=Paper= {{Paper |id=Vol-433/paper-18 |storemode=property |title=The Mathematical in Music Thinking |pdfUrl=https://ceur-ws.org/Vol-433/paper14.pdf |volume=Vol-433 }} ==The Mathematical in Music Thinking== https://ceur-ws.org/Vol-433/paper14.pdf

The Mathematical in Music Thinking

Rudolf Wille and Renate Wille-Henning

Technische Universität Darmstadt, Fachbereich Mathematik,
wille@mathematik.tu-darmstadt.de

Abstract. “The Mathematical in music thinking” is based on Heideg-
ger’s understanding of “the Mathematical” as the basic assumption of
the knowledge of the things. Heidegger’s ideas are combined with Peirce’s
classification of sciences, in particular, to distinguish between the Math-
ematical from the less abstract logical thinking and the more abstract
mathematical thinking. The aim of this paper is to make understandable
the role of the Mathematical in music. The paper concentrates on three
domains: the rhythmic of music, the doctrine of music forms, and the
theory of tonal systems. The theoretical argumentations are assisted by
musical examples: the Adagio of Mozart’s string quartet C major (KV
465), the second movement of Webern’s Symphony op.21, and a cadence
illustrating the problem of the harmony of second degree.

1 Music Thinking and The Mathematical

”Musica est exercitium arithmeticae occultum animi” (”Music is a hidden arith-
metical exercise of the soul”) - this statement was written by the philosopher,
mathematician, and scientist Gottfried Wilhelm Leibniz on April 17, 1712, in a
letter to the mathematician and diplomat Christian von Goldbach. Leibniz re-
ferred with his statement to the astonishing phenomenon of the correspondence
between musical tones and numbers which has been already demonstrated by
the pythagoreans on their monochord. This phenomenon has been extensively
described by the German musicologist Martin Vogel in his book “Die Lehre von
den Tonbeziehungen”; there he writes: “Each interval used in music corresponds
to a certain numerical proportion and, since each melody and each harmonic
connection can be composed by numerically described intervals, each composi-
tion can finally be understood and analytically recognized as an arrangement of
uniquely determined relations of numbers” ([18], p.9).
If one wants to comprehensively understand the role of mathematics in mu-
sic thinking, then the numerical relations in music compositions pointed out by
Vogel do not suffice. In particular, the numerical relations cannot suitably grasp
the more extended set semantics basic for modern mathematics. For our theme
we use the understanding of “the Mathematical” which Martin Heidegger worked
out in his 1935/36 lecture on “Basic Questions of Metaphysics” (published in
[11]). For Heidegger “the Mathematical” is not derivable out of mathematics,
but mathematics itself is at the time a historically, socially, and culturally de-
termined formation abstracted from the Mathematical. Heidegger deduced his

c Radim Belohlavek, Sergei O. Kuznetsov (Eds.): CLA 2008, pp. 167–180,
ISBN 978–80–244–2111–7, Palacký University, Olomouc, 2008.
168 Rudolf Wille, Renate Wille-Henning

