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<title>Musical tuning</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Musical tuning</span></span>

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<p>In <a href="Music" title="Music">music</a>, there are two common meanings for <b>tuning</b>:

</p>

<ul><li><a href="#Tuning_practice">Tuning practice</a>, the act of tuning an instrument or voice.</li>

<li><a href="#Tuning_systems">Tuning systems</a>, the various systems of <a href="Pitch_(music)" title="Pitch (music)">pitches</a> used to tune an instrument, and their theoretical bases.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Tuning_practice">Tuning practice</h2></div>

<p><b>Tuning</b> is the process of adjusting the pitch of one or many tones from musical instruments to establish typical intervals between these tones. Tuning is usually based on a fixed reference, such as <a href="A440_(pitch_standard)" title="A440 (pitch standard)">A = 440 Hz</a>. The term "<i>out of tune</i>" refers to a pitch/tone that is either too high (<a href="Sharp_(music)" title="Sharp (music)">sharp</a>) or too low (<a href="Flat_(music)" title="Flat (music)">flat</a>) in relation to a given reference pitch. While an instrument might be in tune relative to its own range of notes, it may not be considered 'in tune' if it does not match the chosen reference pitch. Some instruments become 'out of tune' with temperature, humidity, damage, or simply time, and must be readjusted or repaired.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Different methods of sound production require different methods of adjustment:

</p>

<ul><li>Tuning to a pitch with one's voice is called <i>matching pitch</i> and is the most basic skill learned in <a href="Ear_training" title="Ear training">ear training</a>.</li>

<li>Turning <a href="Tuning_peg" class="mw-redirect" title="Tuning peg">pegs</a> to increase or decrease the <a href="Tension_(mechanics)" class="mw-redirect" title="Tension (mechanics)">tension</a> on strings so as to control the pitch. Instruments such as the harp, piano, and harpsichord require a wrench to turn the tuning pegs, while others such as the <a href="Violin" title="Violin">violin</a> can be tuned manually.</li>

<li>Modifying the length or width of the tube of a <a href="Wind_instrument" title="Wind instrument">wind instrument</a>, <a href="Brass_instrument" title="Brass instrument">brass instrument</a>, <a href="Pan_pipes" class="mw-redirect" title="Pan pipes">pipe</a>, <a href="Bell_(instrument)" class="mw-redirect" title="Bell (instrument)">bell</a>, or similar instrument to adjust the pitch. In woodwind instruments, this is usually done by adjusting the instrument's <a href="Mouthpiece_(woodwind)" title="Mouthpiece (woodwind)">mouthpiece</a> or neck to change the pitch. In brass instruments, this is usually done by moving a tuning slide.</li></ul>

<p>The sounds of some instruments, notably <a href="Unpitched_percussion_instrument" title="Unpitched percussion instrument">unpitched percussion instrument</a> such as <a href="Cymbal" title="Cymbal">cymbals</a>, are of <a href="Indeterminate_pitch" class="mw-redirect" title="Indeterminate pitch">indeterminate pitch</a>, and have irregular overtones not conforming to the <a href="Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a>. See <a href="#Tuning_of_unpitched_percussion_instruments">§ Tuning of unpitched percussion instruments</a>.

</p><p>Tuning may be done aurally by sounding two pitches and adjusting one of them to match or relate to the other. A <a href="Tuning_fork" title="Tuning fork">tuning fork</a> or electronic tuning device may be used as a reference pitch, though in ensemble rehearsals often a <a href="Piano" title="Piano">piano</a> is used (as its pitch cannot be adjusted for each performance). <a href="Orchestra" title="Orchestra">Symphony orchestras</a> and <a href="Concert_band" title="Concert band">concert bands</a> usually tune to an <a href="A_(musical_note)" title="A (musical note)">A</a>440 or a <a href="B%E2%99%AD_(musical_note)" title="B♭ (musical note)">B♭</a>, respectively, provided by the principal <a href="Oboe" title="Oboe">oboist</a> or <a href="Clarinet" title="Clarinet">clarinetist</a>, who tune to the keyboard if part of the performance.<sup id="cite_ref-Rock_2-0" class="reference"><a href="#cite_note-Rock-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

When only strings are used, then the principal string (violinist) typically has sounded the tuning pitch, but some orchestras have used an electronic tone machine for tuning.<sup id="cite_ref-Rock_2-1" class="reference"><a href="#cite_note-Rock-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Tuning can also be done through a prior recording; this method uses simultaneous audio.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Interference <a href="Beat_(acoustics)" title="Beat (acoustics)">beats</a> are used to objectively measure the accuracy of tuning.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> As the two pitches approach a harmonic relationship, the frequency of beating decreases. When tuning a unison or octave it is desired to reduce the beating frequency until it cannot be detected. For other intervals, this is dependent on the <a href="#Tuning_systems">tuning system</a> being used.

</p><p><a href="Harmonic" title="Harmonic">Harmonics</a> may be used to facilitate tuning of strings that are not themselves tuned to the unison. For example, lightly touching the highest string of a cello at the middle (at a <a href="Node_(physics)" title="Node (physics)">node</a>) while bowing produces the same pitch as doing the same a third of the way down its second-highest string. The resulting unison is more easily and quickly judged than the quality of the perfect fifth between the fundamentals of the two strings.

