Mathematics and Music - November 2006, Stage 4&5

Problems

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Six Notes All Nice Ratios

Stage: 4 Challenge Level: Challenge Level:1

The Pythagoreans noticed that nice simple ratios of string length made nice sounds together.

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Pythagoras’ Comma

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Using an understanding that 1:2 and 2:3 were good ratios, start with a length and keep reducing it to 2/3 of itself. Each time that took the length under 1/2 they doubled it to get back within range.

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Equal Temperament

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

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Tuning and Ratio

Stage: 5 Challenge Level: Challenge Level:1

Why is the modern piano tuned using an equal tempered scale and what has this got to do with logarithms?

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Euclid's Algorithm and Musical Intervals

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Use Euclid's algorithm to get a rational approximation to the number of major thirds in an octave.

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Rarity

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that it is rare for a ratio of ratios to be rational.