=Paper=
{{Paper
|id=Vol-433/paper-5
|storemode=property
|title=A Formal Concept Analysis of Harmonic Forms and Interval Structures
|pdfUrl=https://ceur-ws.org/Vol-433/paper1.pdf
|volume=Vol-433
}}
==A Formal Concept Analysis of Harmonic Forms and Interval Structures==
https://ceur-ws.org/Vol-433/paper1.pdf
A Formal Concept Analysis of Harmonic Forms
and Interval Structures
Tobias Schlemmer and Stefan E. Schmidt
Technische Universität Dresden,
Fachrichtung Mathematik, 01062 Dresden, Germany
Tobias.Schlemmer@mailbox.tu-dresden.de, midt1@msn.com
http://www.math.tu-dresden.de/~schlemme/
Abstract. While small concept lattices are often represented by line
diagrams to better understand their full structure, large diagrams may
be too complex to do this. However, such a diagram may still be used
to receive new ideas about the inherent structure of a concept lattice.
This will be demonstrated for a certain family of formal contexts arising
from mathematical musicology. In particular, we investigate how chord
patterns can be characterised by their interval structure. For such con-
texts of pattern structures, it turns out that each corresponding concept
lattice incorporates two competing building principles, one emanating
from the top the other from the bottom of the lattice.
Key words: formal concept lattice, harmonic form, musicology, interval
1 Introduction
Harmonic forms provide basic notions for the descriptions of chords. Well-known
examples are the harmonic form of a major triad which stands for all major
chords and, similarly, the minor triad which is the harmonic form of minor
chords.
Besides harmonies and chords, harmonic forms play an important role for
tuning software like Mutabor [1]. In general, applications of mathematical mu-
sicology to music software unfold different questions about the mathematical
structure of harmonic forms in tone systems.
In the past, formal concept analysis has been applied to various fields of music
already (e. g. [3], [4], [5], and [6]). Here as well as in other fields of mathematical
musicology (cf. Mazzola et al. [7]), harmonic forms have been analysed to a
certain extent, however, there are still plenty of open problems to address.
The description of the structure of harmonic forms leads to concept lattices
that are often considered as too large to be drawn meaningfully. However, the
diagrams will serve us as a source of information useful for finding more adequate
mathematical models.
c Radim Belohlavek, Sergei O. Kuznetsov (Eds.): CLA 2008, pp. 9–22,
ISBN 978–80–244–2111–7, Palacký University, Olomouc, 2008.
10 Tobias Schlemmer, Stefan E. Schmidt
2 Qualitative analysis of harmonic forms
To describe musical objects, we need mathematical notions of them. A funda-
mental one is that of a tone system, which can be modelled as a collection of
tones and their interval structure (see also [9], [10]):
A triple T = (T, δ, I) is called an (algebraic) tone system, if T is a set,
I = (I, +, 0) is an Abelian group and δ : T × T → I is a map such that for all
t1 , t2 , t3 ∈ T the following hold:
δ(t1 , t2 ) + δ(t2 , t3 ) = δ(t1 , t3 ) and (1)
δ(t1 , t2 ) = 0 iff t1 = t2 . (2)
The elements of the set T are called tones and each subset of T is called a chord.
The elements of I are considered as intervals. For s, t ∈ T , the interval from s to
t is given by i ∈ I if i = δ(s, t) holds; in this case we can agree upon s + i := t.
For a tone t, the set of all intervals from t is given by I(t) := δ[{t}×T ]; it follows
t + I(t) = T and I(s) = δ(s, t) + I(t) for all s, t ∈ T .
We call T homogeneous if I(t) is a subgroup of I for some tone t; in this case,
we observe that I(s) = I(t) = δ[T × T ] holds for all tones s and t in T. We refer
to T as a (freely) n-generated tone system if T is homogeneous, δ[T × T ] = I,
and I is a (freely) n-generated group.
A transposition by the interval i ∈ I is defined as the map τi : T → T
such that t 7→ t + i (if t + i exists for every tone t). Obviously, in case of a
homogeneous tone system, a transposition τi exists for every i ∈ δ[T × T ]. In
particular, an n-generated tone system allows a transposition by any interval
of I, and we observe, that the set of all transpositions forms a transformation
group canonically isomorphic to the interval group I.
