Eduardo Lopes Time Rhythm and Meter in Music the Gravitational Concept

8/12/2019 Eduardo Lopes Time Rhythm and Meter in Music the Gravitational Concept

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TIME, RHYTHM, AND METER IN MUSIC: THE GRAVITATIONAL CONCEPT

Eduardo Lopes, PhD

Aux!ar" Pro#essor

Resear$h U%& % Mus$ a%d Mus$o!o'" (U%IMeM)

Depar&*e%& o# Mus$

U%+ers&" o# E+ora

Por&u'a!

e!ue+ora-p&

Abstract: .or Da+d Eps&e% (Shap%' T*e, /001) &he or'a%sa&o% o# &*e s dua! % %a&ure- Ths dua!&"

"e!ds &2o *odes o# s&ru$&ure, a $!o$34!3e or'a%sa&o% ($hro%o*e&r$ &*e) a%d a% or'a%sa&o% &ha& s

u%5ue &o a par&$u!ar expere%$e (%&e'ra! &*e)- .or h*, $hro%o*e&r$ &*e s perod$ % %a&ure a%d

s&ru$&ured 6" $o%s&a%& #a$&ors, 2hereas %&e'ra! &*e s s&ru$&ured 6" e!e*e%&s %&r%s$ &o &s u%5ue

s&ua&o%- Eps&e% a!so su''es&s &ha& &he dua!&" o# &he &*e or'a%sa&o% s a!so prese%& % &he dura&o%a!

aspe$&s o# *us$- I% &hs *a%%er, *e&er $orrespo%ds &o $hro%o*e&r$ &*e, a%d rh"&h* &o %&e'ra! &*e- I%

&hs paper & 2!! 6e su''es&ed &ha& &he rh"&h*7*e&er 6#ur$a&o% "e!ds dua! &*e s"s&e*s &ha& are

pro$essed % para!!e!- These s"s&e*s %&erse$& o% +arous !e+e!s, *u&ua!!" a##e$&%' &he #or*s a%d

ar&$u!a&o%s &ha& ea$h assu*es8 &hs !eads % &ur% &o $o%#!$&s a%d reso!u&o%s o# &e%so%s 6e&2ee% rh"&h*

a%d *e&er- The %&era$&o% o# &hese *us$ para*e&ers %#er % !s&e%ers a se%se o# *o&o% (3%ess): &he

pe$e depar&s #ro* a po%& arr+%' a& a%o&her s!o2er or #as&er, % a% order!" or dsorder!" #asho%-

A!&hou'h %o& ex$!us+e!", &he per$ep&o% o# 3%ess s % 'rea& par& due &o &he rh"&h*$ s&ru$&ure o# *us$

9 dura&o%s o+er a *e&r$a! $o%&ex&- Us%' &he $o%$ep& o# &he herar$h" % 'ra+&", &hs ar&$!e proposes a%

e*pr$a!!" 6ased &heore&$a! $o%s&ru$&o% &ha& exp!a%s &he opera&o% a%d #u%$&o% o# *e&er 2&h% &he

per$ep&o% o# 3%ess % a pe$e o# *us$- esdes &he *por&a%$e o# &hs ssue &o &he o+era!! *us$

$o**u%&", &he exp!a%a&o% o# &he opera&o% o# 3%ess % *us$ s a!so re!e+a%& #or o&her per#or*a%$e ar&s

&ha& %&era$& 2&h *us$, su$h as dra*a a%d da%$e-

Keywords: Mus$ Theor"- Mus$ Rh"&h* a%d Me&er- Co'%&o%- Mus$ Per$ep&o%-

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Introduction

Mus$a! *e&er s a ps"$ho!o'$a! $o%s&ru$& 6ased o% &he or'a%;a&o% o# sou%ds 2h$h

$o%s&&u&e a *us$ pe$e8 &s expresso% $o*es % &he shape


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6o!o'$a! rh"&h*s (hear&6ea&, pu!se, a!pha 2a+es, a%d o&her %&er%a! $!o$3s) or ex&er%a!

