Maths in music: ratios of pitch
A guitar or sitar string makes a sound when it shakes. The number of shakes in one second is the frequency. High frequency gives a high note (high pitch).
A short string shakes faster than a long one. The rule is simple: pitch goes up when length goes down. Half the length gives double the frequency. The two notes sound like the same note, one higher. This pair is called an octave. Its ratio is 2 : 1.
Two-thirds of the length gives 3/2 = 1.5 times the frequency. This pair is called a fifth. Its ratio is 3 : 2. Notes with small whole-number ratios sound sweet together. So the musical scale is built from ratios.
Example: a note of 440 Hz (the note A that orchestras tune to). One octave up is 880 Hz. A fifth above is 660 Hz.
Maths in visual art: perspective
A flat page cannot hold real depth. Painters trick the eye with perspective. The idea: things that are far away look smaller. A pole 10 m away looks half the size of the same pole 5 m away. In maths: apparent size is in inverse proportion to distance.
Parallel lines that go away from you, like the two rails of a railway, seem to come closer and meet at one point on the horizon. This is the vanishing point. The horizon is at the height of the viewer's eyes. If the viewer is lower or higher, the whole picture changes, but the lines still meet at the horizon.
Drawing with perspective is just drawing lines to a point and keeping ratios right.
Maths in visual art: the golden ratio
Cut a line in two parts, a (long) and b (short), so that whole ÷ long = long ÷ short. This special ratio is about 1.618 and is called the golden ratio (symbol φ, phi). It equals (1 + √5) ÷ 2.
You can find it with Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13... Each number is the sum of the two before it. Divide a number by the one before it: 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, 21 ÷ 13 = 1.615. The answers get closer to 1.618.
Squares of Fibonacci sides fit together into a spiral. Some artists and designers use rectangles in this ratio because they look balanced. It is a helpful guide, not a strict law. Many great works do not use it.
More in the lesson The golden ratio.
Shapes in art: tessellations
A tessellation (tiling) covers a flat surface with shapes. There are no gaps and no overlaps. You see it in floor tiles, honeycombs, rangoli, jaali screens and Islamic and Indian patterns.
The rule: at every corner, the angles must add up to exactly 360°. Equal triangles: each angle is 60°, and 6 × 60° = 360°. Squares: 4 × 90° = 360°. Regular hexagons: 3 × 120° = 360°. Regular pentagons: each angle is 108°, and 360 ÷ 108 is not a whole number, so pentagons alone leave gaps.
Mixed shapes work too. A regular octagon has angles of 135°. Two octagons and one square: 135° + 135° + 90° = 360°.
Maths in literature: patterns in stories and poems
Writers use patterns that can be counted. In a poem, a rhyme scheme uses letters for end sounds: A B A B means lines 1 and 3 rhyme, and lines 2 and 4 rhyme. A haiku counts syllables: 5, 7, 5. A song repeats a chorus again and again.
Stories have patterns too: a beginning, middle and end (three parts), the rule of three (three tries, three wishes), and repeated lines in folk tales. A pattern is something that repeats in a rule. Finding a rule is the heart of maths.
Maths in film: animation and geometry
A film is a set of still pictures, called frames, shown one after another very fast. Our eyes join them into motion. Cinema uses about 24 frames per second (fps). So 1 minute has 24 × 60 = 1,440 frames.
Animators use geometry: a ball that bounces follows a curve, and each frame moves it a little. Computer animation uses coordinates and angles. To turn, slide or resize a drawing, the computer uses maths called transformations: translation, rotation, scaling and reflection.
Try it: find maths at home
- In the 3D above, slide the string length. Check: length 6 gives pitch ×2.
- Stand on a straight road or a train platform. Look at the two lines at the side. Where do they meet?
- Use a ruler: measure the long and short side of a visiting card or a book. Divide. Is it near 1.6?
- Look at a floor tile or rangoli. Which shape repeats? Count its corners and angles.
- Pick a song and mark the rhyme letters of its lines: A, B, A, B...
- Draw a stick figure in a notebook corner on 10 pages. Move it a little each page. Flip the pages fast!
Key formulas and definitions
- Pitch (frequency) ∝ 1 ÷ string length
- Octave = 2 : 1, fifth = 3 : 2
- Apparent size ∝ 1 ÷ distance
- Golden ratio φ = (1 + √5) ÷ 2 ≈ 1.618
- Tiling corner rule: angles at a point add up to 360°
- Frames in a film = frames per second × seconds
Worked examples
1. A string is 60 cm long. It is pressed so that 40 cm vibrates. What is the ratio of the new pitch to the old pitch? If the old note is 440 Hz, what is the new frequency?
Ratio = 60 ÷ 40 = 1.5 (a fifth). New frequency = 440 × 1.5 = 660 Hz.
2. A tree looks 4 cm tall in a photo when you stand 5 m away. How tall would it look from 20 m (same camera)?
Size is inversely proportional to distance. Distance becomes 20 ÷ 5 = 4 times, so the size is 4 ÷ 4 = 1 cm.
3. A ribbon of 26 cm is cut in the golden ratio. Find the long and the short part.
Long part = 26 ÷ 1.618 ≈ 16.1 cm. Short part = 26 − 16.1 = 9.9 cm. Check: 26 ÷ 16.1 ≈ 1.615 and 16.1 ÷ 9.9 ≈ 1.626. Both are about 1.62.
4. Can regular pentagons alone tile a floor? Use angles.
A regular pentagon has interior angle 108°. At a corner, we need a whole number of angles to make 360°. 360 ÷ 108 = 3.33, not a whole number. So there will be a gap. No.
5. A film runs for 90 minutes at 24 frames per second. How many frames is that?
Seconds = 90 × 60 = 5,400. Frames = 5,400 × 24 = 129,600.
6. A 12-line poem repeats the pattern A B A B three times. How many lines end with sound A?
Each A B A B group has 2 A lines. Three groups give 3 × 2 = 6 lines.
Common mistakes
- Thinking a longer string gives a higher note. It is the opposite: shorter string, higher pitch.
- Thinking that things in a painting are really smaller far away. Perspective is how the eye sees them, not their real size.
- Believing the golden ratio is in every great artwork. It is one useful guide among many.
- Forgetting the 360° rule and saying pentagons or circles tile a floor without gaps.