DENSITY DEGREE OF INTERVALS AND CHORDS BY ORLANDO LEGNAME MARYLAND 1998
CONTENTS
INTRODUCTION DENSITY DEGREE CLASSIFICATION OF INTERVALS AND CHORDS ANALYSIS FINAL CONSIDERATIONS APPENDIX BIBLIOGRAPHY
INTRODUCTION Since Pyt!"#$!% !% &#'n( te !$it)etic!* $e*!ti#n% #& inte$+!*%, !t *e!%t -. cent'$ie% !"#, )!ny te#$etic!* /#$0% !#'t Py%ic% #& M'%ic !+e een /$itten2 P!$tic'*!$*y ('$in" te %e+enteent cent'$y, ti% &ie*( #& 0n#/*e("e %t!$te( t# i)3$#+e ec!'%e #& te %t'(y #& +i$!ti#n% !n( tei$ $e*!ti#n% t# )'%ic!* %#'n(%2 Bet/een 1.4. !n( 1.45, Me$%enne en'nci!te( te L!/% /ic c!$$y i% n!)e !n( (e%c$ie te &!ct#$% t!t te +i$!ti#n &$e6'ency #& ! %t$in" (e3en(% '3#n2 Hi% i(e!% in&*'ence( H'y"en% /# !*%# /!% inte$e%te( in te 3$#*e) #& c#n%#n!nce !n( (i%%#n!nce2 Ti% i%%'e t##0 te !ttenti#n e+en #& Ne/t#n, #ne #& te )!7#$ "eni'%e% te ')!n 0in( e+e$ 3$#('ce(2 In i% /#$0, Principia /#$0, Principia,, te )!%te$ %t!te% i% )!te)!tic!* !n!*y%i% !#'t %#'n( 3$#3!"!ti#n2 S#)e ye!$% *!te$, E'*e$ 3'*i%e( i% Dissertatio i% Dissertatio physica de sono, sono, /#$0 t!t /#'*( ec#)e ! c*!%%ic #& Ac#'%tic% Science2 In te nineteent cent'$y, He$)!nn +#n He*)#*t $e!*ie( #ne #& te )#%t i)3#$t!nt $e%e!$ce% e+e$ (#ne !#'t !$)#nic %e$ie% !n( e!t%2 B!%e( #n it, e /#n(e$e( !#'t it% !33*ic!ti#n in )'%ic !n( te c#n%#n!nc c#n%#n!ncee 3en#)en 3en#)en#n2 #n2 Hi% Die Hi% Die Lehre von den Tonempfindungen Tonempfindungen,, /ic /ic En"*i En"*i% % t$!n%*!ti#n i% On the Sensations of Tone, Tone , /!% 3'*i%e( in 1882 1
One )!y %ee in ti% $e3#$t t!t te 3y%ici%t% %ee) t# e !*/!y% inte$e%te( in )'%ic!* 3en#)en!, 't $!te$, te )'%ici!n% ec!)e )#$e inte$e%te( in 3y%ic!* &!ct% !n( tei$ i)3*ic!ti#n% t# )'%ic!* #cc'$$ence% )#%t*y in ti% cent'$y2 One #& te /#$0%, /ic c#n%i(e$ t#%e $e*!ti#n% /it ! 3$!ctic!* +ie/, !n( tei$ 'ti*i!ti#n in )'%ic!* c#)3#%iti#n, i% P!'* Hindemith:% Unterweisung im Tonsatz, 3'*i%e( in 1945, /ic En"*i% t$!n%*!ti#n i% The Craft of Musica Composition! In ti% /#$0, te c#)3#%e$ (e+e*#3e( ! %y%te) #& inte$+!*% !n( c#$(% !%e( in 3y%ic!* *!/%, !cc#$(in" t# tei$ (i%%#n!nce (e"$ee2 Te ('!*i%tic c#nce3t #& t/# 3#*!$itie% ; c#n%#n!nce !n( (i%%#n!nce ; c!n e &#'n( 3!$tic'*!$*y ('$in" te t#n!* 3e$i#( #& )'%ic i%t#$y, !n( i% $e*!te( t# te 3i*#%#3ic %c##*% #& R!ti#n!*i%) !n( P#%iti+i%)2 By c#nt$!%t, te %c#*!$% #& )e(ie+!* ti)e% %ee)e( t# t!0e ! )#$e #*i%tic !33$#!c t# inte$+!