# Scientific pitch notation

For the tuning system using C256, see scientific pitch.
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 one of several methods that name the notes of the standard Western chromatic scale by combining a letter name, accidentals, and a number identifying the pitch's octave. The definition of scientific pitch notation in this article is that proposed to the Acoustical Society of America in 1939,[2] where C0 is about 16.352 Hz.

Scientific pitch notation is a logarithmic frequency scale.

Scientific pitch notation is an example of a note-octave notation (see below).

## Octave number

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). As another example, in ascending the white keys on a keyboard, C4 immediately follows B3, as in the following sequence: "C3 D3 E3 F3 G3 A3 B3 C4 D4"

Scientific pitch notation was originally designed as a companion to scientific pitch, in which C4 was defined as exactly 256 Hz. A different standard pitch system, 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 (ISO 16.[3]) in 1955. However, the use of scientific pitch notation to distinguish octaves does not depend on the pitch standard used (for example, A4 may be tuned to other frequencies under different tuning standards, and SPN octave designations still apply).

## 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).[4]

## 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).[5]

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. One problem with MIDI note notation is that, unlike Scientific Pitch Notation, it has not been universally implemented. Some manufacturers, for example have assigned MIddle C to MIDI note #48, whereas other assign it to MIDI note #60.[6]

### Meantone temperament

The notation is sometimes used in the context of meantone temperament, and does not always assume equal temperament nor the standard concert A of 440 Hz; this is particularly the case in connection with earlier music.

The standard proposed to the Acoustical Society Of America[2] explicitly states a logarithmic scale for frequency, which excludes meantone temperament, and the base frequency it uses gives A4 a frequency of almost exactly 440 Hz (it comes out as approximately 439.96 Hz, which is a difference of 0.01 cents). 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.

## C-flat and B-sharp issues

There is some possible confusion regarding the assignment of an octave to C or B. The convention is that the letter name is first combined with the Arabic numeral to determine a specific pitch, which is then altered by applying accidentals. For example, the symbol C4 means "the pitch one chromatic step below the pitch C4" and not "the pitch-class C in octave 4", so C4 is the same pitch as B3, not B4. Similarly, at the other end of the fourth octave, B4 is the same pitch as C5, not C4.

The matter may be clarified by viewing "" and "" as denoting lowering or raising by a variable amount. In equal temperament, that amount is exactly 21/12 ≈ 1.0594631, or 100 cents, whereas in quarter-comma meantone it is precisely 57/4/16 ≈ 1.0449067, or 76.049 cents, taking other values for other meantone tunings. Hence "C4" is the same as "C4", which in equal temperament is 100 cents below middle C, and equal to B3. In quarter-comma meantone, C4 is 76.049 cents below C4 and 41.059 cents (ratio 128/125 = 1.024) sharper than B3, which is 117.108 cents (ratio 8/55/4 = 57/4/15.625 ≈ 1.0699845) below C4.

## Table of note frequencies

The table below gives notation for pitches based on standard concert pitch.

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)

## Scientific pitch versus scientific pitch notation

A musical initiative called scientific pitch is an absolute pitch standard that sets middle C (or C4) to 256 Hz. 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".

## References

1. ^ International Pitch Notation
2. ^ a b Young, R. W. (1939). "Terminology for Logarithmic Frequency Units". The Journal of the Acoustical Society of America 11 (1): 134–000. Bibcode:1939ASAJ...11..134Y. doi:10.1121/1.1916017. edit
3. ^ ISO 16:1975 Acoustics -- Standard tuning frequency (Standard musical pitch). International Organization for Standardization. 1975.
4. ^ Black Hole Sound Waves “Sound waves 57 octaves lower than middle-C are rumbling away from a supermassive black hole in the Perseus cluster”
5. ^ Robert Guérin, MIDI Power!, ISBN 1-929685-66-1
6. ^ Octave Notation