Editing Equal temperament (section)

=== Mathematics ===
{{anchor|12TET}}
[[File:Monochord ET.png|250px|thumb|One octave of 12&nbsp;{{sc|tet}} on a monochord]]

In 12&nbsp;tone equal temperament, which divides the octave into 12&nbsp;equal parts, the width of a [[semitone]], i.e. the [[Interval ratio|frequency ratio]] of the interval between two adjacent notes, is the [[twelfth root of two]]:

:<math> \sqrt[12]{2\ } = 2^{\tfrac{1}{12}} \approx 1.059463 </math>

This interval is divided into 100&nbsp;cents.

==== Calculating absolute frequencies ====
{{See also|Piano key frequencies}}
To find the frequency, {{math|''P{{sub|n}}''}}, of a note in 12&nbsp;{{sc|TET}}, the following formula may be used:

:<math>\ P_n = P_a\ \cdot\ \Bigl(\ \sqrt[12]{2\ }\ \Bigr)^{ n-a }\ </math>

In this formula {{math|''P{{sub|n}}''}} represents the pitch, or frequency (usually in [[hertz]]), you are trying to find. {{math|''P{{sub|a}}''}} is the frequency of a reference pitch. The indes numbers {{mvar|n}} and {{mvar|a}} are the labels assigned to the desired pitch ({{mvar|n}}) and the reference pitch ({{mvar|a}}). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A{{sub|4}} (the reference pitch) is the 49th&nbsp;key from the left end of a piano (tuned to [[A440 (pitch standard)|440&nbsp;Hz]]), and C{{sub|4}} ([[middle C]]), and F{{music|#}}{{sub|4}} are the 40th and 46th&nbsp;keys, respectively. These numbers can be used to find the frequency of C{{sub|4}} and F{{music|#}}{{sub|4}}:

:<math>P_{40} = 440\ \text{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(40-49)} \approx 261.626\ \text{Hz}\ </math>
:<math>P_{46} = 440\ \text{Hz}\ \cdot\ \Bigl( \sqrt[12]{2}\ \Bigr)^{(46-49)} \approx 369.994\ \text{Hz}\ </math>

==== Converting frequencies to their equal temperament counterparts ====
To convert a frequency (in Hz) to its equal 12&nbsp;{{sc|TET}} counterpart, the following formula can be used:

:<math>\ E_n = E_a\ \cdot\ 2^{\ x}\ \quad </math> where in general <math> \quad\ x\ \equiv\ \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12\log_{2} \left(\frac{\ n\ }{ a }\right) \Biggr) ~.</math>

[[File:12ed2-5Limit.svg|250px|thumb|Comparison of intervals in 12-TET with just intonation]]

{{math|''E{{sub|n}}''}} is the frequency of a pitch in equal temperament, and {{math|''E{{sub|a}}''}} is the frequency of a reference pitch. For example, if we let the reference pitch equal 440&nbsp;Hz, we can see that {{sc|'''E'''}}{{sub|5}} and {{sc|'''C'''}}{{music|#}}{{sub|5}} have the following frequencies, respectively:

: <math>E_{660} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 7 }{\ 12\ }\right)}\ \approx\ 659.255\ \mathsf{Hz}\ \quad </math> where in this case <math> \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl(\ 12 \log_{2}\left(\frac{\ 660\ }{ 440 }\right)\ \Biggr) = \frac{ 7 }{\ 12\ } ~.</math>

: <math>E_{550} = 440\ \mathsf{Hz}\ \cdot\ 2^{\left(\frac{ 1 }{\ 3\ }\right)}\ \approx\ 554.365\ \mathsf{Hz}\ \quad </math>  where in this case <math> \quad x = \frac{ 1 }{\ 12\ }\ \operatorname{round}\!\Biggl( 12 \log_{2}\left(\frac{\ 550\ }{ 440 }\right)\Biggr) = \frac{ 4 }{\ 12\ } = \frac{ 1 }{\ 3\ } ~.</math>

==== Comparison with just intonation ====
The intervals of 12&nbsp;{{sc|TET}} closely approximate some intervals in [[just intonation]].<ref>
{{cite book
 |last=Partch |first=Harry
 |year=1979
 |title=Genesis of a Music |edition=2nd
 |publisher=Da Capo Press
 |isbn=0-306-80106-X
 |page=[https://archive.org/details/genesismusicacco00part/page/n167 134]
 |url=https://archive.org/details/genesismusicacco00part
 |url-access=limited
}}
</ref>
The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away.

