The α (alpha) scale is a non-

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 ...

-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, but without requiring (as

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 normally do) an

(2:1). It may be approximated by dividing 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 nine equal steps (3:2),Carlos, Wendy (1989–96)
"Three Asymmetric Divisions of the Octave"
''WendyCarlos.com''. "9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters)." or by dividing the

minor third In music theory, a minor third is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions (see: interval number). The minor third is one of two com ...

(6:5) into four steps (6:5).Milano, Dominic (November 1986)
"A Many-Colored Jungle of Exotic Tunings"
''Keyboard''. "The idea was to split a minor third into two equal parts. Then that was divided again."Carlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552. The size of this scale step may also be precisely derived from using 9:5 (B, 1017.60 cents, ) to approximate the interval (=6:5, E, 315.64 cents, ). The formula below finds the minimum by setting the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of the mean square deviation with respect to the scale step size to 0.

\frac\approx0.06497082462

and

0.06497082462\times1200=77.965

() At 78 cents per step, this totals approximately 15.385 steps per

, however, more accurately, the alpha

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'', ...

is 77.965 cents and there are 15.3915 per octave.Benson, Dave (2006). ''Music: A Mathematical Offering'', p.232-233. . "This actually differs very slightly from Carlos' figure of 15.385 α-scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents." Sethares, William (2004). ''Tuning, Timbre, Spectrum, Scale'', p.60. . Scale step of 78 cents. Though it does not have an octave, the alpha scale produces "wonderful triads," ( and ) and the

beta scale 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 ...

has similar properties but the sevenths are more in tune. However, the alpha scale has "excellent

harmonic seventh chord The harmonic seventh chord is a major triad plus the harmonic seventh interval (ratio of 7:4, about 968.826 centsBosanquet, Robert Holford Macdowall (1876). ''An elementary treatise on musical intervals and temperament'', pp. 41-42. Diapason ...

s...using the ctaveinversion of , i.e., []."

References

Equal temperaments Non–octave-repeating scales Wendy Carlos {{music-theory-stub

See also

References