**ic (i&&e$ence%, $e"!$(in" te) !% +!$i#'% e<3$e%%i#n% #& =c#*#$e=2 In ! $e*!te( )!nne$, tin0in" %ince te *!te 19 t cent'$y >!i(e( y te %t'(ie% #& He*)#*t? !% !"!in e"'n t# %'%tit'te te c#n%#n!nce @ (i%%#n!nce ('!*ity /it #te$ c#nce3t% )#$e !**;enc#)3!%%in"2 Te te#$y 3$#3#%e( y Hindemith in The Craft of Musica Composition, i% %'c #ne !n( it i% !%e( #n t/# 3y%ic!* 3en#)en!, te #+e$t#ne %e$ie% !n( te c#)in!ti#n t#ne%2 i*e i% i(e!% !+e een %e+e$e*y c$iticie(, tey c!n %ti** e '%e( !% ! (e3!$t'$e 3#int &#$ ! ne/ !33$#!c2 Hindemith% !$)#nic te#$y !% t/# !%ic 3$inci3*e%, (e%i"n!te( !% Series " !n( Series #2 Series ", !%e( #n te #+e$t#ne %e$ie%, i% ! %e6'ence #& te t/e*+e c$#)!tic n#te% in !n #$(e$ %#/in" te $!n0in"% #& tei$ $e*!ti#n%i3% t# ! t#n!* cente$ EXAMPLE 1!2 Series # i% ! %e6'ence #& te t/e*+e inte$+!*% >/itin te %3!ce #& !n #ct!+e? in (ec$e!%in" #$(e$ #& c#n%#n!nce EXAMPLE 12
F$#) te%e 3$inci3*e%, te c#)3#%e$;te#$i%t (e$i+e% ! c#)3*e< %y%te) #& c#$(% in (ec$e!%in" #$(e$ #& c#n%#n!nce, /ic !%t$!ct i% %#/n in T!*e 12 TABLE 1
A. Chords withot Tritone
B. Chords with Tritone
I2 $ithout seconds and sevents
II2 $ithout minor seconds or ma%or sevents
12 R##t !n( !%% t#ne !$e i(entic!* The tritone su&ordinate -2 R##t *ie% !#+e te !%% !? it )in#$ %e+ent #n*y >n# )!7#$ %ec#n(? R##t !n( !%% t#ne !$e i(entic!* ? 2
C#nt!inin"
)!7#$
%ec#n(% #$ )in#$ %e+ent% #$ #t 12 R##t !n( !%% t#ne !$e i(entic!* -2 R##t *ie% !#+e te !%% 42 C#nt!inin" )#$e t!n #ne t$it#ne III2Containing seconds or sevenths or &oth
IV 2 Containing minor seconds or ma%or sevenths or &oth
12 R##t !n( !%% t#ne !$e i(entic!*
One or more tritones su&ordinate
-2 R##t *ie% !#+e te !%%
12 R##t !n( !%% t#ne !$e i(entic!* -2 R##t *ie% !#+e te !%%
V 2 In(ete$)in!te
VI2 In(ete$)in!te
T'%, Hindemith 3$#3#%e% ! )et#( #& !n!*y%i% '%in" t/# c#nce3t% "!Two'voice framewor(2 St'(y #& te $e*!ti#n%i3 et/een te !%% *ine !n( te )#%t 3$#)inent #& te '33e$ 3!$t% >!cc#$(in" t# %e$ie% -?2 #! )armonic fuctuation! St'(y #& ten%i#n c!n"in" in ! c#$( 3$#"$e%%i#n >!cc#$(in" t# c#$( t!*e?2 Hindemith 3$#3#%e% !n #$(e$in" #& te inte$+!*%, 't (#e% n#t (e&ine ! &i!% /e %!** %ee?, !$e c#$$e%3#n(in"*y +i%'!**y (en%e$ t!n c#n%#n!nt inte$+!*%2
DENSITY DEGREE S#'n( i% +i$!ti#n, ! 3e$i#(ic!* #cc'$$ence /ic )!y e $e3$e%ente( "$!3ic!**y &$#) !n #%ci**#%c#3e? !% ! /!+e&#$)2 Te %i)3*e%t /!+e&#$) in n!t'$e i% te %ine /!+e, te ti)$e #& /ic $e%e)*e% ! t'nin" &#$02
