science

Piano key frequencies

Piano key frequencies is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Piano key frequencies rather than just read about it. In short: This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 112-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (referred to as A440). Every octave is made of twelve steps called semitones.

Key takeaways

Piano key frequencies belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Piano key frequencies to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Piano key frequencies from memory before moving on to harder problems.

Reference excerpt

This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 112-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (referred to as A440). Every octave is made of twelve steps called semitones. A jump from the lowest semitone to the highest semitone in one octave doubles the frequency (for example, the fifth A is 440 Hz and the sixth A is 880 Hz). The frequency of a pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the twelfth root of two (approximately 1.059463). For example, to get the frequency one semitone up from A4 (A♯4), multiply 440 Hz by the twelfth root of two. To go from A4 up two semitones (one whole tone) to B4, multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). To go from A4 up three semitones to C5 (a minor third), multiply 440 Hz three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). For other tuning schemes, refer to musical tuning. This list of frequencies is for a theoretically ideal piano. On an actual piano, the ratio between semitones is slightly larger, owing to string stiffness that causes inharmonicity, i.e., the tendency for the harmonic makeup of each note to run sharp. To compensate for this, octaves are tuned slightly wide, stretched according to the inharmonic characteristics of each instrument. This deviation from equal temperament is graphically represented by the Railsback curve. The following equation gives the frequency f (Hz) of the nth key on the idealized standard piano with the 49th key tuned to A4 at 440 Hz:

f ( n ) = ( 2 12 ) n − 49 × 440 Hz = 2 n − 49 12 × 440 Hz {\displaystyle f(n)=\left({\sqrt[{12}]{2}}\,\right)^{n-49}\times 440\,{\text{Hz}}\,=2^{\frac {n-49}{12}}\times 440\,{\text{Hz}}\,}

where n is shown in the table below. Conversely, the key number of a pitch with a frequency f (Hz) on the idealized standard piano is:

n = 12 log 2 ⁡ ( f 440 Hz ) + 49 {\displaystyle n=12\,\log _{2}\left({\frac {f}{440\,{\text{Hz}}}}\right)+49}

List

Values in bold are exact on an idealized standard piano. Keys shaded gray are rare and only appear on extended pianos. The normal 88 keys are numbered 1 to 88; extra low and high keys which only appear on extended pianos are numbered −11 to 0 and 89 to 100 respectively. A 112-key piano that extends from A−1 to C9 was first built between 2022 and 2024 by Stuart & Sons.

See also Piano tuning Scientific pitch notation Music and mathematics

References

Illustrations

Worked examples

Example 1 — a first encounter with Piano key frequencies

Start with the simplest possible case. Write down what Piano key frequencies claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Piano key frequencies before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Piano key frequencies ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Piano key frequencies

In research: Piano key frequencies appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Piano key frequencies in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Piano key frequencies is common in secondary-school and first-year university syllabi. It links to neighbouring topics Musical tuning, Piano, so understanding it makes those chapters shorter.
In everyday life: Look for Piano key frequencies outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Piano key frequencies in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Piano key frequencies means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Piano key frequencies out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Piano key frequencies in simple terms?

Why does Piano key frequencies matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Piano key frequencies?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Piano key frequencies.