# Scientific pitch notation

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Scientific pitch notation (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.

## Contents

Although scientific pitch notation was originally designed as a companion to scientific pitch (see below), the two are not synonymous. Scientific pitch is a pitch standard—a system that defines the specific frequencies of particular pitches (see below). 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 (A to G) 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 (the one-line octave's C or simply 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:

• "B3" and all of its possible variants (B , B, B, B, B ) would properly be designated as being in octave "3".
• "C4" and all of its possible variants (C , C, C, C, C ) would properly be designated as being in octave "4".
• In equal temperament "C4" is same frequency as "B3".

## 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 meantone temperaments C
4
is a lower frequency than B3; 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 [2] 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 four and five primes, as well as the 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 (e.g. C7 and C8) 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 E0 or higher than E
10
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 (B
−53
or 3.235  fHz). [3]

## Similar systems

There are pitch-octave notation conventions that appear similar to scientific pitch notation but are based on an alternative octave convention that differs from scientific pitch notation, usually by one octave. For example, middle C ("C4" in ISPN) appears in some MIDI software as "C5" (MIDI note 60). [4] This convention is probably related to a similar convention in sample-based trackers, where C5 is the basic pitch at which a sample plays (8287.12 Hz in MOD), forcing the musician to treat samples at any other pitch as transposing instruments when using them in songs. Alternately, both Yamaha and the software MaxMSP define middle C as C3. Apple's GarageBand also defines middle C (261.6256 Hz) as C3.

Using scientific pitch notation consistently, the MIDI NoteOn message assigns MIDI note 0 to C−1 (five octaves below C4 or Middle C; lowest note on the two largest organs of the world; about one octave below the human hearing threshold: its overtones, however, are audible), MIDI note 21 to A0 (the bottom key of an 88-key piano), MIDI note 60 to C4 (Middle C), MIDI note 69 to A4 (A440), MIDI note 108 to C8 (the top key of an 88-key piano), and MIDI note 127 to G9 (beyond the piano; one octave above the highest note on some keyboard glockenspiels; some notes above the highest-pitched organ pipes).

This creates a linear pitch space in which an octave spans 12 semitones, where each semitone is the distance between adjacent keys of the piano keyboard. Distance in this space corresponds to musical pitch distance in an equal-tempered scale, 2 semitones being a whole step, and 1 semitone being a half step. An equal-tempered semitone can also be subdivided further into 100 cents. Each cent is 1100 semitone or 11200 octave. This measure of pitch allows the expression of microtones not found on standard piano keyboards.

### 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 [5] 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. [lower-alpha 1]

## 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. Keys which do not appear on any piano (medium gray) or only on an extended 108-key piano (light gray) are highlighted.

Fundamental frequency in hertz (MIDI note number)
Octave
Note
−1012345678910
C8.175799 (0)16.35160 (12)32.70320 (24)65.40639 (36)130.8128 (48)261.6256 (60)523.2511 (72)1046.502 (84)2093.005 (96)4186.009 (108)8372.018 (120)16744.04
C/D8.661957 (1)17.32391 (13)34.64783 (25)69.29566 (37)138.5913 (49)277.1826 (61)554.3653 (73)1108.731 (85)2217.461 (97)4434.922 (109)8869.844 (121)17739.69
D9.177024 (2)18.35405 (14)36.70810 (26)73.41619 (38)146.8324 (50)293.6648 (62)587.3295 (74)1174.659 (86)2349.318 (98)4698.636 (110)9397.273 (122)18794.55
E/D9.722718 (3)19.44544 (15)38.89087 (27)77.78175 (39)155.5635 (51)311.1270 (63)622.2540 (75)1244.508 (87)2489.016 (99)4978.032 (111)9956.063 (123)19912.13
E10.30086 (4)20.60172 (16)41.20344 (28)82.40689 (40)164.8138 (52)329.6276 (64)659.2551 (76)1318.510 (88)2637.020 (100)5274.041 (112)10548.08 (124)21096.16
F10.91338 (5)21.82676 (17)43.65353 (29)87.30706 (41)174.6141 (53)349.2282 (65)698.4565 (77)1396.913 (89)2793.826 (101)5587.652 (113)11175.30 (125)22350.61
F/G11.56233 (6)23.12465 (18)46.24930 (30)92.49861 (42)184.9972 (54)369.9944 (66)739.9888 (78)1479.978 (90)2959.955 (102)5919.911 (114)11839.82 (126)23679.64
G12.24986 (7)24.49971 (19)48.99943 (31)97.99886 (43)195.9977 (55)391.9954 (67)783.9909 (79)1567.982 (91)3135.963 (103)6271.927 (115)12543.85 (127)25087.71
A/G12.97827 (8)25.95654 (20)51.91309 (32)103.8262 (44)207.6523 (56)415.3047 (68)830.6094 (80)1661.219 (92)3322.438 (104)6644.875 (116)13289.75   26579.50
A13.75000 (9)27.50000 (21)55.00000 (33)110.0000 (45)220.0000 (57)440.0000 (69)880.0000 (81)1760.000 (93)3520.000 (105)7040.000 (117)14080.00   28160.00
B/A14.56762 (10)29.13524 (22)58.27047 (34)116.5409 (46)233.0819 (58)466.1638 (70)932.3275 (82)1864.655 (94)3729.310 (106)7458.620 (118)14917.24   29834.48
B15.43385 (11)30.86771 (23)61.73541 (35)123.4708 (47)246.9417 (59)493.8833 (71)987.7666 (83)1975.533 (95)3951.066 (107)7902.133 (119)15804.27   31608.53

Mathematically, given the number n of semitones above middle C, the fundamental frequency in hertz is given by ${\displaystyle 440\cdot 2^{(n-9)/12}}$ (see twelfth root of two). Given the MIDI NoteOn number m, the frequency of the note is normally ${\displaystyle 440\cdot 2^{(m-69)/12}}$ Hz, using standard tuning.