understanding of “the Mathematical” from the ancient Greeks: τ ὰ µαϑήµατ α
means “the learnable”. Learning the learnable is a kind of “taking”, by which
the taker takes only such things which, strictly speaking, he already has. Ac-
cording to Heidegger it follows: “τ ὰ µαϑήµατ α, the Mathematical, is what of
the things we actually already know, which we therefore do not first take out of
the things, but which we already bring with us in a certain way” ([11], p.57); or
phrased in another way: “The Mathematical is that basic position to the things
by which we take on the things according to that which the things have already
been given to us. The Mathematical is therefore the basic assumption of the
knowledge of the things” ([11], p.58). For Heidegger this makes clear the central
significance of the Mathematical for modern thinking, because “a will of refor-
mation and self-foundation of the knowledge form as such” lies in the character
of the Mathematical as distinctive conception ([11], p.75).
But how can we recognize the Mathematical? A promising approach is to
abstract logical forms of thinking to mathematical forms of thinking which gives
rise to rich mathematical theory developments retroacting, in particular, the
logical forms and in this way enriching also the logical thinking (cf. [23]). To
capture the Mathematical in music thinking, it suggests itself to identify first
of all the logical in music, for instance in a manner as articulated by the musi-
cologist Hans-Peter Reineke in referring to musical hearing; he writes: “Certain
regulatives in musical hearing constitute and preserve music as a logical being
that must sound plausibly out of itself if it shall be accepted” [17]. During the
ending 18th century the term “musical logic” was linked to the idea “that mu-
sic is an art which is autonomous, resting in itself, and submitted only to its
own law of form; in particular, its right to exist needs not to be justified extra-
musically” ([3], p.66). But, inspite of numerous efforts (here, first of all, the
musicologist Hugo Riemann has to be named), a musical logic has never been
really established in musicology. Nevertheless, to identify the Mathematical in
music thinking, the connection between logical and mathematical thinking shall
be discussed more extensively.
The philosopher and scientist Charles Sanders Peirce has convincingly de-
scribed the connection between logical and mathematical thinking in the frame
of his philosophy of science. In his classification of sciences from 1903 ([16],
258ff.), in which he ordered the sciences by the degree of their abstractness,
mathematics as the most abstract science of all sciences is positioned at the
most abstract level. As the only hypothetical science, mathematics has the task
to develop a cosmos of forms of potential realities. All other sciences, under which
philosophy is the most abstract, relate to actual realities. According to Peirce’s
classification, philosophy partitions into phenomenology, normative science, and
metaphysics while normative science divides further into esthetics, ethics, and
logic. Musicology has to be classified - such as history - under the descriptive
science. In Peirce’s classification the sciences are ordered in a manner that each
science

– refers, according to its general principles, exclusively to the sciences which
are more abstract than itself, and
The Mathematical in Music Thinking 169

– makes use of examples and specific facts elaborated by sciences which are
less abstract than then the considered science.

For instance, logic as the third part of normative science is supposed to refer to
ethics, esthetics, phenomenology, and mathematics concerning its general prin-
ciples, and gains its actually real contents from metaphysics and the special
sciences, particularly also from musicology. On the other hand, musicology can
benefit from the manifoldness of the forms of logical and mathematical thinking.
As already pointed out, Heidegger does not view “the Mathematical” as part
of mathematics, but views mathematics as an abstraction of the Mathematical,
respectively. Thus, it seems very likely to locate the Mathematical within the
phenomenology which is the initial part of philosophy in Peirce’s classification of
sciences ([16], p.258ff.). According to Peirce, the general task of phenomenology
is to investigate the universal qualities of the phenomenons in their immedi-
ate character. Heidegger’s conceptions of thingness can be understood as such
universal qualities of phenomenons. This becomes more clear by the following
determination of the nature of the Mathematical which has been summarized
by Heidegger in his book [11] on p.71f:

1. The Mathematical is a conception of thingness leaping virtually over its
things.
2. This conception determines what the things are considered for, as what they
and how they should be acknowledged in advance.
3. The conception of the Mathematical is an axiomatic anticipation in the na-
ture of the things tracing out how each thing and each relationship between
those things are formed.
4. This formation offers the scale for delimiting the domain which embraces in
future all things of such nature.
5. The axiomatically determined domain now demands for the things belong-
ing to it an accessibility suitable alone for the axiomatically predetermined
things.

For getting a better understanding of Heidegger’s conception of the Mathe-
matical, it might be helpful to discuss Heidegger’s summary with respect to an
example. Let us choose the space in which we live. Our understanding of the
space is quite supported by our experiences with the bodies in the space so that
we can rephrase Heidegger’s five statements concerning the space of bodies as
follows:

1. The conception of space leaping over its bodies is a model of the Mathemat-
ical (which has been abstracted mathematically to the real vector space).
2. This conception determines what the bodies are considered for, as what they
and how they should be acknowledged in advance (which can be supported
by representing the bodies mathematically using bounded connected subsets
of the real vector space).
3. The conception of the Mathematical is an axiomatic anticipation in the na-
ture of the bodies tracing out how each body and each relationship between
170 Rudolf Wille, Renate Wille-Henning

those bodies are formed (which become mathematically descriptive by alge-
braic terms).
4. This formation offers the scale for delimiting the domain which embraces in
future all bodies of such nature (in particular, this allows to measure bodies
mathematically).
5. The axiomatically determined spacial domain now demands for the bodies
belonging to it an accessibility suitable alone for the axiomatically predeter-
mined bodies (which can be mathematically abstracted within the axiomat-
ically defined real vector space).