A morphism from a tone system T = (T, δ, I) to a tone system T0 = (T 0 , δ 0 , I 0 )
is defined as a pair φ := (φT , φI ), consisting of a map φT : T → T 0 and a group
homomorphism φI : I → I0 , such that for all s, t ∈ T we have
φI δ(s, t) = δ 0 φT (s), φT (t) .
(3)
If, in addition, φT and φI are bijections then φ is called an isomorphism
(from T to T0 ). Every transposition τ induces via (τ, idI ) an automorphism on
T.
For every positive integer n, a freely n-generated tone system is always iso-
morphic to the tone system (Zn , δ, Zn ) where δ(x, y) := y − x for all x, y ∈ Zn .
In the following we consider the 1-generated tone system T = (Z, δ, Z) and
we fix a positive integer O ∈ Z+ , which we consider as an interval called octave.
Let ZO denote the residue ring of integers modulo O and let TO := (ZO , δO , ZO )
be the 1-generated algebraic tone system (where δO (x, y) denotes the difference
y−x in ZO ). Following the language of musicology, TO is called a chroma system,
and its elements are refereed to as chromas. More specifically, we will refer to TO
as O-tone equal tempered chroma system, in short O-tet. The most commonly
used of these are the 12-tet (T12 ) and the 7-tet (T7 ).
A Formal Concept Analysis of Harmonic Forms and Interval Structures 11
Table 1. parameters describing concept lattices of harmonic forms.
Group # of harmonic forms # of irreducibles (rows/columns) # of concepts
Z1 2 1 2
Z2 3 2 3
Z3 4 3 4
Z4 6 4 6
Z5 8 6 9
Z6 14 11 18
Z7 20 13 42
Z8 36 25 142
Z9 60 39 1 460
Z10 108 73 9 325
Z11 188 112 1 798 542
Z12 352 212 208 946 771
The canonical group homomorphism φO : Z → ZO (which maps every integer
x to its residue modulo O, denoted by xO ) induces via (φO , φO ) a morphism from
T onto TO . Every chord X in T is mapped to the chord XO := {xO | x ∈ X} in
TO , which will be called the harmony of X.
Chords and harmonies can efficiently be classified by the occurrence of inter-
vals and chromas. In particular, two chords or harmonies have the same pattern
if they are related by a transposition. The corresponding pattern classes we refer
to as chordal forms or harmonic forms, respectively.
The degree of consonance or dissonance of a harmony is mostly influenced
by its pattern.
Harmonic forms and their hierarchical order have been studied by Rudolf
Wille and other authors ([5]). Though the corresponding concept lattice for
the 7-tet T7 has a nice diagram (see figure 1), the number of concepts of TO
is rapidly growing for increasing octave O. Table 1 shows some statistics about
these concept lattices. For every chroma system there are printed the number of
harmonic forms, the count of rows and columns in the formal context (describing
the order of the harmonic forms), and the number of formal concepts in the
corresponding concept lattice.
In other important chroma systems the hierarchical order of harmonic forms
is too complex to be examined in the fashion above. Therefore, it is interesting
to use other properties to clarify the structure of harmonic forms.
One important property is the interval structure of harmonic forms, since
the intervals contained in a harmonic form have a major impact on their degree
of consonance or dissonance. One example is shown in figure 2. Here, for the
7-tet a formal context is composed of the set of harmonic forms as objects, the
set of intervals as attributes, and the interval occurrence as incidence relation.
Also, in comparison with figure 1, this concept lattice is significantly simpler
(but ordered oppositely).
The other corresponding lattices are relatively small too, as shown in table
2, which enables us to have a view on the 12-tet lattice (fig. 3).
12 Tobias Schlemmer, Stefan E. Schmidt
Fig. 1. Dedekind-McNeille completion of the order of harmonic forms in Z7 .
