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http://www.mtosmt.org/issues/mto.10.16.3/mto.10.16.3.waters_williams.html 

The paper provides methods for modeling harmonies and harmonic progressions based on

diatonic, acoustic, heatonic, and octatonic collections. !t eamines these four spaces in relation to

 "a## harmon$, particularl$ post%1&60s "a## compositions in which harmonic function is suppressed

or absent.

Theories of harmon$ rest on two fundamental 'uestions: 1( )hat are the ob"ects *i.e., the

harmonies( and+ ( )hat are the relationships among the ob"ects *i.e., harmonic

 progressions/successions(- The success of recent wor in neo%iemannian theor$ addresses

situations for which the answer to the first 'uestion involves standard or s$ntactic harmonies, but

the answer to the second 'uestion involves non%standard or non%s$ntactic progressions *hilds

1&&, ohn 1&&a and 1&&b, 2outhett and teinbach 1&&, 4ollin 1&&, 5$er 1&&(. 7 roughl$

analogous situation arises for addressing "a## harmonic progressions, particularl$ for compositions

after 1&60. ompositions of this period often use standard "a## harmonies, but in progressions that

suppress or ignore harmonic function. 8et the analog$ is rough because, in general, "a## harmonies

emplo$ chordal etensions of sevenths, ninths, elevenths, and/or thirteenths. 9urther, harmonies in

a number of "a## compositions after 1&60 also mae use of non%s$ntactic *or non%standard( chords,

man$ of which are derived from the heatonic collection and use pcset 01;< as a subset. *(

=a## compositions of the 1&60s draw primaril$ from harmonies that are subsets of four

fundamental collections: the diatonic, the acoustic *also considered a mode of the ascending

melodic minor scale(, the octatonic, and the heatonic. Example 1 indicates some representative

 "a## harmonies from these four underl$ing collections.

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Example 1. Jazz harmonies from Diatonic, Acoustic, Octatonic, and

Hexatonic Collections

Example 1a. Harmonies

from C Diatonic Collection

(D0 = C, D, E, F, , A, !, C"

Cma#$, Cma#$%&, C'%& etc.

Fma#$( "

Dmin$%&, Dmin$%&%, Dmin$%&%%)

dom$%&, dom$%&%)

$sus

Fma#$%A 

 

Example 1b. Harmonies

from F Acoustic Collection

(A* = F, , A, !, C, D, E , F"

(+ome harmonies include

cset -0/1

C, E , , ! or E , , !, D"

F)( "

Cmin2$, Cmin2$%&, Cmin2$%&%,

Cmin2$%&%%), Cmin'%&, etc.

 A min$%&( *", A min$%&%( *", A min$%&%% )(

*"

!$ altered (=  &, &, , )"

E ma#$( *", E ma#$%&( *"

 

Example 1c. Harmonies from D%E

Octatonic Collection

(Oct 3,) = D, E , F, , A , A, !, C, D"

D$( &", D)( &", D)( &% "

F$( &", F)( &", F)( &% " 

 A $( &", A )( &", A )( &% "

!$( &", !)( &", !)( &% "

Co ma#$, Co&(ma#$", Co(ma#$", Co )(ma#$"

E o ma#$, E o&(ma#$", E o(ma#$", E o )(ma#$"

o ma#$, o&(ma#$", o(ma#$", o )(ma#$"

 A o ma#$, A o&(ma#$", A o(ma#$", A o )(ma#$"

 

Example 1d. -0/ Harmonies from

!%C Hexatonic Collection(H0 = !, C, E , E, , A "

!ma#$( *", E ma#$( *", ma#$( *"

Cmin2$, Emin2$, A min2$

 

Example 1e. +eci4c Hexatonic

Harmonies in 5ost6&'0s Comositions

 A ma#$( *% &"1 7a8ne +horter, 9:onetta;

 A ma#$( *"%E1 Chic< Corea, 9+on of the

7ind;

Emin2$%A  1 Corea, 9+on of the 7ind;

 A ma#$( *% *"1 +horter, 9>ris;

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>ample 1 relates chord t$pe to underl$ing collections, and thus provides some eplanator$ power 

for understanding chords in isolation. !n fact, what ! have done is slightl$ recast the primar$ focus

of "a## pedagog$ and theor$ since at least 1&&: the mapping of underl$ing collections toindividual chords in order to suggest methods for improvisation and/or composition.*;( uch

discussions do not alwa$s sa$ much about principles of chord succession, however, particularl$

for music of the 1&60s whose progressions ma$ avoid or suppress functional harmonic patterns.

9urther, if collections of 6% pitches?particularl$ the four indicated above?do provide the source

sets for post%1&60s harmonies, their relationship ma$ be abstract: in practice all members of the

underl$ing collection need not be stated b$ the harmonic accompan$ing instruments *or b$

improvisers(. Thus two 'uestions remain: how do we address post%1&60s harmonic progressions,

and how do we do it in a wa$ that more clearl$ models harmonic practice- These two 'uestions

can lead us past the chord/scale taonom$ of "a## pedagog$ to consider harmonies instead as

subsets of these collections and eamine the wa$s in which the harmonies connect.