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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 . It may be approximated by splitting the perfect fourth (4:3) into two equal parts , or eight equal parts , totaling approximately 18.8 steps per octave.

The scale step may also precisely be derived from using 11:6 (B↑♭-, 1049.36 cents, Play) to approximate the interval 3:2⁄5:4, which equals 6:5 Play.

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.

${\displaystyle {\frac {11\log _{2}{(3/2)}+6\log _{2}{(5/4)}+5\log _{2}{(6/5)}}{11^{2}+6^{2}+5^{2}}}=0.05319411048}$ and ${\displaystyle 0.05319411048\times 1200=63.832932576}$ (Play)

Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, Play is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents) Play though both have nice triads (Play major triad, minor triad, and 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'."

See also

• 19 equal temperament
• Bohlen–Pierce scale
• Gamma scale

References

1. ^ Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.
2. Carlos, Wendy (2000/1986). "Liner notes", Beauty in the Beast. ESD 81552.
3. ^ 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."
4. Sethares, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. ISBN 1-85233-797-4. Scale step of 63.8 cents.
5. Taruskin, Richard (1996). Stravinsky and the Russian Traditions: A Biography of the Works through Mavra, p.1394. ISBN 0-520-07099-2.

External links

• Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.

v t e Microtonal music
Composers	Richard Barrett Béla Bartók Easley Blackwood Jr. Heinz Bohlen Julián Carrillo Franklin Cox Mildred Couper John Eaton Brian Ferneyhough Michael Finnissy Bjørn Fongaard Alois Hába Christiaan Huygens Charles Ives Ben Johnston György Ligeti Stu Mackenzie Claus-Steffen Mahnkopf Joel Mandelbaum Joe Maneri Roger Redgate John Schneider Sevish Ezra Sims Nicola Vicentino Claude Vivier Elaine Walker Ivan Wyschnegradsky La Monte Young
Inventors	Glenn Branca Wendy Carlos Ivor Darreg Adriaan Fokker Lou Harrison Yuri Landman Harry Partch Tui St. George Tucker Nicola Vicentino
Tunings and scales	Non-octave- repeating scales Alpha scale Beta scale Gamma scale Delta scale Lambda scale (Bohlen–Pierce scale) Equal temperament 15 17 19 22 23 24 31 34 41 53 58 72 96 Just intonation Harry Partch's 43-tone scale Double diatonic
Concepts and techniques	Limit Otonality and Utonality Semitone Sonido 13 Xenharmonicity Tonality diamond
Groups and publications	Boston Microtonal Society Genesis of a Music Huygens-Fokker Foundation
Compositions	Beauty in the Beast quarter tone pieces just pieces Mother Sonata for Microtonal Piano Suite for Microtonal Piano Twelve Microtonal Etudes for Electronic Music Media Flying Microtonal Banana K.G. L.W.
Other topics	Enharmonic keyboard Generalized keyboard Modernism (music)

• Equal temperaments
• Non–octave-repeating scales
• Wendy Carlos