</p>

<div class="mw-heading mw-heading3"><h3 id="Open_strings">Open strings</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Stringed_instrument_tunings" title="Stringed instrument tunings">Stringed instrument tunings</a></div>

<p>In <a href="Music" title="Music">music</a>, the term <b>open string</b> refers to the fundamental note of the unstopped, full string.

</p><p>The strings of a <a href="Guitar" title="Guitar">guitar</a> are normally tuned to <a href="Perfect_fourth" title="Perfect fourth">fourths</a> (excepting the G and B strings in standard tuning, which are tuned to a third), as are the strings of the <a href="Bass_guitar" title="Bass guitar">bass guitar</a> and <a href="Double_bass" title="Double bass">double bass</a>. <a href="Violin" title="Violin">Violin</a>, <a href="Viola" title="Viola">viola</a>, and <a href="Cello" title="Cello">cello</a> strings are tuned to <a href="Perfect_fifth" title="Perfect fifth">fifths</a>. However, non-standard tunings (called <a href="Scordatura" title="Scordatura">scordatura</a>) exist to change the sound of the instrument or create other playing options.

</p><p>To tune an instrument, often only one reference pitch is given. This reference is used to tune one string, to which the other strings are tuned in the desired intervals. On a guitar, often the lowest string is tuned to an E. From this, each successive string can be tuned by fingering the fifth fret of an already tuned string and comparing it with the next higher string played open. This works with the exception of the G string, which must be stopped at the fourth fret to sound B against the open B string above. Alternatively, each string can be tuned to its own reference tone.

</p>

<p>Note that while the guitar and other modern stringed instruments with fixed frets are tuned in <a href="Equal_temperament" title="Equal temperament">equal temperament</a>, string instruments without frets, such as those of the violin family, are not. The violin, viola, and cello are tuned to <a href="Beat_(acoustics)" title="Beat (acoustics)">beatless</a> just perfect fifths and ensembles such as string quartets and orchestras tend to play in fifths based <a href="Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a> or to compensate and play in equal temperament, such as when playing with other instruments such as the piano. For example, the cello, which is tuned down from <a href="A440_(pitch_standard)" title="A440 (pitch standard)">A220</a>, has three more strings (four total) and the just perfect fifth is about two <a href="Cent_(music)" title="Cent (music)">cents</a> off from the equal tempered perfect fifth, making its lowest string, C−, about six cents more flat than the equal tempered C.

</p><p>This table lists open strings on some common string instruments and their standard tunings from low to high unless otherwise noted.

</p>

<th>Instrument

</th>

<th>Tuning

</th></tr>

<tr>

<td><a href="Violin" title="Violin">violin</a>, <a href="Mandolin" title="Mandolin">mandolin</a>, <a href="Banjo#Four-string_banjos" title="Banjo">Irish tenor banjo</a>

</td>

<td>G, D, A, E

</td></tr>

<tr>

<td><a href="Viola" title="Viola">viola</a>, <a href="Cello" title="Cello">cello</a>, <a href="Tenor_banjo" class="mw-redirect" title="Tenor banjo">tenor banjo</a>, <a href="Mandola" title="Mandola">mandola</a>, <a href="Mandocello" title="Mandocello">mandocello</a>, <a href="Tenor_guitar" title="Tenor guitar">tenor guitar</a>

</td>

<td>C, G, D, A

</td></tr>

<tr>

<td><a href="Double_bass" title="Double bass">double bass</a>, <a href="Mando-bass" class="mw-redirect" title="Mando-bass">mando-bass</a>, <a href="Bass_guitar" title="Bass guitar">bass guitar</a>*

</td>

<td>(B*,) E, A, D, G, (C*)

</td></tr>

<tr>

<td><a href="Guitar" title="Guitar">guitar</a>

</td>

<td>E, A, D, G, B, E

</td></tr>

<tr>

<td><a href="Concert_harp" class="mw-redirect" title="Concert harp">concert harp</a>

</td>

<td>C<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, D<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, F<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, A<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span>, B<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> (repeating)

</td></tr>

<tr>

<td><a href="Ukulele" title="Ukulele">ukulele</a>

</td>

<td>G, C, E, A (the G string is higher than the C and E, and two half steps below the A string, known as <a href="Reentrant_tuning" title="Reentrant tuning">reentrant tuning</a>)

</td></tr>

<tr>

<td><a href="5-string_banjo" class="mw-redirect" title="5-string banjo">5-string banjo</a>

</td>

<td>G, D, G, B, D (another <a href="Reentrant_tuning" title="Reentrant tuning">reentrant tuning</a>, with the short 5th string tuned an octave above the 3rd string)

</td></tr>

<tr>

<td><a href="Cavaquinho" title="Cavaquinho">cavaquinho</a>

</td>

<td>D, G, B, D (standard Brazilian tuning)

</td></tr></tbody></table>

<div class="mw-heading mw-heading3"><h3 id="Altered_tunings">Altered tunings</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Scordatura" title="Scordatura">scordatura</a></div>

