# Scientific pitch notation

Ten Cs in scientific pitch notation

Scientific pitch notation (or SPN, also known as American Standard Pitch Notation (ASPN) and International Pitch Notation (IPN))[1] is a method of specifying musical pitch by combining a musical note name (with accidental if needed) and a number identifying the pitch's octave.

Although scientific pitch notation (SPN) was originally designed as a companion to "scientific pitch" (see below), the two are not synonymous, and should not be confused. Scientific pitch is a pitch standard—a system which defines the specific frequencies of particular pitches (see below). SPN 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 SPN to distinguish octaves does not depend on the pitch standard used.

## Nomenclature

The octave number increases by 1 upon an ascension from B to C (and not from G to A, as one might expect). Thus "A4" refers to the first A above C4 (middle C). In describing musical pitches, enharmonic spellings can give rise to anomalies where C4 is a lower frequency than B3; such paradoxes do not arise in a scientific context.

## Usage

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.

Other traditional octave naming systems—where for example C0 is written as ′′C, or C͵͵ or CCC in Helmholtz pitch notation, or referred to as subcontra C, and where C4 is written as c′ or one-lined C—applies to the written notes that may or may not be transposed. For example, a d′ played on a B trumpet is actually a C4 in scientific pitch notation.

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 notation refers to the C below middle C, whereas "c" in ABC Notation refers to middle C itself. With scientific pitch notation, middle C is always C4, and C4 is never any note but middle C. This notation system also avoids the "fussiness" of having to visually distinguish between, say, four and five primes, as well as typographic issues involved in producing acceptable subscripts or substitutes for them. C7 is much easier to quickly distinguish visually from C8, than is, for example, c′′′′ from c′′′′′, and the use of simple integers makes subscripts unnecessary altogether.

Although pitch notation is intended to describe audible sounds, it can also be used to specify the frequency of non-audible phenomena. For example, when the Chandra X-ray Observatory observed the waves of pressure fronts propagating away from a black hole, the one oscillation every 10 million years was described by NASA as corresponding to the B fifty-seven octaves below middle C (or B−53).[2]

## Similar systems

Notation that appears to be scientific pitch notation may actually be based on an alternative octave numbering. While they are still note-octave systems, if they are called "scientific pitch notation", this is certainly an error. For example, MIDI software and hardware often uses C5 or C3 to represent middle C (note 60).[3]

This creates a linear pitch space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and A440 is assigned the number 69. Distance in this space corresponds to musical distance as measured in psychological experiments and understood by musicians. (An equal-tempered semitone is subdivided into 100 cents.) The system is flexible enough to include microtones not found on standard piano keyboards. For example, the pitch halfway between C (60) and C (61) can be labeled 60.5.

### Meantone temperament

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[4] 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, in other words, 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.

Frequency in hertz (semitones above or below middle C)
Octave
Note
0 1 2 3 4 5 6 7 8 9 10
C 16.352 (−48) 32.703 (−36) 65.406 (−24) 130.81 (−12) 261.63 (0) 523.25 (+12) 1046.5 (+24) 2093.0 (+36) 4186.0 (+48) 8372.0 (+60) 16744.0 (+72)
C/D 17.324 (−47) 34.648 (−35) 69.296 (−23) 138.59 (−11) 277.18 (+1) 554.37 (+13) 1108.7 (+25) 2217.5 (+37) 4434.9 (+49) 8869.8 (+61) 17739.7 (+73)
D 18.354 (−46) 36.708 (−34) 73.416 (−22) 146.83 (−10) 293.66 (+2) 587.33 (+14) 1174.7 (+26) 2349.3 (+38) 4698.6 (+50) 9397.3 (+62) 18794.5 (+74)
E/D 19.445 (−45) 38.891 (−33) 77.782 (−21) 155.56 (−9) 311.13 (+3) 622.25 (+15) 1244.5 (+27) 2489.0 (+39) 4978.0 (+51) 9956.1 (+63) 19912.1 (+75)
E 20.602 (−44) 41.203 (−32) 82.407 (−20) 164.81 (−8) 329.63 (+4) 659.26 (+16) 1318.5 (+28) 2637.0 (+40) 5274.0 (+52) 10548.1 (+64) 21096.2 (+76)
F 21.827 (−43) 43.654 (−31) 87.307 (−19) 174.61 (−7) 349.23 (+5) 698.46 (+17) 1396.9 (+29) 2793.8 (+41) 5587.7 (+53) 11175.3 (+65) 22350.6 (+77)
F/G 23.125 (−42) 46.249 (−30) 92.499 (−18) 185.00 (−6) 369.99 (+6) 739.99 (+18) 1480.0 (+30) 2960.0 (+42) 5919.9 (+54) 11839.8 (+66) 23679.6 (+78)
G 24.500 (−41) 48.999 (−29) 97.999 (−17) 196.00 (−5) 392.00 (+7) 783.99 (+19) 1568.0 (+31) 3136.0 (+43) 6271.9 (+55) 12543.9 (+67) 25087.7 (+79)
A/G 25.957 (−40) 51.913 (−28) 103.83 (−16) 207.65 (−4) 415.30 (+8) 830.61 (+20) 1661.2 (+32) 3322.4 (+44) 6644.9 (+56) 13289.8 (+68) 26579.5 (+80)
A 27.500 (−39) 55.000 (−27) 110.00 (−15) 220.00 (−3) 440.00 (+9) 880.00 (+21) 1760.0 (+33) 3520.0 (+45) 7040.0 (+57) 14080.0 (+69) 28160.0 (+81)
B/A 29.135 (−38) 58.270 (−26) 116.54 (−14) 233.08 (−2) 466.16 (+10) 932.33 (+22) 1864.7 (+34) 3729.3 (+46) 7458.6 (+58) 14917.2 (+70) 29834.5 (+82)
B 30.868 (−37) 61.735 (−25) 123.47 (−13) 246.94 (−1) 493.88 (+11) 987.77 (+23) 1975.5 (+35) 3951.1 (+47) 7902.1 (+59) 15804.3 (+71) 31608.5 (+83)

Mathematically, given the number ${\displaystyle n}$ of semitones above middle C, the frequency in hertz is given by ${\displaystyle 440\cdot 2^{\frac {n-9}{12}}}$ (see twelfth root of two).

## Scientific pitch versus scientific pitch notation

Scientific pitch (q.v.) 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 (C4) 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, and A4 may be tuned to other frequencies under different tuning standards as well, and SPN octave designations still apply (ISO 16[5]).

With changes in concert pitch and the widespread adoption of A 440 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 16.352 Hz under the current international standard system.[4]

In very recent times scientific pitch (Verdi pitch) has become associated with calls to reestablish a lower musical pitch standard.