THE ARCANE ARCHIVE
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Subject: A Look at the Musical & Color Scales
                                      
   I have been interested in the attributes of light and sound for a long
   time. The effects of resonance, harmonics, harmony, discord and the
   many other aspects of sound have fed my interest in music for some
   forty years now. Though I have some degree of red-green
   color-blindness I have enjoyed painting and I have made my living in
   the printing trade for twenty five years, developing some expertise in
   the subtle manipulation of colored pigments to achieve a variety of
   results.
   
   When I encountered the system of correspondence between color and
   sound which is presented as "The Queens Scale" in much esoteric
   teaching I was intrigued. Of course my first question was "What is the
   connection?" Why is it said that the pitches of the western
   twelve-tone scale correspond to the twelve primary, secondary and
   tertiary colors? Besides the very nice (and suggestive) fact that
   there are twelve members of each set, what do they have in common? It
   seems to me that the clear answer is that light and sound are both
   conveniently described as waves, albeit of greatly different
   frequencies. Perhaps, I thought, there is an instructive relationship
   between the frequencies of the light of the twelve colors and the
   frequencies of the sounds of the twelve tones. I decided to
   investigate.
     _________________________________________________________________
   
   An octave is an interval defined by a factor of two. For example, the
   generally accepted standard for the frequency of the note A3 (A below
   middle C) is 220 Hz (Hz=cycles per second.) If you double that
   frequency to 440 Hz you have A4 (A above middle C.) Double it again to
   880 Hz and you have A5, and so on. This 1:2 ratio holds for all notes
   said to be an octave apart. The twelve tones of the scale fall within
   a single octave.
   
                       A A# B C C# D D# E F F# G G# A
      220.000 233.082 246.942 261.626 277.183 293.665 311.127 329.628
                  349.228 369.994 391.995 415.305 440.000
                                      
   The ratio of one tone to the next is 1 to the 12th root of 2 (approx.
   1.059463094.) This scheme is called equal temperament, as the change
   in frequency from one note to its octave is made in 12 EQUAL steps.
   Equal temperament is the current conventional standard for tuning
   musical instruments.
   
   The first question I had was whether the frequencies of the colors
   might fall within a one octave range. I figured that it would be most
   convenient if they did, but also that it would be very unlikely. So I
   looked it up. My dictionary gave me a range of wavelengths (in
   nanometers) for the primary and secondary colors. I also consulted a
   color image of the spectrum of visible light and determined my own
   best estimation of the wavelengths of the colors.
   
                                spectrum.jpg
                                      
                 Source Violet Blue Green Yellow Orange Red
     Dictionary 400-450 nm 450-500 nm 500-570 nm 570-590 nm 590-610 nm
                                 610-780 nm
           Observation 395 nm 450 nm 535 nm 580 nm 645 nm 710 nm
                                      
   I was surprised to find that the wavelengths of the colors did fall
   approximately within the range of one octave! Perhaps they actually do
   correspond mathematically to the pitches of the twelve tone scale. The
   next step was to determine what note each of the colors actually is.
     _________________________________________________________________
   
                      Velocity=Wavelength x Frequency
     _________________________________________________________________
   
   This formula describes the relationship between wavelength and
   frequency. Since the speed of light is known and fixed, reliable
   numbers can be generated for the frequencies of the colors. For
   example:
   
   2.997925 x 10+8 m/s (speed of light)/ 3.95 x 10-7 nm (wavelength of
   violet) = 7.58968354 x 10+1 4 Hz (frequency of violet)
   
   Starting from the wavelengths arrived at above, as well as intervening
   numbers for the tertiary colors, and calculating their frequencies in
   this manner provides these results:
   
                   Color Wavelength (nm) Frequency (GHz)
                        Red-Violet 340 881,742.6471
                          Violet 395 758,968.3544
                        Blue-Violet 420 713,791.6667
                           Blue 450 666,205.5556
                        Blue-Green 490 611,821.4286
                           Green 535 560,359.8131
                       Yellow-Green 555 540,166.6667
                          Yellow 580 516,883.6207
                       Yellow-Orange 610 491,463.1148
                          Orange 645 464,794.5736
                        Red-Orange 670 447,451.4925
                            Red 710 422,242.9577
                                      
   Since both sound and light are described as waves, and can be measured
   in terms of frequency they can be thought of as existing at different
   places on a continuum. Admittedly, they are waves of different
   natures. Light is electromagnetic, while sound is a pressure wave in a
   medium such as air or water, so in that sense they cannot be equated.
   But since they do share "waviness," since they are both examples of
   the cyclic nature of reality, it may be interesting to see what
   happens if we think of them as being located at different parts of the
   same spectrum or as different pitches on the same scale.
     _________________________________________________________________
   
   So, now let us "scale down" these light frequencies. By applying the
   idea of octaves (as described above) we can reduce their frequencies
   to the range of musical pitch, by dividing by 2, 40 times (240.)
   
