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Chromatic circle

Comparison of the alpha scale's approximations with the just values

Twelve-tone equal temperament vs. just

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 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 $\ \left({\tfrac {\ 3\ }{2}}\right)^{\tfrac {1}{9}}\ ,$ ^[1] or by dividing the minor third (6:5) into four frequency ratio steps of $\ \left({\tfrac {\ 6\ }{5}}\right)^{\tfrac {1}{4}}~.$ ^[1]^[2]^[3]

The size of this scale step may also be precisely derived from using 9:5 (B♭, 1017.60 cents, Play^ⓘ) to approximate the interval $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠ 3:2 / 5:4 ⁠ = 6:5$   (E♭, 315.64 cents, Play^ⓘ ).^[4]

Carlos'

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.^[4]

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

\ {\frac {\ 9\ \log _{2}\left({\frac {\ 3\ }{2}}\right)+5\log _{2}\left({\frac {\ 5\ }{4}}\right)+4\ \log _{2}\left({\frac {\ 6\ }{5}}\right)\ }{\ 9^{2}+5^{2}+4^{2}\ }}\approx 0.06497082462\

and $\ 0.06497082462\times 1200=77.964989544\$ (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.^[4]^[5]

Though it does not have a perfect 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.^[2] However, the alpha scale has

"excellent harmonic seventh chords ... using the [octave] inversion of ⁠ 7  4 ⁠, i.e., ⁠87⁠ [Play]."^[1]

interval name	size (steps)	size (cents)	just ratio	just (cents)	error
septimal major second	3	233.89	8:7	231.17	+2.72
minor third	4	311.86	6:5	315.64	−3.78
major third	5	389.82	5:4	386.31	+3.51
perfect fifth	9	701.68	3:2	701.96	−0.27
harmonic seventh	octave−3	966.11	7:4	968.83	−2.72
octave	15	1169.47	2:1	1200.00	−30.53
octave	16	1247.44	2:1	1200.00	+47.44

References

^ ^a ^b ^c Carlos, Wendy (1989–1996). Three asymmetric divisions of the octave (Report). Archived from the original on 2017-07-12. Retrieved 2010-06-13 – via WendyCarlos.com. 9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters).
^ ^a ^b Milano, Dominic (November 1986). "A many-colored jungle of exotic tunings" (PDF). Keyboard. Archived (PDF) from the original on 2010-12-02. Retrieved 2010-06-13 – via wendycarlos.com. The idea was to split a minor third into two equal parts. Then that was divided again.
^ Carlos, Wendy (2000) [1986]. Beauty in the Beast (record liner notes). ESD 81552.
^ ^a ^b ^c Benson, Dave (2006). Music: A mathematical offering. Cambridge University Press. pp. 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.
^ Sethares, W. (2004). Tuning, Timbre, Spectrum, Scale. Springer. p. 60. ISBN 1-85233-797-4. ... scale step of 78 cents.

This music theory article is a stub. You can help Wikipedia by expanding it.

[Three-1] Carlos, Wendy (1989–1996). Three asymmetric divisions of the octave (Report). Archived from the original on 2017-07-12. Retrieved 2010-06-13 – via WendyCarlos.com. 9 steps to the perfect (no kidding) fifth." The alpha scale "splits the minor third exactly in half (also into quarters).

[Milano-2] Milano, Dominic (November 1986). "A many-colored jungle of exotic tunings" (PDF). Keyboard. Archived (PDF) from the original on 2010-12-02. Retrieved 2010-06-13 – via wendycarlos.com. The idea was to split a minor third into two equal parts. Then that was divided again.

[Liner-3] Carlos, Wendy (2000) [1986]. Beauty in the Beast (record liner notes). ESD 81552.

[Benson-4] Benson, Dave (2006). Music: A mathematical offering. Cambridge University Press. pp. 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.

[5] Sethares, W. (2004). Tuning, Timbre, Spectrum, Scale. Springer. p. 60. ISBN 1-85233-797-4. ... scale step of 78 cents.