a$&+&es (2a!3%', s3pp%', rd%' a horse, e&$-)- A 'es&a!&4ore%&a&ed approa$h 2ou!d

&he% see3 &o u%$o+er &he %&r%s$ opera&o%s or $o%s&ru$&o%s o# per$ep&o%, su$h as &he

presu*a6!" %%a&e hu*a% &e%de%$" &o 'roup a seres o# s&ead" pu!ses %&o re$urre%& se&s

o# &2o or &hree-

O% &he ph"so!o'$a! sde, Da+d Eps&e% (/001) $&es %u*erous s$e%&s&s #ro*

E%s&e%, Goodd", Lash!e", a%d Poppe!, &o &he $"6er%e&$s o# Fe%er, % de+e!op%' hs

&heor" o# &he 6o!o'$a! $!o$3- I% &hs &heor" &he hu*a% 6od" 2or3s !3e a 6' $!o$3

2h$h &se!# rea$&s &o &he $o%Bu%$&o% o# s*a!!er $!o$3s represe%&ed 6" &he d##ere%&

6od" or'a%s8 &he


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&he se$o%d o%e appears- Thus, &he $o%$!uso% &ha& #or$es are re5ured &o produ$e &he

u%&" o# &he par o# &aps $o*pe!s &he Ges&a!& ps"$ho!o's& &o assu*e &ha&, a!&hou'h &he

#rs& &ap has $eased &o exs&, so*e&h%' *us& ha+e re*a%ed 2h$h ser+es as o%e o# &he

po%&s a& 2h$h &he #or$es app!"- Ths so*e&h%' s 3%o2% as a


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ph!osoph$a! reassura%$e &ha& &he a%a!o'" 6e&2ee% &he *%d a%d a $o*pu&er pro'ra*

pro+des- A!&hou'h a pro'ra* s %o&h%' 6u& a #!o2 o# s"*6o!s, & has rea!&" e%ou'h &o

$o%&ro! &he opera&o% o# +er" &a%'6!e *a$h%er" &ha& exe$u&es +er" ph"s$a! opera&o%s-

.or Nesser, a%o&her ad+a%&a'e o# &he


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/ 9 The per$ep&ua! pro$ess 2!! pr*ar!" 'roup e+e%&s % a 6%ar" or &er%ar" #asho%-

9 esdes ha+%' a% %&r%s$ de%&&", 'roup%'s su$h as #+e or se+e% *a" a!so 6e

per$e+ed as a $o*pos&e o# 6%ar" a%d &er%ar" 'roups-

@ 9 I% &he $ase o# per$ep&ua! dou6& 6e&2ee% 6%ar" a%d &er%ar" or'a%sa&o%, &he #or*er

2!! pre+a!-

" The #ra$itational %oncept

A$$ord%' &o Ges&a!& pr%$p!es &o2ards $o'%&+e or'a%;a&o% o# u%&s, & s proposed

&ha& &he #rs& u%& o# a 'roup s per$e+es as *ore sa!e%&- There#ore, app!"%' &hspr%$p!e &o *us$a! *e&er, &hs Buses &he reaso% 2h" & s a$$ep&ed &ha& &he #rs& 6ea&

o# a *easure s per$e+ed as &he s&ro%'es&- I% &hs 2a", & s a!so u%ders&a%da6!e 2h" &he

se$o%d o# a &2o46ea& *easure $a% 6e per$e+es as &he 2ea3es&-

I% *" +e2, &he 3%ess 9 &he se%se o# *us$a! *o&o% 4 *p!ed 6" a par&$u!ar rh"&h*$

$o%s&ru$& resu!&s #ro* &he d##ere%$e 6e&2ee% &he per$ep&ua! s&re%'&h o# &he 6ea&s- I

2ou!d !3e a& &hs po%& &o %&rodu$e &he ph"s$a! $o%$ep& o# 'ra+&" 9 o#&e% %+o3ed %

&he ds$usso% o# &o%a! s&ru$&ure 4 as a *e&aphor #or rh"&h*$ 3%ess- The d##ere%$e o#