3
T/# (i&&e$ent;3itce( t'nin" &#$0%, %#'n(in" t#"ete$, 3$#('ce !n inte$+!*2 Te $e%'*t!nt inte$+!* /!+e&#$) )!y !*%# e "$!3ic!**y $e3$e%ente(, !% ! %'3e$i)3#%iti#n #& te t/# #$i"in!* %ine /!+e%2 E!c inte$+!* !% ! 3!$tic'*!$ !n( 'ni6'e /!+e&#$) !% %een in EXAMPLE -2
Te !#+e inte$+!* /!+e&#$)% #cc'$ /en t/# %#'n( %#'$ce% !$e c*#%e in 3$#
An!*yin" te $e%'*t% &$#) #t %et% #& /!+e&#$)%, /e %ee t!t the level of graphic complexity coincides with the interval density degree. C#)3!$in" te 3e$&ect &i&t "$!3% /it t#%e #& te )in#$ %ec#n(, &#$ in%t!nce, /e n#te t!t te *!tte$ "$!3% !$e )#$e c#)3*e< tei$ !%3ect% !$e (en%e$2 A% %#)e #& te Li%%!7#'%;c'$+e inte$+!* "$!3% !$e +e$y c#)3*e< >ec#)in" (i&&ic'*t t# !n!*ye +i%'!**y?, /e n#te t!t ! )!te)!tic!* 3$#ce('$e e
F#$ te 8 inte$+!*% in ! %e+en;#ct!+e e+en;te)3e$e( c$#)!tic %c!*e, /e c!n (e+e*#3 ! (en%ity; (e"$ee $!n0in" &$#) t# !#'t 2 F#$ inte$+!*% '3 t# !n #ct!+e, te (en%ity;(e"$ee #$(e$ i% %#/n in EXAMPLE 2
Te %i"ni&ic!nt (i&&e$ence et/een #'$ #$(e$ !n( Hindemith% Series #, i% te t$it#ne 3#%iti#n, /ic i% *#c!te( et/een te )in#$ ti$( !n( te )in#$ %i
4
O'$ c!*c'*!ti#n% (e)#n%t$!te t!t te t$it#ne !% !n( inte$)e(i!$y c!$!cte$i%tic et/een te *e%%; (en%e inte$+!*% >&i&t !n( &#'$t? !n( te (en%e$ >%ec#n( !n( %e+ent?2 One %'%3ect% t!t te $e!%#n Hindemith 3*!ce( te t$it#ne !t te en( #& Series # /!% ec!'%e #& te !$)#nic )e!nin" t!t te t$it#ne !% e*( in te /e%te$n )'%ic2 e 3$#3#%e t!t te t$it#ne $e%#*'ti#n (#e% n#t !+e 3y%ic!* e<3*!n!ti#n, !% Hindemith %'33#%e(, 't $!te$ ! 3%yc#*#"ic!* #ne2 I& ti% i% t$'e, #ne #& t#n!*ity:% &#'n(!ti#n% ; te t$it#ne $e%#*'ti#n ; i% ! %'7ecti+e, !n( n#t ! 3y%ic!*, 3en#)en#n2 O'$ c!*c'*!ti#n% !*%# $e+e!* t!t te c#n&i"'$!ti#n% %#/n in EXAMPLE (# n#t $e3e!t te)%e*+e% /en te '33e$ n#te #& te inte$+!* i% t$!n%3#%e( '3 #ne #$ )#$e #ct!+e%2 Den%e$ inte$+!*% /itin #ne #ct!+e, c!n ec#)e *e%% (en%e$ 'n(e$ te%e c#n(iti#n%, !% te &#**#/in" in%t!nce% ; )!7#$ %ec#n( *e%% dense t!n )in#$ ti$( EXAMPLE ! ; )!7#$ ti$( ess dense t!n 3e$&ect &#'$t EXAMPLE ; )!7#$ %e+ent ess dense t!n )in#$ %e+ent EXAMPLE c2 en /e %'$3!%% t$ee #ct!+e%, te $e%'*t% !$e %'$3$i%in", !% &#$ in%t!nce ; )!7#$ %e+ent ec#)e% ess dense t!n ne
T# +i%'!*ie te (en%ity (e"$ee )#(i&ic!ti#n !t te c#$$e%3#n(in" #ct!+e%, /e (e+e*#3e( ! t$ee; (i)en%i#n!* "$!3ic %#/e( in EXAMPLE .2
