Music Through Computation

Transcription

1 Music Through Computation Carl M c Tague July 7, 2003 International Mathematica Symposium

2 Objective: To develop powerful mathematical structures in order to compose interesting new music. (not to analyze existing music although inspiration often comes from existing music and analytical techniques)

3 Sound Spaces 6 Integers (2000) Schenkerian Analysis Xi-Operator Schenkerian Synthesis Helix of Fifths Model of Functional Harmony (ii-v-i) 7 (2002) Dissonance Curves Ripples Through Pitch Space (2003) Lament (work in progress)

4 6 Integers (2000) 2x as many notes time (ca. 5 min.)

5 Sound Spaces a sound function c ( B )m m-channel sounds a sound space B [ (L p ) m or whatever ]

6 Example: Piano intensity attack time duration B = Z R + piano ( R R )m

7 But why bother with sound spaces at all? Why not just work directly within (L p ) m (or whatever)?

8 Why use sound spaces? (L p ) m is dauntingly HUGE! Want to avoid the ultimate writer s block how do you ever get started in the space of all possible sounds? Want nice little representations of sounds inside the computer.

9 Why use sound spaces? Want musical topologies! the standard metric on (L p ) m is too rigid, unmusical. Natural (e.g. continuous) operations on the spaces should correspond to musical processes. E.g. variations might lie within neighborhoods: The Goldberg Variations

10 General approach to composition Inductively construct increasingly complex and specialized sound spaces until an entire piece of music is the image of a single, conspicuous point. Think of it as building increasingly powerful musical instruments.

11 Simple motivating example: a a a D a a the Brahms rhythm

12 Simple motivating example: a a a D a a the Brahms rhythm

13 Simple motivating example: a a a D a a the Brahms rhythm

14 Simple motivating example: a a a D a a the Brahms rhythm

15 The General Xi (X) Construction

16 copies of an existing sound space B (each equipped with a potentially distinct sound function) The General Xi (X) Construction

17 The General Xi (X) Construction a new space A copies of an existing sound space B (each equipped with a potentially distinct sound function)

18 The General Xi (X) Construction a new space A a new inheritance function f 3 : A Æ B 3 copies of an existing sound space B (each equipped with a potentially distinct sound function)

19 The General Xi (X) Construction a new space A a new inheritance function f 3 : A Æ B 3 copies of an existing sound space B (each equipped with a potentially distinct sound function) ( ( ( )m )m )m

20 The General Xi (X) Construction a new space A a new inheritance function f 3 : A Æ B 3 copies of an existing sound space B (each equipped with a potentially distinct sound function) + ( ( ( ( )m )m )m )m

21 The General Xi (X) Construction a new space A a new inheritance function f 3 : A Æ B 3 copies of an existing sound space B (each equipped with a potentially distinct sound function) ( )m induced map making A into a sound space ( )m + ( ( )m )m

22 Xi for diagram chasers: Given a list of sound functions { c i : Bö( )m } i=1, N and a family of inheritance functions { f n : AöB n } n make A into a sound space via the induced map: f N c 1 L c N + A B N [( ) ] m N induced ( )m

23 So, with Xi in hand, we can build new sound spaces by constructing a few: fundamental sound spaces families of inheritance functions and arranging them into hierarchies. This is precisely what we do, next

24 A simple sound space: Consider the plane 2 as a sound space by regarding the point (t,d) as a hum at time t with duration d. (t,d) Hum ( t t+d )m The so-called time-vector approach.

25 Two Useful Families of Inheritance Functions: The diagonal maps for simultaneity: A a n A n (a,,a) just make n copies For successiveness: 2 (t,d) a n ( 2 ) n ( ) a 3 ( )( )( ) [(t,d ), (t+d,d ),, (t+(n-1)d,d )] where d =d/n even subdivision of an interval into n subintervals

26 Application: Rhythm Trees a a a X(a,(H,X(a,(H,X(a,(H,H,H)))))) where H=Hum D a a X(D,(X(a,(H,H)),X(a,(H,H,H)))) the Brahms rhythm These functions, evaluated at (0,1) give the corresponding rhythms performed in the time interval (0,1).

27 Instead of Time Vectors, Functions of the Unit Interval [0,1]T Instead of (t,d): f(1)=t+d f(0)=t [ ] 0 1 But, we can use nonlinear functions to achieve accel and deceleration and expressive rhythms!

28 Inheritance Functions for [0,1]T D just as before. Generalized a: ( 0 1 a 3 ) ( ) ( ) ( )

29 But why not just use time vectors and apply a global time map at the end? The hierarchical [0,1]T approach permits local modification of the time map. Furthermore, different simultaneous components of a piece can have distinct time maps!

30 Products of Inheritance Functions We can form products of inheritance functions and thus pass several attributes of sound through the tree at once in parallel. E.g. rhythm, pitch, harmony, dynamics

31 Sound Spaces 6 Integers (2000) Schenkerian Analysis Xi-Operator Schenkerian Synthesis Helix of Fifths Model of Functional Harmony (ii-v-i) 7 (2002) Dissonance Curves Ripples Through Pitch Space (2003) Lament (work in progress)

32 # E #B#Fx A D # # G # C # F # A E B C G D F B b E b A b D b G b F b C b b G FC D A E B B b E b G # C # F # Circle of Fifths

33 # E #B#Fx A D # # G # C # F # A E B C G D F B b E b A b D b G b F b C b b G FC D A E B B b E b G # C # F # Circle of Fifths Enharmonic equivalence : =?

