View source for Scientific pitch notation

{{Short description|Musical notation system to describe pitch and relative frequency}}
[[File:Scientific pitch notation octaves of C.png|thumb|300px|Ten Cs in scientific pitch notation]]
'''Scientific pitch notation''' ('''SPN'''), also known as '''American standard pitch notation''' ('''ASPN''') and '''international pitch notation''' ('''IPN'''), is a method of specifying musical [[Pitch (music)|pitch]] by combining a musical [[Note (music)|note]] name (with [[accidental (music)|accidental]] if needed) and a number identifying the pitch's [[octave]].<ref>[http://www.flutopedia.com/octave_notation.htm International Pitch Notation]</ref><ref>Robert W. Young. "[https://pubs.aip.org/asa/jasa/article-abstract/11/1/134/551706/Terminology-for-Logarithmic-Frequency-Units Terminology for Logarithmic Frequency Units]". ''[[Journal of the Acoustical Society of America]]''. 1 July 1939; 11 (1): 134–139.</ref>

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 standard)|A440]] pitch [[Equal temperament#Twelve-tone equal temperament|equal temperament]], the system begins at a frequency of 16.35160 [[Hertz|Hz]], which is assigned the value '''C<sub>0</sub>'''. 

The octave 0 of the scientific pitch notation is traditionally called the [[sub-contra octave]], and the tone marked C<sub>0</sub> 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 [[Audio frequency|hear frequencies]] no lower than 20 Hz as pitches.

The octave number increases by 1 upon an ascension from B to C. Thus, ''A<sub>0</sub>'' refers to the first A ''above'' C<sub>0</sub> and [[middle C]] (the [[one-line octave]]'s C or simply {{nowrap|'''c′'''}})  is denoted as ''C<sub>4</sub>'' in SPN.  For example, C<sub>4</sub> is one note above B<sub>3</sub>, and A<sub>5</sub> is one note above G<sub>5</sub>.

The octave number is tied to the alphabetic character used to describe the pitch, with the division between note letters ‘B’ and ‘C’, thus:
:* "B<sub>3</sub>" and all of its possible variants (B{{music|bb}}, B{{music|b}}, B, B{{music|#}}, B{{music|x}}) would properly be designated as being in octave "3".
:* "C<sub>4</sub>" and all of its possible variants (C{{music|bb}}, C{{music|b}}, C, C{{music|#}}, C{{music|x}}) would properly be designated as being in octave "4".
:* In equal temperament "C{{music|b}}{{sub|4}}" is same frequency as "B{{sub|3}}".

==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 convention]]s 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]] C{{su|p={{music|b}}|b=4}} is a lower frequency than B{{sub|3}}; 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<ref>{{cite book |title=Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik |trans-title=On the Sensations of Tone as a Physiological Basis for the Theory of Music |author-link=Hermann von Helmholtz |first=Hermann |last=von Helmholtz |orig-year=1870 |year=1912 |publisher=Whitefish, MT. Kellinger |edition=4 |isbn=978-1-4191-7893-1 |lang=en |translator=[[Alexander J. Ellis|Ellis, A.J.]] |via=[[Internet Archive]] |url=https://archive.org/stream/onsensationston01helmgoog#page/n5/mode/2up}}</ref> 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{{sub|4}}, and C{{sub|4}} 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{{sub|7}} is much easier to quickly distinguish visually from C{{sub|8}}, than is, for example, {{nowrap|c′′′′}} from {{nowrap|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 E{{sub|0}} or higher than E{{su|p={{music|b}}|b=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&nbsp;million years was described by [[NASA]] as corresponding to the B{{music|b}} fifty-seven octaves below middle C (B{{su|p={{music|b}}|b=−53}} or 3.235&nbsp;[[femto-|fHz]]).<ref name="nasa">{{cite press release |publisher=NASA |url=https://science.nasa.gov/science-news/science-at-nasa/2003/09sep_blackholesounds/ |title=Black hole sound waves |quote=Sound waves 57 octaves lower than middle-C are rumbling away from a supermassive black hole in the Perseus cluster.}}</ref>


The notation is sometimes used in the context of [[meantone temperament]], and does not always assume [[equal temperament]] nor the standard concert A<sub>4</sub> of 440&nbsp;[[Hertz|Hz]]; this is particularly the case in connection with earlier music.