A Formal Concept Analysis of Harmonic Forms and Interval Structures 13
second
unison
fourth
third
rest
0 ×
0, 1 ××
0, 2 × ×
0, 1, 2 ×××
0, 3 × ×
0, 1, 3 ××××
0, 2, 3 ××××
0, 1, 2, 3 ××××
0, 1, 4 ×× ×
0, 2, 4 × ××
0, 1, 2, 4 ××××
0, 1, 3, 4 ××××
0, 2, 3, 4 ××××
0, 1, 2, 3, 4 ××××
0, 1, 3, 5 ××××
0, 1, 2, 3, 5 ××××
0, 1, 2, 4, 5 ××××
0, 1, 2, 3, 4, 5 ××××
0, 1, 2, 3, 4, 5, 6 × × × ×
Fig. 2. Formal context and concept lattice which qualitatively describes the contained
intervals of harmonic forms in the 7-tet
Table 2. Statistics of harmony interval concept lattices
Group # of harmonic forms # forms clar. # forms red. # intervals # int. red. #concepts
Z1 2 2 1 1 1 2
Z2 3 3 2 2 2 3
Z3 4 3 2 2 2 3
Z4 6 5 3 3 3 5
Z5 8 5 3 3 3 5
Z6 14 7 5 5 4 7
Z7 20 9 4 4 4 9
Z8 36 12 6 5 5 13
Z9 60 13 7 5 5 16
Z10 108 20 6 6 6 33
Z11 188 23 6 6 6 33
Z12 352 32 7 6 7 65
14 Tobias Schlemmer, Stefan E. Schmidt
Figures 2 and 3 are largely Boolean lattices (except the node named “rest”).
It is also visible in figure 3 that not every node has a label. In the language of
music this means that the intervals cannot be combined freely. They have to
fulfil certain restrictions.
On the other hand, in this approach we consider neither the order nor the
multiplicity of intervals. So with this method some sets of harmonic forms are
identified. For example the major triad and the minor triad share the same
label since they consist of a minor third, a major third and a fourth as chroma
intervals.
In comparison with the concept lattices of the 7-tet and the 12-tet, the ones
of the 6-tet and the 8-tet are less symmetric (figure 4). This means, the lattices
have a more complex underlying structure.
3 Analysis reflecting interval multiplicities
The Dedekind-McNeille completion of the ordering of harmonic forms does
not reflect the notion of an interval. However, it describes a much finer granular-
ity than given by the previously discussed type of lattice (derived from interval
occurrence). The gap between these two lattice types can be filled by considering
many-valued contexts, which reflect multiplicities of intervals within harmonic
forms.
Figure 5 shows such a context in the 7-tet. Its ordinary scaled version is
shown in figure 6 and the corresponding concept lattice appears in figure 7. This
lattice has more concepts than the one presented in figure 2, but it contains
less information than the one in figure 1. Though the multiplicity of intervals
is reflected by the lattice, their internal arrangement remains neglected. For
example, the harmonic forms of {0, 1, 3} and {0, 2, 3} are different but share the
same node. This impacts the musical interpretation, as some harmonic forms
(like the major triad and the minor triad in the 12-tet) are indistinguishable.
Furthermore, the lattice of figure 7 has a very regular structure. In the upper
part, the free distributive lattice with three generating elements is visible. The
concepts below it form a typical configuration for all lattices of this family, which
can be seen more clearly in the lattices investigated in the sequel.
After observing the beautiful structure provided by the 7-tet, we investigated
higher orders in a similar fashion. Our suggestive approach was to draw the
diagrams by ordering the intervals from left to right according to their sizes. For
the 8-tet this can be seen in the left diagram of figure 8.
Aiming for further insight, the following question arises: Which lattices allow
nice diagrams and how can these diagrams be realised? To get a clue how to
answer this question we focus again on the left diagram in figure 8. The right
hand side of this diagram shows some interesting structural specialities: There
are shorter chains from top to bottom of the lattice than on the left hand side.
This phenomenon has the following cause: The concepts on the right hand side
share the fifth (distance 4), which is an element of order 2 in the interval group.
So these intervals occur only in pairs. This differs from the other intervals, which
A Formal Concept Analysis of Harmonic Forms and Interval Structures 15
Fig. 3. Concept lattice qualitatively describing the contained intervals of harmonic
forms in the 12-tet. The list of harmonic forms in the lower node is truncated.