*ass 6e&2ee% o6Be$&s $rea&es 2ha& s $a!!ed a d##ere%$e o# po&e%&a!8 % &hs *a%%er, &he

s*a!!es& o6Be$& s a&&ra$&ed &o2ards &he !ar'er- E+e% &hou'h & s ps"$ho!o'$a! % %a&ure,

&he d##ere%$e 6e&2ee% &he 2e'h& o# &he #rs& 6ea& a%d &ha& o# &he se$o%d see*s &o $rea&e

a s*!ar d##ere%$e o# po&e%&a! 6e&2ee% &he*- The #rs& 6ea& $a% &he% 6e assessed as

s&ro%' a%d s&a6!e, 2h!e &he se$o%d 6ea& s 2ea3 a%d u%s&a6!e as a resu!& o# &s 3%e&$

po&e%&a!8 % o&her 2ords, &he 3%e&$ 5ua!&es o# 6ea&s are %+erse &o &her per$ep&ua!2e'h&- (.'- s a s$he*a&$ represe%&a&o% o# &hs 'ra+&a&o%a! *o&o% % a 6%ar"

*easure-) Moreo+er, &he 'ra+&a&o%a! $o%$ep& *p!es &ha& &here s a $o%ss&e%& de$rease

o# 2e'h& #ro* &he #rs& &o &he !as& 6ea& o# a *easure8 as 2!! 6e see%, &hs pr%$p!e &he%

%&era$&s 2&h &he pr%$p!e o# herar$h$a! or'a%sa&o% 2&h% *easures-


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4 'ra+&a&o%a!73%e&$*o+e*e%&

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Gra+&a&o%a! a&&ra$&o%&o2ards &he #rs& 6ea&

A!&hou'h I ha+e exp!a%ed &he dea o# 3%e&$ po&e%&a! as a ps"$ho!o'$a! 'ra+&a&o%a!

#e!d 2&h% a *easure, &he pr%$p!e a!so app!es 6e&2ee% *easures: % a rh"&h*$

se5ue%$e, &he 3%e&$ *o&o% s &o2ards &he #rs& 6ea& o# &he %ex& *easure, 2&h &he

3%e&$ po&e%&a! o# &he 2ea3 6ea&s 6e%' rea!sed o% &he #o!!o2%' s&ro%' 6ea&s (.'- @)-

.ro* &he per#or*er=s po%& o# +e2, dru**er4ex&raord%are o6 Moses (/0?)

expresses &hs 6" sa"%' &ha& p!a"%' a %o&e o% &he #rs& 6ea& o# a *easure se&&!es &he

*e&er 6u& &e%ds &o s&op &he rh"&h*$ *o&o%, 2hereas a %o&e p!a"ed o% &he se$o%d 6ea&

2!! &e%d &o prope! &he *o&o%-

/ / /

.'- @Gra+&a&o%a! a&&ra$&o%

% a *easure se5ue%$e

.ro* a *us$a%s= po%& o# +e2, &he dru**er o6 Moses (/0?) rea!;es &he pre+ous

$o%$ep& 6" sa"%' &ha& p!a"%' a %o&e o% &he #rs& 6ea& o# a *easure s&ops &he rh"&h*$

&he *o&o%, 2h!e p!a"%' a %o&e o% 6ea& &2o &e%ds &o *o+e &he pe$e #or2ard-

The pre+ous 'ra+&a&o%a! $o%$ep& 2as e*pr$a!!" &es&ed % Lopes (@), depar&%'

#ro* &he #o!!o2%' h"po&heses: a) % !%e 2&h &he 'ra+&a&o%a! $o%$ep&, &here s % a&er%ar" *e&re a de$rease o# per$ep&ua! 2e'h& #ro* &he #rs& &o &he !as& 6ea&8 () % a

5ua&er%ar" *e&re &here 2!! 6e e&her (a) a de$rease o# per$ep&ua! 2e'h& #ro* &he #rs& &o

