{{Short description|Musical scale invented by Wendy Carlos}} Wendy Carlos devised several musical scales. Several are non-octave repeating scales, which Carlos named alpha, beta, and gamma. Each approximates just intervals using multiples of a single interval. She also used the upper partials of the harmonic series to tune a chromatic scale. Carlos showcased these scales on her 1986 album ''Beauty in the Beast''.
==Background== As a teenager, Wendy Carlos was fascinated by alternate tunings and experimented on her parents' piano.<ref name=EM/> She admired Harry Partch's microtonal music but felt he made a mistake by designing instruments that had rapid decay and weak overtones.<ref name=Milano/><ref name=CMJ/>{{rp|37}}
A 1980 study devised a method of comparing alternate tunings to consonant intervals.<ref> Yunik, M. and G. W. Swift. "[https://quod.lib.umich.edu/cache//b/b/p/bbp2372.1982.046/bbp2372.1982.046.pdf A Microprocessor Based Keyboard Instrument for Microtonal Music]", ''Proceedings of the International Computer Music Conference''. San Francisco: Computer Music Association, 1982. 589.</ref><ref>Yunik, M. and G. W. Swift. "[https://doi.org/10.2307/3679466 Tempered Music Scales for Sound Synthesis]", ''Computer Music Journal'' 4, no. 4, 1980. [https://www.jstor.org/stable/3679466 60–65].</ref> Carlos plugged asymmetric ratios into the model and noticed several distinct peaks of consonance. When each of the three peaks was doubled, Carlos found similar consonance.<ref name=CMJ>Carlos, Wendy. "[https://doi.org/10.2307/3680176 Tuning: At the Crossroads]", ''Computer Music Journal'', vol. 11, no. 1. 1987. [https://www.jstor.org/stable/3680176 29–43]."</ref>{{rp|42}} Carlos recalled, "since this is virgin territory, like Christopher Columbus I hereby christen these three peaks in the plot Carlos Alpha, Beta, and Gamma."<ref name=EM/>
The practical breakthrough came when Stoney Stockwell explained to Carlos how to access a synthesizer's tuning table, where pitch frequencies are stored. She was able to reverse engineer the equally tempered table into a new software with a 1.5 cent resolution, which enabled her to implement the microtunings.<ref name=EM/>
Carlos' 1984 album, ''Digital Moonscapes'', showcased the virtual orchestra she dubbed the "LSI Philharmonic" in reference to large-scale integration computer chips. It represented a peak of what she felt she could do with standard synthesizer timbres. For her next album, Carlos developed alternate tunings in order to generate new timbres. ''Beauty in the Beast'' features music written in these tuning systems.<ref name=Sewell>Sewell, Amanda. ''Wendy Carlos: A Biography''. Oxford University Press, 2020.</ref>{{rp|156}}
In 1987, Carlos wrote about her tuning experiments in ''Computer Music Journal'' (CMJ). The issue included a vinyl flexi disc with several recordings that demonstrated the unique timbres she created. The soundsheet also included excerpts from ''Beauty in the Beast'' and her 1987 album ''Secrets of Synthesis''.<ref name=Soundsheet>"[http://www.jstor.org/stable/3680180 Soundsheet Examples]", ''Computer Music Journal'', vol. 11, no. 1. 1987. 76.</ref> In 2006, CMJ reissued the contents of its vinyl inserts on DVD.<ref>"[http://www.jstor.org/stable/4618012 DVD Program Notes]", ''Computer Music Journal'' 30, no. 4. 2006. 135–41.</ref>
==Alpha scale== The steps of the alpha scale are 78 cents.<ref name=CMJ/> In a traditional octave with a 2:1 tuning ratio, the alpha scale yields 15.385 steps. There are four steps to the minor third, five to the major third, and nine to the perfect fifth.<ref name=Three>Carlos, Wendy. "[http://www.wendycarlos.com/resources/pitch.html Three asymmetric divisions of the octave]", wendycarlos.com. Accessed April 8, 2026.</ref><ref name=Milano>Milano, Dominic. "[http://www.wendycarlos.com/other/PDF-Files/Kbd86Tunings*.pdf A many-colored jungle of exotic tunings]", ''Keyboard''. November, 1986. 64f.</ref><ref name=Liner>Carlos, Wendy. ''Beauty in the Beast''. Liner notes. East Side Digital Records, 2000.</ref><ref name=Benson>Benson, David J. ''[https://logosfoundation.org/kursus/music_math.pdf Music: a Mathematical Offering]''. Cambridge University Press, 2007. 221–4.</ref><ref name=Sethares>Sethares, William. ''[https://archive.org/details/tuning-timbre-spectrum-scale-2nd-ed/mode/2up Tuning, Timbre, Spectrum, Scale]'' New York: Springer., 2005. 60.</ref>
The alpha scale produces "wonderful triads".<ref name=Milano/> Initially, Carlos overlooked inversions of the alpha scale. She discovered that they yield "excellent harmonic seventh chords", which is part of why the alpha scale was one of Carlos' favorite tunings.<ref name=Three/>
==Beta scale== The steps of the beta scale are 63.8 cents. That yields 18.809 steps per traditional octave. The beta scale has five steps to the minor third, six to the major third, and eleven to the perfect fifth.<ref name=CMJ/> Carlos developed the beta scale by splitting a perfect fourth evenly in half. The scale is very similar to alpha, but its sevenths are more in tune.<ref name=Milano/>