16 Tobias Schlemmer, Stefan E. Schmidt
Fig. 4. Concept lattice qualitatively describing the contained intervals of harmonic
forms in the 6-tet and the 8-tet
pattern unison second third fourth
rest
0 1
0, 1 2 1
0, 2 2 1
0, 1, 2 3 2 1
0, 3 2 1
0, 1, 3 3 1 1 1
0, 2, 3 3 1 1 1
0, 1, 2, 3 4 3 2 1
0, 1, 4 3 1 2
0, 2, 4 3 2 1
0, 1, 2, 4 4 2 2 2
0, 1, 3, 4 4 2 2 3
0, 2, 3, 4 4 2 2 2
0, 1, 2, 3, 4 5 4 3 3
0, 1, 3, 5 4 1 3 2
0, 1, 2, 3, 5 5 3 4 3
0, 1, 2, 4, 5 5 3 3 4
0, 1, 2, 3, 4, 5 6 5 5 5
0, 1, 2, 3, 4, 5, 6 7 7 7 7
Fig. 5. Many-valued context describing the contained intervals of the 7-tet
A Formal Concept Analysis of Harmonic Forms and Interval Structures 17
2 × second (d = 1)
3 × second (d = 1)
4 × second (d = 1)
5 × second (d = 1)
6 × second (d = 1)
7 × second (d = 1)
2 × fourth (d = 3)
3 × fourth (d = 3)
4 × fourth (d = 3)
5 × fourth (d = 3)
2 × third (d = 2)
3 × third (d = 2)
4 × third (d = 2)
5 × third (d = 2)
6 × third (d = 2)
7 × third (d = 2)
second (d = 1)
unison (d = 0)
fourth (d = 3)
third (d = 2)
rest
0 ×
0, 1 × ×
0, 2 × ×
0, 1, 2 × × × ×
0, 3 × ×
0, 1, 3 × × × ×
0, 2, 3 × × × ×
0, 1, 2, 3 × × × × × × ×
0, 1, 4 × × × ×
0, 2, 4 × × × ×
0, 1, 2, 4 × × × × × × ×
0, 1, 3, 4 × × × × × × ×
0, 2, 3, 4 × × × × × × ×
0, 1, 2, 3, 4 × × × × × × × × × × ×
0, 1, 3, 5 × × × × × × ×
0, 1, 2, 3, 5 × × × × × × × × × × ×
0, 1, 2, 4, 5 × × × × × × × × × × ×
0, 1, 2, 3, 4, 5 × × × × × × × × × × × × × × × ×
0, 1, 2, 3, 4, 5, 6 × × × × × × × × × × × × × × × × × × × × × ×
Fig. 6. Scaled formal context quantitatively describing the contained intervals of har-
mony patterns in the 7-tet
18 Tobias Schlemmer, Stefan E. Schmidt
Fig. 7. Concept lattices of the context counting the intervals in the 7-tet
Fig. 8. Concept lattice describing the contained intervals in the 8-tet in two different
chain decompositions
A Formal Concept Analysis of Harmonic Forms and Interval Structures 19
can be combined more freely. For example the harmonic form represented by the
harmony {0, 1, 2, 3, 4} has only three (unordered) pairs of chromas containing a
third, namely {0, 2}, {1, 3}, {2, 4}.
Thus, divisibility of the group order has an impact on the (potential) layout
and also the aesthetics of the generated diagram. On the other hand, harmonic
forms of T4 can be embedded into T8 in various ways. The most important ones
are defined by the mappings of chromas f1 : t 7→ t and f2 : t 7→ 2t. Each of these
maps preserves to a reasonable extent the interval structure.
This suggests to rearrange the order of the intervals in the diagram to better
unfold the lattice diagram. Coprime intervals (where coprime is meant in the
number theoretic sense) should be positioned far apart from each other, while
those with small greatest common divisor not equal to 1, should be in close
proximity. The unfolded lattice is shown on the right hand side of figure 8.
It turns out that divisibility is the main property which makes the upper part
of such a pattern lattice deviate from a product of chains. The lower part of the
pattern lattice is of a significantly different shape. To discuss this, we point out
the following general fact: In the chroma system TO intervals can be described
as Lee distances (see [11]) between chromas.
The lowest point of such a diagram (as given in figure 8) represents the
harmonic form of the complete chroma set ZO . Here, the Lee distances between
unordered pairs of chromas form the set {0, . . . , bO/2c}. In the diagram the
upper neighbour of the concept of the complete harmonic form represents an
almost complete chroma set of size one less than ZO . That means, all intervals
occur with the same frequency O − 2. Next, selecting O − 2 chromas, results in
a harmonic form with two chromas omitted. Thus, each interval which ends in
one of these two points will occur only O − 4 times in the pattern. But there
is one exception: The interval between the two deleted points has been counted
twice. Consequently, this interval occurs (still) O − 3 times in the harmonic
form. Obviously, with an increasing number of deleted chromas, the number of
intervals additionally vanishing, decreases further.
Because of the above, the lattice is divided into several levels of object con-
cepts according to the number of chromas in the harmonic forms. This is not
hard to understand since every harmonic form with k chromas contains k(k−1) 2
intervals (where multiplicities are respected). The latter means that the concept
of such a harmonic form has k(k−1) 2 + 1 attribute concepts of interval sets above
it. Adding a chroma increases the number of interval concepts by k.
