beta scale

{{Short description|Musical scale}}

File:Perfect fourth on C.png

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| 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

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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 intervals using multiples of a single interval without, as is standard in equal temperaments, 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}}}}],Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], Keyboard. or eight equal parts [(4:3){{sup|{{frac|8}}}} = 64 cents],Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552. totaling approximately 18.8 steps per octave.

The scale step may also precisely be derived from using 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|5:4}}, which equals 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.}}

$\backslash frac\{11\backslash log\_2\{(3/2)\}+6\backslash log\_2\{(5/4)\}+5\backslash log\_2\{(6/5)\}\}\{11^2+6^2+5^2\}=0.05319411048$ and $0.05319411048\backslash times1200=63.832932576$ ({{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)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."Sethares, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. {{ISBN|1-85233-797-4}}. Scale step of 63.8 cents. {{audio|Harmonic seventh on C.mid|Play}} though both have nice triads ({{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."

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

class="wikitable" |align=center bgcolor="#ffffb4"|interval name |align=center bgcolor="#ffffb4"|size (steps) |align=center bgcolor="#ffffb4"|size (cents) |align=center bgcolor="#ffffb4"|just ratio |align=center bgcolor="#ffffb4"|just (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|−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|−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|−11.33

align=center|octave |align=center|18 |align=center|1148.99 |align=center|2:1 |align=center|1200.00 |align=center|−51.01

align=center|octave |align=center|19 |align=center|1212.83 |align=center|2:1 |align=center|1200.00 |align=center| +12.83