    Color Wavelength (nm) Frequency (GHz) "Reduced" Frequency (Pitch) Hz
                     Stand. Pitch (Equal Temp.) Hz NOTE
              Red-Violet 340 881,742.6471 801.9403 739.989 F#
                 Violet 395 758,968.3544 690.2777 698.457 F
              Blue-Violet 420 713,791.6667 649.1897 659.255 E
                 Blue 450 666,205.5556 605.9104 622.254 D#
               Blue-Green 490 611,821.4286 556.4483 587.330 D
                 Green 535 560,359.8131 509.6443 554.365 C#
              Yellow-Green 555 540,166.6667 491.2787 523.251 C
                 Yellow 580 516,883.6207 470.1029 493.883 B
             Yellow-Orange 610 491,463.1148 446.9831 466.164 A#
                 Orange 645 464,794.5736 422.7282 440.000 A
              Red-Orange 670 447,451.4925 406.9548 415.305 G#
                  Red 710 422,242.9577 384.0277 391.996 G
                                      
   Well, that works out fairly well. The pitches seem to be generally in
   the range of the western twelve tone scale, and they are more or less
   within one octave. In some cases, there is a fairly close coincidence
   of frequencies, Violet (690 Hz) and F (698 Hz) for example, but
   overall you would be hard pressed to say that it is clear that certain
   colors correspond to certain pitches. The wavelengths for the colors
   with which we started, however, were purely subjective and fairly
   arbitrary. Other people may well have selected other wavelengths for
   the various colors. My selections may or may not be "correct." They
   certainly can't be said to be in "equal temperament" as described
   above.
     _________________________________________________________________
   
   What would happen if we "went the other way?" That is to say, to start
   with the standard pitches and step them UP by octaves into the range
   of visible light - to reverse the process we have just used.
   
   Stand. Pitch (Equal Temp.) Hz NOTE Frequency (GHz) Wavelength (nm) My
                        original guesses (nm) Color
               739.989 F# 813626.5101 368.4645 340 Red-Violet
                 698.457 F 767961.1633 390.3746 395 Violet
               659.255 E 724858.8154 413.5874 420 Blue-Violet
                  622.254 D# 684175.6164 438.1806 450 Blue
               587.330 D 645775.7899 464.2362 490 Blue-Green
                 554.365 C# 609531.1800 491.8411 535 Green
              523.251 C 575320.8238 521.0875 555 Yellow-Green
                 493.883 B 543030.5473 552.0730 580 Yellow
             466.164 A# 512552.5848 584.9010 610 Yellow-Orange
                 440.000 A 483785.2189 619.6810 645 Orange
               415.305 G# 456632.4411 656.5291 670 Red-Orange
                   391.996 G 431003.6316 695.5684 710 Red
                                      
   This gives us a "scale" of wavelengths which are absolutely within a
   one octave range and are in equal temperament. But, they don't match
   up exactly with the "accepted" wavelengths of the colors. Perhaps we
   can make one more adjustment to see what happens. We can shift the
   standard musical scale down a bit to better align with the frequencies
   of the colors.
     _________________________________________________________________
   
   We can apply our octave ratio to the musical notes and scale them DOWN
   just as well as we can scale them UP. This can give us very low
   frequencies - on the order of cycles per MINUTE, or DAY or YEAR! We
   might say that the orbit of the moon is a G#! Well, by scaling down C
   Natural (523.2511305981 Hz) we reach 1.022 Hz. Let us shift that very
   low C to exactly 1 Hz; that's one cycle per second and gives us a
   pitch of 512 Hz for C, instead of 523.25, a drop of about a third of a
   semi-tone.
   
