The β (beta) scale is a non-octave-repeating

musical scale In music theory, a scale is any set of musical notes ordered by fundamental frequency or pitch. A scale ordered by increasing pitch is an ascending scale, and a scale ordered by decreasing pitch is a descending scale. Often, especially in the ...

invented by

Wendy Carlos Wendy Carlos (born Walter Carlos, November 14, 1939) is an American musician and composer best known for her electronic music and film scores. Born and raised in Rhode Island, Carlos studied physics and music at Brown University before moving ...

and first used on her album ''

Beauty in the Beast ''Beauty in the Beast'' is a studio album from the American keyboardist and composer Wendy Carlos, released in 1986, on Audion Records, her first for a label other than Columbia Records since 1968. The album uses alternate musical tunings and s ...

'' (1986). It is derived from approximating

just interval In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and c ...

s using multiples of a single interval without, as is standard in

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

s, requiring an

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

(2:1). It may be approximated by splitting the

perfect fifth In music theory, a perfect fifth is the Interval (music), musical interval corresponding to a pair of pitch (music), pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval fro ...

(3:2) into eleven equal parts 3:2) ≈ 63.8 cents It may be approximated by splitting the

perfect fourth A fourth is a musical interval encompassing four staff positions in the music notation of Western culture, and a perfect fourth () is the fourth spanning five semitones (half steps, or half tones). For example, the ascending interval from C to ...

(4:3) into two equal parts 4:3)Milano, Dominic (November 1986)
"A Many-Colored Jungle of Exotic Tunings"
''Keyboard''. or eight equal parts 4:3) = 64 centsCarlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552. totaling approximately 18.8 steps per

. The

scale step In Schenkerian theory, a scale-step (german: Stufe) is a triad (based on one of the diatonic scale degrees) that is perceived as an organizing force for a passage of music (in accordance with the principle of composing-out). In ''Harmony'', ...

may also precisely be derived from using 11:6 (B, 1049.36 cents, ) to approximate the interval , which equals 6:5 .

\frac=0.05319411048

and

0.05319411048\times1200=63.832932576

() Although neither has an octave, one advantage to the beta scale over the

alpha scale The α (alpha) 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, but without re ...

is that 15 steps, 957.494 cents, is a reasonable approximation to the

seventh harmonic The harmonic seventh interval, also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio (about 969 cents). This is somewhat narrower than and is, "particularly sweet", "sweeter in quality" than an "ordinar ...

(7:4, 968.826 cents)Benson, Dave (2006). ''Music: A Mathematical Offering'', p.232-233. . "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. . Scale step of 63.8 cents. though both have nice triads (, , and ). "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 The δ (delta) scale is a non-octave repeating musical scale. It 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 Tradi ...

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. .

References

External links

*Carlos, Wendy (1989–96)
"Three Asymmetric Divisions of the Octave"
''WendyCarlos.com''. {{DEFAULTSORT:Beta Scale Equal temperaments Non–octave-repeating scales Wendy Carlos

See also

References

External links