Scientific pitch notation

Scientific pitch notation, also known as American standard pitch notation and international pitch notation, is a method of specifying musical pitch by combining a musical note name and a number identifying the pitch's octave.
Although scientific pitch notation was originally designed as a companion to scientific pitch, the two are not synonymous. Scientific pitch is a pitch standard—a system that defines the specific frequencies of particular pitches. Scientific pitch notation concerns only how pitch names are notated, that is, how they are designated in printed and written text, and does not inherently specify actual frequencies. Thus, the use of scientific pitch notation to distinguish octaves does not depend on the pitch standard used.

Nomenclature

The notation makes use of the traditional tone names which are followed by numbers showing which octave they are part of.
For standard A440 pitch equal temperament, the system begins at a frequency of 16.35160 Hz, which is assigned the value C₀.
The octave 0 of the scientific pitch notation is traditionally called the sub-contra octave, and the tone marked C₀ in SPN is written as ,,C or C,, or CCC in traditional systems, such as Helmholtz notation. Octave 0 of SPN marks the low end of what humans can actually perceive, with the average person being able to hear frequencies no lower than 20 Hz as pitches.
The octave number increases by 1 upon an ascension from B to C. Thus, A₀ refers to the first A above C₀ and middle C is denoted as C₄ in SPN. For example, C₄ is one note above B₃, and A₅ is one note above G₅.
The octave number is tied to the alphabetic character used to describe the pitch, with the division between note letters 'B' and 'C', thus:

Use

Scientific pitch notation is often used to specify the range of an instrument. It provides an unambiguous means of identifying a note in terms of textual notation rather than frequency, while at the same time avoiding the transposition conventions that are used in writing the music for instruments such as the clarinet and guitar. It is also easily translated into staff notation, as needed. In describing musical pitches, nominally enharmonic spellings can give rise to anomalies where, for example in Pythagorean intonation C₄ is a lower frequency than B; but such paradoxes usually do not arise in a scientific context.
Scientific pitch notation avoids possible confusion between various derivatives of Helmholtz notation which use similar symbols to refer to different notes. For example, "C" in Helmholtz's original notation refers to the C two octaves below middle C, whereas "C" in ABC Notation refers to middle C itself. With scientific pitch notation, middle C is always C, and C is never any note but middle C. This notation system also avoids the "fussiness" of having to visually distinguish between four and five primes, as well as the typographic issues involved in producing acceptable subscripts or substitutes for them. C is much easier to quickly distinguish visually from C, than is, for example, from, and the use of simple integers makes subscripts unnecessary altogether.
Although pitch notation is intended to describe sounds audibly perceptible as pitches, it can also be used to specify the frequency of non-pitch phenomena. Notes below E or higher than E₁₀ are outside most humans' hearing range, although notes slightly outside the hearing range on the low end may still be indirectly perceptible as pitches due to their overtones falling within the hearing range. For an example of truly inaudible frequencies, when the Chandra X-ray Observatory observed the waves of pressure fronts propagating away from a black hole, their one oscillation every 10 million years was described by NASA as corresponding to the B fifty-seven octaves below middle C.
The notation is sometimes used in the context of meantone temperament, and does not always assume equal temperament nor the standard concert A₄ of 440 Hz; this is particularly the case in connection with earlier music.
The standard proposed to the Acoustical Society of America explicitly states a logarithmic scale for frequency, which excludes meantone temperament, and the base frequency it uses gives A₄ a frequency of exactly 440 Hz. However, when dealing with earlier music that did not use equal temperament, it is understandably easier to simply refer to notes by their closest modern equivalent, as opposed to specifying the difference using cents every time.

Table of note frequencies

The table below gives notation for pitches based on standard piano key frequencies: standard concert pitch and twelve-tone equal temperament. When a piano is tuned to just intonation, C₄ refers to the same key on the keyboard, but a slightly different frequency. Notes not produced by any piano are highlighted in medium gray, and those produced only by an extended 108-key piano, light gray.

	−1	0	1	2	3	4	5	6	7	8	9	10
C	8.175799	16.35160	32.70320	65.40639	130.8128	261.6256	523.2511	1046.502	2093.005	4186.009	8372.018	16744.04
C/D	8.661957	17.32391	34.64783	69.29566	138.5913	277.1826	554.3653	1108.731	2217.461	4434.922	8869.844	17739.69
D	9.177024	18.35405	36.70810	73.41619	146.8324	293.6648	587.3295	1174.659	2349.318	4698.636	9397.273	18794.55
E/D	9.722718	19.44544	38.89087	77.78175	155.5635	311.1270	622.2540	1244.508	2489.016	4978.032	9956.063	19912.13
E	10.30086	20.60172	41.20344	82.40689	164.8138	329.6276	659.2551	1318.510	2637.020	5274.041	10548.08	21096.16
F	10.91338	21.82676	43.65353	87.30706	174.6141	349.2282	698.4565	1396.913	2793.826	5587.652	11175.30	22350.61
F/G	11.56233	23.12465	46.24930	92.49861	184.9972	369.9944	739.9888	1479.978	2959.955	5919.911	11839.82	23679.64
G	12.24986	24.49971	48.99943	97.99886	195.9977	391.9954	783.9909	1567.982	3135.963	6271.927	12543.85	25087.71
A/G	12.97827	25.95654	51.91309	103.8262	207.6523	415.3047	830.6094	1661.219	3322.438	6644.875	13289.75	26579.50
A	13.75000	27.50000	55.00000	110.0000	220.0000	440.0000	880.0000	1760.000	3520.000	7040.000	14080.00	28160.00
B/A	14.56762	29.13524	58.27047	116.5409	233.0819	466.1638	932.3275	1864.655	3729.310	7458.620	14917.24	29834.48
B	15.43385	30.86771	61.73541	123.4708	246.9417	493.8833	987.7666	1975.533	3951.066	7902.133	15804.27	31608.53

Mathematically, given the number of semitones above middle C, the fundamental frequency in hertz is given by . Given the MIDI NoteOn number, the frequency of the note is normally Hz, using standard tuning.

Scientific pitch versus scientific pitch ''notation''

Scientific pitch is an absolute pitch standard, first proposed in 1713 by French physicist Joseph Sauveur. It was defined so that all Cs are integer powers of 2, with middle C at 256 hertz. As already noted, it is not dependent upon, nor a part of scientific pitch notation described here. To avoid the confusion in names, scientific pitch is sometimes also called "Verdi tuning" or "philosophical pitch".
The current international pitch standard, using A₄ as exactly 440 Hz, had been informally adopted by the music industry as far back as 1926, and A440 became the official international pitch standard in 1955. SPN is routinely used to designate pitch in this system. A₄ may be tuned to other frequencies under different tuning standards, and SPN octave designations still apply.
With changes in concert pitch and the widespread adoption of A440 as a musical standard, new scientific frequency tables were published by the Acoustical Society of America in 1939, and adopted by the International Organization for Standardization in 1955. C₀, which was exactly 16 Hz under the scientific pitch standard, is now 16.35160 Hz under the current international standard system.