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THE SEMIOTICS OF MUSIC

Robert N. St. Clair

University of Louisville, USA

-INTRODUCTION

The original model of semiotics by Ferdinand de Saussure, created the concept of a sign by conflating

meaning and expression. This model was based within the context of grammatical space. In visual

semiotics, it was found that the model was inadequate because it could not account for the context of

visual space. Grammatical space is linear and based on syntagmatic and paradigmatic contrasts (3D

mathematical lattice) but visual space differs because it is simultaneous. What this says, in essence, is

that mathematically, semiotics is context sensitive. This is what Marshall McLuhan meant when he

stated that “the medium is the message.” It should not be surprising to learn that the semiotics of

music is also context sensitive in that it operates within tonal space. Hence, the new framework for

semiotics should be the revised concept of the sign in which meanings belong to the epistemological

realm and expression resides in the ontological realm. The mapping of meaning onto form (semiosis)

differs mathematically for language, pictures, and sound.

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While grammatical space is linear and visual space is simultaneous (2D, 3D), musical space is modular

and fourth dimensional as it includes time as a dimension. With the advent of modern science, tones

are analyzed in terms of their frequencies. However, at the time of Euclid and Protagoras, tones were

studied in terms of their musical intervals.

A good modern example of this concept can be found by investigating the length of the strings in a guitar and the placement of the frets along that string. One half the length of the string is an octave. Two thirds the length is a perfect fifth. Four thirds the length is a fourth, and so on.

Hence, ancient musicians were very familiar with this concept of musical intervals. The piano is based

on this concept.

The sequential order of the black and white keys together make up an octave on the piano. All of these keys are separated from each other by a half tone (semitone). The movement from C to E is five semitones or a major third. It is separated by three whole notes (white keys) The distance from C to G is a perfect fourth. It is just four whole notes away (white keys). The key that is farthest away is the major seventh. It is seven keys away from C. The keys C, E, and G are harmonious and constitute the C chord.

Mathematically, if an octave (2:1 ratio) is divided by a fifth (2/3 ratio) the result is a fourth. In other

words 2:1/3:2-4:3, Using logarithmic ratios, one arrives at the following (Tymoczko, 2011):

octave - fifth = fourth 2:1 / 3:2 = 4:3 (1)

fifth - fourth = tone 3:2 / 4:3 = 9:8 (2)

fourth - 2 tones = semitone

(e.g. E to F) 4:3 / (9:8)2 = 256:243 (3)

tone - 2 semitones = comma 9:8 / (256:243)2 = 531441:524288 (4)

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The piano keyboard, which has 88 keys, is made up a linear sequence of octaves. The black keys represent sharps (#) and flats (b); the white keys designate whole notes. When the keys are played sequentially, the represent the C scale. For purposes of orientation, there is a middle C. The equivalent notes above it are line octaves and the notes below small, great, and contra octaves.

As early as 230 BCE, Euclid's had an algorithm that made a reasonable approximation of the number of

fifths in an octave. It is 7/12, because a fifth is roughly 7 semitones and an octave is roughly 12 of them.

These twelve pitches are inherently cyclic. As noted earlier, the piano represents 88 notes that are

arranged in octaves. Every octave repeats itself. Hence the octave, which is a linear representation of

12 notes, can be represented as a circle. This analog schema is captured by the clock which contains 12

hours that repeat themselves. The movement around the face of the clock is clockwise, i.e. towards the

right. Hence, the representation is geometrical.

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There is much more to say about this mathematical representation of music. The next section of this

discussion has to do with musical chords and how they form another kind of geometric schema. After

discussing chords and their components, the discussion will turn to the concept of how chords form a

network or Tonnenz (German: network of chords).

THE STRUCTURE OF MUSCIAL CHORDS

There are many chords in music. These are grouped into major and minor chords. The major chords

consist of the root (I), the third (III) and the fifth (V). For the C chords, this would be CEG.

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The minor chords differ by have the third (III) reduced by a semitone. Whereas C major is CEG, C minor

becomes CEᵇG (C, E flat, G).