5
By te "$!3ic !n!*y%i%, /e +e$i&y t!t te (en%ity #& )!7#$ %e+ent i% !*)#%t te %!)e in te &#'$ #ct!+e%2 On te #te$ !n(, te #ct!+e i% te *e%% (en%e in te &i$%t 't ten it% (en%ity %t!$t% t# inc$e!%e ec#)in" i""e$ t!n te %e+ent% in te &#'$t #ct!+e2 Ti% !n!*y%i% )!y %'""e%t t!t te c#nce3t #& te #ct!+e, !% ! $e3etiti#n #& te %!)e n#te in !n#te$ $e"i#n, i% !n !33$#
Den%ity;(e"$ee $!n0in" e<3*!in% /y te inte$+!* #& tent i% %# #&ten 'ti*ie( in )'%ic!* c#)3#%iti#n2 A&te$ te )!7#$;ti$(;t/#;#ct!+e%;'3, te tent i% te *e!%t (en%e inte$+!* /ic !% !$)#nic &'ncti#n2 It i% #&ten 'ti*ie( !% te #$(e$% #& ! c#$(, !% in te &i$%t )e!%'$e #& te B!c% The $e'tempered Cavier P$e*'(e I EXAMPLE 8 !n( )e!%'$e% , 19 !n( -2
A %i)i*!$ 3$#ce('$e t# t!t #'t*ine( !#+e &#$ inte$+!*% )!y e '%e( t# (ete$)ine te (en%ity (e"$ee% #& c#$(%, te net $e%'*t ein" (en%ity;(e"$ee c#$( $!tin"% &$#) !#'t 1. t# .2 E
6
Hindemith% te#$y (#e% n#t $ec#"nie (i&&e$ence% !)#n" c#$( 3#%iti#n%2 Acc#$(in" t# #'$ 3$#3#%!*, #/e+e$, /e %ee t!t te &i$%t t/# #3en;&#$)!ti#n )!7#$ c#$(% EXAMPLE 9c !n( 9( !$e *e%% (en%e t!n *!%t c*#%e;&#$)!ti#n )!7#$ c#$( EXAMPLE 9&2 F'$te$)#$e, !)#n" te%e *#/;(en%ity c#$(% i% te %'3e$i)3#%e(;&i&t% c#$( EXAMPLE 9e, /ic Hindemith c#n%i(e$e( +e$y ten%e2 Te ten%e%t c#$(% in Hindemith% te#$y !$e te %'3e$i)3#%e( &#'$t% >C;F;B?, %'3e$i)3#%e( )!7#$ ti$(% >C;E;G? !n( !** 0in(% #& (i)ini%e( c#$(%2 A** #& te%e, #/e+e$, !cc#$(in" t# #'$ te#$y, e*#n" t# te *e%%;(en%e "$#'3 #& c#$(%, !*t#'" tey ce$t!in*y !$e (en%e$ t!n te c#$(% #& EXAMPLE 92 Te (en%ity #& %'3e$i)3#%e( &#'$t%, %'3e$i)3#%e( )!7#$ ti$(%, !n( t/# ty3e% #& (i)ini%e( c#$(% >$##t !n( .; 3#%iti#n? !$e %#/n in EXAMPLE 12
Te%e &#'$ c#)3!$!ti+e*y *e%%;(en%e c#$(% !$e c*!%%i&ie( !)#n" te )#%t (i%%#n!nt in Hindemith:% te#$y ec!'%e #& tei$ $e*!ti#n t# t#n!*ity2 S'3e$i)3#%e(;)!7#$;ti$(% !n( %'3e$i)3#%e(;&#'$t% c#$(% !+e *itt*e &'ncti#n in t#n!*ity !n( !$e c#n%i(e$e( In(ete$)in!te2 Di)ini%e( c#$(%, /ic !+e (#)in!nt &'ncti#n, !$e !*%# In(ete$)in!te !n( !+e te *#/e%t t#n!* +!*'e ec!'%e, $!te$ t!n (e&inin" ! t#n!*ity, tey !**#/ %e+e$!* 3#%%ii*itie% &#$ $e%#*'ti#n2 In #'$ 3$#3#%!*, te c#$(% /ic !+e te "$e!te%t (en%ity !$e %e+e$!* &#$)!ti#n% #& custers, te (en%e%t in #3en 3#%iti#n% EXAMPLE 112
7
