The tempered scale is the musical tuning system that divides the octave into 12 strictly equal semitones. Each semitone corresponds to multiplying the frequency by the 12th root of 2, approximately 1.0595. This system, gradually adopted from the 18th century, allows playing in all keys on the same instrument.
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How to measure it in class
With the FizziQ app, it is possible to verify the frequencies of the tempered scale.
Steps:
- Open FizziQ and select the frequency synthesizer
- Generate the frequency of A4 (440 Hz)
- Calculate the theoretical frequency of A#4: 440 x 1.0595 = approximately 466 Hz
- Generate this frequency and compare with a real keyboard
- Repeat to construct the entire chromatic scale
- Verify that after 12 semitones, you find 880 Hz (octave)
Scientific activities on this topic
- Build a scale with the synthesizer: https://www.fizziq.org/en/activities/the-range/
- Compare tempered scale and natural scale
- Analyze the frequencies of a piano or xylophone
Learn more
Why “tempered”?
The word comes from “temper,” to moderate. In natural scales (Pythagorean or just intonation), some intervals are perfectly consonant but others are very out of tune. The tempered scale “tempers” these deviations by distributing the error uniformly: no interval is perfect, but none is really wrong.
The problem of fifths:
If you chain 12 perfect fifths (ratio 3/2), you should return to the starting note (7 octaves higher). But (3/2)^12 is approximately 129.75 while 2^7 = 128. This difference, the “Pythagorean comma,” makes it impossible to have a scale where all fifths are perfect AND where octaves are in tune.
Calculating frequencies:
In the tempered scale, the frequency of a note located n semitones above A4 is:
f = 440 x (12th root of 2)^n
| Note | n | Frequency (Hz) |
|---|---|---|
| A4 | 0 | 440.00 |
| B4 | 2 | 493.88 |
| C5 | 3 | 523.25 |
| D5 | 5 | 587.33 |
| E5 | 7 | 659.26 |
Formula
f(n) = 440 x 2^(n/12)
where:
- f(n): frequency of the note (Hz)
- n: number of semitones relative to A4 (440 Hz)
- The coefficient 2^(1/12), approximately 1.0595, is the tempered semitone ratio
Application examples
- A piano is tuned in tempered scale to allow modulations
- Guitars have frets spaced according to the tempered ratio
- Synthesizers generate tempered frequencies by calculation
- Bach composed “The Well-Tempered Clavier” to promote this system
FAQ
Q: Is the tempered scale out of tune? A: It is a compromise: thirds are slightly off (plus or minus 14 cents), fifths too (plus or minus 2 cents). But these deviations are acceptable to the ear and allow transposition.
Q: Do all instruments use the tempered scale? A: Fixed-pitch instruments (piano, guitar) yes. Variable-pitch instruments (violin, voice, trombone) can adjust to play more accurate intervals.
Q: How can I verify tempered tuning with FizziQ? A: Measure the frequency of each note on an instrument and compare to theoretical values. Good tempered tuning will be within plus or minus 3 Hz.
Related concepts
Octave - Frequency - Pitch - Semitone - Musical tuning - Harmonics