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→top: {{clarify|date=December 2018|reason=A just minor third is 315.64 cents, and thus one fourth is 78.91 cents, not 78.}}
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[[Image:Minor third on C.png|thumb|right|250px|Minor third (just: 315.64 cents {{audio|Just minor third on C.mid|Play}},<br/> 12-tet: 300 cents {{audio|Minor third on C.mid|Play}},<br/> Alpha scale: 312 cents {{audio|Alpha scale minor third on C.mid|Play}}]]
[[Image:Minor third on C.png|thumb|right|250px|Minor third (just: 315.64 cents {{audio|Just minor third on C.mid|Play}},<br/> 12-tet: 300 cents {{audio|Minor third on C.mid|Play}},<br/> Alpha scale: 312 cents {{audio|Alpha scale minor third on C.mid|Play}}]]


The '''α (alpha) scale''' is a non-octave-repeating [[musical scale]]. In one version it splits the [[perfect fifth]] (3:2) into nine equal parts of approximately 78.0 cents.{{fact|date=January 2012}} In another it splits the [[minor third]] into two equal parts,<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 four equal parts of approximately 78 cents each<ref name="Liner">Carlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552.</ref> {{audio|Alpha scale step on C.mid|Play}}. At 78 cents per step, this totals approximately 15.385 steps per [[octave]]. The scale step may be precisely derived from using [[minor seventh|9:5]] {{audio|Greater just minor seventh on C.mid|Play}} to approximate the interval {{frac|3:2|[[major third|5:4]]}},<ref name="Benson"/> which equals 6:5 {{audio|Just minor third on C.mid|Play}}.
The '''α (alpha) scale''' is a non-octave-repeating [[musical scale]]. In one version it splits the [[perfect fifth]] (3:2) into nine equal parts of approximately 78.0 cents.{{fact|date=January 2012}} In another it splits the [[minor third]] into two equal parts,<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 four equal parts of approximately 78 cents each<ref name="Liner">Carlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552.</ref> {{audio|Alpha scale step on C.mid|Play}}.{{clarify|date=December 2018|reason=A just minor third is 315.64 cents, and thus one fourth is 78.91 cents, not 78.}} At 78 cents per step, this totals approximately 15.385 steps per [[octave]]. The scale step may be precisely derived from using [[minor seventh|9:5]] {{audio|Greater just minor seventh on C.mid|Play}} to approximate the interval {{frac|3:2|[[major third|5:4]]}},<ref name="Benson"/> which equals 6:5 {{audio|Just minor third on C.mid|Play}}.


It was invented by [[Wendy Carlos]] and used on her album ''[[Beauty in the Beast]]'' (1986).
It was invented by [[Wendy Carlos]] and used on her album ''[[Beauty in the Beast]]'' (1986).


Though it does not have an octave, the alpha scale produces, "wonderful [[triad (music)|triads]]," ({{audio|Alpha scale major triad on C.mid|Play major}} and {{audio|Alpha scale minor triad on C.mid|minor triad}}) and the [[beta scale]] has similar properties but the [[Harmonic seventh|sevenths]] are more in tune.<ref name="Milano"/> However, the alpha scale has, "excellent [[harmonic seventh chord]]s...using the [octave] inversion of {{frac|7|4}}, i.e., [[septimal whole tone|{{frac|8|7}}]]."<ref>Carlos, Wendy (1989–96). [http://www.wendycarlos.com/resources/pitch.html "Three Asymmetric Divisions of the Octave"], ''WendyCarlos.com''.
Though it does not have an octave, the alpha scale produces, "wonderful [[triad (music)|triads]]," ({{audio|Alpha scale major triad on C.mid|Play major}} and {{audio|Alpha scale minor triad on C.mid|minor triad}}) and the [[beta scale]] has similar properties but the [[Harmonic seventh|sevenths]] are more in tune.<ref name="Milano"/> However, the alpha scale has, "excellent [[harmonic seventh chord]]s...using the [octave] inversion of {{frac|7|4}}, i.e., [[septimal whole tone|{{frac|8|7}}]]."<ref>Carlos, Wendy (1989–96). [http://www.wendycarlos.com/resources/pitch.html "Three Asymmetric Divisions of the Octave"], ''WendyCarlos.com''.
</ref> {{audio|Alpha scale harmonic seventh chord on C.mid|Play}} More accurately the alpha [[scale step]] is 77.965 cents and there are 15.3915 per octave.<ref name="Benson">Benson, Dave (2006). ''Music: A Mathematical Offering'', p.232-233. {{ISBN|0-521-85387-7}}. "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."</ref><ref>Sethares, William (2004). ''Tuning, Timbre, Spectrum, Scale'', p.60. {{ISBN|1-85233-797-4}}. Scale step of 78 cents.</ref>
</ref> {{audio|Alpha scale harmonic seventh chord on C.mid|Play}} More accurately the alpha [[scale step]] is 77.965 cents and there are 15.3915 per octave.<ref name="Benson">Benson, Dave (2006). ''Music: A Mathematical Offering'', p.232-233. {{ISBN|0-521-85387-7}}. "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."</ref><ref>Sethares, William (2004). ''Tuning, Timbre, Spectrum, Scale'', p.60. {{ISBN|1-85233-797-4}}. Scale step of 78 cents.</ref>{{clarify|date=December 2018|reason=How is 77.965 cents or 15.3915 determined?}}


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Revision as of 05:13, 27 December 2018

Minor third (just: 315.64 cents Playⓘ,
12-tet: 300 cents Playⓘ,
Alpha scale: 312 cents Playⓘ

The α (alpha) scale is a non-octave-repeating musical scale. In one version it splits the perfect fifth (3:2) into nine equal parts of approximately 78.0 cents.[citation needed] In another it splits the minor third into two equal parts,[1] or four equal parts of approximately 78 cents each[2] Playⓘ.[clarification needed] At 78 cents per step, this totals approximately 15.385 steps per octave. The scale step may be precisely derived from using 9:5 Playⓘ to approximate the interval 3:2⁄5:4,[3] which equals 6:5 Playⓘ.

It was invented by Wendy Carlos and used on her album Beauty in the Beast (1986).

Though it does not have an octave, the alpha scale produces, "wonderful triads," (Play majorⓘ and minor triadⓘ) and the beta scale has similar properties but the sevenths are more in tune.[1] However, the alpha scale has, "excellent harmonic seventh chords...using the [octave] inversion of 7⁄4, i.e., 8⁄7."[4] Playⓘ More accurately the alpha scale step is 77.965 cents and there are 15.3915 per octave.[3][5][clarification needed]

interval name size
(steps) 		size
(cents) 		just ratio just
(cents) 		error
septimal major second 3 233.90 8:7 231.17 +2.72
major third 5 389.83 5:4 386.31 +3.51
perfect fifth 9 701.69 3:2 701.96 −0.27
harmonic seventh 12 935.58 7:4 968.83 −33.25
octave 15 1169.48 2:1 1200.00 −30.52
octave 16 1247.44 2:1 1200.00 +47.44

See also

Sources

  1. ^ a b 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. ^ a b Benson, Dave (2006). Music: A Mathematical Offering, p.232-233. ISBN 0-521-85387-7. "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."
  4. ^ Carlos, Wendy (1989–96). "Three Asymmetric Divisions of the Octave", WendyCarlos.com.
  5. ^ Sethares, William (2004). Tuning, Timbre, Spectrum, Scale, p.60. ISBN 1-85233-797-4. Scale step of 78 cents.
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