Den%ity;(e"$ee !tt$i'te( t# inte$+!*% !n( c#$(% !$e n#t !%#*'te 3y%ic!* )e!%'$e)ent%2 S#)e $e;inte$3$et!ti#n )!y e nece%%!$y in *i"t #& &'$te$ in+e%ti"!ti#n%2 B't )!ny (en%ity;(e"$ee n#ti#n% c#$$e%3#n( t# t$!(iti#n!* !$)#nic !n!*y%e%, !n( c!n c#nt$i'te t# ! )#$e 3$#&#'n( c#)3$een%i#n #& )'%ic!* /#$0%2
CLASSIFICATION OF INTERVALS AND CHORDS Te /!y ti% te#$y t'$ne( #'t, it i% nece%%!$y ! c#)3'te$ t# c#$( (en%ity c!*c'*!ti#n, /ic t'$n% it t# %#)etin" '%e*e%% t# )'%ici!n%2 Te c#)3*e&#$ eIIA !n( IIB &#$ eIA? c#nt!in% c#$(% /it #n*y 3e$&ect inte$+!*% in te &i$%t #ct!+e >P, P, !n( #ct!+e? 3*'% t/# 0in(% #& ti$(% >M1 !n( M4 J -?2 Te$e !$e +e$y &e/ c#$(% t!t &'*&i** te%e c#n(iti#n% !n( %#)e #& te) !$e n#t e+en c!**e( c#$(% y t$!(iti#n!* !$)#ny EXAMPLE 1- ; IA2 T# te %ec#n( %'"$#'3, e*#n" c#$(% /it !** #te$ i)3e$&ect c#n%#n!nce% >ti$(% !n( %iP11 !n( P J -?2 it te%e inte$+!*%, #ne c!n 'i*( !** 0in(% #& t$i!(% !n( tei$ in+e$%i#n% EXAMPLE 1- ; IB2 S'"$#'3 IIA inc*'(e% inte$+!*% *i0e !'")ente( &#'$t, )in#$ %e+ent, !n( )!7#$ ti$teent, 3*'% t$ee 0in(% #& )!7#$ %ec#n(% >M9, M- J -, !n( M- J 4? eM1, M5 J -, !n( M5 J 4? /it te eC;D;E?, %#)e &#$)!ti#n% #& t$i!(% !n( %e+ent c#$(% EXAMPLE 1- ; IIB2 S'"$#'3 IIIA inc*'(e% te )!7#$ %e+ent, )in#$ ti$teent !n( te &i$%t t/# 0in(% #& )in#$ %ec#n(% >)- J 4 !n( )- J ?2 Te $e!* )in#$ %ec#n(, te (en%e%t inte$+!* in te &i$%t #ct!+e, !33e!$% #n*y in S'"$#'3 IIIB, /ic inc*'(e% !*%# te )in#$ &#'$teent2 it te%e inc*'%i#n%, /e !+e &$#) te )!7#$ %e+ent c#$( t# )#$e (en%e #ne% t!t in )!ny c!%e% c!nn#t e #$"!nie( in %'3e$i)3#%e( ti$(% EXAMPLE 1- ; III A ; B2 S'"$#'3 IVA c#nt!in% te )in#$ %i)5 J , M5 J, #ct!+e%, )- J , M- J ? /i(enin" te $!n"e #& te c#$(%2 S'"$#'3 IVB inc*'(e% ! +e$y (en%e inte$+!*, te )in#$ nint, /ic te *!%t #ne in te &i$%t t/# #ct!+e% $!n"e2
8
S'"$#'3 V c#nt!in% te *!%t )in#$ %ec#n( >)- J -?, !n( !** #te$ inte$+!*% '3 t# %i< #ct!+e% /it te eA J ). J ?, /ic !$e inc*'(e( in te *!%t S'"$#'3 >VB? /it !** 3#%%i*e #te$%2 Ti% c#nc*'(e% te t!*e c#+e$in" !** 3#%%ii*itie% #& inte$+!*% !n( c#$(%2 F#$ te 3'$3#%e #& !n!*y%i%, #ne )!y '%e te %!)e T!*e - t# c*!%%i&y te inte$+!*% #& ! t/#;+#ice &$!)e/#$02 In ti% c!%e te inte$+!*% /#'*( c!$$y te %!)e "$#'3 c*!%%i&ic!ti#n >*i0e IIA? !n( )!y e c#)3!$e( t# te $e%3ecti+e c#$(2 T/# (en%ity;(e"$ee !n!*y%e% e*#/ 'ti*ie Hindemith:% c#nce3t% #& t/#;+#ice &$!)e/#$0 !n( !$)#nic &*'ct'!ti#n t# $e+e!* ne/ 3e$%3ecti+e%2 T#()e *
A
B
P, P, #ct!+e, P1-, - #ct!+e%
M4, M4, )., M.