&he !as& 6ea&, or (6) &he 6ea& herar$h" /, @, , (#ro* s&ro%'es& &o 2ea3es&)-

The exper*e%&s $o%#r*ed &he proposed %&er%a! or'a%sa&o% #or &er%ar" *e&ers- .or

5ua&er%ar" *e&ers, &he +er" s*a!! per$e%&a'e repor&%' a /44@4 2e'h& or'a%sa&o%

see*s &o 6e $o%$!us+e % ex$!ud%' & as a per$ep&ua! rea!&"8 a!&hou'h &he resu!&s #or

6ea&s a%d 2ere %o& as pred$&ed % h"po&hess (6), a%d 2ere % a%" $ase %o& +er"


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$!ear4$u&, &here 2as s&ro%' $o%#r*a&o% o# &he pred$&ed s&re%'&h o# &he &hrd 6ea&- Ths

%d$a&es &ha& 5ua&er%ar" *e&ers &e%d &o 6e heard herar$h$a!!", 6ased o% super*posed

6%ar" *e&ers- I sha!! ds$uss &hs #ur&her o%- The #o!!o2%' #or*u!a&o% o# &he *e&re

s&ra&u* s 6ased o% &hese #%d%'s a%d #ur&her $ase s&udes $a% 6e #ou%d % Lopes

(?a8 ?6)-

& Binary 'eters

%ar" *e&ers p!a" a pr*ar" ro!e % &he pro$ess o# *e&r$a! or'a%sa&o% % *us$-Me&ers o# &hs 3%d are $o*posed o# &2o 6ea&s, 2&h &he #rs& 6ea& per$ep&ua!!"

a$$e%&ua&ed (.'- )-

No&a&o% Le'e%d:

4 Measure L%e

4 ea&

!4 Per$ep&ua! Fe'h& (&he

*ore do&s &he s&ro%'er &he 6ea&, as% Lerdah! a%d >a$3e%do##=s

(/0?@) %o&a&o%)-

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I%&er%a! or'a%sa&o% o#a 6%ar" *e&er-

I% order &o address &he 5ua!&a&+e ssues %here%& % 6%ar" *e&ers, & s he!p#u! &o re++e

&he Gree3 $o%$ep&s o# thesisa%d arsis, 2h$h 2ere app!ed &o *e&r$a! #ee&- Cur& Sa$hs

(/01@:/?) exp!a%s &hese as #o!!o2s: Ko# &he &2o se$&o%s % e+er" *e&r$a! #oo&, o%e

had a s&ro%' a%d o%e a 2ea3er 2e'h&, ho2e+er su$h 2e'h& *a" ha+e 6ee% rea!;ed or

s*p!" su''es&ed---- The Gree3s $a!!ed &he s&ro%'er o# &hese 2e'h&s &he thesis or basis,


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$a!!ed arsis,

5ua!&es o# 6%ar" *e&ers- Fhe% $o%du$&%' a 6%ar" *e&re, a *oder% $o%du$&or s&!!

per#or*s a s*!ar a$&o% 2&h &he do2% a%d up *o+e*e%& o# &he 6a&o%- Us%' &he

'ra+&a&o%a! $o%$ep&, 2e $a% des$r6e &hs as #o!!o2s: &he 2e'h& o# &he #rs& 6ea& &e%ds

&o s&a6!se &he *o&o%, as sho2% % &he #oo& s&a*p%' or do2%2ard *o+e*e%& o# &he

6a&o%8 &he 2ea3%ess o# &he se$o%d 6ea& $rea&es a 3%e&$ po&e%&a! 2h$h see3s s&a6!&",

a%d s rea!sed % &he up2ard *o+e*e%& o# &he #oo& or 6a&o%-

( Ternary 'eters

I su''es&ed pre+ous!" &ha& &er%ar" pa&&er%s represe%& a 3%d o# e!a6ora&o% o# &he

pr*ar" 6%ar" or'a%sa&o%- Fe *a" &here#ore $o%$e+e a &er%ar" *e&re as a 6%ar"

*e&re &o 2h$h a% ex&ra 6ea& has 6ee% added (.'- 1)-

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Der+a&o% o# a &er%ar" *e&er

Fha& 2e'h& shou!d &he% 6e ass'%ed &o &he ex&ra 6ea& The pr%$p!e o# su$$ess+e!"

