In all technicality, music can be composed of notes at any arbitrary physical^{[clarification needed]} frequency. Since the physical causes of music are vibrations of mechanical systems, they are often measured in hertz (Hz), with 1 Hz meaning one vibration per second. For historical and other reasons, especially in Western music, only twelve notes of fixed frequencies are used. These fixed frequencies are mathematically related to each other, and are defined around the central note, A₄. The current "standard pitch" or modern "concert pitch" for this note is 440 Hz, although this varies in actual practice (see History of pitch standards).

The note-naming convention specifies a letter, any accidentals, and an octave number. Each note is an integer number of half-steps away from concert A (A₄). Let this distance be denoted n. If the note is above A₄, then n is positive; if it is below A₄, then n is negative. The frequency of the note (f) (assuming equal temperament) is then:

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In a score, each note is assigned a specific vertical position on a staff position (a line or space) on the staff, as determined by the clef. Each line or space is assigned a note name. These names are memorized by musicians and allow them to know at a glance the proper pitch to play on their instruments.

The staff above shows the notes C, D, E, F, G, A, B, C and then in reverse order, with no key signature or accidentals.

[clarification needed] frequency. Since the physical causes of music are vibrations of mechanical systems, they are often measured in hertz (Hz), with 1 Hz meaning one vibration per second. For historical and other reasons, especially in Western music, only twelve notes of fixed frequencies are used. These fixed frequencies are mathematically related to each other, and are defined around the central note, A₄. The current "standard pitch" or modern "concert pitch" for this note is 440 Hz, although this varies in actual practice (see History of pitch standards).

The note-naming convention specifies a letter, any accidentals, and an octave number. Each note is an integer number of half-steps away from concert A (A₄). Let this distance be denoted n. If the note is above A₄, then n is positive; if it is below A₄, then n is accidentals, and an octave number. Each note is an integer number of half-steps away from concert A (A₄). Let this distance be denoted n. If the note is above A₄, then n is positive; if it is below A₄, then n is negative. The frequency of the note (f) (assuming equal temperament) is then:

For example, one can find the frequency of C₅, the first C above A₄. There are 3 half-steps between A₄ and C₅ (A₄ → A♯₄ → B₄ → C₅), and the note is above A₄, so n = 3. The note's frequency is:

${\displaystyle f=2^{\frac {3}{12}}\times 440{\text{ Hz}}\approx 523.2{\text{ Hz}}}$♭₄ → G₄ → G♭₄ → F₄), and the note is below A₄, so n = −4. The note's frequency is:

${\displaystyle f=2^{-{\frac {4}{12}}}\times 440{\text{ Hz}}\approx 349.2{\text{ Hz}}}$powers of two times the original frequency, since n is a multiple of 12 (12k, where k is the number of octaves up or down), and so the formula reduces to:

${\displaystyle f=2^{\frac {12k}{12}}\times 440{\text{ Hz}}=2^{k}\times 440{\text{ Hz}}}$factor of 2. In fact, this is the means by which this formula is derived, combined with the notion of equally-spaced intervals.

The distance of an equally tempered semitone is divided into 100 cents. So 1200 cents are equal to one octave – a frequency ratio of 2:1. This means that a cent is precisely equal to ¹²⁰⁰√2, which is approximately 1.000578.

For use with the MIDI (Musical Instrument Digital Interface) standard, a frequency mapping is defined by:

¹²⁰⁰√2, which is approximately 1.000578.

For use with the MIDI (Musical Instrument Digital Interface) standard, a frequency mapping is defined by:

where p is the MIDI note number (and 69 is the number of semitones between C₋₁ (note 0) and A₄). And in the opposite direction, to obtain the frequency from a MIDI note p, the formula is defined as:

${\displaystyle f=2^{\frac {p-69}{12}}\times 440{\text{ Hz}}}$

For notes in an A440 equal temperament, this formula delivers the standard MIDI note number (p). Any other frequencies fill

For notes in an A440 equal temperament, this formula delivers the standard MIDI note number (p). Any other frequencies fill the space between the whole numbers evenly. This lets MIDI instruments be tuned accurately in any microtuning scale, including non-western traditional tunings.

Note names and their history

Music notation systems have used letters of the alphabet for centuries. The 6th-century philosopher Boethius is known to have used the first fourteen letters of the classical Latin alphabet (the letter J did not exist until the 16th century),

A B C D E F G H I K L M N O,

to signify the notes of the two-octave range that was in use at the time^[7] and in modern scientific pitch notation are represented as

A₂ B₂ C₃ D₃ E₃ F₃ G₃ A₃ B₃ C₄ D₄ E₄ F₄ G₄.