Let us record for this paper that the Mathematical as part of phenomenology
is less abstract than mathematics, but is more abstract than logic, the third sub-
part of normative science. For investigating the Mathematical in music thinking,
it is important to understand the relationships between Mathematical and log-
ical thinking. Peirce convincingly explains the close connection between logical
and mathematical thinking in his Cambridge Conferences Lectures from 1898,
which have only completely be published, with 100 pages introduction and com-
mentary, in 1992 under the title “Reasoning and the Logics of Things” [15].
Without pointing in details to Peirce’s explanations, it shall be attempted in the
following to demonstrate an analogous connection between the forms of music
thinking and the forms of the Mathematical with its abstractions in mathemat-
ics thinking. The manifoldness of music thinking, in which we would have to
investigate the Mathematical, cannot exhaustively be discussed in this contribu-
tion. Therefore we shall concentrate on forms of thinking about the rhythmic of
music, the doctrine of music forms, and the theory of tone systems.

2 The Mathematical in the Rhythmic of Music

By Riemann’s Music Encyclopaedia, rhythm has to be understood as an au-
tonomous principle of form and order which is characterized on the one hand
by regularity and relationship to a fixed tempo, on the other hand by grouping,
subdivision, and alternation. In this conceptual characterization, first of all

– the “uniformity of parts” ,
– the “succession of parts”, and
– the “distinctness of parts”

have entered in music thinking as basic forms of thinking of the Mathematical.
In the case of rhythmic, these forms of thought become forms of mathematics if
uniformity, succession, and distinctness of rhythm-parts are defined in the sense
of an established semantics of mathematics. The metric fixation of rhythmic in
musical notation may definitely be understood as such a semantically abstracting
mathematization. However the musical interpretations usually liberate from the
rigid mathematical structure by their agogics and accentuations. Therefore the
Mathematical does not disappear by mathematizing the rhythmic, but keeps
preserved in its autonomous independence.
The Mathematical in Music Thinking 171

Fig. 1. The Adagio of Mozart’s string quartet C major (KV 465)
172 Rudolf Wille, Renate Wille-Henning
24 c’”

g” Violino I
18
e”
12 c”

g’
6
e’
Violino II
0 c’

g Viola
−6
e

−12 c Violoncello

−18 G
E
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
0 1 2 3 4 5 6 7 8

Fig. 2. A mathematical representation of the first eight bars of the Adagio of Mozart’s
string quartet C major (KV 465)

The interplay between the Mathematical in music and the mathematization
of music shall be demonstrated here by the Adagio of Mozart’s string quartet C
major (KV 465). The score of the Adagio - presented in Fig. 1 - shows that the
Adagio consists only of 22 bars in which astonishingly many dissonances occur,
but which finally leads to the light C major clearness of the following Allegro.
The result of a mathematization of the first eight bars of the Adagio is
shown in Fig. 2. The presented mathematical structure shall be considered
as embedded into a two-dimensional real vector space. The part of its vertical
axis from -20 to 25 is visible on the left of the diagram (the numbers -18, -12,
-6, 0, 6, 12, 18, 24 shall help to identify the integer locations on the vertical
axis). There is a one-to-one correspondence between the integers of the vertical
axis from -20 to 25 and the tones of the chromatic scale from E to c000 #. Fig. 2
indicates the part of this correspondence which horizontally links the numbers
−20 < −17 < −12 < −8 < −5 < 0 < 4 < 7 < 12 < 16 < 19 < 24 to the tones of
the C major triad E < G < c < e < g < c0 < e0 < g 0 < c00 < e00 < g 00 < c000 #
The location of the integers 0, 1, ... , 8 on the (imaginary) horizontal axis are
indicated by the numbers on the bottom of the diagram (the smallest unit for
the horizontal numbers is one sixth, in numerals: 1/6). The horizontal straight
line segments on the right of the vertical axis, closed on the left end and open on
the right end, represent the sounds of the four instruments with their pitches,
respectively (the pitch of such a line segment is determined by the height of the
The Mathematical in Music Thinking 173

