Question

Twelve-tone music requires that the 12 notes of the chromatic scale be played before any tone is repeated. How many different ways can the 12 tones be played? How long will it take to play all possible sequences of 12 tones if one sequence can be played in four seconds?

Short answer

There are 479,001,600 ways, taking about 60.8 years.

Step-by-step solution

Calculate Possible Sequences

To find the number of different ways the 12 tones can be arranged, determine the number of permutations of 12 distinct items. This is given by the factorial of 12, or \(12!\).

Calculate 12-Factorial

Calculate \(12!\) which is the product of all positive integers up to 12. This is:\[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479,001,600 \]

Estimate Total Time

Given that each sequence takes 4 seconds to play, the total time required to play all possible sequences is the product of the number of sequences and the time per sequence. Thus:\( 4 \text{ seconds/sequence} \times 479,001,600 \text{ sequences} = 1,916,006,400 \text{ seconds} \)

Convert Time Units

Convert the total time from seconds to years for better comprehension. First, calculate the total time in minutes: divide by 60. Then convert to hours: divide by 60 again. Finally, convert to years: divide by 24 to get days, and then divide by 365.25 to account for leap years approximately:\[ \frac{1,916,006,400}{60 \times 60 \times 24 \times 365.25} \approx 60.8 \text{ years} \]

Key concepts

Factorials

Factorials are a way of multiplying a series of numbers in descending order. It's a concept that is often used in mathematics to calculate permutations and combinations. If you see the notation \( n! \), it's read as "n factorial."
For example, \( 12! \) means that you take all the whole numbers from 12 down to 1 and multiply them together.

• \( 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479,001,600 \)

This large number represents all the different ways you can arrange 12 distinct items.
Factorials grow really fast as you increase the number.
They are crucial in problems involving ordering or arranging items, like in permutations.

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and grouping items. It's the math behind solving problems like, "How many ways can these items be arranged?" or "How many groups of these items can you form?"
A critical part of combinatorics is understanding permutations, which involve arranging items in a specific order.
When calculating permutations for a set of items where order matters, like a 12-tone music sequence, we use factorials.

• The formula for permutations of \( n \) items is \( n! \).
• In this context, the music sequence problem is essentially about finding all possible unique arrangements, giving us \( 12! \), or 479,001,600 arrangements.

Understanding combinatorics can significantly simplify how you tackle problems of arrangements and groupings.

Twelve-Tone Music

Twelve-tone music is a method of composition that uses all 12 notes of the chromatic scale before any are repeated. This musical approach ensures that no particular note is given more importance than another, creating an even texture.
In practice, each sequence of notes must be unique and cover all 12 notes. Given the nature of permutations, you can imagine the vast number of sequences possible when dealing with 12 notes—all based on permutations of these notes.
Understanding and solving how to play these 12 tones differently each time requires a grasp of combinatorics and factorials. Each sequence becomes a unique permutation of the 12 tones, not unlike a puzzle that must always be solved anew in a different order.