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Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Quantum Music: Applying Quantum Theory to Music Theory and Composition Yes´ e J. Felipe California State University, Los Angeles QMATH 13, Georgia Tech, October 10, 2016 Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Table of Contents Introduction: What is Music? 1 How to quantize Music? 2 Melody: A simple example 3 Harmony 4 Challenges and Future Work 5 Bibliography 6 Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Introduction: What is Music? Music is a collection of sounds, or tones , that feature pitch, beat, and volume. Music consists of 3 elements: Melody : A set of musical notes, or tones, that are played in succession. Harmony : A set of musical notes that are played simultaneously. Rhythm : Duration of these sounds (or lack of sounds) in time. Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Traditionally, sounds called whole tones are represented as follows: For a single octave, all of the sounds used in music are represented by the Chromatic scale : Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Two elements of music notation that are used to denote pitch are ♭ , the flat and ♯ , the sharp . An alternate way to represent tones is by using the number 0 to represent no sound, and the numbers 1 − 12 as follows: C → 1, C ♯ → 2, D → 3, . . . , A ♯ → 11, B → 12 Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography How to quantize Music? A quantum tone can be represented be a quantum musical state | ψ � . A quantum musical state would be defined as: | ψ � = α 0 | ψ 0 � + α 1 | ψ 1 � + α 2 | ψ 2 � + . . . + α 12 | ψ 12 � (1) With | ψ � ∈ C 13 and � i | α i | 2 = 1. Here, | ψ i � are the orthogonal unit vectors:       1 0 0 0 1 0       | ψ 0 � =  , | ψ 1 � =  , . . . , | ψ 12 � =  .   .   .  . . .       . . .     0 0 1 Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography We can also define a flat and sharp operators such that: ˆ ♭ | ψ i � = ψ i − 1 | ψ i − 1 � (2) ˆ ♯ | ψ i � = ψ i +1 | ψ i +1 � (3) Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Melody: A simple example We start by constructing a simple system of two quantum tone states 1 | ψ 0 � + 1 | ψ � 1 = √ √ | ψ 1 � (4) 2 2 1 | ψ 0 � + 1 | ψ � 2 = √ √ | ψ 1 � (5) 2 2 Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography The system would have the following 4 possible outcomes, with equal probabilities: Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Similarly for the following system quantum tone states, 1 | ψ 0 � + 1 | ψ 1 � + 1 | ψ � 3 = √ √ √ | ψ 6 � (6) 3 3 3 1 | ψ 0 � + 1 | ψ 1 � + 1 √ √ √ | ψ � 4 = | ψ 6 � (7) 3 3 3 Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography The system would have the following 9 possible outcomes, with equal probabilities: Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography An advantage of this representation is that melodies can be composed by applying transformations, to tone states. In the previous example, | ψ � 4 = ˆ 1 | ψ � 3 (8) Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Harmony: Entanglement? Quantum Harmony can be thought as an entangled state of more than one quantum note, and could be represented using quantum chords. Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Challenges and Future Work Incorporating rhythm. Incorporating volume. Finally, how to implement all of these ideas? Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory

Introduction: What is Music? How to quantize Music? Melody: A simple example Harmony Challenges and Future Work Bibliography Bibliography V. Putz, K. Svozil Quantum Music . arXiv:1503.09045v2 D.E. Parson Quantum Composition and Improvisation . aaai.org/ocs/index.php/AIIDE/AIIDE12/paper/view/5473 C. Cohen-Tannoudji, B. Diu, F. Laloe Quantum Mechanics . Wiley VCH, 1st edition, 1992. J.P. Clendinning, E.W. Marvin The Musician’s Guide to Theory and Analysis . W.W. Norton, 1st edition, 2004. Yes´ e J. Felipe Quantum Music: Applying Quantum Theory to Music Theory