line segment measured by the vertical axis). The small vertical line segments on
the right of the vertical axis and the line segment between the points (0,-12)
and (0,-11) indicate the beginning of the sound belonging to the horizontal line
segment connected at the bottom of that small vertical line segment. The union
of all those line segments can be divided into four disjoint subsets corresponding
exactly to the four instruments Violino I, Violino II, Viola, and Violoncello
(notice that the representations of the sounds beginning at the points (18/6,2),
(19/6,8), (20/6,10) and (42/6,0), (43/6,6), (44/6,8) belong to the Viola subset,
but not to the Violino II subset). Thus, the structure of those four subsets
determines the mathematical representation of the first eight bars of the Adagio.
It is not difficult to extend this representation to a mathematical representation
of the whole Adagio.
Although the discussed mathematical description of the tones of the Adagio
by their pitch, length, location, and instrument are in one-to-one correspon-
dence to the notes of the score presented in Fig. 1, there are more signatures
in the score concerning tempo, loudness, crescendo, and bows which are not
mathematized. Above all the expressive interpretations of a score by rhythm,
agogics, accentuations etc. are far away from a meaningful mathematization.
That, in particular, the rhythm evades any mathematical description becomes
clear by the following quotation: “The rhythm comprises the order, division,
and meaningful arrangement of the time development of sound events. In spite
of the tendency, created by the rhythm, to return to the same or the similar,
the rhythm should not be confused with the metre and beat because just the
vivid differences of the courses of time make possible the musical manifoldness
of the rhythms which first of all appear through graded durations of sounds and
accents, but also through melodic movements, changing sounds and tone colours,
changes of tempo and loudness, phrasing and articulation” ([2], p.656).

3 The Mathematical in the Doctrine of Music Forms

According to the Composer György Ligeti: “The combination of association, ab-
straction, remembrance, and prevision let only actually achieve the suggestive-
ness which makes possible the conception of a musical form” ([1], p.9). Without
the principle of order, the musical forms would be neither communicable nor
apperceivable. Clear orders and relationships are a criterion of its conceivabil-
ity and indispensible assumption for its understanding. The smallest units of
musical sense are the so-called “motives” which are understood as the smallest
meaningful elements of musical compositions. Motives join up with their own
transformations and other motives to larger parts which might be again only
parts of a larger whole (cf. [1], p.16f).
For understanding the Mathematical in music forms, it might be helpful to
analyse the multitude of music forms in the Adagio presented in Fig. 1 (cf. ([14],
p.446). As a whole the music form is an introduction to the Allegro, the first
movement of the C major string quartet (KV 465). The introduction divides
into two parts each of which has 11 bars; the first part is polyphonic, the second
174 Rudolf Wille, Renate Wille-Henning

part is homophonic. The violoncello starts the Adagio with eighth notes repeated
through all the eleven bars of the first part, interrupted only by a four notes
motive chromatically ascending at the end of the fourth bar and the eighth
bar, respectively. After the first four eigth notes of the violoncello the other
three instruments present a theme which divides into two motives each of which
consisting of four notes, where the viola starts at the end of the first bar, the
violino II one quarter note later, and the violino I again one quarter note later.
The first chord of the four instruments combining the notes c - g - e’[ - a” contains
the two surprising dissonances g - a” and e’[ - a” and allowed in the following
further dissonances until the second motive occurs in combing consonant chords.
Starting from the fifth bar, the first four bars are repeated always a major note
downwards. The last three bars of the first part of the Adagio function as a bridge
to the second part in which the four instruments play the same role between each
other in diminishing the motives.
The example shows that the mathematization of music forms can use in
addition to the descriptive dimensions pitch, length, location, and instrument
also the dimension “music form”. In our example Fig. 1 we can consider as
music forms the whole Adagio, the disjunctive two parts of the Adagio which
cover the Adagio, smaller meaningful parts such as periods, themes, phrases,
motives, scales, harmonies, chords, tones etc. Many of those music forms of the
Adagio can be mathematically represented by a subset of the two-dimensional
vector space sketched in Fig. 2; for instance:
– the first motive of the theme presented first for the viola,
– the first theme presented first for the viola,
– the first motive of the theme presented first for the violino II,
– the first theme presented first for the violino II,
– the first motive of the theme presented first for the violino I,
– the first theme presented first for the violino I,
– the first motive of the theme presented secondly for the viola,
– the first theme presented first for the viola,
– the first motive of the theme presented secondly for the violino II,
– the first theme presented first for the violino II,
– the first motive of the theme presented secondly for the violino I,
– the first theme presented first for the violino I,
– the first four tone motive ending with B presented for the violoncello,
– the second four tone motive ending with B presented for the violoncello.
The mathematical description of music forms may extend the mathemati-
zation of structures determined by the dimensions of pitch, length, location,
and instrument as, for example, presented in Fig. 2. Nevertheless, the expressive
interpretations of musical scores are still not in reach to be completely math-
ematized. Thus, there is still quite a distance between the Mathematical and
the more abstract mathematization, but further attempts of diminishing the
distance can be elaborated of which two approaches shall be briefly mentioned.
In the doctrine of music forms, symmetries play a special role for which the
form of thinking “equality of parts as expression of a whole” (cf. [19]) can be as-
sumed to belong to the Mathematical. This phenomenological form of thinking
The Mathematical in Music Thinking 175

