Musical Note Frequencies

A musical note is a symbol that indicates the pitch and duration of a musical sound. As the basic element of musical notation, it is used to transcribe and communicate music to musicians. Each note corresponds to a sound of a well-determined frequency.

Discover FizziQ

How to measure it in class

FizziQ’s frequency meter and spectrogram display the fundamental frequency of a note played in front of the microphone, which makes it possible to reconstruct the tempered scale from measurements.

Steps:

Open FizziQ’s frequency meter and play the A4 of a tuning fork or a tuned instrument: check that the device does display 440 Hz.
Then play each chromatic note going up, from that A to the A one octave above, and record the fundamental frequency of each in the experiment notebook.
Calculate the ratio between each note and the previous one: you should find approximately 1.059 each time, that is 2^(1/12).
Check the ratio between the two A notes: it must be exactly 2, whatever path is followed.
Compare the measured fifth A-E to the 3/2 ratio of a pure fifth: the deviation on the order of 0.1% is the signature of equal temperament.
Repeat the measurements with the spectrogram to observe, above the fundamental, the harmonics that give the instrument its timbre.

Scientific activities on this topic

To understand the frequencies of musical notes, analyze the frequencies of the notes of a scale with the sound library and the frequency meter of the FizziQ app (see the activity)

Air columns and pitch - vary the length of a tube and relate the note obtained to the length of the air column.

An article complements these activities: Six science experiments with the tone generator.

Learn more

The frequency of musical notes in Western music is determined by the musical notation system called the tempered scale. It is based on equal temperament, formulated mathematically as early as the end of the 16th century: the Chinese prince Zhu Zaiyu gave the exact calculation in 1584, and in Europe Vincenzo Galilei (father of Galileo) and then Simon Stevin proposed equivalent constructions in the same years. Its generalization in Western musical practice, however, came much later and spread over the 18th and 19th centuries.

The equal temperament system divides the octave into 12 equal intervals, each representing a “semitone”. The frequency of a given note is determined using the following mathematical relationship:

f_n = f_0 * 2^(n/12)

where f_n is the frequency of the note, f_0 is the frequency of the reference note (usually A), and n is the number of semitones separating the reference note from the target note.

The reference A is generally set at 440 Hz (hertz), meaning this note vibrates at 440 cycles per second. Here are the approximate frequencies of musical notes in the tempered scale starting from the reference A at 440 Hz:

A: 440 Hz
A# or Bb: 466.16 Hz
B: 493.88 Hz
C: 523.25 Hz
C# or Db: 554.37 Hz
D: 587.33 Hz
D# or Eb: 622.25 Hz
E: 659.26 Hz
F: 698.46 Hz
F# or Gb: 739.99 Hz
G: 783.99 Hz
G# or Ab: 830.61 Hz

Equal temperament is a compromise, not a perfect solution

The ear judges as consonant those intervals whose frequency ratio is simple: 2/1 for the octave, 3/2 for the fifth, 5/4 for the major third. Equal temperament, on the other hand, requires the twelve semitones to be identical, so that every interval is written 2^(n/12). Yet 2^(7/12) = 1.4983, whereas the pure fifth equals 3/2 = 1.5000. The tempered fifth is therefore slightly too short, by about 0.11%, that is 2 cents. The deviation is even more pronounced for the major third: 2^(4/12) = 1.2599 versus 5/4 = 1.2500, that is 0.8%, a 14-cent difference clearly audible on sustained sounds.

This is not a manufacturing defect but an arithmetic impossibility. Stacking twelve pure fifths gives (3/2)¹² ≈ 129.7, while seven octaves give 2⁷ = 128. The ratio 129.7/128 ≈ 1.0136 is the Pythagorean comma: pure fifths never land exactly back on the octave, no matter how many are stacked. A choice must therefore be made. Older temperaments concentrated the deviation on a few intervals, very pure in some keys and frankly out of tune in others. Equal temperament spreads it uniformly: no interval is perfectly in tune except the octave, but none is unusable, and all keys become equivalent. This is what makes free modulation possible.

Before the invention of this system, musicians used temperament systems that produced notes slightly different from current notes, making it difficult to play with instruments from different makers or different temperament systems.

The tempered scale system was developed to allow modulation and playing in different keys without the need to constantly retune instruments. It is widely used in Western classical music, popular music, and many other musical genres in the Western world.

However, there are other tuning systems in the world, such as the Chinese pentatonic scale, the maqam scale in Arabic music, the Pythagorean scale, and many others. These tuning systems are based on different frequency ratios and can produce distinct musical colors.