2ea3er 6ea&s #ro* &he #rs& &o &he !as& 2as $o%#r*ed 6" &he exper*e%& repor&ed ear!er

% &hs $hap&er, a%d '+es rse &o &he or'a%sa&o% o# a re'u!ar &er%ar" *e&re sho2% %

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.'- I%&er%a! or'a%sa&o%

o# a &er%ar" *e&er

) * The Emancipation o+ ,uaternary 'eter

A s*p!e app!$a&o% o# &he 'ra+&a&o%a! $o%$ep&, a!o%' &he sa*e !%es o# 6%ar" a%d

&er%ar" *e&ers, 2ou!d '+e rse &o a 5ua&er%ar" *e&re % 2h$h ea$h 6ea& 2as 2ea3er

&ha% &he pre$ed%' o%e (.'- )-

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.'- I%&er%a! or'a%sa&o% o# a 5ua&er%ar" *e&er

6ased o% &he 'ra+&a&o%a! $o%$ep&

Ho2e+er, &he da&a der+ed #ro* &he exper*e%& re'ard%' &he or'a%sa&o% o# 5ua&er%ar"

*e&re 2as % a$$orda%$e 2&h &he $o**o% dea &ha& 5ua&er%ar" *e&re s a herar$h$a!

e!a6ora&o% o# &he pr*ar" 6%ar" *e&re (.'- ?)- O%e shou!d 6ear % *%d &ha& &he

$o'%&+e pr%$p!e o# 6%ar" a%d &er%ar" or'a%sa&o% 2or3s a& a!! s&a'es o# &he rh"&h*$

%#ere%$e pro$ess8 a%" or'a%sa&o% der+ed #ro* ea$h %e2 s&ep o# &he rh"&h*$ *ode!

s a& #rs& a!2a"s *easured a'a%s& a pre#era6!e 6%ar" or &er%ar" or'a%sa&o%-

There#ore, o%!" # & s %o& poss6!e &o 6ase a% or'a%sa&o% o% a 6%ar" a%d &er%ar" u%&

s a%o&her $o%sdered (a 'ood exa*p!e o# &hs s &he pre#erred 6%ar" su64su6d+so%

!e+e! o# a &er%ar" su6d+so% rh"&h* $e!!)- I& $o*es as %o surprse, &he%, &ha& a%" *e&er

a6o+e &he pr*ar" 6%ar" a%d &er%ar" *e&ers s !a6!e &o 6e $o%s&ru$&ed as a% e!a6ora&o%

o# &hese-

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-.'- ?

A 5ua&er%ar" *e&re as a herar$h$a!e!a6ora&o% o# a 6%ar" *e&er

A!&hou'h & $a% % &hs 2a" 6e see% as a% e!a6ora&o% o# a 6%ar" *e&re, I 6e!e+e &ha&

2despread use o# &he 5ua&er%ar" *e&re % d##ere%& *us$a! $u!&ures a%d 'e%res has !ed

&o &s separa&o% #ro* &he 6%ar" *e&re a%d &he rse o# &s o2% de%&&"- A#&er a!!, a

5ua&er%ar" *e&re s d##ere%& #ro* a 6%ar" *e&re pre$se!" % &he a!&era&o% o# s&ro%'er

a%d 2ea3er par&s o# 6ea&s, 2h$h '+e & a% add&o%a! d*e%so% a%d $o*p!ex&" o# &s

o2%- Co%sder%' o%$e *ore &he 'ra+&a&o%a! $o%$ep&, a!! *e&r$a! po%&s 2h$h are a&

d##ere%& ds&a%$es #ro* &he 'ra+&a&o%a! $e%&er shou!d ha+e d##ere%& 2e'h&s- I% &he

pre+ous exa*p!e, 6ea&s &2o a%d #our $a%%o& ha+e &he sa*e per$ep&ua! 2e'h& (as

sho2% % .'- ?)- There#ore, $o*6%%' &he exper*e%&a! da&a 2&h e*pr$a!