## 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. A4 may be tuned to other frequencies under different tuning standards, and SPN octave designations still apply (ISO 16). [6]

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 16.352 Hz under the current international standard system. [5]

## Footnotes

1. The conventions of musical pitch notation require the use of sharps and flats on the circle of fifths closest to the key currently in use, and forbid substitution of notes with the same frequency in equal temperament, such as A and B. These rules have the effect of (usually) producing more nearly consonant pitches when using meantone systems, and other non-equal temperaments. In almost all meantone temperaments, the so-called enharmonic notes, such as A and B, are a different pitch, with A at a lower frequency than the enharmonic B. With the single exception of equal temperament (which fits in among meantone systems as a special case) enharmonic notes always have slightly different frequencies.

## Related Research Articles

An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size as pitch is perceived roughly as the logarithm of frequency.

In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G3 and C4 are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C4 and G3 is therefore 4:3, a just fourth.

In music, a note is the representation of a musical sound.

In music, an octave or perfect octave is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music," the use of which is "common in most musical systems." The interval between the first and second harmonics of the harmonic series is an octave.

Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈702 cents wide.

In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord.

C or Do is the first note and semitone of the C major scale, the third note of the A minor scale, and the fourth note of the Guidonian hand, commonly pitched around 261.63 Hz. The actual frequency has depended on historical pitch standards, and for transposing instruments a distinction is made between written and sounding or concert pitch. It has enharmonic equivalents of B and D.

Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. The enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. The term is derived from Latin enharmonicus, from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν (en) and ἁρμονία (harmonía).

In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set

A440 (also known as Stuttgart pitch) is the musical pitch corresponding to an audio frequency of 440 Hz, which serves as a tuning standard for the musical note of A above middle C, or A4 in scientific pitch notation. It is standardized by the International Organization for Standardization as ISO 16. While other frequencies have been (and occasionally still are) used to tune the first A above middle C, A440 is now commonly used as a reference frequency to calibrate acoustic equipment and to tune pianos, violins, and other musical instruments.

Piano tuning is the act of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.

The twelfth root of two or is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio of a semitone in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals as consisting of different numbers of a single interval, the equal tempered semitone. A semitone itself is divided into 100 cents.

This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (referred to as A440). Since every octave is made of twelve steps and since a jump of one octave doubles the frequency (for example, the fifth A is 440 Hz and the sixth A is 880 Hz), each successive pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the twelfth root of two (approximately 1.059463). For example, to get the frequency one semitone up from A4 (A4), multiply 440 by the twelfth root of two. To go from A4 to B4 (up one whole tone, or two semitones), multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). To go from A4 to C5 (which is a minor third), multiply 440 three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). For other tuning schemes refer to musical tuning.

In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F tuned using the D-based Pythagorean tuning system, and another F tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B and A are both approximated by the same interval although they are a septimal kleisma apart.

Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, 112 the width of an octave, is called a semitone or half step.

MIDI Tuning Standard (MTS) is a specification of precise musical pitch agreed to by the MIDI Manufacturers Association in the MIDI protocol. MTS allows for both a bulk tuning dump message, giving a tuning for each of 128 notes, and a tuning message for individual notes as they are played.

Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory.

C (C-sharp) is a musical note lying a chromatic semitone above C and a diatonic semitone below D; it is the second semitone of the solfège. C-sharp is thus enharmonic to D. It is the second semitone in the French solfège and is known there as do dièse. In some European notations, it is known as Cis. In equal temperament it is also enharmonic with B.

A is the ninth semitone of the solfège.

## References

1. von Helmholtz, Hermann (1912) [1870]. Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik [On the Sensations of Tone as a Physiological Basis for the Theory of Music]. Translated by Ellis, A.J. (4 ed.) via Internet Archive.
2. "Black hole sound waves" (Press release). NASA. Sound waves 57 octaves lower than middle-C are rumbling away from a supermassive black hole in the Perseus cluster.
3. Guérin, Robert (2002). MIDI Power!. ISBN   1-929685-66-1.
4. Young, Robert W. (1939). "Terminology for Logarithmic Frequency Units". Journal of the Acoustical Society of America . 11 (1): 134–000. Bibcode:1939ASAJ...11..134Y. doi:10.1121/1.1916017.
5. ISO 16:1975 Acoustics – Standard tuning frequency (Standard musical pitch). International Organization for Standardization. 1975.
• English Octave-Naming Convention – Dolmetsch Music Theory Online
• Notefreqs – A complete table of note frequencies and ratios for midi, piano, guitar, bass, and violin. Includes fret measurements (in cm and inches) for building instruments.