In case of example T12 , the above mentioned levels of harmonic forms become
increasingly apparent towards the bottom part of the lattice T12 (see figure 9).
The right hand side of the diagram shows the “lightning rod chains”, which
result from the even group order leading to the pairwise occurrence of tritone
intervals (distance 6) as described above.
The structure of the left hand side in the diagram of T12 is induced by the
divisibility of 1, 2, 3 and 4. Though nesting can simplify the diagram in certain
cases, the resulting lattices are still too complex for us to analyse. The reason
is that every interval chain < n2 generates all the levels described above. An
20 Tobias Schlemmer, Stefan E. Schmidt
Fig. 9. Concept lattice of the harmony pattern vs. interval count context of the 12-tet
A Formal Concept Analysis of Harmonic Forms and Interval Structures 21
Fig. 10. Concept lattice created omitting all but the intervals 1, 2 and 4 in the 12-tet
example of such a large projection which does not allow a sensible nesting is
demonstrated in figure 10.
4 Conclusions and further research topics
The current work shows how one can overcome the obstacles of getting a mean-
ingful interpretation of concept lattices of increasing complexity. In particular,
we analyse concept lattices describing harmonic forms and their intervals in
different ways, focusing on the 7-tet and the 12-tet.
For future analysis we propose that the information hidden in complex dia-
grams (e. g. as given in figure 9) may be used to further investigate the inherent
structure of the concept lattice.
22 Tobias Schlemmer, Stefan E. Schmidt
Ongoing work is concerned with a description of the interval structure of a
tone system and its influence on the structure of the concept lattices of harmonic
forms in case of a totally ordered (or, more generally, lattice ordered) interval
group.
Another extension of this work will be the investigation of tone systems
with more complicated interval structures, for example the diatonic scale and
Leonhard Euler’s Tonnetz.
This area of research also aims to have an impact on the further development
of the tuning software Mutabor.
References
1. Mutabor team: Mutabor. Software http://www.math.tu-dresden.de/~mutabor/.
2. Ganter, B., Wille, R.: Formal concept analysis: mathematical foundations.
Springer, Berlin (1999)
3. Wille, R., Wille-Henning, R.: Towards a semantology of music. In Priss, U.,
Polovina, S., Hill, R., eds.: ICCS. Volume 4604 of Lecture Notes in Computer
Science., Springer (2007) 269–282
4. Wille, R., Wille-Henning, R.: Beurteilung von Musikstücken durch Adjektive: Eine
begriffsanalytische Exploration. In: Von Intentionalität zur Bedeutung konvention-
alisierter Zeichen: Festschrift für Gisela Harras zum 65. Geburtstag. Volume Bd.
35. G. Narr, Tübingen (2006) 453
5. Wille, R.: Musiktheorie und Mathematik. In: Musik und Mathematik: Salzburger
Musikgespräch 1984 unter Vorsitz von Herbert von Karajan. Springer-Verlag,
Berlin (1985) 4–31
6. Wille, R.: Mathematische Sprache in der Musiktheorie. In: Jahrbuch Überblicke
Mathematik 1980. Bibliographisches Institut, Mannheim (1980) 167–184
7. Mazzola, G., Göller, S., Müller, S.: The topos of music: geometric logic of concepts,
theory, and performance. Birkhauser Verlag, Boston, MA (2002)
8. Wille, R.: Triadic concept graphs. In Mugnier, M.L., Chein, M., eds.: Concep-
tual Structures: Theory, Tools and Applications. Volume 1453 of Lecture Notes in
Computer Science., Springer (1998) 194–208
9. Neumaier, W., Wille, R.: Extensionale Standardsprache in der Musiktheorie – eine
Schnittstelle zwischen Musik und Informatik. In Hesse, H., ed.: Mikrotöne. Volume
III. Edition Helbing edn. Helbing, Insbruck (1990) 139–167
10. Neumaier, W.: Was ist ein Tonsystem? : eine historisch-systematische Theorie der
abendländischen Tonsysteme, gegründet auf die antiken Theoretiker Aristoxenos,
Eukleides und Ptolemaios, dargestellt mit Mitteln der modernen Algebra. Vol-
ume 9. Lang, Frankfurt am Main (1986)
11. Lee, C.: Some properties of nonbinary error-correcting codes. Information Theory,
IEEE Transactions on 4(2) (Jun 1958) 77–82