M1
P11
+, - *
P J-
I
, o%t#&es A, )5
M-
M9, M14
)1, A11
M- J -, M- J 4
M1, M5 J -, + -, >n# M5?
+, - ,
P J 4
M5
)-
)14
)1
P J -
)4 J -, A J -, M. J -
#ct!+e%
M4 J
)- J 4, )- J
0 - /
II
III
+* - / ). J -
)9
P J 4
)5 J -
)5 J , M5 J
)4 J4, M. J 4
#ct!+e%
M4 J, A J
)- J
0 -
IV
+* -
9
)- J -
A J , !n( ). J
A J 4, ). J 4, )5 J 4
A** #te$ inte$+!*%
V
PJ A** J 2 o%t#&es Te inte$+!* in #*( i% te i""e%t in te %'"$#'3
ANALYSIS Te &i$%t P$e*'(e #& B!c% e**;te)3e$e( C*!+ie$ i% ! %e6'ence #& $#0en c#$(%, /it !n i)3*icit )e*#(y /itin it% %t$'ct'$e2 It i% &#$)e( y %e+e$!* c!(ence% t!t (e+e*#3, !$tic'*!tin" &#'$ c*e!$ 3!$t% !% /e c!n %ee in T!*e 42 T#()e ,
+e#sres
1st 3#rt
*nd 3#rt
,rd 3#rt
/th 3#rt
1 t#
t# 11
1- t# 19
- t# 4 G 3e(!* 3#int
•
H#rmon!
M#('*!ti#n ; G •
Ret'$n t# C C 3e(!* 3#int
C!(ence T;T •
O(s4 •
Di)ini%e( c#$(%
M#$e in+e$%i# n%
M#$e c#$(%
(i)
F#$ ! ette$ +i%'!*i!ti#n #& te c#)3#%iti#n 3$#ce%% in ti% e
10
One c!n n#tice !*%# t!t te c#)3#%e$ $e3e!t% te e
11
F#$ ! ette$ +i%'!*i!ti#n #& te c#)3#%iti#n 3$#ce%% in ti% e
An !n!*y%i% %#/% t!t te$e i% ti)$!* &*'ct'!ti#n in te &i$%t )e!%'$e, 't n# (en%ity;(e"$ee c!n"e%2 A&te$ ! %)!** &*'ct'!ti#n in te %ec#n( )e!%'$e, ! 3$e3!$!ti#n &#$ te c*i)!< %t!$t%2 A i" (i&&e$ence i% #3ene( et/een t/#;+#ice &$!)e/#$0 !n( !$)#nic &*'ct'!ti#n in te e"innin" #& ti$( )e!%'$e, !n( ten #t *ine% !*)#%t t#"ete$ $e!c ! i"e$ 3#int in te e"innin" #& &i&t )e!%'$e2
12
Ten #t *ine% )!0e ! 6'ic0 ($#3 !n( ten $e!c te $e!* c*i)!<, !"!in #t t#"ete$ !t te )!