34 # E #B#Fx A D # # G # C # F # A E B C G D F B b E b A b D b G b F b C b b G FC D A E B B b E b G # C # F # Circle of Fifths Enharmonic equivalence : =?

35 # Helix of Fifths E #B#Fx A D # # G # C # F # A E B C G D F B b E b A b D b G b F b C b b G FC D A E B B b E b G # C # F # Circle of Fifths Enharmonic equivalence : =?

36 # Helix of Fifths Strongly inspired by the work of Eric Regener E #B#Fx A D # # G # C # F # A E B C G D F B b E b A b D b G b F b C b b G FC D A E B B b E b G # C # F # Circle of Fifths Enharmonic equivalence : =?

37 # E #B#Fx A D # # G # C # F # A E B C G D F B b E b A b D b G b F b C b b G FC D A E B B b E b G # C # F # Helix of Fifths Circle of Fifths Strongly inspired by the work of Eric Regener Give it the algebraic structure (Z,+). ( Î ) nota(n) := n mod 7, n / 7 (letter name, accidental) Enharmonic equivalence : =?

38 Then, look at the sublattice: H := {(h, p) Œ Z 2 : 4h - p Œ 7Z} à (Z 2,+) where h is helix position and p is staff position. It has a positive cone: P := (h, p) Œ H : p 0 { } and a corresponding absolute value: (h, p) := ( h sign(p), p ).

39 Brief Introduction to Functional Harmony

40 Brief Introduction to Functional Harmony

41 Brief Introduction to Functional Harmony I II III IV V VI VII I

42 Brief Introduction to Functional Harmony I II III IV V VI VII I II V I

43 Brief Introduction to Functional Harmony I II III IV V VI VII I II V I II V I II V I II V I II V I

44 (ii-v-i) 7 (2002) I ii V I ii V I ii V I ii V I ii V I ii V I ii V I ii V I ii V I 3 3 = = = = =2187 total progession length: 3280

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61 Here, the helix was fitted with 3-limit tuning. More generally: use factorization of rationals to biject into O : a a a p i n i (finite support) p i the i th prime and lift a nice metric from.

62 Sound Spaces 6 Integers (2000) Schenkerian Analysis Xi-Operator Schenkerian Synthesis Helix of Fifths Model of Functional Harmony (ii-v-i) 7 (2002) Dissonance Curves Ripples Through Pitch Space (2003) Lament (work in progress)

63 Dissonance of 2 Pure Sine Tones Kameoka, Kuriyagawa & Sethares Dissonance Empirical Psychoacoustics Frequency difference (not ratio hence interval corresponding to maximum dissonance depends on register)

64 Dissonance of 2 Harmonic Buzzes Dissonance Frequency ratio (of 2 harmonic buzzes)

65 Ripples Through Pitch Space (2003) 4 Mvts Potential Field 20 evenly-spaced particles Pitch space

66 Sound Spaces 6 Integers (2000) Schenkerian Analysis Xi-Operator Schenkerian Synthesis Helix of Fifths Model of Functional Harmony (ii-v-i) 7 (2002) Dissonance Curves Ripples Through Pitch Space (2003) Lament (work in progress)

67 But what about melodies?

68 But what about melodies? Idea: Do Schenkerian analysis in reverse via Xi Schenkerian synthesis!

69 But what about melodies? Idea: Do Schenkerian analysis in reverse via Xi Schenkerian synthesis! But what is Schenkerian analysis?

70 Introduction to Schenkerian Analysis in One Page!

71 Introduction to Schenkerian Analysis in One Page! Happy Birthday!

72 Introduction to Schenkerian Analysis in One Page! Happy Birthday! Relative structural significance?

73 Introduction to Schenkerian Analysis in One Page! Happy Birthday! Relative structural significance?

74 Introduction to Schenkerian Analysis in One Page! Happy Birthday! Relative structural significance?

75 Introduction to Schenkerian Analysis in One Page! Happy Birthday! Relative structural significance?

76 Introduction to Schenkerian Analysis in One Page! Happy Birthday! Relative structural significance?

77 Inheritance Functions for Schenkerian Synthesis ascending descending to from

78 Lament (work in progress) 2 Mvts (so far) Lyre from Ur (from ca B.C.) Source: Oriental Institute Melodic line created with Schenkerian Synthesis: embedded within self.

79 Sound Spaces 6 Integers (2000) Schenkerian Analysis Xi-Operator Schenkerian Synthesis Helix of Fifths Model of Functional Harmony (ii-v-i) 7 (2002) Dissonance Curves Ripples Through Pitch Space (2003) Lament (work in progress)

80 Summary: Mathematical structures were described which can be used to produce music through computation. Most important was the versatile Xi Operator, which may be used to construct models for expressive rhythm, functional harmony and melody.

81 Please visit my web page to hear these pieces and others.

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84 Want the mathematical structures to be musically meaningful (whatever that means) at least inspired or informed by musical experience, intuition or theory.

85 Can also use [0,1]T to control continuous parameters of sound. E.g. loudness

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87 I call this construction the Xi-Operator (X) Given a family of inheritance functions and an ordered list of sound spaces, it produces a new sound space: ( { f n : A Æ B n } n ) X A Æ (L p ) m ( c i : B Æ (L p ) m ) i=1,kn ( )

88 An alternate view; inductive use of Xi as information propagating through a tree: f 2 g 2 h 3 i 2 Information flows down the tree, manipulated at each branch by the local inheritance function until it reaches the Os, which denote possibly distinct, existing sound spaces.