finds its abstraction in mathematics by the mathematical concepts of symmetry
transformation” and “symmetry group”, respectively. A direct correspondence
between the phenomenological and the mathematical form of thinking regarding
compositions is almost only given by strong canons. But if one weakens the math-
ematical concept of symmetry transformation to a concept of partial symmetry
transformation, then considerably more correspondencies could be identified.

Fig. 3. The symmetry structure of the second movement of Anton Webern’s Symphony
op. 21

As another generalization of the mathematical form of symmetry, the twelve-
tone music used more general symmetries which view octave tones to be struc-
turally identified. The example shown in Fig. 3 represents twelve tone rows by
a sequence of eleven straight sections on a circle. Each circle presents at least
one symmetry and all circles together are arranged in such a way that a 180◦
rotation maps the total picture onto itself. Musically this indicates that the total
symphony is a transposition of its retrogression.
The composer Fred Lerdahl and the linguist Ray Jackendoff have elaborated
a much more far-reaching approach to formally grasping forms of music which
was published in their book “A generative theory of tonal music” [13]. For this,
they developed a generative grammar of music, which was inspired by Chomsky’s
linguistic transformation grammar, but developed purely within music thinking.
176 Rudolf Wille, Renate Wille-Henning

As fundamental components of the musical understanding of a composition they
considered grouping structures of subunits of the composition. For these group-
ing structures the form of thinking “division of a whole into subunits” can be
assumed to belong to the Mathematical and abstracted to a mathematical struc-
ture of a weighted ordered set. Lerdahl and Jackendoff impressively demonstrate
their theory by many examples, as fore instance by the beginning of Mozart’s
Symphony G minor, KV 550.

4 The Mathematical in the Theory of Tonal Systems

Tonal systems, which serve as foundation of music thinking, rest thoroughly on
different forms of thinking of the Mathematical:

– Behind the tonal system of the equal-tempered keyboard, there is the form of
thinking of a musical scale consisting of 7 white keys with the steps whole-
whole-half-whole-whole-whole-half which are completed by 5 black keys to a
musical scale with 12 half steps.
– The tonal systems of musical instruments with finger-board suggest a form of
thinking which relates to finger positions; for example, the player of a violin
thinks especially which finger has to be placed on which string in which
position .
– The tonal system of the names of tones obtains its form by the names of the
12 octave tones c - c# - d - e[ - e - f - f# - g - a[ - a - b[ - b which are rising
by half-tone steps; adding # or [ to a tone name yields the name of a tone
which is a half-tone higher or lower, respectively.
– The tonal system of the standard notation is founded on the form of the 5+5-
line system with additional ledger lines, in which the tones are represented
by note-heads with and without accidentals on and between the lines; the
tone distances describable in this way are multiples of half-tone steps.
– The harmonic tone system extends the form of the tone system of tone names
by adding integer exponents to the tone names; a tone name tz represents
a tone which is z-many syntonic commas higher or lower than the tone
t0 , respectively (syntonic comma := 4 fifth – 2 octaves – 1 major third;
multiplicatively, the syntonic comma is the frequency ratio 81 : 80 obtained
by computing ((3 : 2)4 : (2 : 1)2 ) : (5 : 4)) where the frequency ratio 3 : 2
represents the fifth, the ratio 2 : 1 the octave and the ratio 5 : 4 the major
third).