<p>Violin scordatura was employed in the 17th and 18th centuries by Italian and German composers, namely, <a href="Biagio_Marini" title="Biagio Marini">Biagio Marini</a>, <a href="Antonio_Vivaldi" title="Antonio Vivaldi">Antonio Vivaldi</a>, <a href="Heinrich_Ignaz_Franz_Biber" title="Heinrich Ignaz Franz Biber">Heinrich Ignaz Franz Biber</a> (who in the <i>Rosary Sonatas</i> prescribes a great variety of scordaturas, including crossing the middle strings), <a href="Johann_Pachelbel" title="Johann Pachelbel">Johann Pachelbel</a> and <a href="Johann_Sebastian_Bach" title="Johann Sebastian Bach">Johann Sebastian Bach</a>, whose <i><a href="Cello_Suites_(Bach)" title="Cello Suites (Bach)">Fifth Suite For Unaccompanied Cello</a></i> calls for the lowering of the A string to G. In <a href="Wolfgang_Amadeus_Mozart" title="Wolfgang Amadeus Mozart">Mozart</a>'s <i><a href="Sinfonia_Concertante_for_Violin%2C_Viola_and_Orchestra" class="mw-redirect" title="Sinfonia Concertante for Violin, Viola and Orchestra">Sinfonia Concertante</a></i> in E-flat major (K. 364), all the strings of the solo viola are raised one half-step, ostensibly to give the instrument a brighter tone so the solo violin does not overshadow it.

</p><p>Scordatura for the violin was also used in the 19th and 20th centuries in works by <a href="Niccol%C3%B2_Paganini" title="Niccolò Paganini">Niccolò Paganini</a>, <a href="Robert_Schumann" title="Robert Schumann">Robert Schumann</a>, <a href="Camille_Saint-Sa%C3%ABns" title="Camille Saint-Saëns">Camille Saint-Saëns</a>, <a href="Gustav_Mahler" title="Gustav Mahler">Gustav Mahler</a>, and <a href="B%C3%A9la_Bart%C3%B3k" title="Béla Bartók">Béla Bartók</a>. In Saint-Saëns' "<a href="Danse_Macabre_(Saint-Sa%C3%ABns)" class="mw-redirect" title="Danse Macabre (Saint-Saëns)">Danse Macabre</a>", the high string of the violin is lower half a tone to the E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> so as to have the most accented note of the main theme sound on an open string. In Mahler's <a href="Symphony_No._4_(Mahler)" title="Symphony No. 4 (Mahler)">Symphony No. 4</a>, the solo violin is tuned one whole step high to produce a harsh sound evoking Death as the Fiddler. In Bartók's <i>Contrasts</i>, the violin is tuned G<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span>-D-A-E<span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span> to facilitate the playing of tritones on open strings.

</p><p>American folk violinists of the <a href="Appalachians" class="mw-redirect" title="Appalachians">Appalachians</a> and <a href="Ozarks" title="Ozarks">Ozarks</a> often employ alternate tunings for dance songs and ballads. The most commonly used tuning is A-E-A-E. Likewise banjo players in this tradition use many tunings to play melody in different keys. A common alternative banjo tuning for playing in D is A-D-A-D-E. Many Folk guitar players also used different tunings from standard, such as D-A-D-G-A-D, which is very popular for Irish music.

</p><p>A musical instrument that has had its pitch deliberately lowered during tuning is said to be <i>down-tuned</i> or <i>tuned down</i>. Common examples include the electric guitar and electric bass in contemporary <a href="Heavy_metal_music" title="Heavy metal music">heavy metal music</a>, whereby one or more strings are often tuned lower than <a href="Concert_pitch" title="Concert pitch">concert pitch</a>. This is not to be confused with electronically changing the <a href="Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a>, which is referred to as <a href="Pitch_shifting" title="Pitch shifting">pitch shifting</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Tuning_of_unpitched_percussion_instruments">Tuning of unpitched percussion instruments</h3></div>

<p>Many percussion instruments are <i>tuned</i> by the player, including <a href="Pitched_percussion_instrument" title="Pitched percussion instrument">pitched percussion instruments</a> such as <a href="Timpani" title="Timpani">timpani</a> and <a href="Tabla" title="Tabla">tabla</a>, and <a href="Unpitched_percussion_instrument" title="Unpitched percussion instrument">unpitched percussion instruments</a> such as the <a href="Snare_drum" title="Snare drum">snare drum</a>.

</p><p>Tuning pitched percussion follows the same patterns as tuning any other instrument, but tuning unpitched percussion does not produce a specific <a href="Pitch_(music)" title="Pitch (music)">pitch</a>. For this reason and others, the traditional terms <i><a href="Tuned_percussion" class="mw-redirect" title="Tuned percussion">tuned percussion</a></i> and <i><a href="Untuned_percussion" class="mw-redirect" title="Untuned percussion">untuned percussion</a></i> are avoided in recent <a href="Organology" title="Organology">organology</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tuning_systems">Tuning systems</h2></div>

<p>

A <i>tuning system</i> is the system used to define which <a href="Semitone" title="Semitone">tones</a>, or <a href="Pitch_(music)" title="Pitch (music)">pitches</a>, to use when playing <a href="Music" title="Music">music</a>. In other words, it is the choice of number and spacing of <a href="Frequency" title="Frequency">frequency</a> values used.