TONNENTZ

Leonard Euler (1739) developed a conceptual lattice diagram to represent tonal space. This

mathematical framework was described in Latin as Tentamen novae theoriae musicae ex certissismis

harmoniae principliis dilucide expostiate. In modern terms it is called Tonnentz (German: tone

network).

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.Euler investigated the relationship between the perfect fifth and the major third. Euler noted that at

the top of the diagram is the note F and underneath it and to the left is C, a perfect fifth of F. To the

right of F and slightly over it is A which is a major third of F. This musical relationship was discovered by

Ernst Neurmann (1866). It was Hugo Rienman, a noted musicologist of that era, who explored this

relationship between chords and the modulation of keys. One can extend the horizontal rows of the

Tonnetz indefinitely to form further sequences of perfect fifths: F-C-G-D-A-E-B-F#-C#(Db)-Ab-Eb-Bb-F-C

etc. Starting with F, one reaches another F after 12 perfect fifths, and so on.

The Tonnetz appealed to 19th century German theorists because it allowed not only tonal relationships

but also the spatial representations of tonal distance

In the modern version of the Tonnentz, this can be seen, for example, by looking at the dark blue A

minor triad in the graphic and its parallel major triad (A-C#-E) is the triangle right below. They both share

the same vertices A and E. As Richard Cohn (1998) noted, the relative major of A minor, C major (C-E-G)

is the upper-right adjacent triangle, sharing the C and the E vertices. The dominant triad of A minor, E

major (E-G#-B) is diagonally across the E vertex, and shares no other vertices. One important point is

that every shared vertex between a pair of triangles is a shared pitch between chords - the more shared

vertices, the more shared pitches the chord will have. This provides a visualization of the principle of

parsimonious voice-leading, in which motions between chords are considered smoother when fewer

pitches change. This principle is especially important in analyzing the music of late-19th century

composers like Wagner, who frequently avoided traditional tonal relationships. The never ending series

of ascending fifths become a cycle and is mathematically comparable to a torus.

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CHORD PROGRESSIONS

The study of chord progressions is interesting because there are many chord sequences that can be

arranged. Just as plot motivators move a story along within a plot, chords move the harmony in music.

Chord progressions are what give a piece of music its harmonic movement. Some of these chord

movements are uplifting, others are somber, and some have the melodic sensation of ocean waves. In

addition some of the progressions of chords express stability and even tension.

The way that chords are placed one after the other in music is called a chord progression. These chords

have different harmonic functions. As noted above, these are departure, dynamic tension, and stability.

The diagram below shows the formulas of the more common chord progressions in major and minor

keys.

The Roman numbers in a chord progression formula represents the triad form of the chord. Additional

diatonic tones be used to extend these triads. The Roman numerals refer to the position of each chord

in the diatonic scale. The diagram below shows how the Roman numeral scale degree can be interpreted

with different chords. All of the examples below can be interpreted from the same chord formula.

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The chords indicated by the Roman numerals also have names. For instance, the first chord of the scale

is the tonic. The fifth chord is the dominant. The diagram below shows the functional names and scale

degree of the diatonic scale. Beneath this are notes from several common keys that match the function

and degree. Further chord nomenclature is provided below for the different keys of a chord.

There are many ways to create harmonic movement. One may substitute a dominant chord in lace of

a minor chord or vice versa. One could play a diminished chord instead of a dominant. There are

many ways to create harmonic movement. One of the most common progressions, however, is I, IV,

V (one, four, five). If one wants to explore this key in C major, it will have C as its tonic chord, F as the

Subdominant and G as the Dominant.

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One of the most common progressions in music is the I, IV, V (one, four, five) and say we want to

explore this progression in the key of C major. Here are some chords and their extensions.

CIRCLE OF FIFTHS

The circle of fifths was created by Pythagoras. He favored the number three and originally looked at

notes in the formation of triads. However, when it came to semitones, he arranged them into a circle of

twelve notes that he referred to as “cents.” He attributed 100 cents to each of the twelve positions.

