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Table 1: Harmonic functions related to scale degrees.
Degree
Function
Name Symbol
Shift
minor major
1 Tonic T 0 0
2 Subdominant 2nd S
2
2 2
3 Tonic 3rd T
3
4 3
4 Subdominant S 5 5
5 Dominant D 7 7
6 Tonic 6th T
6
9 8
It is important to stress that in most cases there
is no single, ultimate harmony solution for a given
melody, usually there are many possibilities, rang-
ing from the simplest and obvious to the compli-
cated and non-trivial ones. The choice depends on
the style of music, as well as the capabilities of in-
strument/orchestration and the abilities of perform-
ers/harmonizers. There are also cases, when no addi-
tional information, except for the leading melody and
harmonic functions, is needed to perform the music.
Common example is improvised
`
a vista accompani-
ment to songs, when playing the guitar or piano.
For the purposes of this paper we have focused on
a tonal system with two basic scales: major and mi-
nor (natural minor as well as harmonic and melodic
modifications). Our considerations and experiments
are based on six harmonic functions (built on 1
st
, 2
nd
,
3
rd
, 4
th
, 5
th
and 6
th
grade of major/minor scales) with
basic modifications: adding seventh (minor and ma-
jor), ninth (minor and major), sixth (minor and ma-
jor). We have omitted incomplete Dominant seventh
built on grade 7
th
as being equivalent to Dominant
seventh with omitted root, alterations and higher in-
tervals (alike eleventh, thirteenth), that may be seen as
dissonances that are meant to resolve (and enrich har-
mony), rather than integral harmonic modifiers. The
tonal harmonic functions along with names and sym-
bols are presented in Table 1.
3 PROPOSED HARMONIZATION
MODEL
The proposed harmonization model is based on sev-
eral mechanisms, that are supposed to closely follow
music theory:
1. Particular notes have various harmonic impor-
tance,
2. Each note may excite (fit to) one or several har-
monic functions based on its pitch and consid-
ered function components,
3. Some harmonic functions are more likely to oc-
cur than others (the most popular example being
Tonic - a base and consolation for vast majority of
tonal music), therefore it is preferable to prioritize
functions being more frequently used,
4. Specific harmonic functions’ successions are
more or less probable, therefore it is possible
to support more likeable (frequent) successions
(alike Dominant → Tonic or Tonic → Subdomi-
nant).
The mechanisms are implemented in possibly in-
dependent way and weighted (using a range [0;1]) in
order to easily configure the degree of influence (or
eliminate the influence at all) on the final harmoniza-
tion, in comparison with the other mechanisms.
3.1 Data Representation of Harmonic
Functions
Each harmonic function is stored in a structure that
contains a vector (of length 6, as we consider 6
different diatonic functions) of Harmonic Function
Strengths and a corresponding vector of modifiers.
In our experiments the modifiers are mostly limited
to sevenths, with occasional ninths and in rare cases
to sixths (sixth is present in so-called Chopin chord:
Dominant with seventh and natural or augmented
sixth instead of fifth, resolving down to first degree of
scale). The diatonic function with modifiers is equiv-
alent to chords i.e. Tonic with modifier seventh in key
G-major is equivalent to G
7
. Some modifiers cover
others, i.e. a chord with ninth is supposed to gorge a
chord with seventh.
Musical piece is decomposed into elementary har-
monic fragments (defined as indivisible musical piece
unit with one diatonic function attached to). For ex-
perimental evaluation we have chosen fragments of
fixed lengths at the most two beats or less depend-
ing on the time of musical work, i.e. for the time
3
4
it would be two beats for the first harmonic frag-
ment of measure and one beat for the remaining frag-
ment of measure; for the time
2
4
one harmonic frag-
ment per measure would most probably be sufficient.
Each harmonic unit is processed independently - with
exception of supporting specific successions of har-
monic functions (see Section 3.6) - therefore it does
not require additional modifications, as shorter har-
monic fragments will also naturally have lower influ-
ence on larger neighbors. In most cases this simplis-
tic division would be sufficient, as harmonic functions
rarely change in practice more frequently, while over-
fragmentation could result in lower accuracy of har-
monization.
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