   Adjusted Pitch (Eq. Temp.) Hz NOTE Frequency (GHz) Wavelength (nm) My
                        original guesses (nm) Color
               724.077 F# 796131.0811 376.5617 340 Red-Violet
                 683.438 F 751447.6772 398.9533 395 Violet
               645.079 E 709272.1601 422.6763 420 Blue-Violet
                  608.874 D# 669463.7728 447.8099 450 Blue
               574.700 D 631889.6587 474.4381 490 Blue-Green
                 542.445 C# 596424.4175 502.6496 535 Green
              512.000 C 562949.6873 532.5387 555 Yellow-Green
                 483.263 B 531353.7493 564.2051 580 Yellow
             456.140 A# 501531.1551 597.7545 610 Yellow-Orange
                 430.539 A 473382.3745 633.2988 645 Orange
               406.374 G# 446813.4635 670.9567 670 Red-Orange
                   383.566 G 421735.7509 710.8539 710 Red
                                      
   The scheme detailed in this chart has these properties:
   
     The musical scale is in equal temperament.
   
     It is derived from a value of a very low C Natural of 1 Hz.
   
     The color scale is in equal temperament.
   
     It is derived from the values of the musical scale.
   
     The colors so derived come very close to the primary, secondary and
   tertiary colors seen on the spectrum of visible light. (The only one
   that seems a bit off to me is yellow. 564 seems a little on the green
   side, but not much.)
     _________________________________________________________________
   
   An additional area of inquiry is a musical scale in other than equal
   temperament. Perhaps a Pythagorean temperament (or some other
   historical temperament) would yield further interesting results.
   
   This Chart show the results of beginning with a Pythagorean scale.
   Since it is not in equal temperament, it is significant which note is
   used for the Do of the scale. Since we're finding that G seems to come
   closest to the frequency of Red which is the lowest of the "color
   notes" in the visible spectrum I've decided to use it as the Do, the
   first note of the scale, and again setting C to 512 Hz (starting from
   1 Hz.)
   
    Adjusted Pitch (Pyth. Temp.) Hz NOTE Frequency (GHz) Wavelength (nm)
                       My original guesses (nm) Color
              729.0000 F# 801543.9768 374.0188 340 Red-Violet
                 682.6667 F 750599.9380 399.4038 395 Violet
              648.0000 E 712483.5349 420.7711 420 Blue-Violet
                 606.8148 D# 667199.9449 449.3293 450 Blue
               576.0000 D 633318.6977 473.3675 490 Blue-Green
                 546.7500 C# 601157.9826 498.6917 535 Green
              512.0000 C 562949.9535 532.5385 555 Yellow-Green
                 486.0000 B 534362.6512 561.0282 580 Yellow
             455.1111 A# 500399.9587 599.1058 610 Yellow-Orange
                 432.0000 A 474989.0233 631.1567 645 Orange
              404.5432 G# 444799.9633 673.9940 670 Red-Orange
                  384.0000 G 422212.4652 710.0513 710 Red
     _________________________________________________________________
   
   Here it is setting the Do at D# (Blue). It gives the best value for
   Yellow and for Green. A# (Yellow -Orange) gives a similar result.
   
    Adjusted Pitch (Pyth. Temp.) Hz NOTE Frequency (GHz) Wavelength (nm)
                       My original guesses (nm) Color
              719.1879 F# 790755.4710 379.1216 340 Red-Violet
                 682.6667 F 750599.9197 399.4039 395 Violet
              639.2781 E 702893.7520 426.5118 420 Blue-Violet
                 606.8148 D# 667199.9286 449.3293 450 Blue
               576.0000 D 633318.6823 473.3675 490 Blue-Green
                 539.3909 C# 593066.6032 505.4955 535 Green
              512.0000 C 562949.9398 532.5385 555 Yellow-Green
                 479.4586 B 527170.3140 568.6824 580 Yellow
             455.1111 A# 500399.9465 599.1058 610 Yellow-Orange
                 432.0000 A 474989.0117 631.1567 645 Orange
              404.5432 G# 444799.9524 673.9940 670 Red-Orange
                  384.0000 G 422212.4548 710.0513 710 Red
     _________________________________________________________________
   
   Do you have any comments, critisisms, additions or observations? If
   you'll email them to me I'll post them here for others to see.
   
                                   email
                               Jeff Williams
                             teliesen@ihot.com