Though it is

Music notation systems have used letters of the alphabet for centuries. The 6th-century philosopher Boethius is known to have used the first fourteen letters of the classical Latin alphabet (the letter J did not exist until the 16th century),

A B C D E F G H I K L M N O,

to signify the notes of the two-octave range that was in use at the time^[7] and in modern scientific pitch notation are represented as

Ato signify the notes of the two-octave range that was in use at the time^[7] and in modern scientific pitch notation are represented as

A₂ B₂ C3Though it is not known whether this was his devising or common usage at the time, this is nonetheless called Boethian notation. Although Boethius is the first author known to use this nomenclature in the literature, Ptolemy wrote of the two-octave range five centuries before, calling it the perfect system or complete system – as opposed to other, smaller-range note systems that did not contain all possible species of octave (i.e., the seven octaves starting from A, B, C, D, E, F, and G).

Following this, the range (or compass) of used notes was extended to three octaves, and the system of repeating letters A–G in each octave was introduced, these being written as Following this, the range (or compass) of used notes was extended to three octaves, and the system of repeating letters A–G in each octave was introduced, these being written as lower-case for the second octave (a–g) and double lower-case letters for the third (aa–gg). When the range was extended down by one note, to a G, that note was denoted using the Greek letter gamma (Γ). (It is from this that the French word for scale, gamme derives, and the English word gamut, from "Gamma-Ut", the lowest note in Medieval music notation.)

The remaining five notes of the chromatic scale (the black keys on a piano keyboard) were added gradually; the first being B♭, since B was flattened in certain modes to avoid the dissonant tritone interval. This change was not always shown in notation, but when written, B♭ (B-flat) was written as a Latin, round "b", and B♮ (B-natural) a Gothic script (known as Blackletter) or "hard-edged" b. These evolved into the modern flat (♭) and natural (♮) symbols respectively. The sharp symbol arose from a barred b, called the "cancelled b".

In parts of Europe, including Germany, the Czech Republic, Slovakia, Poland, Hungary, Norway, Denmark, Serbia, Croatia, Slovenia, Finland and Iceland (and Sweden before about 1990s), the Gothic b transformed into the letter H (possibly for hart, German for hard, or just because the Gothic b resembled an H). Therefore, in German music notation, H is used instead of B♮ (B-natural), and B instead of B♭ (B-flat). Occasionally, music written in German for international use will use H for B-natural and B^b for B-flat (with a modern-script lower-case b instead of a flat sign). Since a Bes or B♭ in Northern Europe (i.e., a B elsewhere) is both rare and unorthodox (more likely to be expressed as Heses), it is generally clear what this notation means.

In Italian, Portuguese, Spanish, French, Romanian, Greek, Albanian, Russian, Mongolian, Flemish, Persian, Arabic, Hebrew, Ukrainian, Bulgarian, Turkish and Vietnam the note names are do–re–mi–fa–sol–la–si rather than C–D–E–F–G–A–B. These names follow the original names reputedly given by Guido d'Arezzo, who had taken them from the first syllables of the first six musical phrases of a Gregorian chant melody "Ut queant laxis", which began on the appropriate scale degrees. These became the basis of the solfège system. For ease of singing, the name ut was largely replaced by do (most likely from the beginning of Dominus, Lord), though ut is still used in some places. For the seventh degree, the name si (from Sancte Iohannes, St. John, to whom the hymn is dedicated), though in some regions the seventh is named ti.

The two notation systems most commonly used today are the Helmholtz pitch notation system and the scientific pitch notation system. As shown in the table above, they both include several octaves, each starting from C rather than A. The reason is that the most commonly used scale in Western music is the major scale, and the sequence C–D–E–F–G–A–B–C (the C major scale) is the simplest example of a major scale. Indeed, it is the only major scale that can be obtained using natural notes (the white keys on the piano keyboard) and is typically the first musical scale taught in music schools.

In a newly developed system, primarily in use in the United States, notes of scales become independent of music notation. In this system the natural symbols C–D–E–F–G–A–B refer to the absolute notes, while the names do–re–mi–fa–so–la–ti are relativized and show only the relationship between pitches, where do is the name of the base pitch of the scale (the tonic), re is the name of the second degree, etc. The idea of this so-called "movable do," first suggested by John Curwen in the 19th century, was fully developed and involved into a whole educational system by Zoltán Kodály in the middle of the 20th century, which system is known as the Kodály method or Kodály concept.