</p><p>Due to the <a href="Psychoacoustics" title="Psychoacoustics">psychoacoustic</a> interaction of tones and <a href="Timbre" title="Timbre">timbres</a>, various tone combinations sound more or less "natural" in combination with various timbres. For example, using harmonic timbres:

</p>

<ul><li>A tone caused by a vibration twice the frequency of another (the ratio of 1:2) forms the natural sounding <a href="Octave" title="Octave">octave</a>.</li>

<li>A tone caused by a vibration three times the frequency of another (the ratio of 1:3) forms the natural sounding <a href="Perfect_twelfth" class="mw-redirect" title="Perfect twelfth">perfect twelfth</a>, or <a href="Perfect_fifth" title="Perfect fifth">perfect fifth</a> (ratio of 2:3) when octave-reduced.</li></ul>

<p>More complex musical effects can be created through other relationships.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>The creation of a tuning system is complicated because musicians want to make music with more than just a few differing tones. As the number of tones is increased, conflicts arise in how each tone combines with every other. Finding a successful combination of tunings has been the cause of debate, and has led to the creation of many different tuning systems across the world. Each tuning system has its own characteristics, strengths and weaknesses.

</p>

<div class="mw-heading mw-heading3"><h3 id="Systems_for_the_twelve-note_chromatic_scale">Systems for the twelve-note chromatic scale</h3></div>

<p>It is impossible to tune the twelve-note <a href="Chromatic_scale" title="Chromatic scale">chromatic scale</a> so that all <a href="Interval_(music)" title="Interval (music)">intervals</a> are pure. For instance, three pure major thirds stack up to <span style="font-size: 85%;"><style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num"> 125 </span><span class="sr-only">/</span><span class="den"> 64 </span></span>⁠</span></span>, which at <span class="nowrap"> <span style="white-space:nowrap">1<span style="margin-left:0.25em">159</span></span> cents</span> is nearly a quarter tone away from the octave (1200 cents). So there is no way to have both the octave and the major third in just intonation for all the intervals in the same twelve-tone system. Similar issues arise with the fifth <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>, and the minor third <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 6 </span><span class="sr-only">/</span><span class="den"> 5 </span></span>⁠</span></span>, or any other choice of harmonic-series based pure intervals.

</p><p>Many different compromise methods are used to deal with this, each with its own characteristics, and advantages and disadvantages.

</p><p>The main ones are:

</p>

<dl><dt><a href="Just_intonation" title="Just intonation">Just intonation</a></dt>

<dd>In just intonation, the frequencies of the scale notes are related to one another by simple numeric ratios, a common example of this being <span class="texhtml"> <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 1 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 9 </span><span class="sr-only">/</span><span class="den"> 8 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 5 </span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 4 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 3 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 5 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 15 </span><span class="sr-only">/</span><span class="den"> 8 </span></span>⁠</span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 2 </span><span class="sr-only">/</span><span class="den"> 1 </span></span>⁠</span></span> </span> to define the ratios for the seven notes in a C major scale, plus the return to the tonic an octave up on the 8th (<a href="Octave" title="Octave">"perfect 8th" or octave</a>). In this example, though many intervals are pure, the interval from D to A (<span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 5 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span></span> to <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 9 </span><span class="sr-only">/</span><span class="den"> 8 </span></span>⁠</span></span>) is <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 40 </span><span class="sr-only">/</span><span class="den"> 27 </span></span>⁠</span></span> instead of the expected <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 3 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span></span>. The same issue occurs with most just intonation tunings. This can be dealt with to some extent using alternative pitches for the notes. Even that, however, is only a partial solution, as an example makes clear: If one plays the sequence C G D A E C in just intonation, using the intervals <span style="font-size: 85%;"><span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"> 3 </span><span class="sr-only">/</span><span class="den"> 2 </span></span>⁠</span></span></span>, <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 3 </span><span class="sr-only">/</span><span class="den"> 4 </span></span>⁠</span></span>, and <span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 4 </span><span class="sr-only">/</span><span class="den"> 5 </span></span>⁠</span></span>, then the second C in the sequence is higher than the first by a <a href="Syntonic_comma" title="Syntonic comma">syntonic comma</a> of <span style="font-size: 85%;"><span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"> 81 </span><span class="sr-only">/</span><span class="den"> 80 </span></span>⁠</span></span></span>. This is the infamous "<a href="Comma_pump" title="Comma pump">comma pump</a>". Each time around the comma pump, the pitch continues to spiral upwards. This shows that it is impossible to keep to any small fixed system of pitches if one wants to stack musical intervals this way. So, even with <i>adaptive tuning</i>, the musical context may sometimes require playing musical intervals that are not pure. Instrumentalists with the ability to vary the pitch of their instrument may micro-adjust some of the intervals naturally; there are also systems for adaptive tuning in software (<a href="Microtuner" title="Microtuner">microtuners</a>). Harmonic fragment scales form a rare exception to this issue. In tunings such as <span class="texhtml">1:1, 9:8, 5:4, 3:2, 7:4, 2:1</span>, all the pitches are chosen from the harmonic series (divided by powers of 2 to reduce them to the same octave), so all the intervals are related to each other by simple numeric ratios.</dd>