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[2]

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[4]

[5]

@@ Line 1: / Line 1: @@
+{{Short description|Musical scale invented by Wendy Carlos}}
-[[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&nbsp;cents {{audio|Just minor third on C.mid|Play}},<br/> [[12-tone equal temperament|12&nbsp;TET]]: 300&nbsp;cents {{audio|Minor third on C.mid|Play}},<br/> Alpha scale: 312&nbsp;cents {{audio|Alpha scale minor third on C.mid|Play}}]]
 {{multiple image
  | align =
@@ Line 6: / Line 7: @@
  | header = [[Chromatic circle]]
  | image1 = Alpha scale chromatic circle.png
- | caption1 = Comparing the alpha scale's approximations with the just values
+ | 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 '''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 interval]]s using multiples of a single interval without, as is standard in [[equal temperament]]s, requiring an [[octave]] (2:1). It may be approximated by dividing the [[perfect fifth]] (3:2) into nine equal steps (3:2{{sup|1/9}}),<ref name="Three">Carlos, Wendy (1989–96). [http://www.wendycarlos.com/resources/pitch.html "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)."</ref> or by dividing the [[minor third]] (6:5) into four steps (6:5{{sup|1/4}}).<ref name="Three"/><ref name="Milano">Milano, Dominic (November 1986). [http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf "A Many-Colored Jungle of Exotic Tunings"], ''Keyboard''. "The idea was to split a minor third into tow equal parts. Then that was divided again."</ref><ref name="Liner">Carlos, Wendy (2000/1986). "Liner notes", ''Beauty in the Beast''. ESD 81552.</ref>
+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 interval]]s using multiples of a single interval, but without requiring (as [[temperament (music)|temperament]]s normally do) an [[octave]] (2:1). It may be approximated by dividing the [[perfect fifth]] (3:2) into nine equal steps, with frequency ratio <math>\ \left( \tfrac{\ 3\ }{ 2 } \right)^{\tfrac{1}{9} }\ ,</math><ref name=Three>{{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&nbsp;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 }}</ref> or by dividing the [[minor third]] (6:5) into four frequency ratio steps of <math>\ \left( \tfrac{\ 6\ }{ 5 } \right)^{\tfrac{1}{4} } ~.</math><ref name=Three/><ref name=Milano>{{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 }}</ref><ref name=Liner>{{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&nbsp;81552}}</ref>
-The scale step may also be precisely derived from using [[minor seventh|9:5]] (B{{music|b}}, 1017.60 cents, {{audio|Greater just minor seventh on C.mid|Play}}) to approximate the interval {{frac|3:2|[[major third|5:4]]}} (=6:5, E{{music|b}}, 315.64 cents, {{audio|Just minor third on C.mid|Play}}).<ref name="Benson"/>
+The size of this scale step may also be precisely derived from using [[minor seventh|9:5]] {{big|(}}B{{music|b}}, 1017.60&nbsp;cents, {{audio|Greater just minor seventh on C.mid|Play}}{{big|)}} to approximate the interval {{nobr|{{math| {{sfrac| [[perfect fifth|3:2]] | [[major third|5:4]] }} {{=}} 6:5 }} }} {{big|(}}E{{music|b}}, 315.64&nbsp;cents, {{audio|Just minor third on C.mid|Play}} {{big|)}}.<ref name=Benson/>
-{{Quote|Carlos' 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]].<ref name="Benson"/>}}
+: 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&nbsp;third, and four of them approximate a 6:5 minor&nbsp;third. In order to make the approximation as good as possible we minimize the [[mean square deviation]].<ref name=Benson/>
+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 .}}
