LOGICAL FOUNDATION OF MUSIC

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1 LOGICAL FOUNDATION OF MUSIC philosophicl pproch Im Anfng wr die Tt Goethe, Fust CARMINE EMANUELE CELLA

2 NATURE OF MUSICAL KNOWLEDGE Musicl knowledge cn e thought s complex system with dul nture: intuitive nd formlized Formlized nture is ctully logicl structure, sed on underlying lgers with wellstructured opertors Logicl structures involved with music (musicl logics) re not only truth-logics nd don t elong to single discipline Contriutes to musicl logics come from: philosophy, mthemtics, rtificil intelligence, musicl theory, computer music, etc. LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 2

3 SUSANNE LANGER S S APPROACH (1) In 1929 the Americn review The Monist pulished smll rticle y Susnne K. Lnger titled A set of postultes for the logicl structure of music Every system hs finite numer of possile configurtion For reltively simple systems (for exmple the chess gme) n exhustive serch for ech configurtion is possile, lthough difficult For complex systems however, this could e not possile (for exmple sciences, rts, etc.) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 3

4 SUSANNE LANGER S S APPROACH (2) The only possile thing in such systems is to find forml reltions mong some sic elements Lnger s hypothesis: music is system mde of some sic elements linked y definite principles A such set of principles constitutes the strct form of the music or its logicl structure nd is itself specil lger neither numericl nor Boolen ut of eqully mthemticl form nd menle to t lest one interprettion This logicl structure is descried y set of postultes LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 4

5 LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 5 BASIC POSTULATES (EXCERPTS) BASIC POSTULATES (EXCERPTS) Let K e set of elements, nd _ two inry opertions, C mondic reltion (property) nd < didic reltion. Then hold:...etc..., / 7. ) ( ) ( ) ( ) K / ( d,,, 6. ) ( ),(,, 5.,, 4. If 3. 2., If 1. r K K r d c d c K c c c K c K K K K K K = = = = = =

6 MUSICAL INTERPRETATION (EXCERPTS) The interprettion of the descried lger leds to the cretion of the forml structure of music: 1. If, re musicl elements, the intervl -with- is musicl element 2. If is musicl element, the unison -with- is musicl element 3. If, re musicl elements, the musicl progression -to- is musicl element 4. If, re musicl elements, nd if -to- = -to- then nd re the sme musicl element 5. If,, c re musicl elements then the intervl (-with-)-with-c is the sme intervl of -with-(-with-c) 6. If,, c re musicl elements the exists t lest musicl element d such s the intervl of the progression (-to-)-with-(c-to-d) is equl to the progression of the intervl (-with-c)-to-(-with-d) [counterpoint principle] etc LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 6

7 NOTES ON THE NEW ALGEBRA The postultes descrie new lger tht is not Boolen lger for the following resons: 1. _ it is non-commuttive 2. the zero of the lger hs n incomplete nture 3. there isn t the one of the lger All essentil reltions mong musicl elements cn e demonstrted from the postultes, for exmple: the repetitionl chrcter of the order of tones within the octve, the equivlence of consonnce-vlues of ny intervl nd ny repetition of itself, etc. LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 7

8 POSSIBLE EXTENSIONS Mny other reltions mong musicl elements cn e derived from the postulte-set Even complete development of it cn give us only the generl musicl possiilities The structures employed in Europen music require further specifictions s next-memer postulte for the series generted y <, determintion of the consonnt intervls other thn unisons nd repetitions, the introduction of T-function # nd, nd so on. Alterntive sets of restrictions upon originl K cn e used to derive different types of music (Hwiin, Gelic, etc.) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 8

9 A SET-THEORETICAL APPROACH Lnger s pproch suffers from n overemphsis on hrmony t the expense of contrpuntl texture It lcks of the temporl dimension: it s lmost impossile to pply Lnger s postultes to rel world exmple A more suitle pproch involves set-theory Our concern will then e to tke few steps towrd n dequte chrcteriztion of the musicl system int set-theoreticl terms: towrd strct musicl systems LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 9

10 ABSTRACT MUSICAL SYSTEMS (1) A temporl frme is n oredered qudruple <T, t-, -t, > stysfying the following xioms: T1. T T2. t-, -t T T3. t- -t T4. T X T T5. t- t (t- - first in T) T6. t -t (-t - lst in T) T7. t t (reflexivity in T of ) T8. se t t' e t' t'' llor t t'' (trnsitivity in T of ) T9. se t t' e t ' t llor t = t' (nti-simmetry in T of ) T10. t t' oppure t' t (strong connexity in T of ) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 10

11 ABSTRACT MUSICAL SYSTEMS (2) In the sme wy pitch frme is n oredered quintuple <P, p-, -p,, > stysfying the sme set of xiom P1- P10 otined in perfect nlogy with the set T1-T10 ove, s well s the dditionl xiom: P11. P ( null-pitch is not in P) A musicl frme is structure: <<T, t-, -t, >, <P, p-, -p,, >, V> such s hold: (i). <T, t-, -t, > is temporl frme (ii). <P, p-, -p,, > is pitch frme (iii). V is non-empty set of voices LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 11

12 ABSTRACT MUSICAL SYSTEMS (3) A musicl frme with voice-indexed temporl prtitions is structure: F = <<T, t-, -t, >, <P, p-, -p,, >, V, S> such s hold: (i). <<T, t-, -t, >, <P, p-, -p,, >, V> is musicl frme (ii). S is point-selector over tht frme in the sense of eing funcion from V to the power-set of T such s for ech v V: (ii.i). S v is finite suset of T (ii.ii) t- nd -t re oth in S v LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 12

