Editing Twelfth root of two (section)

==The equal-tempered chromatic scale==

A [[interval (music)|musical interval]] is a ratio of frequencies and the [[Equal temperament|equal-tempered]] chromatic scale divides the [[octave]] (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 2{{sup|{{frac|1|12}}}} times that of the one below it.<ref>{{Cite web |title=Equal temperament {{!}} Definition & Facts {{!}} Britannica |url=https://www.britannica.com/art/equal-temperament |access-date=2024-06-03 |website=www.britannica.com |language=en}}</ref>

Applying this value successively to the tones of a chromatic scale, starting from '''A''' above [[middle C|middle '''C''']] (known as [[A440 (pitch standard)|A<sub>4</sub>]]) with a frequency of 440&nbsp;Hz, produces the following sequence of [[pitch (music)|pitch]]es:
{| class="wikitable" style="text-align: center;"
! Note
! Standard interval name(s)<br />relating to A 440
! Frequency<br /> (Hz)
! Multiplier
! Coefficient<br />(to six decimal places)
! {{abbr|Just intonation|for comparison}}<br /> ratio
!{{abbr|Difference|between equal-tempered scale and just intonation}}<br />(± [[Cent_(music)|cents]])
|-
| A     || [[Unison]] || 440.00 || 2{{sup|{{frac|0|12}}}} || {{val|1.000000}}
|1
|align=right|0
|-
| A{{music|#}}/B{{music|b}} || [[Minor second|Minor second/Half step/Semitone]] || 466.16 || 2{{sup|{{frac|1|12}}}} || {{val|1.059463}}
|≈ {{frac|16|15}} <!-- +1.2% -->
|align=right|+11.73
|-
| B     || [[Major second|Major second/Full step/Whole tone]] || 493.88 || 2{{sup|{{frac|2|12}}}} || {{val|1.122462}}
|≈ {{frac|9|8}} <!-- +0.3% -->
|align=right| −3.91
|-
| C     || [[Minor third]] || 523.25 || 2{{sup|{{frac|3|12}}}} || {{val|1.189207}}
|≈ {{frac|6|5}} <!-- +1.1% -->
|align=right| +15.64
|-
| C{{music|#}}/D{{music|b}} || [[Major third]] || 554.37 || 2{{sup|{{frac|4|12}}}} || [[cube root of two#In music theory|{{val|1.259921}}]]
|≈ {{frac|5|4}} <!-- -1.0% -->
|align=right| −13.69
|-
| D     || [[Perfect fourth]] || 587.33 || 2{{sup|{{frac|5|12}}}} || {{val|1.334839}}
|≈ {{frac|4|3}} <!-- -0.2% -->
|align=right| −1.96
|-
| D{{music|#}}/E{{music|b}} || [[Tritone|Augmented fourth/Diminished fifth/Tritone]] || 622.25 || 2{{sup|{{frac|6|12}}}} || [[square root of two|{{val|1.414213}}]]
|≈ {{frac|7|5}} <!-- -1.4% -->
|align=right| +17.49
|-
| E     || [[Perfect fifth]] || 659.26 || 2{{sup|{{frac|7|12}}}} || {{val|1.498307}}
|≈ {{frac|3|2}} <!-- +0.2% -->
|align=right| +1.96
|-
| F     || [[Minor sixth]] || 698.46 || 2{{sup|{{frac|8|12}}}} || {{val|1.587401}}
|≈ {{frac|8|5}} <!-- +1.3% -->
|align=right| +13.69
|-
| F{{music|#}}/G{{music|b}} || [[Major sixth]] || 739.99 || 2{{sup|{{frac|9|12}}}} || {{val|1.681792}}
|≈ {{frac|5|3}} <!-- +1.5% -->
|align=right| −15.64
|-
| G     || [[Minor seventh]] || 783.99 || 2{{sup|{{frac|10|12}}}} || {{val|1.781797}}
|≈ {{frac|16|9}} <!-- +1.8% -->
|align=right| +3.91
|-
| G{{music|#}}/A{{music|b}} || [[Major seventh]] || 830.61 || 2{{sup|{{frac|11|12}}}} || {{val|1.887748}}
|≈ {{frac|15|8}} <!-- -1.3% -->
|align=right| −11.73
|-
| A     || [[Octave]] || 880.00 || 2{{sup|{{frac|12|12}}}} || {{val|2.000000}}
|2
|align=right|0
|}

The final '''A''' (A<sub>5</sub>: 880&nbsp;Hz) is exactly twice the frequency of the lower '''A''' (A<sub>4</sub>: 440&nbsp;Hz), that is, one octave higher.

===Other tuning scales===
Other tuning scales use slightly different interval ratios:
* The [[Just intonation|just]] or [[Pythagorean tuning|Pythagorean]] perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a [[grad (musical interval)|grad]], the twelfth root of the [[Pythagorean comma]] (<math display=inline>\sqrt[12]{531441/524288}</math>).
* The equal tempered [[Bohlen–Pierce scale]] uses the interval of the thirteenth root of three (<math display=inline>\sqrt[13]{3}</math>).
* Stockhausen's ''[[Studie II]]'' (1954) makes use of the twenty-fifth root of five (<math display=inline>\sqrt[25]{5}</math>), a compound major third divided into 5×5 parts.
* The [[delta scale]] is based on ≈<math display=inline>\sqrt[50]{3/2}</math>.
* The [[gamma scale]] is based on ≈<math display=inline>\sqrt[20]{3/2}</math>.
* The [[beta scale]] is based on ≈<math display=inline>\sqrt[11]{3/2}</math>.
* The [[alpha scale]] is based on ≈<math display=inline>\sqrt[9]{3/2}</math>.