Today, one refers to these positions in terms of musical notes separated by semitones. The result is the

modern circle of fifths. There are many regular relationships that exist between major and minor chords

and the circle of fifths is one of the best musical devices to reveal this relationship. It also provides the

frequently used intervals for harmonic movement. In the following chart (supra), one finds the major

key names and chord roots. As one moves clockwise to the right, each chord takes on an additional

sharp. The chord of G has one sharp; the chord of D has two, and so on. As one moves

counterclockwise, major chord takes on takes on one flat. The chord of F has one flat; the chord of B flat

has two, and so on. Notice that the movement clockwise involves every fifth note. Hence, the name of

the circle of fifths is given to this chart. However, the movement counterclockwise also involves every

fifth note, but it is called the circle of fourths. This is because if one moves clockwise to the fourth, it is

the same as moving counterclockwise to the fifth.

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Typically the "circle of fifths" is used in the analysis of classical music, whereas the "circle of fourths" is

used in the analysis of jazz music, but this distinction is not exclusive. This chart provides an expedient

instrument for musicians it that it represents notes, chords or keys. How is this useful? If one is on the

C note and you want to move the its perfect fifth, you merely move clockwise to the next note on the

chart which is a G, the fifth of C. Next, if one is doing a chord progression in C, the next chord in the

progression is based on the perfect fifth of C. Hence, the next chord in the progression is G. The circle

of fifths tells a song writer which chords are available to him in a major chord. If he is composing in the

major C chord, then he has the F and G chords available to him. These are the chords on either side of

the C chord on the circle. If he wants to know what minor chords are available to him, he continues

clockwise to the next three chords, D, A, and E which are minor chords. The final available chord in the

key of C is the diminished chord, B. This means that the chords available to one composing in the key

of C is C major, D minor, E minor, F major, G major, A minor, and B diminished. The reason why these

chords are in the key of C major is because there are no sharps or flats in that key and the

aforementioned chords available in that key do not contain either sharps or flats. Finally, if one needs

to shift keys, the same concept applies. If one is in the key of C, the next key in the series is the key of G.

In this case, the circle of firths tells the musician how many sharps or flats are in a key. If one is writing a

musical piece in the chord of C and he wants to rewrite it in the chord of E, he can use the circle of fifth

and move from C to E which is four positions away moving clockwise. At the chord of E, he looks for the

new major chords, the new minor chords, and the new diminished chord. He now has rewritten his song

in the key of E. On the circle of fifths, some chords are more closely related to other chords. The more

notes that a chord share, the more closely related they are to each other. Not surprisingly, major chords

and minor chords share the most notes. To demonstrate this relationship, musicians have added this

information in an inner circle of the circle of fifths.

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On the chart (supra), the chord that is closest to C major is A minor. Notice that they share the same

keys and that there are no sharps or flats in A minor. Another thing that musicians need to know about

the circle of fifths is that it can help the composer modulate between chords in a composition.

Consider, for example, modulating from a C chord to a G chord, which are adjacent to each other in the

circle. They share all the same notes except for C sharp. In order to modulate from the chord of C to G,

the composer must trick listener to hear a smooth transition. This done by notes referred to as “pivot

chord” It consists of a chord that is available to both C and G. One can ascertain this information by

comparing both C and D chords:

Chord of C: C major, D minor, E minor, F major, G major, A minor, and B diminished

Chord of G: G major, A minor, B minor, C major, G major, E minor, and F# diminished

The following chords exist in both keys: C major, E minor, G major, and A minor. Therefore, any of

these chords can be used as pivot chords in modulating from the chords of C to the chord of G. For

example, one may begin with C major, move to E minor, and end with G major as a way of modulating

from C to G. Before leaving the topic of the circle of fifths, it should be noted that it really displays

vector equilibrium around its center. This is the same concept that Buckminster Fuller argued for in his

depiction of the geodesic dome (infra). As a dodecagon, it depicts the circle of fifths. It also has two 6-

sided hexagons that depict whole tone scales. It has three 4-sided squares, the four sets of diminished

chords. It also depicts four 3-sided triangles, the augmented major chords. As a 1-sided dodecagon it

represents the chromatic sequence of intervals, and finally it contains six lines showing the triton pairs.