<dt><a href="Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a></dt>

<dd>A Pythagorean tuning is technically both a type of just intonation and a zero-<a href="Syntonic_comma" title="Syntonic comma">comma</a> meantone tuning, in which the frequency ratios of the notes are all derived from the number ratio 3:2. Using this approach for example, the 12 notes of the Western chromatic scale would be tuned to the following ratios: <span class="texhtml"> 1:1, 256:243, 9:8, 32:27, 81:64, 4:3, 729:512, 3:2, 128:81, 27:16, 16:9, 243:128, 2:1 .</span> Also called "3-limit" because it uses no prime factors other than 2 and 3, this Pythagorean system was of primary importance in Western musical development in the Medieval and Renaissance periods. As with nearly all just intonation systems, it has a <a href="Wolf_interval" title="Wolf interval">wolf interval</a>. In the example given, it is the interval between the <span class="texhtml"> 729:512 </span> and the<span class="texhtml"> 256:243 </span> (F<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-sharp">♯</span></span></sup> to D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>, if one tunes the <span style="font-size: 85%;"><span class="texhtml"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 1 </span></span>⁠</span></span></span> to C). The major and minor thirds are also impure, but at the time when this system was at its zenith, the third was considered a dissonance, so this was of no concern. See also: <a href="Sh%C3%AD-%C3%A8r-l%C7%9C" class="mw-redirect" title="Shí-èr-lǜ">Shí-èr-lǜ</a>.</dd>

<dt><a href="Meantone_temperament" title="Meantone temperament">Meantone temperament</a></dt>

<dd>A system of tuning that averages out pairs of ratios used for the same interval (such as 9:8 and 10:9). The best known form of this temperament is <a href="Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma meantone</a>, which tunes major thirds justly in the ratio of 5:4 and divides them into two whole tones of equal size – this is achieved by flattening the fifths of the Pythagorean system slightly (by a quarter of a <a href="Syntonic_comma" title="Syntonic comma">syntonic comma</a>). However, the fifth may be flattened to a greater or lesser degree than this and the tuning system retains the essential qualities of <a href="Meantone_temperament" title="Meantone temperament">meantone temperament</a>. Historical examples include <a href="19_equal_temperament" title="19 equal temperament"><span class="nowrap"><span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span></span> comma</span></a> and <span class="nowrap"><span style="font-size: 85%;"><span class="sfrac">⁠<span class="tion"><span class="num"> 2 </span><span class="sr-only">/</span><span class="den"> 7 </span></span>⁠</span></span> comma</span> meantone.</dd>

<dt><a href="Well_temperament" title="Well temperament">Well temperament</a></dt>

<dd>Any one of a number of systems where the ratios between intervals are unequal, but approximate to ratios used in just intonation. Unlike meantone temperament, the amount of divergence from just ratios varies according to the exact notes being tuned, so that C–E is probably tuned closer to a 5:4 ratio than, say, D<sup><span class="music-symbol" style="font-family: Arial Unicode MS, Lucida Sans Unicode;"><span class="music-flat">♭</span></span></sup>–F. Because of this, well temperaments have no <a href="Wolf_interval" title="Wolf interval">wolf intervals</a>.</dd>

<dt><a href="Equal_temperament" title="Equal temperament">Equal temperament</a></dt>

<dd>The standard twelve-tone equal temperament is a special case of meantone temperament (extended eleventh-comma), in which the twelve notes are separated by <a href="Logarithm" title="Logarithm">logarithmically</a> equal distances (100 cents): <a class="external text external" href="https://upload.wikimedia.org/wikipedia/commons/b/b3/Et_scale.ogg">A harmonized C major scale in equal temperament</a> (.ogg format, 96.9 KB). This is the most common tuning system used in Western music, and is the standard system used as a basis for <a href="Piano_tuning" title="Piano tuning">tuning a piano</a>. Since this scale divides an octave into twelve equal-ratio steps and an octave has a frequency ratio of two, the frequency ratio between adjacent notes is then the <a href="Twelfth_root_of_two" title="Twelfth root of two">twelfth root of two</a>, <span class="nowrap"><span class="texhtml"> 2<sup> <span style="font-size: 85%;">1/12</span> </sup> ≋ 1.05946309</span> ... .</span> However, the octave can be divided into other than 12 equal divisions, some of which may be more harmonically pleasing, as far as thirds, sixths, and <a href="Harmonic_seventh" title="Harmonic seventh">harmonic sevenths</a> (via augmented sixths) are concerned, such as <a href="19_equal_temperament" title="19 equal temperament">19 equal temperament</a> (extended <span style="font-size: 85%;"><span class="nowrap"><span class="sfrac">⁠<span class="tion"><span class="num"> 1 </span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span></span> comma</span> meantone), <a href="31_equal_temperament" title="31 equal temperament">31 equal temperament</a> (extended quarter-comma meantone) and <a href="53_equal_temperament" title="53 equal temperament">53 equal temperament</a> (extended Pythagorean tuning).</dd></dl>

<p>Tuning systems that are not produced with exclusively just intervals are usually referred to as <i><a href="Musical_temperament" title="Musical temperament">temperaments</a></i>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Other_scale_systems">Other scale systems</h3></div>

<ul><li>Natural overtone scale, a scale derived from the harmonic series</li>

<li><a href="Slendro" title="Slendro">Slendro</a>, a <a href="Pentatonic_scale" title="Pentatonic scale">pentatonic scale</a> used in <a href="Gamelan" title="Gamelan">Indonesian gamelan music</a></li>