Te )!te)!tic!* 3$#ce('$e %'""e%te( 3e$)it% te c!*c'*!ti#n #& te "$!3ic *ine *en"t, /ic i% 3$#3#$ti#n!* t# it% c#)3*e#& t'nin";&#$0% &#$ e
Te %'3e$3#%iti#n #& t/# #& te%e %#'n(% i% e6'i+!*ent t# te !*"e$!ic %') #& tei$ e6'!ti#n%2 C#n%i(e$in" t/# %#'n(% /it te %!)e !)3*it'(e !n( 3!%e, !n( 'ti*iin" te )!te)!tic!* t$ic0 t!t t$!n%&#$)% te %') #& t/# c#%ine% in 3$#('ct, te $e%'*t i%
Ti% )e!n% t!t te $e%'*t!nt i% ! /!+e /it (#'*e !)3*it'(e !n( &$e6'ency e6'!* t# te %') #& te #$i"in!*% (i+i(e( 3e$ t/#, )#('*!te( y ! /!+e /ic &$e6'ency i% te (i&&e$ence #& te #$i"in!*% (i+i(e( 3e$ t/#2 13
In te c!%e #& Li%%!7#'% c'$+e, te /!+e *ine% #& 3$#3!"!ti#n &#$) ! 9 (e"$ee !n"*e2 Te$e&#$e, 3!$!)et$ic &#$) i% te e%t /!y t# $e3$e%ent tei$ e6'!ti#n%, #ne #& te) #%ci**!tin" in te < !
In ti% c!%e, #ne !% t# c#n%i(e$ !** inte$+!*% /it ! $!ti#n!* 3$#3#$ti#n, ec!'%e #n*y in ti% /!y te$e i% ! c#)3*ete Li%%!7#'% &i"'$e2 U%in" i$$!ti#n!* n')e$%, !% in te c!%e #& /e**;te)3e$e( %y%te), ti% "$!3ic% ne+e$ en( !n(, #& c#'$%e, it t'$n% '%e*e%% t# #'$ 3$#ce('$e2 T# c!*c'*!te te c'$+e *en"t, te 3$#ce%% in+#*+e% inte"$!* #& )!"nit'(e #& te 3!$!)et$ic e6'!ti#n% (i&&e$enti!ti#n, #$,
$e%#*+in" te (i&&e$enti!ti#n, /e !+e,
!% te )!"nit'(e i% c!*c'*!te( y te %6'!$e $##t #& te %6'!$e c##$(in!te%,
(#in" te %'%tit'ti#n%, /e !+e
Ti% inte"$!* (#e% n#t !+e ! (i$ect %#*'ti#n, #n*y n')e$ic2 It% *i)it% (e3en( #n te &$e6'ency +!*'e2 T# %#*+e ti% 3$#*e) /!% nece%%!$y te 'ti*i!ti#n #& ! c#)3'te$ 3$#"$!) in C t!t +e$i&ie% in /!t )#)ent te < !n( y +!*'e% e e6'!* t# 1, !t ti% )#)ent /e !+e te 3#int /en te Li%%!7#'% c'$+e c#)3*ete( te /!y t/# ti)e%, ten te *i)it #& te inte"$!* i% te !*& #& ti% +!*'e2 T# %i)3*i&y /e c#%e &#**#/in" e<3$e%%i#n
1, $e%'*tin" in te c!%e #& te &i&t inte$+!* >
1
-
12? &#$ e
t!t $e%'*t%,
Te +!*'e #& te inte"$!* i% n')e$ic!**y c!*c'*!te(, y !((in" te !$e! #& ! *i)it n')e$ #& $ect!n"*e%2
14
A% te $!in 3e$ce3ti#n% '%'!**y #cc'$ *#"!$it)ic!**y, #ne )!y &in( ette$ t# c!*c'*!te te &in!* (en%ity +!*'e in te %!)e /!y #& (B inten%ity, y c!*c'*!tin" te *#"!$it) #& te $e*!ti#n et/een te inte$+!* !n( te *e!%t (en%e, /ic i% te 'ni%#n2 S#, te &#$)'*! /#'*( e
