Song Image of the Song of Time JP Marceaux December 20, 2017

Dedicated to and written for Conway LeBleu One of the most interesting innovations of the 20th century abstract art movement was the development of pictorial representations of music, seen particularly in the correspondence between Wassily Kandinsky and Arnold Schoenberg. Here, I use quantum wavefunctions to achieve the same goal of creating an image that reflects the qualities of a piece of music. I do this by defining wavefunctions for the different notes present in a song and graphing the superposition of all notes. The song chosen in this case was the "Song of Time". The central object of the art is The Ledgend of Zelda Song of Time, depicted in musical notation below.

Figure 1: https://www.youtube.com/watch?v=SEBybsl_k1U). A construction of a numeric encodage of the song is necessary in order to arrange the song’s information in a form that can be interpreted by the computer. I choose to do this with a structural definition of each note, wherein each note is associated with a block that contains numeric description of its essence. Memory blocks containing a note name, a note index, a chromatic index, a temporal location, a temporal length, and a frequency represent a note. The algorithm uses this information to construct the image that is displayed on the monitor. Here, I choose to omit the frequency of a note and give only its mathematical definition. Note A D F A D F A C’ B G F G A D C E D Note index 5 1 3 5 1 3 5 7 6 4 3 4 5 1 0 2 1 Chromatic index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Time 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 1 1 1 1 Length 1 2 1 1 2 1 12 1 1 1 1 1 2 2 2 2 2 The prime over the C denotes that the note occurs an octave above. The note number is an integer enumeration defined by setting the first C equal to 0 and continuing in integer steps such that the second C is equal to 7. The note length is in units of quarter-notes. The frequency is determined according to the equation: n

fn ≡ f0 2 7 , where n is the note number and f0 is an arbitrary definition of the center frequency of the scale. The form is such that we are working over a scale of 7 notes. The chromatic index of a note of index n is defined by: cn ≡ mod7 (n), (1) 1

such that the chromatic index takes on the 7 integer values between 0 and 6 inclusive. We map each integer cn ∈ [0, 6] to a color. In the final image, I use the concept of overtones to add detail and interest. Overtones, or harmonics, are physical phenomenon that occur in all real instruments. When a musician plucks a string, blows a note, or strikes a membrane, the resulting waveform contains a superposition harmonics or basis states, normally taken as sines and cosines. This superposition can be defined by its fundamental note, which is of lowest frequency and often of highest amplitude. We assume that all harmonics of the notes occur in integer multiples of the fundamental frequency. Thus, an overtone of the note f0 is mf0 , where m = 1, 2, 3.... To a good approximation, all the integer multiples of a given note will be equal in frequency to notes of higher frequency. However, the note name will change. For example, in the chromatic scale, the 3rd harmonic of the note C0 is G2. The note name, not the frequency, is used to define the color index. We now consider the chromatic index of an overtone. Let us label the j th overtone of a note of index n by its frequency with the notation: fnj , where

(2) n

fnj = jf0 2 7 .

(3)

The chromatic index of a note is defined in terms of its note index. The frequency of an overtone can be used to define the overtone’s note index. We may associate the frequency of an overtone with a new note index m and impose the equivalence: fm = fnj , which is not valid for all j because

n

m

2 7 = j2 7

is not true for all {n, j, m} ∈ Z3 . We shall impose the equivalence by rounding fnj to the nearest fm . This frequency fnj will be near the frequency of some other note of index m. Setting the two frequency equal: m n f0 2 7 = jf0 2 7 m n = + log(j), 7 7 where the passage between the two lines was accomplished with a logarithm of base 2. The note index of the jth overtone of a note of index m is thus: n = 7logm + j,

(4)

where the RHS is rounded to the nearest integer. We employ the notation: j Im

(5)

to represent the note index of the j th overtone of the note of index m. We use Equation (1) to find the chromatic index, remembering that we must round the index to the nearest whole integer: c(fnj ) = mod7 (round(7log(j) + m)).

(6)

In words, the above equation says that the chromatic index of the j th overtone of the note of index n is give by the modulus base 2 of the sum of the note index plus 8 times the logarithm base 2 of the harmonic number. We will make use of the notation cjn ≡ c(fnj ) 2

(7)

We now consider the image of the song, and we invoke techniques used in the definition of a quantum mechanical wavefunction. We will consider the screen as a space-time graph, where the space values x are graphed on the ordinate and the time values t are graphed on the abscissa. We seek to define an overall wavefunction Ψ that represents the space-time development of the song. A convenient analogy is that the image produced is a graph of the temporal development of a string that plays the song of time. We associate each note index with a quantum number n, where n = 0, 1, 2.... We associate each chromatic index with a quantum number l, such that l takes on integer values ∈ [0, 6]. This choice of definition of l is similar to a numbering of the states of a particle of spin 3. The n quantum number will define the shape of the wavefunction and the l quantum number will define the color. The background of the image begins as black. The computer samples the overall wavefunction Ψ, which produces a color that the computer creates on the monitor. Ψ is composed of superposition of individual wavefunctions. I write the 3 wavelets plus a gabor wavelet that I will use in my definition of Ψ. I will pass from wavefunctions defined in only space to a space-time wavefunction. The first wavelet forms the fundamental basis upon which all superposition are drawn. It is: ϕln (x) = sin(nπx)χl , (8) where x has a domain ∈ [0, 1] and χl is the quantum color vector associated with the color value l. The second wavelet defines the harmonic superposition. It is: ψn (x, t) =

X 1 ck n k=1

k

ϕI k (x, t),

(9)

n

where we use Equations (5) and (7) to define the note index and color index of the overtone respectively. Let us define the gabor wavelet of the block b by: gb = e

(t−τb )2 αγb

,

where α is the base width of each note and τb and γb are the temporal displacement and the augmented width of the block of index b respectively. This wavelet will be used to define the temporal description of the wave. The third wavelet defines the wavefunction of a single block Φb (x, t) = ψb.n (x) ∗ gb (t) =

X 1 ck b.n k=1

k

!

ϕI k (x) e

(t−τb )2 αγb

,

(10)

b.n

where the symbol b.n denotes the note index contained in the block of index b. This is C++ code for the content n of the structure b. The wavefunction for the entire piece of music is finally the sum of each individual block’s wavefunction: Ψ(x, t) =

16 X b=0

3

Φb (x, t)

(11)

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