<li><a href="Pelog" title="Pelog">Pelog</a>, the other main gamelan scale</li>

<li><a href="Harry_Partch's_43-tone_scale" title="Harry Partch's 43-tone scale">Harry Partch's 43-tone scale</a></li>

<li><a href="Bohlen%E2%80%93Pierce_scale" title="Bohlen–Pierce scale">Bohlen–Pierce scale</a></li>

<li><a href="Alpha_scale" title="Alpha scale">Alpha</a>, <a href="Beta_scale" title="Beta scale">beta</a>, <a href="Delta_scale" title="Delta scale">delta</a>, and <a href="Gamma_scale" title="Gamma scale">gamma scales</a> of <a href="Wendy_Carlos" title="Wendy Carlos">Wendy Carlos</a></li>

<li><a href="Quarter_tone_scale" class="mw-redirect" title="Quarter tone scale">Quarter tone scale</a></li>

<li><a href="Thirteenth_Sound" class="mw-redirect" title="Thirteenth Sound">Thirteenth Sound</a></li>

<li><a href="19_equal_temperament" title="19 equal temperament">19 equal temperament</a></li>

<li><a href="22_equal_temperament" title="22 equal temperament">22 equal temperament</a></li>

<li><a href="31_equal_temperament" title="31 equal temperament">31 equal temperament</a></li>

<li><a href="53_equal_temperament" title="53 equal temperament">53 equal temperament</a></li>

<li><a href="Schismatic_temperament" title="Schismatic temperament">Schismatic temperament</a></li>

<li><a href="Miracle_temperament" class="mw-redirect" title="Miracle temperament">Miracle temperament</a></li>

<li><a href="Hexany" title="Hexany">Hexany</a></li></ul>

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<p><br>

</p>

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<ul><li><a href="3rd_bridge" title="3rd bridge">3rd bridge</a></li>

<li><i><a href="Beauty_in_the_Beast" title="Beauty in the Beast">Beauty in the Beast</a></i></li>

<li><a href="Chinese_musicology" title="Chinese musicology">Chinese musicology</a></li>

<li><a href="Dynamic_tonality" title="Dynamic tonality">Dynamic tonality</a></li>

<li><a href="Electronic_tuner" title="Electronic tuner">Electronic tuner</a></li>

<li><a href="Ethnomusicology" title="Ethnomusicology">Ethnomusicology</a></li>

<li><a href="Mathematics_of_musical_scales" class="mw-redirect" title="Mathematics of musical scales">Mathematics of musical scales</a></li>

<li><a href="Mersenne's_laws" title="Mersenne's laws">Mersenne's laws</a></li>

<li><a href="Microtonal_music" class="mw-redirect" title="Microtonal music">Microtonal music</a></li>

<li><a href="Microtuner" title="Microtuner">Microtuner</a></li>

<li><a href="MIDI_Tuning_Standard" class="mw-redirect" title="MIDI Tuning Standard">MIDI tuning standard</a></li>

<li><a href="Musical_theory" class="mw-redirect" title="Musical theory">Musical theory</a></li>

<li><a href="Open_chord" title="Open chord">Open chord</a></li>

<li><a href="Physics_of_music" class="mw-redirect" title="Physics of music">Physics of music</a></li>

<li><a href="Pseudo-octave" title="Pseudo-octave">Pseudo-octave</a></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustics</a></li>

<li><a href="Standard_tuning" title="Standard tuning">Standard tuning</a></li>

<li><a href="Stretched_tuning" title="Stretched tuning">Stretched tuning</a></li>

<li><a href="Vibrating_string" class="mw-redirect" title="Vibrating string">Vibrating string</a></li>

<li><a href="Xenharmonic" class="mw-redirect" title="Xenharmonic">Xenharmonic</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFObataya2017" class="citation journal cs1">Obataya, Eiichi (2017-10-01). <a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S1296207416300541">"Effects of natural and artificial ageing on the physical and acoustic properties of wood in musical instruments"</a>. <i>Journal of Cultural Heritage</i>. Wooden Musical Instruments Special Issue. <b>27</b>: <span class="nowrap">S63 –</span> <span class="nowrap">S69</span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.culher.2016.02.011">10.1016/j.culher.2016.02.011</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1296-2074">1296-2074</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:138058574">138058574</a>.</cite></span>

</li>

<li id="cite_note-Rock-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rock_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rock_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.rockfordsymphony.com/faqs/why-does-the-orchestra-tune-to-the-oboe">"Why does the orchestra tune to the oboe?"</a>. <i>RockfordSymphony.com</i>. 2019-03-11<span class="reference-accessdate">. Retrieved <span class="nowrap">2022-08-29</span></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBozkurt2012" class="citation journal cs1">Bozkurt, Barış (2012-09-01). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://doi.org/10.1162/COMJ_a_00128">"A System for Tuning Instruments Using Recorded Music Instead of Theory-Based Frequency Presets"</a></span>. <i>Computer Music Journal</i>. <b>36</b> (3): <span class="nowrap">43–</span>56. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1162%2FCOMJ_a_00128">10.1162/COMJ_a_00128</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0148-9267">0148-9267</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:1576941">1576941</a>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFSfetcu2014" class="citation book cs1">Sfetcu, Nicolae (2014-05-07). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kXyFAwAAQBAJ&q=Interference+beats+are+used+to+objectively+measure+the+accuracy+of+tuning&pg=PT332"><i>The Music Sound</i></a>. Nicolae Sfetcu.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">W. A. Mathieu (1997) <i>Harmonic Experience: Tonal Harmony from Its Natural Origins to Its Modern Expression</i>. Inner Traditions.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li>Barbour, J. Murray (1951). <i>Tuning and Temperament: A Historical Survey</i>. East Lansing: Michigan State College Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-43406-0</bdi>.</li></ul>