-<math>\frac{9\log_2(3/2)+5\log_2(5/4)+4\log_2(6/5)}{9^2+5^2+4^2}\approx0.06497082462</math> and <math>0.06497082462\times1200=77.965</math> ({{audio|Alpha scale step on C.mid|Play}})
+: <math>\ \frac{\ 9\ \log_2\left( \frac{\ 3\ }{ 2 } \right) + 5\log_2\left( \frac{\ 5\ }{ 4 } \right) + 4\ \log_2\left( \frac{\ 6\ }{ 5 } \right)\ }{\ 9^2 + 5^2 + 4^2\ } \approx 0.06497082462\ </math>
-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 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>[[William Sethares|Sethares, William]] (2004). ''Tuning, Timbre, Spectrum, Scale'', p.60. {{ISBN|1-85233-797-4}}. Scale step of 78 cents.</ref>
-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}}]] [{{audio|Alpha scale harmonic seventh chord on C.mid|Play}}]."<ref name="Three"/>
+and <math>\ 0.06497082462 \times 1200 = 77.964989544\ </math> ({{audio|Alpha scale step on C.mid|Play}})
+At 78&nbsp;cents per step, this totals approximately 15.385 steps per [[octave]], however, more accurately, the alpha [[scale step]] is 77.965&nbsp;cents and there are 15.3915&nbsp;steps per octave.<ref name=Benson>{{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&nbsp;{{mvar|α}}-scale degrees to the octave. This is obtained by approximating the scale degree to 78.0&nbsp;cents.}}</ref><ref>{{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=...&nbsp;scale step of 78&nbsp;cents.}}</ref>
+Though it does not have a [[perfect 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 {{sfrac| 7 | 4 }}, i.e., [[septimal whole tone|{{sfrac|8|7}}]] [{{audio|Alpha scale harmonic seventh chord on C.mid|Play}}]."<ref name=Three/>
 {| class="wikitable"
@@ Line 32: / Line 39: @@
 |align=center|septimal major second
 |align=center|3
-|align=center|233.90
+|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|&minus;3.78
 |-
 |align=center|major third
 |align=center|5
-|align=center|389.83
+|align=center|389.82
 |align=center|5:4
 |align=center|386.31
@@ Line 46: / Line 60: @@
 |align=center|perfect fifth
 |align=center|9
-|align=center|701.69
+|align=center|701.68
 |align=center|3:2
 |align=center|701.96
@@ Line 52: / Line 66: @@
 |-
 |align=center|harmonic seventh
-|align=center|12
+|align=center|octave&minus;3
-|align=center|935.58
+|align=center|966.11
 |align=center|7:4
 |align=center|968.83
-|align=center|&minus;33.25
+|align=center|&minus;2.72
 |-
 |align=center|octave
 |align=center|15
-|align=center|1169.48
+|align=center|1169.47
 |align=center|2:1
 |align=center|1200.00
-|align=center|&minus;30.52
+|align=center|&minus;30.53
 |-
 |align=center|octave
@@ Line 74: / Line 88: @@
 ==See also==
-*[[Bohlen–Pierce scale]]
+* [[Bohlen–Pierce scale]]
-*[[Beta scale]]
+* [[Beta scale]]
-*[[Gamma scale]]
+* [[Gamma scale]]
-*[[Delta scale]]
+* [[Delta scale]]
+* [[Harmonic scale]]
-==Sources==
+==References==
-{{reflist}}
+{{reflist|25em}}
 {{Microtonal music}}
@@ Line 86: / Line 101: @@
 {{Scales}}
 {{Wendy Carlos}}
 {{DEFAULTSORT:Alpha Scale}}
 [[Category:Equal temperaments]]
 [[Category:Non–octave-repeating scales]]
 [[Category:Wendy Carlos]]
 {{music-theory-stub}}

v t e Wendy Carlos
Albums	Switched-On Bach (1968) The Well-Tempered Synthesizer (1969) Sonic Seasonings (1972) Switched-On Bach II (1973) Switched-On Brandenburgs (1980) Digital Moonscapes (1984) Beauty in the Beast (1986) Peter & the Wolf/Carnival of the Animals – Part II (1988)
Soundtrack albums	Stanley Kubrick's A Clockwork Orange (1972) Walter Carlos's Clockwork Orange (1972) The Shining (Original Sound Track) (1980) Tron: Original Motion Picture Soundtrack (1982)
Musical tunings	Alpha scale Beta scale Gamma scale Harmonic scale