Each culture has its own musical traditions and its own scales, making the music of the world diverse and rich in sonic textures. Examples include:

Chinese pentatonic scale: The Chinese pentatonic scale is widely used in traditional Chinese music. It consists of five notes per octave and is characterized by the absence of semitones between its degrees. The notes are often notated as do, re, mi, sol, and la, and they are used to create melancholic and meditative melodies.
Maqam scale in Arabic music: The maqam scale is used in Arabic, Persian, and Turkish music. It includes a set of notes with specific intervals and allows great flexibility in ornamentation and musical expression. Each maqam has its own characteristic scale.
Javanese gamelan scale: The gamelan is a traditional instrumental ensemble from Java, Indonesia. It uses specific scales for each instrument, thus creating a complex and harmonious sound. Gamelan scales vary from one region of Java to another.
Blues pentatonic scale: The blues pentatonic scale is used in Western blues and rock. It consists of five notes per octave and is famous for its bluesy and expressive character.
Inuit scale (Inuit Qilaut scale): The Inuit peoples of northern Canada use a specific scale in their traditional music. It is based on the sounds of nature and is played on instruments such as Inuit drums and flutes.
Pythagorean scale: The Pythagorean scale is based on simple frequency ratios and was used in ancient Greece. It is built by stacking pure fifths with ratio 3/2, hence pure fourths with ratio 4/3, and it is the scale that brings out the Pythagorean comma.
African pentatonic scales: Many African cultures use pentatonic scales in their traditional music. These scales vary from one region of the continent to another and contribute to the richness of African music.

Formula

In the tempered scale, the frequency of a note located n semitones above a reference note is:

f_n = f_0 × 2^(n/12)

where:

f_n: frequency of the target note (Hz)
f_0: frequency of the reference note, generally the A at 440 Hz (Hz)
n: number of semitones separating the two notes, negative going down (dimensionless)

The ratio between two consecutive semitones is therefore constant:

f_(n+1)/f_n = 2^(1/12) ≈ 1.0595

For n = 12, we recover the octave:

f_12/f_0 = 2^(12/12) = 2

To compare two intervals precisely, the cent is used, one hundredth of a tempered semitone:

number of cents = 1200 × log₂(f₂/f₁)

where:

f₁, f₂: the two frequencies being compared (Hz)
1200: number of cents in an octave

Application examples

The A one octave above the 440 Hz A equals 440 × 2 = 880 Hz. The A one octave below equals 220 Hz.
The C above is three semitones above the 440 Hz A: 440 × 2^(3/12) = 440 × 1.1892 ≈ 523.3 Hz.
Tuning a guitar: the low E string vibrates at 82.4 Hz, the high E string at 329.6 Hz, exactly two octaves higher (ratio 4).
The reference pitch has varied throughout history: approximately 415 Hz in the Baroque era, 435 Hz in France in the 19th century, 440 Hz since the standardization of 1939. A Baroque orchestra therefore plays nearly a semitone lower than a modern orchestra.
Beats are used for tuning: two strings at 440 Hz and 442 Hz produce 2 beats per second. The string is tightened until these beats disappear.
A flute goes out of tune as it warms up: the speed of sound in air increases with temperature, so the resonance frequency of the pipe rises. This is why wind instruments go sharp during a concert while strings go flat.

FAQ

Q: Why 2^(1/12) and not some other number? A: Because we want twelve identical semitones in an octave, and the octave corresponds to a doubling of frequency. We therefore need a ratio r such that r¹² = 2, hence r = 2^(1/12) ≈ 1.0595. Musical intervals multiply, they do not add.

Q: Why is the tempered fifth not exactly 3/2? A: Because 2^(7/12) = 1.4983 and not 1.5. This is mathematically unavoidable: a power of 2^(1/12) cannot equal exactly 3/2, since that would amount to writing 3 as a rational power of 2. Equal temperament accepts this 2-cent deviation, imperceptible, in order to make all keys playable.

Q: What is the comma? A: It is the small residual gap between intervals built by different paths. The Pythagorean comma is the ratio between twelve pure fifths and seven octaves, approximately 1.0136 or 23 cents. It is the concrete manifestation of the impossibility of tuning an instrument with all intervals pure.

Q: Do two instruments playing the same note have the same sound? A: No. They have the same fundamental frequency, hence the same pitch, but harmonics of different intensities. It is this distribution of harmonics, visible on the spectrogram, that constitutes the timbre and makes it possible to tell a flute from a violin.

Q: The frequency meter displays an unstable value, is that normal? A: Yes for a note played on an instrument, especially during the attack and on instruments with vibrato. Read the value during the sustained, stable part of the sound, and work in a quiet place: intense background noise can make the algorithm jump to a harmonic.

Frequency - Pitch (Sound Height) - Octave - Harmonic Sound - Timbre - Spectrogram - Acoustic Beat - Resonance Frequency - Wavelength