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</style><div id="Musical_tunings123" style="font-size:114%;margin:0 4em"></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurement</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pitch_(music)" title="Pitch (music)">Pitch</a></li>

<li><a href="Millioctave" title="Millioctave">Millioctave</a></li>

<li><a href="Savart" title="Savart">Savart</a></li>

<li><a href="Interval_(music)" title="Interval (music)">Interval</a></li>

<li><a href="Interval_ratio" title="Interval ratio">Interval ratio</a></li>

<li><a href="Pitch_class" title="Pitch class">Pitch class</a></li>

<li><a href="Consonance_and_dissonance" title="Consonance and dissonance">Consonance and dissonance</a></li>

<li><a href="List_of_pitch_intervals" title="List of pitch intervals">List of musical intervals</a></li>

<li><a href="List_of_intervals_in_5-limit_just_intonation" title="List of intervals in 5-limit just intonation">List of intervals in 5-limit just intonation</a></li>

<li><a href="List_of_meantone_intervals" title="List of meantone intervals">List of meantone intervals</a></li>

<li><a href="Microtonal_music" class="mw-redirect" title="Microtonal music">Microtone</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Just_intonation" title="Just intonation">Just intonation</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euler%E2%80%93Fokker_genus" title="Euler–Fokker genus">Euler–Fokker genus</a></li>

<li><a href="Harmonic_scale" title="Harmonic scale">Harmonic scale</a></li>

<li><a href="Harry_Partch's_43-tone_scale" title="Harry Partch's 43-tone scale">Harry Partch's 43-tone scale</a></li>

<li><a href="Hexany" title="Hexany">Hexany</a></li>

<li><a href="Limit_(music)" title="Limit (music)">Limit</a>

<ul><li><a href="Pythagorean_tuning" title="Pythagorean tuning">3-limit</a></li>

<li><a href="Five-limit_tuning" title="Five-limit tuning">5-limit</a></li>

<li><a href="7-limit_tuning" title="7-limit tuning">7-limit</a></li></ul></li>

<li><a href="List_of_compositions_in_just_intonation" title="List of compositions in just intonation">List of compositions</a></li>

<li><a href="Otonality_and_Utonality" class="mw-redirect" title="Otonality and Utonality">Otonality</a></li>

<li><a href="Ptolemy's_intense_diatonic_scale" title="Ptolemy's intense diatonic scale">Ptolemy's intense diatonic scale</a></li>

<li><a href="Pythagorean_tuning" title="Pythagorean tuning">Pythagorean tuning</a></li>

<li><a href="Scale_of_harmonics" title="Scale of harmonics">Scale of harmonics</a></li>

<li><a href="Tonality_diamond" title="Tonality diamond">Tonality diamond</a></li>

<li><a href="Tonality_flux" title="Tonality flux">Tonality flux</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Musical_temperament" title="Musical temperament">Temperaments</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Equal_temperament" title="Equal temperament">Equal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Quarter_tone" title="Quarter tone">24-tone</a> (<a href="List_of_quarter_tone_pieces" title="List of quarter tone pieces">pieces</a>)</li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Regular_temperament" title="Regular temperament">Linear</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Meantone_temperament" title="Meantone temperament">Meantone</a> (half-comma, <a href="19_equal_temperament" title="19 equal temperament">third-comma</a>, <a href="Quarter-comma_meantone" title="Quarter-comma meantone">quarter-comma</a>, fifth-comma, sixth-comma, <a href="Septimal_meantone_temperament" title="Septimal meantone temperament">septimal</a>)</li>

<li><a href="Schismatic_temperament" title="Schismatic temperament">Schismatic</a></li>

<li><a href="Miracle_temperament" class="mw-redirect" title="Miracle temperament">Miracle</a></li>

<li><a href="Magic_temperament" title="Magic temperament">Magic</a></li>

<li><a href="Regular_diatonic_tuning" title="Regular diatonic tuning">Regular diatonic</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Irregular</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><a href="Well_temperament" title="Well temperament">Well temperament</a>/<a href="Temperament_ordinaire" title="Temperament ordinaire">Temperament ordinaire</a> (<a href="Kirnberger_temperament" title="Kirnberger temperament">Kirnberger</a>, <a href="Vallotti_temperament" title="Vallotti temperament">Vallotti</a>, <a href="Werckmeister_temperament" title="Werckmeister temperament">Werckmeister</a>, <a href="Young_temperament" title="Young temperament">Young</a>)</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Traditional<br> non-Western</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chinese_musicology" title="Chinese musicology">Chinese musicology</a></li>