==Gamma scale== The steps of the gamma scale are 35.1 cents. That yields 34.188 steps per traditional octave.<ref name=CMJ/> The gamma scale divides the minor third into 9 steps, the major third into 11, and the perfect fifth into 20 steps. Carlos joked that gamma has "too many notes".<ref name=Three/>
Carlos felt the gamma scale produced "nearly perfect triads", but it was too unwieldy to use on ''Beauty in the Beast''.<ref name=Milano/><ref name=EM/> She described it as a "third flavor" that fell between alpha and beta.<ref name=Three/>
==Harmonic scale== [[File:Harmonics to 32.png|thumb|left|Harmonic series above C.<ref>Blatter, Alfred. ''Revisiting Music Theory: Basic Principles''. Taylor & Francis Group, 2016. 5p.</ref><ref>Sauveur, Joseph. ''[https://ks15.imslp.org/files/imglnks/usimg/7/78/IMSLP480903-PMLP779371-Principes_d'acoustique_et_de_musique_-...-Sauveur_Joseph_bpt6k1510877z.pdf Principes d'acoustique et de musique, ou système général des intervalles des sons]''. Paris: Academie Royal des Sciences, 1701. 52.</ref>]] {| class="wikitable" style="float: right;" |+ Harmonic scale<ref name=CMJ/>{{rp|38}} ! Note !! Ratio !Cents |- | C || 1:1 |0.000 |- |D{{music|flat}} |17/16 |104.955 |- |D |9/8 |203.910 |- |E{{music|flat}} |19/16 |297.513 |- |E |5/4 |386.314 |- |F |21/16 |470.781 |- |F{{music|sharp}} |11/8 |551.318 |- |G |3/2 |701.955 |- |A{{music|flat}} |13/8 |840.528 |- |A |27/16 |905.865 |- |B{{music|flat}} |7/4 |968.826 |- |B |15/8 |1088.269 |} Carlos derived a chromatic scale from the fifth octave of the harmonic series. The scale begins on the 16th partial and runs to the 32nd, omitting numbers 23, 25, 29, and 31. Carlos called this a harmonic scale.<ref name=CMJ/>{{rp|37f}}
To enable modulation, Carlos transposed the harmonic scale on all 12 chromatic pitches to generate a theoretical division of the octave into 144 steps.<ref name=Milano/><ref name=EM>Freff. "[https://www.worldradiohistory.com/Archive-All-Music/Electronic-Musician/1986/Electronic-Musician-1986-11.pdf Tuning In To Wendy Carlos]", ''Electronic Musician''. November, 1986. 30–7.</ref>
In practice, she would retune the scale by triggering different fundamentals on a keyboard controller. The harmonic scale is employed in "That's Just It" and "Just Imaginings" on ''Beauty in the Beast''.<ref name=Milano/><ref name=CMJ/>{{rp|37f}} The track ends with a cycle of fifths in perfect tuning.<ref name=EM/>
Versions of Carlos' harmonic scale have also been used by Ezra Sims and Franz Richter Herf.<ref>Sims, Ezra. "[https://doi.org/10.2307/3680228 Observations on Microtonality Issue]", ''Computer Music Journal'', vol. 11, no. 4, 1987. 8f.</ref>
==Approximations== Carlos derived the alpha, beta, and gamma scales by dividing the octave asymmetrically. She found the results to be a pleasing iteration of just intonation and decided to create artificial octaves with various hardware.<ref name=CMJ/><ref name=EM/> Ignoring the octave's natural ratio allowed her to focus on generating justly tuned triads.<ref name=Sills>Sills, Andrew V. "[https://arxiv.org/pdf/2408.14551 Generalized Carlos scales]", ''Journal of Mathematics and Music'' 19.3 (2025): [https://arxiv.org/abs/2408.14551 243-249].</ref>
Mathematicians have approximated Carlos' scales by different means. Minimizing the mean square deviation makes the approximation more accurate. The beta scale's first five steps approximate a minor third in its traditional 6:5 ratio, six steps yield the 5:4 major third, and the 3:2 perfect fifth has eleven steps. The proportion of each scale degree can be expressed mathematically:
<math>\frac{11\log_2{(3/2)}+6\log_2{(5/4)}+5\log_2{(6/5)}}{11^2+6^2+5^2}\approx0.05319411048</math> and <math>0.05319411048\times1200=63.832932576</math><ref name="Benson"/>
==Audio examples== {{Unreferenced section|date=April 2026}} ;Alpha scale *{{audio|Alpha scale minor third on C.mid}} *{{audio|Alpha scale step on C.mid}} *{{audio|Alpha scale major triad on C.mid}} *{{audio|Alpha scale minor triad on C.mid}} *{{audio|Alpha scale harmonic seventh chord on C.mid}} ;Beta scale *{{audio|Beta scale step on C.mid}} *{{audio|Beta scale perfect fourth on C.mid}} *{{audio|Beta scale 15 steps on C.mid}} *{{audio|Beta scale major triad on C.mid}} *{{audio|Beta scale minor triad on C.mid}} *{{audio|Beta scale dominant seventh on C.mid}} ;Gamma scale *{{audio|Gamma scale step on C.mid}} *{{audio|Gamma scale neutral third on C.mid}}
==See also== *19 equal temperament * Bohlen–Pierce scale
==References== {{reflist}}
==External links== ;Wendy Carlos scales at [https://en.xen.wiki/w/Main_Page Xenharmonic Wiki]: * [https://en.xen.wiki/w/Carlos_Alpha Alpha]. *[https://en.xen.wiki/w/Carlos_Beta Beta]. *[https://en.xen.wiki/w/Carlos_Gamma Gamma]. *[https://en.xen.wiki/w/Carlos_harmonic_scale Harmonic].
*''[https://archive.org/details/beauty-in-the-beast/ Beauty in the Beast]'' at Internet Archive. {{Microtonal music}} {{Musical tuning}} {{Scales}} {{Wendy Carlos}}
Category:Equal temperaments Category:Non–octave-repeating scales Category:Wendy Carlos