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Practice sheet · Piano Path · Lens 1: Math

Fractions & exponents

Practice for The Math of Piano: semitones, scales, chords, frequency, rhythm and the circle of fifths (pp.8–16), plus Algebra A1–A2 (pp.17–18).
Warm-upCoreLevel Up
Name Date Time I spent
Remember

12 semitones make an octave. Major scale steps: 2 2 1 2 2 2 1. Beats: whole 4, half 2, quarter 1, eighth ½. One beat lasts 60 ÷ BPM seconds.

Frequency: up an octave × 2, down ÷ 2. A4 = 440 Hz. Each semitone multiplies by 21/12 ≈ 1.0595, so n semitones above A4 is 440 × 2n/12.

Warm-up

One step

Count keys, beats and octaves.
1

Count up

How many semitones from D up to A? From A up to the next D? What do they add up to?

2

Octaves

A4 is 440 Hz. What is A2? What is A6?

3

Fill a bar

How many eighth notes fill one bar of 4/4? Of 3/4? Of 6/8?

4

Beat length

At 100 BPM, how long is one beat? One bar of 4/4?

Core

Two or three steps

Use the keyboard on p.8 to count.
5

Song length

A song has 48 bars of 4/4 at 96 BPM. How long does it last, in seconds and minutes?

6

Build a scale

Use 2 2 1 2 2 2 1 from E to build E major. Which notes do you play, and how many black keys?

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Piano Path · Unizon · PracticeFractions, ratios & exponents
7

Chord menu

How many different sets of 4 notes can you choose from the 12 in an octave? Use 12 × 11 × 10 × 9 ÷ (4 × 3 × 2 × 1).

8

Pure or equal?

A pure fifth above A3 (220 Hz) is 3/2 × 220. Equal temperament gives E4 = 440 × 2−5/12. Find both. How far apart are they?

9

Circle of fifths

Start on G and go up a fifth (7 semitones) three times. Which notes do you visit? How many semitones is that in total?

10

Missing notes

Solve: a 3/4 bar has one quarter note plus x eighth notes. A 6/8 bar has one dotted quarter (3 eighths) plus x eighths.

Level Up

Equations & exponents

A1–A2 (pp.17–18).
11

Practice plan

Start a passage at 72 BPM and add 4 BPM a day: T = 72 + 4d. When do you reach 120 BPM?

12

Solve for tempo

A piece has 40 bars of 3/4 and lasts 80 s. Use t = 60 × beats ÷ BPM to find the tempo.

13

Low A

Solve 440 × 2k = 55. Which A is it?

14

Five up

D5 is 5 semitones above A4. Find its frequency. Then use n = 12 × log₂(f ÷ 440) to check that 880 Hz is 12 semitones up.

Make your own problem

Your turn to be the composer

  1. Pick a piece you know, or invent 4 bars.
  2. Write a question about its rhythm, tempo, scale or pitch.
  3. Solve it, then play it, if you can, to check by ear.

My problem, and my worked answer:

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Piano Path · Unizon · PracticeFractions, ratios & exponents

Try every problem first, then check here. Wrong answers are useful: find the step where your work and the key split apart.

The key never prints with Print my work or Print blank. Open it to get its own Print the answer key button.

Answer key

Worked solutions. Frequencies use equal temperament with A4 = 440 Hz.

Warm-up
1
7 · 5 · 12 D → A is a perfect 5th (7). A → D is a 4th (5). 7 + 5 = 12, one octave.
2
110 Hz · 1,760 Hz Two octaves down: 440 ÷ 2 ÷ 2 = 110. Two up: 440 × 4 = 1,760.
3
8 · 6 · 6 An eighth is ½ a beat: 4 ÷ ½ = 8; 3 ÷ ½ = 6. 6/8 means 6 eighths.
4
0.6 s · 2.4 s 60 ÷ 100 = 0.6. 4 × 0.6 = 2.4.
Core
5
120 s = 2 min 48 × 4 = 192 beats. 192 × 60 ÷ 96 = 120 s.
6
E F♯ G♯ A B C♯ D♯ E · 4 black keys E +2 F♯ +2 G♯ +1 A +2 B +2 C♯ +2 D♯ +1 E. The black keys are F♯, G♯, C♯ and D♯.
7
495 11,880 ÷ 24 = 495. Order doesn’t matter in a chord, so divide by 4 × 3 × 2 × 1.
8
330 Hz · ≈ 329.63 Hz · ≈ 0.37 Hz 220 × 1.5 = 330. 440 × 2−5/12 ≈ 329.63 (table, p.13). The equal fifth is slightly flat, about 2 cents (p.14).
9
D, A, E · 21 semitones G + 7 = D, + 7 = A, + 7 = E. 3 × 7 = 21 (an octave and a 6th up).
10
x = 4 · x = 3 1 + ½x = 3 → ½x = 2 → x = 4. 3 + x = 6 → x = 3.
Level Up
11
day 12 72 + 4d = 120 → 4d = 48 → d = 12. Only speed up when it’s clean.
12
90 BPM 40 × 3 = 120 beats. 80 = 60 × 120 ÷ BPM → BPM = 7,200 ÷ 80 = 90.
13
k = −3 · A1 2k = 55 ÷ 440 = 1/8 = 2−3. Three octaves below A4 is A1.
14
≈ 587.33 Hz · n = 12 440 × 25/12 ≈ 440 × 1.3348 ≈ 587.33. 12 × log₂(2) = 12.
Make your own: count semitones on a drawn keyboard, write beats as fractions, and remember that pitch multiplies while keys add.
Music Maths Practice · Piano Path · UnizonAnswer key