View source for Beta scale

{{Short description|Musical scale}}
[[File:Perfect fourth on C.png|thumb|right|250px|Perfect fourth (just: 498.04 cents {{audio|Just perfect fourth on C.mid|Play}}, 12-tet: 500 cents {{audio|Perfect fourth on C.mid|Play}}, Beta scale: 512 cents {{audio|Beta scale perfect fourth on C.mid|Play}})]]
{{multiple image
 | align = 
 | direction = vertical
 | width = 
 | header = [[Chromatic circle]]
 | image1 = Beta scale chromatic circle.png
 | caption1 = Comparing the beta scale's approximations with the just values
 | image2 = 12-tet scale chromatic circle for comparison with alpha scale et al.png
 | caption2 = Twelve-tone equal temperament vs. just
}}

The '''β''' ('''beta''') '''scale''' is a non-octave-repeating [[musical scale]] invented by [[Wendy Carlos]] and first used on her album ''[[Beauty in the Beast]]'' (1986). It is derived from approximating [[just interval]]s using multiples of a single interval without, as is standard in [[equal temperament]]s, requiring an [[octave]] (2:1). It may be approximated by splitting the [[perfect fifth]] (3:2) into eleven equal parts [(3:2){{sup|{{frac|11}}}} ≈ 63.8 cents]. It may be approximated by splitting the [[perfect fourth]] (4:3) into two equal parts [(4:3){{sup|{{1/2}}}}],<ref name="Milano">Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], ''Keyboard''.</ref> or eight equal parts [(4:3){{sup|{{frac|8}}}} = 64 cents],<ref name="Liner">Carlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552.</ref> totaling approximately 18.8 steps per [[octave]].

The [[scale step]] may also precisely be derived from using [[neutral seventh|11:6]] (B{{music|11}}{{music|b}}{{music|minus}}, 1049.36 cents, {{audio|Neutral seventh on C.mid|Play}}) to approximate the interval {{frac|3:2|[[major third|5:4]]}},<ref name="Benson"/> which equals [[minor third|6:5]] {{audio|Just minor third on C.mid|Play}}.
{{Blockquote|In order to make the approximation as good as possible we minimize the [[mean square deviation]]. ... We choose a value of the scale degree so that eleven of them approximate a 3:2 perfect fifth, six of them approximate a 5:4 major third, and five of them approximate a 6:5 minor third.<ref name="Benson"/>}}

<math>\frac{11\log_2{(3/2)}+6\log_2{(5/4)}+5\log_2{(6/5)}}{11^2+6^2+5^2}=0.05319411048</math> and <math>0.05319411048\times1200=63.832932576</math> ({{audio|Beta scale step on C.mid|Play}})

Although neither has an octave, one advantage to the beta scale over the [[alpha scale]] is that 15 steps, 957.494 cents, {{audio|Beta scale 15 steps on C.mid|Play}} is a reasonable approximation to the [[seventh harmonic]] (7:4, 968.826 cents)<ref name="Benson">Benson, Dave (2006). ''Music: A Mathematical Offering'', p.232-233. {{ISBN|0-521-85387-7}}. "Carlos has 18.809 β-scale degrees to the octave, corresponding to a scale degree of 63.8 cents."</ref><ref>[[William Sethares|Sethares, William]] (2004). ''Tuning, Timbre, Spectrum, Scale'', p.60. {{ISBN|1-85233-797-4}}. Scale step of 63.8 cents.</ref> {{audio|Harmonic seventh on C.mid|Play}} though both have nice [[triad (music)|triads]]<ref name="Milano"/> ({{audio|Beta scale major triad on C.mid|Play major triad}}, {{audio|Beta scale minor triad on C.mid|minor triad}}, and {{audio|Beta scale dominant seventh on C.mid|dominant seventh}}). "According to Carlos, beta has almost the same properties as the alpha scale, except that the sevenths are slightly more in tune."<ref name="Milano"/>

The [[delta scale]] may be regarded as the beta scale's reciprocal since it is "as far 'down' the ([[diminished seventh chord|0 3 6 9]]) circle from α as β is 'up'."<ref name="Taruskin">[[Richard Taruskin|Taruskin, Richard]] (1996). ''Stravinsky and the Russian Traditions: A Biography of the Works through Mavra'', p.1394. {{ISBN|0-520-07099-2}}.</ref>

{| class="wikitable"
|align=center bgcolor="#ffffb4"|interval name
|align=center bgcolor="#ffffb4"|size<br/> (steps)
|align=center bgcolor="#ffffb4"|size<br/> (cents)
|align=center bgcolor="#ffffb4"|just ratio
|align=center bgcolor="#ffffb4"|just<br/> (cents)
|align=center bgcolor="#ffffb4"|error
|-
|align=center|major second
|align=center|3
|align=center|191.50
|align=center|9:8
|align=center|203.91
|align=center|&minus;12.41
|-
|align=center|minor third
|align=center|5
|align=center|319.16
|align=center|6:5
|align=center|315.64
|align=center| +3.52
|-
|align=center|major third
|align=center|6
|align=center|383.00
|align=center|5:4
|align=center|386.31
|align=center|&minus;3.32
|-
|align=center|perfect fifth
|align=center|11
|align=center|702.16
|align=center|3:2
|align=center|701.96
|align=center| +0.21
|-
|align=center|harmonic seventh
|align=center|15
|align=center|957.49
|align=center|7:4
|align=center|968.83
|align=center|&minus;11.33
|-
|align=center|octave
|align=center|18
|align=center|1148.99
|align=center|2:1
|align=center|1200.00
|align=center|&minus;51.01
|-
|align=center|octave
|align=center|19
|align=center|1212.83
|align=center|2:1
|align=center|1200.00
|align=center| +12.83
|}

==See also==
*[[19 equal temperament]]
*[[Bohlen–Pierce scale]]
*[[Gamma scale]]

==References==
{{reflist}}

==External links==
*Carlos, Wendy (1989–96). [http://www.wendycarlos.com/resources/pitch.html "Three Asymmetric Divisions of the Octave"], ''WendyCarlos.com''.

{{Microtonal music}}
{{Musical tuning}}
{{Scales}}
{{Wendy Carlos}}

{{DEFAULTSORT:Beta Scale}}
[[Category:Equal temperaments]]
[[Category:Non–octave-repeating scales]]
[[Category:Wendy Carlos]]