In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents.
{{clear}}
:{| class="wikitable"  style="margin:auto;text-align:center;"
|-
! Interval Name
! Exact value in 12&nbsp;{{sc|TET}}
! Decimal value in 12&nbsp;{{sc|TET}}
! Pitch in 
! Just intonation interval
! Cents in just intonation
! 12&nbsp;{{sc|TET}} cents<br/>tuning error
|-
| Unison ([[C (musical note)|{{sc|'''C'''}}]])
| {{big|2}}{{sup|{{frac|0|12}}}} = {{big|1}}
| 1
| 0
| {{sfrac|1|1}} = {{big|1}}
| 0
| 0
|-
| Minor second ([[D♭ (musical note)|{{sc|'''D'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|1|12}}}} = <math>\sqrt[12]{2}</math>
| {{#expr:2^(1/12) round 6}}
| 100
| {{sfrac|16|15}} = {{big|1.06666...}}
| {{#expr:1200*ln(16/15)/ln2 round 2}}
| {{#expr:100-1200*ln(16/15)/ln2 round 2}}
|-
| Major second ([[D (musical note)|{{sc|'''D'''}}]])
| {{big|2}}{{sup|{{frac|2|12}}}} = <math>\sqrt[6]{2}</math>
| {{#expr:2^(1/6) round 6}}
| 200
| {{sfrac|9|8}} = {{big|1.125}}
| {{#expr:1200*ln(9/8)/ln2 round 2}}
| {{#expr:200-1200*ln(9/8)/ln2 round 2}}
|-
| Minor third ([[E♭ (musical note)|{{sc|'''E'''}}{{music|flat}}]])
| {{big|2}}{{sup|{{frac|3|12}}}} = <math>\sqrt[4]{2}</math>
| {{#expr:2^(1/4) round 6}}
| 300
| {{sfrac|6|5}} = {{big|1.2}}
| {{#expr:1200*ln(6/5)/ln2 round 2}}
| {{#expr:300-1200*ln(6/5)/ln2 round 2}}
|-
| Major third ([[E (musical note)|{{sc|'''E'''}}]])
| {{big|2}}{{sup|{{frac|4|12}}}} = <math>\sqrt[3]{2}</math>
| {{#expr:2^(1/3) round 6}}
| 400
| {{sfrac|5|4}} = {{big|1.25}}
| {{#expr:1200*ln(5/4)/ln2 round 2}}
| +{{#expr:400-1200*ln(5/4)/ln2 round 2}}
|-
| Perfect fourth ([[F (musical note)|{{sc|'''F'''}}]])
| {{big|2}}{{sup|{{frac|5|12}}}} = <math>\sqrt[12]{32}</math>
| {{#expr:2^(5/12) round 6}}
| 500
| {{sfrac|4|3}} = {{big|1.33333...}}
| {{#expr:1200*ln(4/3)/ln2 round 2}}
| +{{#expr:500-1200*ln(4/3)/ln2 round 2}}
|-
| Tritone ([[G♭ (musical note)|{{sc|'''G'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|6|12}}}} = <math>\sqrt{2}</math>
| {{#expr:2^(1/2) round 6}}
| 600
| {{sfrac|64|45}}= {{big|1.42222...}}
| {{#expr:1200*ln(64/45)/ln2 round 2}}
| {{#expr:600-1200*ln(64/45)/ln2 round 2}}
|-
| Perfect fifth ([[G (musical note)|{{sc|'''G'''}}]])
| {{big|2}}{{sup|{{frac|7|12}}}} = <math>\sqrt[12]{128}</math>
| {{#expr:2^(7/12) round 6}}
| 700
| {{sfrac|3|2}} = {{big|1.5}}
| {{#expr:1200*ln(3/2)/ln2 round 2}}
| {{#expr:700-1200*ln(3/2)/ln2 round 2}}
|-
| Minor sixth ([[A♭ (musical note)|{{sc|'''A'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|8|12}}}} = <math>\sqrt[3]{4}</math>
| {{#expr:2^(2/3) round 6}}
| 800
| {{sfrac|8|5}} = {{big|1.6}}
| {{#expr:1200*ln(8/5)/ln2 round 2}}
| {{#expr:800-1200*ln(8/5)/ln2 round 2}}
|-
| Major sixth ([[A (musical note)|{{sc|'''A'''}}]])
| {{big|2}}{{sup|{{frac|9|12}}}} = <math>\sqrt[4]{8}</math>
| {{#expr:2^(3/4) round 6}}
| 900
| {{sfrac|5|3}} = {{big|1.66666...}}
| {{#expr:1200*ln(5/3)/ln2 round 2}}
| +{{#expr:900-1200*ln(5/3)/ln2 round 2}}
|-
| Minor seventh ([[B♭ (musical note)|{{sc|'''B'''}}{{music|b}}]])
| {{big|2}}{{sup|{{frac|10|12}}}} = <math>\sqrt[6]{32}</math>
| {{#expr:2^(5/6) round 6}}
| 1000
| {{sfrac|16|9}} = {{big|1.77777...}}
| {{#expr:1200*ln(16/9)/ln2 round 2}}
| +{{#expr:1000-1200*ln(16/9)/ln2 round 2}}
|-
| Major seventh ([[B (musical note)|{{sc|'''B'''}}]])
| {{big|2}}{{sup|{{frac|11|12}}}} = <math>\sqrt[12]{2048}</math>
| {{#expr:2^(11/12) round 6}}
| 1100
| {{sfrac|15|8}} = {{big|1.875}}
| {{#expr:1200*ln(15/8)/ln2 round 2}}0
| +{{#expr:1100-1200*ln(15/8)/ln2 round 2}}
|-
| Octave ([[C (musical note)|{{sc|'''c'''}}]])
| {{big|2}}{{sup|{{frac|12|12}}}} = {{big|2}}
| {{big|2}}
| 1200
| {{sfrac|2|1}} = {{big|2}}
| 1200.00
| 0
|}