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THE GEOMETRY OF TONAL SPACE

Dmitri Tymoczko, a professor of music at Princeton University, has studied the nature of tonal space.

He has investigated the geometry of music in depth. He points out that the geometry of music began

with Euclid and that that led to other mathematicians continuing this line of research. Pythagoras was

noted for his research on ratios of vibrating strings in music. Leonard Euler developed the concept

further as noted earlier. Tymoczko is continuing within this rich and varied tradition of the geometry of

music. What makes him different, however, is that he is not only a professor of music, but also a

composer. How he uses geometry to compose is provided in the following vide (click the link).

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Dmitri Tymoczko was born in 1969 in Northampton, Massachusetts. He studied music and philosophy at Harvard University, and philosophy at Oxford University. He received his Ph.D in music composition from the University of California, Berkeley. He is currently a Professor of Music at Princeton, where he has taught composition and theory since 2002. He lives in Philadelphia with his wife, Elisabeth Camp, who teaches philosophy at Rutgers University, their son Lukas, who was born in 2008, and their daughter Katya, born 2012.

Link: https://youtu.be/XUyx31f-U3M

The final point of discussion has to do with the frequency of middle A. In an article on the foundations of

musical timing, Jonathan Tennenbaum (1991-1992) argued that no musical timing is acceptable which is

not based on a pitch value for middle C of 256 Hz (cycles per second), corresponding to A no higher than

432 Hz. Tennenbaum essentially rejects the notion that A-440 Hz. As any musician knows, the

frequency of A is 432 Hz. It was changed to 440Hz by Hermann Helmholtz, a nineteenth-century

physicist and physiologist, and that change remains controversial among professional musicians.

Tennenbaum rejects this change as unscientific, invalid, and arbitrary. Helmholtz argued that the science

of music was based on the periodic properties of a vibrating string. He saw fundamental tones as sine

waves of various frequencies. These tones are superimposed on other waves to create overtones or

harmonics. The constant musical intervals are determined by this overtone series. Hence, Helmholtz

argued, musicians should give up well-tempered music and return to natural tuning based on whole-

number ratios. He attacked J. S. Bach and Beethoven for using unnatural tuning. Helmholtz based his

theory on human hearing. He claimed that the ear works as a passive resonator and he insisted that all

musical tonalities are identical regardless of the chosen fundamental pitch. He believed that all physical

scales can be measured as lengths along a straight line, scalar music theory. The problem with this is

that the human voice and universe as a whole is nonlinear. Sound, it can be argued, is not a vibration of

the air. It is an electromagnetic process in which geometric configurations of molecules are rapidly

assembled and disassembled. In modern physics, this is known as a soliton which are distinct geometries

on the microscopic or quantum level of organization. Hence, Helmholtz theory of sound has been

proven to be fallacious. The tiny resonators in the human ear that he postulated do not exist. The

human ear functions in ways similar to a laser amplification device. It is not linear and it is not passive.

Furthermore, the standard musical tone is the human voice. All musical instruments were designed to

imitate the human voice. In particular, the bel canto human voice is an acoustic laser generating the

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maximum density of electromagnetic singularities. It is the human voice that defines the basis for

musical tuning and for all forms of music. Many scholars understood this except for Helmholtz. This is

because the human voice is the basic instrument of music.