Te net $e%'*t ein" (en%ity;(e"$ee inte$+!* $!tin"% &$#) !#'t t# !#'t 2 A %i)i*!$ 3$#ce('$e t# t!t #'t*ine( !#+e &#$ inte$+!*% )!y e '%e( t# (ete$)ine (en%ity (e"$ee% #& c#$(%2 F#$ "$#'3% #& )#$e t!n t/# n#te%, te c*!%%ic!* 3$#ce('$e i% t# %') te $!te% e&#$e te *#"!$it) c!*c'*!ti#n2 A** inte$n!* inte$+!*% #& te c#$( )'%t e inc*'(e(2 S# te &#$)'*! i%
Te net $e%'*t ein" (en%ity;(e"$ee c#$( $!tin"% &$#) !#'t 1. t# .2
BIBLIOGRAPHY
B!c0'%, #n 2 The -custica .oundation of Music 2 Ne/ Y#$0, N#$t#n, 19.92 Bent, I!n D2 An!*y%i% in The ew /rove Dictionary of Music and Musicians 2 !%in"t#n, M!cMi**!n, 32 4;488, 1982 C!(en, N#$)!n2 Hindemith !n( N!t'$e in Music *eview 2 XV, n#2 , 32 -88;4., 192 He*)#*t, He$)!nn2 On the Sensations of Tone 2 Ne/ Y#$0, D#+e$, 192 Hindemith, P!'*2 The Craft of Musica Composition 2 L#n(#n, Sc#t, 192 H#/e, H'e$t S2 0ectronic Music Synthesis 2 Ne/ Y#$0, N#$t#n, 1952 e!n%, Si$ !)e%2 Science and Music 2 Ne/ Y#$0, D#+e$2 19.82 e)3, I!n2 Hindemith in The ew /rove Dictionary of Music and Musicians 2 !%in"t#n, M!cMi**!n, 198, 54;8532 L!), H#$!ce2 The Dynamica Theory of Sound 2 Ne/ Y#$0, D#+e$, 19.2 L!n(!', 1ictor2 The )armonic Theories of Pau Hindemith in *eation to his Paractice as a Composer of Cham&er Music ! P2D (i%%e$tti#n, Ne/ Y#$0 Uni+e$%ity, 1952 15
2 P!'* Hindemith ! C!%e St'(y in Te#$y !n( P$!tice in Music *eview 2 XXI, 32 48;, 19.2 Le"n!)e, O$*!n(# 2 Den%ity De"$ee #& Inte$+!*% !n( C#$(% in #2th Century Music, +#*2 , n#2 11, 32 8;1, N#+e)e$ 19952 Le$(!*, F !c0en(#&&, R2 - /enerative Theory of Tona Music 2 MIT P$e%% C!)$i"e, M!%%, 1992 L'n(in, R2 2 T#/!$(% ! C'*t'$!* Te#$y #& C#n%#n!nce in3 4ourna of Psychoogy, XXIII, 1952 O*%#n, H!$$y F2 Music Physics and 0ngeneering 2 Ne/ Y#$0, D#+e$, 19.52 P!*i%c!, C2 V2 S3en(e$, N2 C#n%#n!nce in The ew /rove Dictionary of Music and Musicians 2 !%in"t#n, M!cMi**!n, 32 ..8;.51, 1982 Se!%#$e, C!$*, Psychoogy of Music 2 Ne/ Y#$0, 19.52 S3en(e$, N2 S'te$;Dy%#n, R2 P%yc#*#"y #& M'%ic in3 The ew /rove Dictionary of Music and Musicians 2 !%in"t#n, M!cMi**!n, 32 488;-5, 1982 Ste3!n, R'(#*& 2 Hindemith:% M!$ien*een in Music *eview 2 +#*2 1, 32-5;-85, 192 T!y*#$, C*i&&#$(2 Te Hindemith Te#$i% in Music *eview 2 +#* , n#2 41, , 32 -.;-.-, !'"@n#+ 19842 inc0e*, F$it2 Music Sound and Sensation 2 Ne/ Y#$0, D#+e$, 19.52 ##(:%, A*e
O$*!n(# Le"n!)e H#)e ; C#)3#%e$ ; C#n('ct#$ ; Te#$i%t ; G'it!$$i%t ; P$#&e%%#$ ; Pe$%#n ; Bi# ; C#nt!ct
16
