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← Why Twelve Notes

Why Twelve Notes

Four short lessons on the arithmetic inside music: a bar that has to add up, tempo as a rate, why the octave has twelve steps, and why loudness is measured on a log scale.

Spark & Whetstone · sparkwhetstone.org/learn/why-twelve-notes · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. A bar has to add up

In 4/4 time every bar holds four beats. A whole note fills the bar, a half note takes two beats, a quarter note one, an eighth note half a beat. Writing a bar is adding fractions until they make exactly one whole.

Children meet this as a puzzle: two quarters and a half fill a bar; three quarters leave a gap; a half and three quarters spill over. The notes are fractions with a sound attached.

Question. A bar in 3/4 time holds three beats. Which set of notes fills it exactly?

  1. A half note and two eighth notes
  2. Three half notes
  3. A half note and two quarter notes

Say it back. In your own words: why is writing a bar of music a fraction problem?

Lesson 2. Tempo is a rate

Tempo is written in beats per minute. At 60 beats per minute each beat lasts one second; at 120 each beat lasts half a second. Doubling the tempo halves the time of every note.

That makes a song’s length a division problem: count the beats in the whole piece, then divide by the tempo.

Question. A piece has 300 beats and is played at 100 beats per minute. How long does it last?

  1. 30 minutes
  2. 3 minutes
  3. 30 seconds

Say it back. In your own words: what happens to each note when the tempo doubles?

Lesson 3. Why twelve

Two notes an octave apart vibrate in the ratio 2 : 1; a fifth is close to 3 : 2. Stack twelve fifths and you almost land seven octaves up — almost. (3/2)¹² is about 129.75, and 2⁷ is 128, a gap of about 1.4% called the Pythagorean comma.

Equal temperament closes the gap by making all twelve steps the same ratio, the twelfth root of two, about 1.0595. Every fifth comes out a hair narrow, and every key on the piano works.

Question. Stepping round the twelve keys by 5 semitones at a time, how many different keys do you visit before coming back to the start?

  1. Only three
  2. Only four
  3. All twelve

Say it back. In your own words: why do twelve pure fifths not quite come back to the starting note?

Lesson 4. Loud is logarithmic

Sound level in decibels compares powers on a logarithmic scale: ten times the power is +10 dB, and doubling the power adds only about 3 dB.

Distance matters too. In the open, the sound spreads over a sphere, so twice as far away is a quarter of the power: about 6 dB quieter.

Question. A speaker is turned up so its power is 100 times what it was. How many decibels louder?

  1. +10 dB
  2. +20 dB
  3. +100 dB

Say it back. In your own words: why does a second speaker not make music sound twice as loud?

At the table: Tune the water glasses

Players: Two or more, ages 6 and up with a grown-up. You'll need: Four or five matching glasses, a jug of water, a spoon.

  1. Fill the glasses to different heights and tap each one gently with the spoon.
  2. Before tapping, guess: does more water make the note higher or lower?
  3. Add or pour off water until two glasses sound the same note, then try to make one sound an octave above another.
  4. Talk about what changed: the water, the length of air, or how the glass rings.

Answers

  1. A half note and two eighth notes — A half note (2) and two eighth notes (½ + ½) make exactly 3 beats. A half and two quarters make 4, too many for 3/4.
  2. 3 minutes — 300 ÷ 100 = 3 minutes. At 150 beats per minute the same piece would take 2 minutes.
  3. All twelve — Because 5 and 12 share no factor, stepping by 5 visits every key once before returning — the circle of fourths, the circle of fifths backwards.
  4. +20 dB — 10 × log₁₀(100) = 10 × 2 = 20 dB. Each factor of ten in power adds 10 dB.