Soliton Waves: https://youtu.be/w2s2fZr8sqQ

It is well known that physicists in the past were interested in the music of the spheres. Each planet, they

argued had its own sound. Johannes Kepler also investigated this phenomenon and derived what is

known as the Keplerian interval in the solar system. It is a period of CV=256 or 1/256 of a second. He

arrived at this by taking the rotation of the earth (2 x 3 x 4) to get one hour. His divided this by 60 ( 3 x

4 x 5) to arrive at a minute and again he divided this by t0 to obtain a second. Next, he divided that

second by 256 (2 to the eight power). He deduced from this that the rotation of the Earth is a wave (G)

what is twenty-four octavbes lower than C=256. Why is this important? It is important because 256 is

the value attributed to the motion of the complete planetary system. It is also important because this is

a natural value whereas A=44- Hz is not a natural value. It bears no coherent relationship with the

universe or its reality. Circular motions are the only complete representations of creative action in the

universe. Carl Friedrich Gauss was also interested in circular action but his focus was on conical spiral

action. His work is important because spiral action combines the isoperimetric principle of the circle

with the principle of growth expressed by the Golden Section. This can be demonstrated by a bel canto

voice. When a soprano sings a scale upward beginning at middle C (=356), the frequency increases and

so does the intensity of the sound. This increase is not linear; it is scalar. In order to maintain the

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“isoperimetric,” makes a register shift at F-sharp. Another action occurs when she arrives at C=512.

One can hear that one cycle of action has been completed. This proves that there is a rotational

component of action as frequency and flux density increases. Hence, Helmholtz was wrong. Sound is

not linear. The Helmholtzian straight-line does not exist (Tennenbaum, 1991-1992).

This is the true geometry of the singer's action. It is simply represented by spiral action upward

on a cone. The cone's axis represents frequency. Each circular cross-section of the cone

represents a bel canto musical tone. The spiral makes one complete rotation in passing from

C=256 to C=512, and one more cycle would bring it from C=512 to the next higher octave,

C=1024. Thus, the interval of an octave corresponds to one complete 360° cycle of conical spiral

action.

All musical intervals, correspond to specific angles on conical-spiral action. if one projects the

conical spiral onto a plane perpendicular to the axis, it can be divided into a full 360° rotation

into twelve equal angles. The radial lengths at the indicated twelve angles are exactly

proportional to the frequencies of the equal-tempered musical scale. As Tannenbaum notes,

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C=256 corresponds to the perihelion of the ellipse C=512 corresponds to the aphelion F

corresponds to the semi-latus rectum F-sharp corresponds to the semi-minor axis G corresponds

to the semi-major axis. At the same time, F. F-sharp and G correspond to harmonic, geometric,

and arithmetic means, respectively, of the octave. C=256 corresponds to the perihelion of the

ellipse C=512 corresponds to the aphelion F corresponds to the semi-latus rectum F-sharp

corresponds to the semi-minor axis G corresponds to the semi-major axis This scale consists of

two congruent tetrachords: C—D—E—F and G—A—B—C. The dividing-tone is F-sharp.

CONCLUDING REMARKS

This discussion on the semiotics of music began with the concept of tonal space. It was argued that the

grammatical space of traditional semiotics could not account for either the visual space of visual

semiotics or the tonal space of musical semiotics. This is why the concept of the sign needed to be

revised into two different realms (the epistemological and the ontological) with mapping functions

between them. These mapping functions are context sensitive. Furthermore, the nature of these

mapping operations differs when one moves from meaning to form (semiosis) or from form to meaning

(structural semiotics). The next are of research worthy of investigation has to do with the semiotics of

human movement (the semiotics of dance and the semiotics of sign language). More will be said about

this at a later time.

REFERENCES

Cohn, Richard (1998). "Introduction to Neo-Riemannian Theory: A Survey and a Historical Perspective".

Journal of Music Theory. 42 (2 Autumn)

Euler, Leonhard (1739). Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide

expositae (in Latin). Saint Petersburg Academy.

Helmholtz, Hermann. (1863). Die Lehre von den Tonempfindungen als physiologische Grundlage für

die Theorie der Musik (The Theory of the Sensations of Tone as a Foundation of Music Theory). Longinus

Green.

Tennenbaum, Jonathan. (1991-92). The Foundations of Scientific Musical Tuning. FIDELIO Magazine, Vol

.1 No.1 , Winter 1991-92

Tymoczko, Dmitri. (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common

Practice (Oxford Studies in Music Theory) Oxford, UK: Oxford University Press, Inc.