Here only the harmonic tone system shall be further discussed. In Fig. 4,
this system is represented by a tone net in just intonation which is freely gen-
erated by the perfect fifth 3 : 2 and the perfect major third 5 : 4 (modulo
the octave 2 : 1). Leonhard Euler was the first who published such a tone net
which he named speculum musicum [6]. Following Euler’s idea, realizations of
the harmonic tone system on musical instruments have been approached again
and again (for an overview about those attempts see [18]). In particular, the
The Mathematical in Music Thinking 177

e−2 b−2 f #−2 c#−2 g#−2 d#−2 a#−2 e#−2 b#−2

c−1 g −1 d−1 a−1 e−1 b−1 f #−1 c#−1 g#−1

a[0 e[0 b[0 f0 c0 g0 d0 a0 e0

f [+1 c[+1 g[+1 d[+1 a[+1 e[+1 b[+1 f +1 c+1

d[[+2 a[[+2 e[[+2 b[[+2 f [+2 c[+2 g[+2 d[+2 a[+2

Fig. 4. The tone net of the harmonic tone system

instrument MUTABOR should be mentioned which even allows to realize arbi-
trary mutating pitches of tones in just intonation, but also in any other form of
intonation (see [7], [22]).
Although performing music pieces in just intonation is an ideal for many
music ensembles (for instance for a string quartet), there are problems of being
consistent with the intonation. This shall be briefly explained by the so-called
Problem of the Harmony of Second Degree illustrated in the harmonic tone sys-
tem shown in Fig. 5 (cf. [21], p.197f). The figure represents a musical cadence
formed by five perfect triads starting with the major triad c0 and ending with
the major triad c−1 . More precisely,

– the major triad c0 meets the major triad f 0 in the note c,
– the major triad f 0 meets the minor triad d−1 in the notes f 0 and a−1 ,
– the minor triad d−1 meets the major triad g −1 in the note d−1 , and
– the major triad g −1 meets the major triad c−1 in the note g −1 .

Playing a cadence as described above, musicians usually have the tendency to
end with the same chord as they started with, i.e. with the major triad c0 . Then,
of course, they have to modify the pitches in between, but still to produce perfect
triads. Cadences with such intonations defy convincing mathematization so that
it would be interesting to find out how much the Mathematical could contribute
to overcome those vaguenesses.
178 Rudolf Wille, Renate Wille-Henning

e−2 b−2 f #−2 c#−2 g#−2 d#−2 a#−2 e#−2 b#−2
A A
A A
c−1 g −1 d−1 a−1 e−1 b−1 f #−1 c#−1 g#−1
A A A
A A A
a[0 e[0 b[0 f0 c0 g0 d0 a0 e0

f [+1 c[+1 g[+1 d[+1 a[+1 e[+1 b[+1 f +1 c+1

d[[+2 a[[+2 e[[+2 b[[+2 f [+2 c[+2 g[+2 d[+2 a[+2

Fig. 5. A musical cadence leading from the major triad c0 in four steps via the major
triad f 0 , minor triad d−1 , and the major triad g −1 to the major triad c−1

5 Semantic Logic in Music Thinking and Its Semantology
A basic question is how to support our understanding of the Mathematical in
music. Since the Mathematical is more abstract than logic which itself is more
abstract than music, the study of logic in music may particularly contribute to
a better understanding of the Mathematical in music thinking.
It is common sense that humans may be affected by music so that it reaches
human feelings, emotions, and thought. Humans can even be deeply moved by
music, particularly by its musical senses and meanings which may be repre-
sented by semantic structures in music (cf. [25]). Now, such structures could be
abstracted to semantic structures in logic. For instance, the chords of a well-
tempered piano can be abstracted to a logic structure which represents the pos-
sible interactions and relationships between those chords.
The result of all such abstractions has been named by the musicologist
C. Dahlhaus “musical logic” which he characterized by the compositional, tech-
nical and esthetic moments which made the automation of instruments possible.
Dahlhaus saw the musical logic closely related to the idea of the “language char-
acter” of music. That music is presented as sounding discourse, as development
of musical thought, is the justification of its esthetic claim, that music is there to
be heard for the sake of itself (see [3], p.105f). The richness of this understanding
of musical logic is an important assumption for a better understanding of the
Mathematical in music thinking.
The Mathematical in Music Thinking 179

To obtain even more insights into the Mathematical in music thinking, a
further development of the recently introduced “semantology of music” could
be helpful (cf. [25]). In particular, its philosophic-logical level is basic for the
analysis of the Mathematical because philosophical concepts with their objects,
their attributes, and their relationships” are highly abstract, but still deduced
from actual realities (cf. [10], [5]). The supportive mathematical level is already
elaborated to a great extent by methods of Formal Concept Analysis (see [8],
[9], [24]).

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