math and music is seen here. Howard Gardner in his book Frames of Intelligence says different modes of thought are used in mathematics- spatial-visual and verbal logical. Verbal-logical and spatial-visual thinking are believed to be carried out in opposite hemispheres of the brain. So when you do math the right as well as the left brain are actively involved just as it is when you play music.

Martin Gardiner, a biophysicist explains this with greater clarity – “Both pitches in a musical scale and numbers increase from step to step from lower to higher. The representations are different, of course, but they require a similar way of understanding and using information.”

number-spiral In three studies involving children, Rauscher found that learning music improved the spatial-temporal reasoning abilities of four – to six-year-olds. She looked for improvements in other areas like verbal abilities, memory and vocabulary but couldn’t find them; however she found that children who learn music score up to 35 per cent higher on spatial tasks.

Gottfried Liebnitz says the experience of listening to music and counting is similar, only we are not aware of the counting when listening to music. Now we know why the written form of western music is called a musical score!

Did we say music and math were strange bedfellows? But it turns out that they are more like close cousins after all! What if one went to a music school to study math and study music with a math teacher? Well all one can say is, play on, little Einsteins!

Always fun, number 1

one

57 + 54 = 111
557 + 554 = 1111
5557 + 5554 = 11111
55557 + 55554 = 111111
5555557 + 5555554 = 11111111

0 x 9 + 1 = 1
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 + 10 = 11111111

Magical number 37

37 x 3 = 111
37 x 6 = 222
37 x 9 = 333
37 x 12 = 444
37 x 15 = 555
37 x 18 = 666
37 x 21 = 777
37 x 24 = 888
37 x 27 = 999

Some more 37 magic

Take any three identical numbers e.g. 777.
Divide the original number by the sum of the number i.e. 777 divided by 7+7+7.
The answer is always 37.
Try it with any other number like 111, 222, 333 etc.

More Number Patterns

4 x 4 = 16
34 x 34 = 1156
334 x 334 = 111556
3334 x 3334 = 11115556

Peculiar number

2520 is a peculiar number.
It is divisible by all numbers from 1 to 9.
Try it yourself.

Even Steven

The product of consecutive numbers is always an even number
3 x 4 = 12
5 x 6 = 30
8 x 9 = 72
23 x 24 = 552
99 x 100 = 9900

nine

The author is a freelance writer based in Hyderabad. She can be reached at sujata117@yahoo.co.uk.