The standard proposed to the Acoustical Society of America<ref name="JASA"/> explicitly states a logarithmic scale for frequency, which excludes meantone temperament, and the base frequency it uses gives A<sub>4</sub> a frequency of exactly 440&nbsp;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.{{efn|The conventions of musical pitch notation require the use of sharps and flats on the [[circle of fifths]] closest to the [[key (music)|key]] currently in use, and forbid substitution of notes with the same frequency in equal temperament, such as A{{music|#}} and B{{music|b}}. These rules have the effect of (usually) producing more nearly [[Just intonation|consonant pitches]] when using [[meantone temperament|meantone]] systems, and other non-[[equal temperament]]s. In almost all [[meantone temperament]]s, the so-called [[enharmonic]] notes, such as A{{music|#}} and B{{music|b}}, are a different pitch, with A{{music|#}} at a ''lower'' frequency than the enharmonic B{{music|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.}}

==Table of note frequencies==

[[File:Piano Frequencies.svg|frame|alt=Piano Keyboard|An 88-key piano keyboard, with the octaves numbered and [[C (musical note)#Middle C|middle C]] (cyan) and [[A440 (pitch standard)|A440]] (yellow) highlighted]]

The table below gives notation for pitches based on standard [[piano key frequencies]]: [[A440 (pitch standard)|standard concert pitch]] and [[twelve-tone equal temperament]]. When a piano is tuned to [[just intonation]], C<sub>4</sub> 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.

{|class="wikitable" style="line-height: 125%;"
|+ [[Fundamental frequency]] in [[hertz]] (MIDI note number)
!{{diagonal split header|Note|Octave}} !!−1 !!0 !!1 !!2 !!3 !!4 !!5 !!6 !!7 !!8 !!9 !!10
|-
!C
|style="background:#bbb;"|8.175799 (0) ||style="background:#ddd;"|16.35160 (12) ||32.70320 (24) ||65.40639 (36) ||130.8128 (48) ||style="background:#0ff;"|261.6256 (60) ||523.2511 (72) ||1046.502 (84) ||2093.005 (96) ||4186.009 (108) ||style="background:#bbb;"|8372.018 (120) ||style="background:#bbb;"|16744.04 &nbsp; &nbsp;
|-
!C{{music|#}}/D{{music|b}}
|style="background:#bbb;"|8.661957 (1) ||style="background:#ddd;"|17.32391 (13) ||34.64783 (25) ||69.29566 (37) ||138.5913 (49) ||277.1826 (61) ||554.3653 (73) ||1108.731 (85) ||2217.461 (97) ||style="background:#ddd;"|4434.922 (109) ||style="background:#bbb;"|8869.844 (121) ||style="background:#bbb;"|17739.69 &nbsp; &nbsp; 
|-
!D
|style="background:#bbb;"|9.177024 (2) ||style="background:#ddd;"|18.35405 (14) ||36.70810 (26) ||73.41619 (38) ||146.8324 (50) ||293.6648 (62) ||587.3295 (74) ||1174.659 (86) ||2349.318 (98) ||style="background:#ddd;"|4698.636 (110) ||style="background:#bbb;"|9397.273 (122) ||style="background:#bbb;"|18794.55 &nbsp; &nbsp; 
|-
!E{{music|b}}/D{{music|#}}
|style="background:#bbb;"|9.722718 (3) ||style="background:#ddd;"|19.44544 (15) ||38.89087 (27) ||77.78175 (39) ||155.5635 (51) ||311.1270 (63) ||622.2540 (75) ||1244.508 (87) ||2489.016 (99) ||style="background:#ddd;"|4978.032 (111) ||style="background:#bbb;"|9956.063 (123) ||style="background:#bbb;"|19912.13 &nbsp; &nbsp; 
|-
!E
|style="background:#bbb;"|10.30086 (4) ||style="background:#ddd;"|20.60172 (16) ||41.20344 (28) ||82.40689 (40) ||164.8138 (52) ||329.6276 (64) ||659.2551 (76) ||1318.510 (88) ||2637.020 (100) ||style="background:#ddd;"|5274.041 (112) ||style="background:#bbb;"|10548.08 (124) ||style="background:#bbb;"|21096.16 &nbsp; &nbsp; 
|-
!F
|style="background:#bbb;"|10.91338 (5) ||style="background:#ddd;"|21.82676 (17) ||43.65353 (29) ||87.30706 (41) ||174.6141 (53) ||349.2282 (65) ||698.4565 (77) ||1396.913 (89) ||2793.826 (101) ||style="background:#ddd;"|5587.652 (113) ||style="background:#bbb;"|11175.30 (125) ||style="background:#bbb;"|22350.61 &nbsp; &nbsp; 
|-
!F{{music|#}}/G{{music|b}}
|style="background:#bbb;"|11.56233 (6) ||style="background:#ddd;"|23.12465 (18) ||46.24930 (30) ||92.49861 (42) ||184.9972 (54) ||369.9944 (66) ||739.9888 (78) ||1479.978 (90) ||2959.955 (102) ||style="background:#ddd;"|5919.911 (114) ||style="background:#bbb;"|11839.82 (126) ||style="background:#bbb;"|23679.64 &nbsp; &nbsp; 
|-
!G
|style="background:#bbb;"|12.24986 (7) ||style="background:#ddd;"|24.49971 (19) ||48.99943 (31) ||97.99886 (43) ||195.9977 (55) ||391.9954 (67) ||783.9909 (79) ||1567.982 (91) ||3135.963 (103) ||style="background:#ddd;"|6271.927 (115) ||style="background:#bbb;"|12543.85 (127) ||style="background:#bbb;"|25087.71 &nbsp; &nbsp; 
|-
!A{{music|b}}/G{{music|#}}
|style="background:#bbb;"|12.97827 (8) ||style="background:#ddd;"|25.95654 (20) ||51.91309 (32) ||103.8262 (44) ||207.6523 (56) ||415.3047 (68) ||830.6094 (80) ||1661.219 (92) ||3322.438 (104) ||style="background:#ddd;"|6644.875 (116) ||style="background:#bbb;"|13289.75 &nbsp; &nbsp; ||style="background:#bbb;"|26579.50 &nbsp; &nbsp; 
|-
!A
|style="background:#bbb;"|13.75000 (9) ||27.50000 (21) ||55.00000 (33) ||110.0000 (45) ||220.0000 (57) ||style="background:#ff0;"|440.0000 (69) ||880.0000 (81) ||1760.000 (93) ||3520.000 (105) ||style="background:#ddd;"|7040.000 (117) ||style="background:#bbb;"|14080.00 &nbsp; &nbsp; ||style="background:#bbb;"|28160.00 &nbsp; &nbsp; 
|-
!B{{music|b}}/A{{music|#}}
|style="background:#bbb;"|14.56762 (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) ||style="background:#ddd;"|7458.620 (118) ||style="background:#bbb;"|14917.24 &nbsp; &nbsp; ||style="background:#bbb;"|29834.48 &nbsp; &nbsp; 
|-
!B
|style="background:#bbb;"|15.43385 (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) ||style="background:#ddd;"|7902.133 (119) ||style="background:#bbb;"|15804.27 &nbsp; &nbsp; ||style="background:#bbb;"|31608.53 &nbsp; &nbsp; 
|}