<li><a href="Dastgah" class="mw-redirect" title="Dastgah">Dastgah</a></li>

<li>Maqam

<ul><li><a href="Arabic_maqam" title="Arabic maqam">Arabic maqam</a></li>

<li><a href="Turkish_makam" title="Turkish makam">Turkish makam</a></li>

<li><a href="Mugham" title="Mugham">Mugham</a></li>

<li><a href="Muqam" title="Muqam">Muqam</a></li></ul></li>

<li><a href="Octoechos" title="Octoechos">Octoechos</a></li>

<li><a href="Pelog" title="Pelog">Pelog</a></li>

<li><a href="Raga" title="Raga">Raga</a> (<a href="Carnatic_raga" title="Carnatic raga">Carnatic raga</a>)</li>

<li><a href="Slendro" title="Slendro">Slendro</a></li>

<li><a href="Tetrachord" title="Tetrachord">Tetrachord</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Non-octave</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Subdividing</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="833_cents_scale" title="833 cents scale">833 cents scale</a></li>

<li><a href="A12_scale" title="A12 scale">A12 scale</a></li>

<li><a href="Bohlen%E2%80%93Pierce_scale" title="Bohlen–Pierce scale">Bohlen–Pierce scale</a> (<a href="Bohlen%E2%80%93Pierce_scale#Just_tuning" title="Bohlen–Pierce scale">Lambda</a>, <a href="Bohlen%E2%80%93Pierce_scale#Bohlen–Pierce_temperament" title="Bohlen–Pierce scale">13ED3</a>, <a href="Bohlen%E2%80%93Pierce_scale#39-tone_equal_division_of_the_tritave" title="Bohlen–Pierce scale">39ED3</a>, <a href="Bohlen%E2%80%93Pierce_scale#65-tone_equal_division_of_the_tritave" title="Bohlen–Pierce scale">65ED3</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Single ratio</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alpha_scale" title="Alpha scale">Alpha (9EDF)</a></li>

<li><a href="Gamma_scale" title="Gamma scale">Gamma (20EDF)</a></li>

<li><a href="Delta_scale" title="Delta scale">Delta (14ED16/15)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Music_theory43" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Music_theory43" style="font-size:114%;margin:0 4em"><a href="Music_theory" title="Music theory">Music theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aesthetics_of_music" title="Aesthetics of music">Aesthetics</a></li>

<li><a href="Musical_analysis" title="Musical analysis">Analysis</a></li>

<li><a href="Elements_of_music" title="Elements of music">Aspects</a></li>

<li><a href="Musical_composition" title="Musical composition">Composition</a></li>

<li><a href="Definition_of_music" title="Definition of music">Definition</a><br></li>

<li><a href="Music_and_mathematics" title="Music and mathematics">Mathematics</a></li>

<li><a href="Musicology" title="Musicology">Musicology</a></li>

<li><a href="Philosophy_of_music" title="Philosophy of music">Philosophy</a></li>

<li><a href="Music_psychology" class="mw-redirect" title="Music psychology">Psychology</a></li>

<li><a href="Set_theory_(music)" title="Set theory (music)">Set theory</a></li>

</ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="List_of_music_theorists" title="List of music theorists">List of music theorists</a></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Frequency_and_pitch67" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Frequency_and_pitch67" style="font-size:114%;margin:0 4em"><a href="Frequency" title="Frequency">Frequency</a> and <a href="Pitch_(music)" title="Pitch (music)">pitch</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Concert_pitch" title="Concert pitch">Concert pitch</a>

<li><a href="Enharmonic" class="mw-redirect" title="Enharmonic">Enharmonic</a></li>

<li><a href="Helmholtz_pitch_notation" title="Helmholtz pitch notation">Helmholtz pitch notation</a></li>

<li><a href="Letter_notation" title="Letter notation">Letter notation</a></li>

<li><a href="Piano_key_frequencies" title="Piano key frequencies">Piano key frequencies</a></li>

<li><a href="Pitch_class" title="Pitch class">Pitch class</a></li>

<li><a href="Scientific_pitch_notation" title="Scientific pitch notation">Scientific pitch notation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Perception</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_pitch" title="Absolute pitch">Absolute pitch</a></li>

<li><a href="Ear_training" title="Ear training">Ear training</a></li>

<li><a href="Pitch_circularity" title="Pitch circularity">Pitch circularity</a></li>

<li><a href="Relative_pitch" title="Relative pitch">Relative pitch</a></li>

<li><a href="Tonal_memory" title="Tonal memory">Tonal memory</a></li>

<li><a href="Tone_deafness" class="mw-redirect" title="Tone deafness">Tone deafness</a></li>

<li><a href="Virtual_pitch" class="mw-redirect" title="Virtual pitch">Virtual pitch</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Electronic_tuner" title="Electronic tuner">Electronic tuner</a></li>

<li><a href="Mersenne's_laws" title="Mersenne's laws">Mersenne's laws</a></li>

<li><a href="Microtuner" title="Microtuner">Microtuner</a></li>

<li>

<ul><li><a href="Beat_(acoustics)" title="Beat (acoustics)">Beating</a></li></ul></li>

<li><a href="Pitch_pipe" title="Pitch pipe">Pitch pipe</a></li>

<li><a href="Savart_wheel" title="Savart wheel">Savart wheel</a></li>

<li><a href="Tuning_fork" title="Tuning fork">Tuning fork</a></li></ul>

</div></td></tr></tbody></table></div>

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