Alpha scale

{{Short description|Musical scale invented by Wendy Carlos}}

Image:Minor third on C.png: 300 cents {{audio|Minor third on C.mid|Play}},
Alpha scale: 312 cents {{audio|Alpha scale minor third on C.mid|Play}}]]

{{multiple image

| align =

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| header = Chromatic circle

| image1 = Alpha scale chromatic circle.png

| caption1 = Comparison of the alpha 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

}}

The {{mvar|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 requiring (as temperaments normally do) an octave (2:1). It may be approximated by dividing the perfect fifth (3:2) into nine equal steps, with frequency ratio $\backslash \; \backslash left(\; \backslash tfrac\{\backslash \; 3\backslash \; \}\{\; 2\; \}\; \backslash right)^\{\backslash tfrac\{1\}\{9\}\; \}\backslash \; ,${{cite report |last=Carlos |first=Wendy |author-link=Wendy Carlos |year=1989–1996 |url=http://www.wendycarlos.com/resources/pitch.html |title=Three asymmetric divisions of the octave |via=WendyCarlos.com |quote=9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters). |access-date=2010-06-13 |archive-date=2017-07-12 |archive-url=https://web.archive.org/web/20170712065803/http://www.wendycarlos.com/resources/pitch.html |url-status=live }} or by dividing the minor third (6:5) into four frequency ratio steps of $\backslash \; \backslash left(\; \backslash tfrac\{\backslash \; 6\backslash \; \}\{\; 5\; \}\; \backslash right)^\{\backslash tfrac\{1\}\{4\}\; \}\; ~.${{cite magazine |last=Milano |first=Dominic |date=November 1986 |url=http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf |via=wendycarlos.com |title=A many-colored jungle of exotic tunings |magazine=Keyboard |quote=The idea was to split a minor third into two equal parts. Then that was divided again. |access-date=2010-06-13 |archive-date=2010-12-02 |archive-url=https://web.archive.org/web/20101202134950/http://wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf |url-status=live }}{{cite AV media |last=Carlos |first=Wendy |author-link=Wendy Carlos |year=2000 |orig-year=1986 |medium=record liner notes |title=Beauty in the Beast |id=ESD 81552}}

The size of this scale step may also be precisely derived from using 9:5 {{big|(}}B{{music|b}}, 1017.60 cents, {{audio|Greater just minor seventh on C.mid|Play}}{{big|)}} to approximate the interval {{nobr|{{math| {{sfrac| 3:2 | 5:4 }} {{=}} 6:5 }} }} {{big|(}}E{{music|b}}, 315.64 cents, {{audio|Just minor third on C.mid|Play}} {{big|)}}.

: Carlos' {{big|{{math|α}}}} (alpha) scale arises from ... taking a value for the scale degree so that nine of them approximate a 3:2 perfect fifth, five of them approximate a 5:4 major third, and four of them approximate a 6:5 minor third. In order to make the approximation as good as possible we minimize the mean square deviation.

The formula below finds the minimum by setting the derivative of the mean square deviation with respect to the {{nobr|scale step size to 0 .}}

: $\backslash \; \backslash frac\{\backslash \; 9\backslash \; \backslash log\_2\backslash left(\; \backslash frac\{\backslash \; 3\backslash \; \}\{\; 2\; \}\; \backslash right)\; +\; 5\backslash log\_2\backslash left(\; \backslash frac\{\backslash \; 5\backslash \; \}\{\; 4\; \}\; \backslash right)\; +\; 4\backslash \; \backslash log\_2\backslash left(\; \backslash frac\{\backslash \; 6\backslash \; \}\{\; 5\; \}\; \backslash right)\backslash \; \}\{\backslash \; 9^2\; +\; 5^2\; +\; 4^2\backslash \; \}\; \backslash approx\; 0.06497082462\backslash$

and $\backslash \; 0.06497082462\; \backslash times\; 1200\; =\; 77.964989544\backslash$ ({{audio|Alpha scale step on C.mid|Play}})

At 78 cents per step, this totals approximately 15.385 steps per octave, however, more accurately, the alpha scale step is 77.965 cents and there are 15.3915 steps per octave.{{cite book |last=Benson |first=Dave |year=2006 |title=Music: A mathematical offering |pages=232–233 |publisher=Cambridge University Press |isbn=0-521-85387-7 |quote=This actually differs very slightly from Carlos' figure of 15.385 {{mvar|α}}-scale degrees to the octave. This is obtained by approximating the scale degree to 78.0 cents.}}{{cite book |last=Sethares |first=W. |author-link=William Sethares |year=2004 |title=Tuning, Timbre, Spectrum, Scale |page=60 |publisher=Springer |isbn=1-85233-797-4 |quote=... scale step of 78 cents.}}

Though it does not have a perfect octave, the alpha scale produces "wonderful 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 sevenths are more in tune. However, the alpha scale has

: "excellent harmonic seventh chords ... using the [octave] inversion of {{sfrac| 7 | 4 }}, i.e., septimal whole tone [{{audio|Alpha scale harmonic seventh chord on C.mid|Play}}]."

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|septimal major second |align=center|3 |align=center|233.89 |align=center|8:7 |align=center|231.17 |align=center|+2.72

align=center|minor third |align=center|4 |align=center|311.86 |align=center|6:5 |align=center|315.64 |align=center|−3.78

align=center|major third |align=center|5 |align=center|389.82 |align=center|5:4 |align=center|386.31 |align=center|+3.51

align=center|perfect fifth |align=center|9 |align=center|701.68 |align=center|3:2 |align=center|701.96 |align=center|−0.27

align=center|harmonic seventh |align=center|octave−3 |align=center|966.11 |align=center|7:4 |align=center|968.83 |align=center|−2.72

align=center|octave |align=center|15 |align=center|1169.47 |align=center|2:1 |align=center|1200.00 |align=center|−30.53

align=center|octave |align=center|16 |align=center|1247.44 |align=center|2:1 |align=center|1200.00 |align=center| +47.44