Mathematically, given the number {{mvar|n}} of semitones above middle C, the fundamental frequency in hertz is given by <math>440 \cdot 2^{(n-9)/12}</math> (see [[twelfth root of two]]). Given the MIDI NoteOn number {{mvar|m}}, the frequency of the note is normally <math>440 \cdot 2^{(m-69)/12}</math> 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 [[C (music)|Cs]] are integer powers of 2, with [[middle C]] (C<sub>4</sub>) at 256&nbsp;[[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 A<sub>4</sub> as exactly 440&nbsp;[[Hertz|Hz]], had been informally adopted by the music industry as far back as 1926, and [[A440 (pitch standard)|A440]] became the official international pitch standard in 1955. SPN is routinely used to designate pitch in this system. A<sub>4</sub> may be tuned to other frequencies under different tuning standards, and SPN octave designations still apply (ISO&nbsp;16).<ref>{{cite book | title = ISO 16:1975 Acoustics – Standard tuning frequency (Standard musical pitch) | publisher = International Organization for Standardization | year = 1975 | url = http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=3601 }}</ref>

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. C<sub>0</sub>, which was exactly 16&nbsp;Hz under the scientific pitch standard, is now 16.35160&nbsp;Hz under the current international standard system.<ref name="JASA">{{Cite journal|first=Robert W.|last=Young|author-link=Robert W. Young|title=Terminology for Logarithmic Frequency Units|journal=[[Journal of the Acoustical Society of America]]|volume=11|issue=1|pages=134–000|year=1939|doi=10.1121/1.1916017|bibcode=1939ASAJ...11..134Y}}</ref>

==See also==
*[[Music and mathematics]]
*[[Helmholtz pitch notation]]
*[[MIDI]]
*[[MIDI tuning standard]]
*[[Piano key frequencies]]
*[[Keyboard tablature]]
*[[Letter notation]]

==Footnotes==
{{notelist}}

==References==
{{reflist}}

==External links==
*[http://www.dolmetsch.com/musictheory1.htm#uspitch English Octave-Naming Convention] – Dolmetsch Music Theory Online
*[http://www.deimos.ca/notefreqs 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.
{{Musical notation}}
{{Pitch (music)}}

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[[Category:Musical notation]]
[[Category:Pitch (music)]]