3%o2!ed'e, o%e $a% propose a% %&er%a! or'a%;a&o% #or &he 5ua&er%ar" *e&er as &he o%e

sho2% % .'- 0 9 % 2h$h &he !as& 6ea& o# &he *easure s &he 2ea3es&-

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The proposed %&er%a! or'a%sa&o%

o# a 5ua&er%ar" *e&er

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/ * Other 'eters 0( /

As pre+ous!" exp!a%ed, *e&ers &ha& do %o& *p!" a%" 3%d o# 6%ar" or &er%ar"

or'a%sa&o% 2&h% &he &a$&us ra%'e are hard &o %#er8 *oreo+er, e+e% 2&h% &he &a$&us

ra%'e, &he su$$ess#u! %#ere%$e o# a 5u%&up!e *e&re s o%!" a$he+ed # &he rh"&h*

s&ra&u*, or a%o&her *us$ para*e&er, *a%&a%s a $!ear a%d o%'o%' a$$e%&ua&o% o# &he

#rs& 6ea&- Me&ers 6ased o% pr*e %u*6ers su$h as 1 a%d *a" &he% 6e de#%ed % &er*s

o# &he 'ra+&a&o%a! $o%$ep&, % o&her 2ords a $o%&%uous de$rease #ro* &he #rs& &o &he

!as& 6ea&, as sho2% % .'- /-

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I%&er%a! or'a%sa&o%o# a 5u%&up!e *e&er

Ne+er&he!ess, &he e##e$&s o# &he per$ep&ua! pr*a$" o# 6%ar" a%d &er%ar" or'a%sa&o%

app!" espe$a!!" &o *e&ers $o*prs%' a% u%e+e% %u*6er o# 6ea&s, su$h as 1 or - The

$o'%&+e *por&a%$e o# 6%ar" a%d &er%ar" or'a%sa&o% s su$h &ha& &he" are !a6!e &o

6rea3 up %&o &2o or &hree46ea& par&s: #or %s&a%$e, &o2ards &he !o2 ra%'e o# &he &a$&us,

&he 5u%&up!e *e&re 2!! &a3e &he shape o# e&her a 6%ar" *e&re #o!!o2ed 6" a &er%ar"

*e&re (.'- // a)), or a &er%ar" *e&re #o!!o2ed 6" a 6%ar" *e&re (.'- // 6))-

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- -a) 6)

.'- //T2o poss6!&es der+ed #ro*

a re!a&+e!" s!o2 5u%&up!e *e&er

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.ro* &he $o'%&+e po%& o# +e2, *e&ers &ha& % &her *os& 6as$ #or* are %o& 6%ar",

&er%ar", or 5ua&er%ar" 2!! a!2a"s 6e d##$u!& &o 6e %#erred- The su$$ess % &her

%#ere%$e 2!! depe%d upo% o&hers #a$&ors 6esdes &he s*p!e rh"&h*$ s&ru$&ure o#

*us$- .'- //a) sho2s a 'ood exa*p!e o# a *e&er &ha& # p!a"ed a& a re!a&+e!" #as&

&e*po 2!! &e%d &o 6e %#erred as 6%ar", 2&h d##ere%& s;ed 6ea&s (% a @ re!a&o%)-

Co%sder%' &ha& &he *us$a! *e&er s !3e a $!o$3 &ha& or'a%;es &he &*e aspe$&s %

*us$, o%e &ha& $o%ss&s o# u%&s 2&h d##ere%& s;es s 2e!! 6e"o%d 2ha& our $o'%&+e

s"s&e*s !oo3s #or- Ths 2ou!d 6e !3e ha+%' our !#e or'a%;ed 6" a 2a&$h % 2h$h &s

se$o%ds 2ou!d +ar" % s;e (2e a!! expere%$e ho2 hard so*e&*es & s &o adBus& &o &he

? da"s o# .e6ruar")- A!&hou'h o%e $a% #%d % Lopes (@) a 6roader ds$usso% o%

rre'u!ar *e&ers, &he $o*p!ex&" o# &hs ssue s 2e!! 6e"o%d &he s$ope o# &hs ar&$!e 9

