Scientific Pitch Notation: What C4 Actually Means

Scientific pitch notation writes a letter, an optional accidental and an octave number: C4, B♭3, F♯5. The letter and the accidental are uncontroversial. The number is where documentation, instrument charts and software quietly disagree, and a single octave of disagreement is enough to send a part into the wrong register.

#The octave number changes at C

In scientific pitch notation the octave number increments at C, not at A. C4, D4, E4, F4, G4, A4 and B4 all belong to the same octave, and the next number up begins at C5. A4 and B4 come before C5, which is why a scale counted from A crosses an octave boundary in the middle.

The choice follows the instrument. On a piano keyboard the pattern of seven letter names restarts at each C, and the register boundary sits where the pattern restarts. Numbering the octaves from A would place a boundary between G and A, where the layout offers no such break. Every convention that numbers octaves from the note the pattern restarts on agrees with the keyboard; the disagreements are about which number sits at the bottom, not about where the boundary is.

The reference note of the whole system is middle C, the C on the first ledger line below the treble staff and the first ledger line above the bass staff. On a standard piano it is the C nearest the middle of the keyboard, sounding at about 261.6 Hz.

#Middle C has three common names

• C4 in scientific pitch notation. This is the convention used in most modern English-language reference material, instrument range charts and academic writing.
• MIDI note 60, an integer in the 0 to 127 range that the MIDI standard uses to identify pitches. The MIDI numbering does not use octave names at all, which is one reason it survives as an interchange format.
• C3 in the labelling convention associated with Yamaha and followed by a number of other manufacturers and software developers, where every octave number is one lower than scientific pitch notation. Some sequencers and samplers print the same key as C5 instead.

The practical consequence is that a label cannot be trusted on its own. A sample library whose documentation says its range starts at C3 may mean C3 or C4 in scientific pitch notation, and a patch labelled C4 may sound an octave away from the C4 you wrote in the score. The sounding pitch is the fact; the printed number is a naming choice. When they conflict, check the instrument’s key mapping or measure the output rather than the label.

Helmholtz notation is the older continental system, and it uses capital and lowercase letters with primes rather than numbers. It still appears in nineteenth-century editions and in some German-language reference works, and the primes are easy to miscount in small print.

#From a note number to a frequency

Modern pitch reference is anchored at A4 = 440 Hz, the tuning frequency specified by ISO 16. Orchestras do not always follow it exactly: many European ensembles tune a little higher, and period-instrument groups working in Baroque repertoire often use 415 Hz, about a semitone below modern pitch, while some Classical-era projects sit near 430 Hz. For documentation purposes, 440 Hz for A4 is the default and the exceptions are stated explicitly.

The relationship between a MIDI note number and its frequency is logarithmic in one direction and exponential in the other. With n as the MIDI note number:

f = 440 × 2^((n − 69)/12)

Work one case carefully. C4 is MIDI note 60, so the exponent is (60 − 69) / 12, which is −0.75. Two to the power of −0.75 is approximately 0.5946, and 440 multiplied by 0.5946 gives 261.6 Hz, the familiar frequency of middle C.

The formula also handles octaves without special cases, because an octave in MIDI numbers is 12 steps, and 2 raised to the power of 12/12 is exactly 2. A5 is MIDI 81, one octave above A4, so it doubles to 880 Hz. C5 is MIDI 72, which is 2^0.25 × 440, or 523.25 Hz.

The inverse is worth having as well:

n = 69 + 12 × log₂(f / 440)

That version turns a measured frequency into a note number, which is what a tuner does when it reports both a pitch name and a deviation in cents. A semitone is 100 cents, and the conversion between a frequency ratio and cents is 1200 × log₂(ratio), so a measured 441 Hz against a target of 440 Hz is about 3.9 cents sharp. Most musicians cannot tune a sustained interval reliably to better than a few cents by ear, which is a useful thing to know before chasing a reading on a meter.

#Where the ambiguity actually bites

Notation software. Many programs have an octave setting that decides whether middle C is called C3 or C4, and it affects note entry by name, MIDI import and the octave numbers printed in range warnings. It is worth setting once per project and leaving it alone, because mixed conventions inside a single file are far harder to debug than a consistent wrong one.

Samplers and virtual instruments. A patch that claims to play C3 may sound an octave under or over the C3 in your score, since the developer chose a labelling convention before choosing the sample root. The reliable check is a single held note against a tuner, not the library’s documentation.

Instrument ranges. A range chart that says an instrument descends to C3 conveys nothing until it also says whether that is sounding or written pitch and which octave convention is in use. Reference material is only usable when the convention is printed with it. Assume the chart is wrong until the convention is stated.

Transposing instruments. A written C on a B♭ clarinet sounds B♭3. Written and sounding octaves differ by more than the transposition interval in the case of instruments such as the tenor saxophone, which sounds a major ninth below its written pitch, so a written C4 emerges as B♭2. Documenting a range means stating the frame of reference as well as the pitches.

Guitar and double bass. The lowest open string of a guitar sounds E2, about 82.4 Hz, and is written E3, because guitar music is notated an octave above sounding pitch. Both numbers are correct: one for the player’s page, one for the concert-pitch measurement. Exported MIDI depends on the software’s convention, which is why the same tablature can arrive in a DAW an octave away from what the player hears.

#Getting the number right without a table

• Choose one convention per project and write it down. Scientific pitch notation is the safest choice for prose, documentation and range charts, because it is explicit about the octave boundary.
• Anchor on two frequencies: C4 is about 261.6 Hz and A4 is 440 Hz. Any reference that contradicts those anchors is using a different system, not a different fact.
• Measure rather than assume. Play C4 on the instrument or library in question and read the result with the chromatic tuner at /tools/chromatic-tuner/, which reports both the pitch name and the deviation in cents.
• Quote ranges in the player’s own frame and add the sounding octave in parentheses. The extra three words save a rehearsal.