13 ABSTRACT MUSICAL SYSTEMS (4) Let F e musicl frme with voice-indexed temporl prtitions. By melodic-rhythmic specifiction on F we understnd n ordered pir <On, FrAtt> of functions on V such s for ech v V: (i). On v T x (P { }) (ii). FrAtt v T x (P { }) ( on function) ( freshly ttcked func.) NB: The pir must stisfy lso specil set of xioms MR1-5 LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 13

14 ABSTRACT MUSICAL SYSTEMS (5) By n strct musicl system we now understnd structure M = <F, <On, FrAtt>> such s: (i). F is musicl frme with voice-indexed temporl prtitions (ii). <On, FrAtt> is melodic-rhythmic spec. on F With the sme formlism we cn define lso: the musicl course of events in v in M (mce), the texture of M (Texture), nd the totl chord progression in M (Chord) Finlly: counterpoint is the study of Texture structure while hrmony is the study of Chord structure LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 14

15 DIFFERENT POINTS OF VIEW Lnger postultes Set-theoreticl. m. s. STATICALLY TYPED SYSTEM DINAMICALLY TYPED SYSTEM (temporlly quntified) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 15

16 A PHILOSOPHICAL PERSPECTIVE In 1910 E. Cssirer ( ) pulished n essy titled Sustnzegriff und Funktionsegriff (Sustnce nd function) Through solid cquintnce of history of science, Cssirer conducts n inquiry into mthemticl, geometric, nd physicl knowledge Cssirer shows how these different forms of knowledge don t look for the common (sustnce) ut for the generl lws, the reltions ( functions) Scientific knowledge leds us to move from the concept of sustnce to the concept of function LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 16

17 A-PRIORI KNOWLEDGE Mthemticl functions re not strctions from sustnces ut re creted y thought In the sme wy, scientific theories nd functionl reltions mong knowledge ojects re creted y thought The knowledge is -priori: the humn ct of knowing is the milestone of knowledge nd not the sustnce per sè In this sense the humn eing is niml symolicum LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 17

18 SUPREMACY OF ACTION Cssirer s ides on sustnce/function dulity hve roots in the philosophy of Pul Ntorp ( ), former Cssirer s techer Following Ntorp, relity is not mde y the ojects discovered y knowledge ut is the sme discovering process We move from the structure to the process (ction) Ntorp quotes Goethe: Im Anfng wr die Tt (At the eginning there ws the Action) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 18

19 THE SIMPLE SYSTEM (INFORMALLY) Music cn e thought s simple system orgnized into two distinct ctegories: stte nd trnsition A stte is n idel configurtion in which the prmeters of music re in rest A trnsition, on the contrry, is possile configurtion in which the prmeters re in tension, continuously evolving Following Cssirer, the former cn e thought s sustnce, the ltter s function LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 19

20 THE GENERATION FUNCTION (INFORMALLY) Let e S 1 nd S 2 two different sttes. Then we cn define function Φ: S 1 _ S 2 clled genertor, such s: (i). Φ cretes trnsformtion of S 1 into S 2 throught finite numer of steps clled orits (temporl evolution) (ii). Φ holds for ech prmeter of the musicl system, such s melody, hrmony nd rhythm It is very importnt to think music s dinmicllytyped system, y defining proper genertors for ech needed prmeter LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 20

21 MELODIC REGIONS Let e S the set of the twelve distinct pitch-clsses. Then P 0, P 1,, P n will e clled specil ordering of S. Φ is permuttion from P n to P n+1 Ech P n is stte while the orits creted y Φ re trnsitions The whole set of trnsitions will e clled melodic region LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 21

22 HARMONIC REGIONS Let O e set of distinct pitch-clsses, clled orit. If some elements of O occurs simultneously the O will e clled hrmonic field Every orit cn hve finite numer of hrmonic fields; the set of fields of single orit is clled hrmonic orit The set of the hrmonic horits will e clled hrmonic region A single pitch orit is n hrmonic trnsition, while field is stte Hrmony nd melody will never e in the sme configurtion LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 22

23 LEWIN S S PERSPECTIVE SET THEORY CLASSICAL (A. FORTE) TRANSFORMATIONAL (D. LEWIN) Music cn e represented through forml structure clled GIS (Generlized Intervl System) nd through trnsformtion function clled IFUNC (Intervl function) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 23

24 CLOSING THE CIRCLE A GIS cn e thought s stte? The IFUNC cn e thought s trnsition? Φ (genertor) must hold for ll the prmeters in the system nd must hppen in temporl frme Does IFUNC stisfy these requirements? LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 24

25 A VISUAL SUMMARY MUSIC SYSTEM INTUITIVE FORMALIZED STATICALLY TYPED DINAMICALLY TYPED IFUNC STATE/GIS (sustnce) TRANSITION (Φ-function) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 25

26 MUSICAL EXAMPLES Vectoril synthesis from two sets of prtils in dditive synthesis (SineWrp 1.0) Trichordl genertors of hexchords s explined y Steve Rouse in 1985: (excerpts from Prcelso y l ros, 2005) LOGICAL FOUNDATION OF MUSIC: A PHILOSOPHICAL APPROACH 26

27 CARMINE EMANUELE CELLA Vi Finli 25/ Pesro (PU) - ITALY Phone: Moile: Mil: We:

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