&here#ore, 2e 2!! !ea+e &hs ssue #or a%o&her &*e-

%onclusions

The #or*u!a&o% proposed % &hs ar&$!e 2h$h de#%es &he opera&o% o# &he *a%

5ua!&es o# &he *os& $o**o% *e&ers % &rad&o%a! 2es&er% *us$ s re!e+a%& %o& o%!" &o

&he *us$ $o**u%&" 6u& a!so &o o&her ar&s- I& 2as proposed &ha& &he %%er or'a%;a&o%

o# *e&er p!a"s a s'%#$a%& ro!e % &he su''es&o% o# 3%ess- I% &hs 2a", &he 3%o2!ed'e

o# &he d##ere%& or'a%;a&o%s o# *e&ers a!!o2s &he $o%&ro! a%d ex&rapo!a&o% o# &hs

*por&a%& per$ep&ua! 5ua!&" (-e- &he %o&o% o# *o+e*e%&) 6" $o*posers or *us$a%s-

O%e $a% $o%$!ude, &he%, &ha& 6" p!a$%' e+e%&s (-e- *us$a! sou%ds) % 2ea3er par&s o# a

par&$u!ar *e&er a $er&a% se%se o# 3%ess s re!eased- Co%+erse!", &he p!a$e*e%& o#

sou%ds o% s&ro%'er *e&r$a! po%&s &e%ds &o s&a6!;e &he *o&o% o# a pe$e- A!&hou'h &he

$o%$ep&s % &hs ar&$!e 2ere 6ase o% e*pr$a! da&a, &he resu!&s are 2e!! % a$$orda%$e

2&h &he pra$&$a! 3%o2!ed'e- O%e $a% eas!" o6ser+e &ha& % Ba;; *us$ 9 a 'e%re &ha& s

3%o2% #or &s 3%e&$ 5ua!&es 9 &here are re$urr%' a$$e%&ua&o%s o# 6ea&s &2o a%d #our

o# a 5ua&er%ar" *e&er- A!so, h'h!" 3%e&$ 5ua!&" a%d s"%$opa&o% prese%& % *a%"

popu!ar *us$ 'e%res s a!so due &o &he re$urr%' p!a$e*e%& o# sou%ds o% 2ea3 *e&r$a!

po%&s-

The 3%o2!ed'e o# &he 3%e&$ 5ua!&es o# *us$ s a!so o# %&eres& #or o&her per#or*a%$ear&s- I% da%$e (as 2e!! % dra*a), &he re!a&o% 6e&2ee% per#or*er a%d &he 3%ess o#

/@


14/14

*us$ $a% 6e +er" e##$e%& #ro* a% aude%$e=s per$ep&ua! po%& o# +e2- The

u%ders&a%d%' o# &he opera&o% o# &hs 5ua!&" $a% &he% 6e o# 'rea& use #or per#or*ers,

$horeo'raphers, as 2e!! as o&her $rea&ors o# ar&-

Bibliography

Epstein 3!Shaping Time- Ne2 Yor3: S$hr*er oo3s, /001-

Ko++4a K!Principles of Gestalt Psychology.Ne2 Yor3: Har$our& a%d ra$e, /0@1-

5asty %!Meteras Rhythm- Ne2 Yor3: Ox#ord U%+ers&" Press, /00-

6erdahl 78 9ac4endo++ !A Generative Theory of Tonal Music- Ca*6rd'e, Mass-:

MIT Press, /0?@-

6opes E!Just in Time to!ar"s a theory of rhythm an" metre. Sou&ha*p&o%, U, @-

@0p- PhD Thess 9 .a$u!&" o# Ar&s 9 U%+ers&" o# Sou&ha*p&o%-

6opes E!Rh"&h* a%d Me&er Co*pos&o%a! Too!s % a Chop%=s Fa!&;- A" Parnassum

Journal- +o!- , ssue //- Apr! ?a)- 4?-

6opes E!.ro* !ues &o La&% >us& % T*e: A rh"&h*$ a%a!"ss o#

Eduardo Lopes Time Rhythm and Meter in Music the Gravitational Concept

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