Physics and Music: Essential Connections and Illuminating Excursions

By Kinko Tsuji and Stefan C. Müller

This book explores the fascinating and intimate relationship between music and physics. Over millennia, the playing of, and listening to music have stimulated creativity and curiosity in people all around the globe. Beginning with the basics, the authors first address the tonal systems of European-type music, comparing them with those of other, distant cultures. They analyze the physical principles of common musical instruments with emphasis on sound creation and particularly charisma. Modern research on the psychology of musical perception – the field known as psychoacoustics – is also described. The sound of orchestras in concert halls is discussed, and its psychoacoustic effects are explained. Finally, the authors touch upon the role of music for our mind and society. Throughout the book, interesting stories and anecdotes give insights into the musical activities of physicists and their interaction with composers and musicians.

• Language: English
• Publisher: Springer
• Release date: May 31, 2021
• ISBN: 9783030686765

Book preview

Part IPhysics + Music = .....

Musik ist

ein Teil des schwingenden Weltalls.

Music is

a part of the vibrating cosmos.

—Busoni

© Springer Nature Switzerland AG 2021

K. Tsuji, S. C. MüllerPhysics and Musichttps://doi.org/10.1007/978-3-030-68676-5_1

1. Introduction

Kinko Tsuji¹   and Stefan C. Müller²

(1)

Shimadzu Europa GmbH (retired), Duisburg, Germany

(2)

Professor emeritus, Institute of Physics, Otto von Guericke University Magdeburg, Magdeburg, Germany

Kinko Tsuji

Email: [e-mail]

1.1 Physics & Music - Quiz of the Year

Let us start this book with the following quiz. Can you identify the persons in Fig. 1.1?

If you have answered with the correct names of all six persons, you are an expert in the hybrid field of Physics & Music. Congratulations! If you can further say what the mathematical/physical symbols signify, maybe you do not need to read this book at all. (Answers are given at the end of this chapter.)

1.2 What Is the Hybrid Field of Physics & Music?

1.2.1 Human Activities with Tones and Movements

Many animals and early humans have created tones and movements for communication: with the practical purpose to live and to survive. They must have sent signals for imminent dangers, wishes for mating, or just expression of certain moods connected with the search of food.

About 100,000–70,000 years ago the structure of the human larynx has gradually changed so as to produce vowels, leading to the ability to speak [1, 2]. In parallel, the size of the human brain increased, and humans became able to develop tools, make fire and create language [3, 4].

Fig. 1.1

Who are they?

(picture by Manuel Covarrubias)

In the Swabian Jura region of Southern Germany an ice age musical instrument of about 40,000 years old was found [5]. This is a flute made from a wing bone of a griffon vulture, and there are some hand-born finger holes, as shown in Fig. 1.2. It suggests that this flute created tones with different pitches and that our ancestors of 40,000 years ago played some melodies with this flute.

Similar to the animal world, for our ancestors both tones and movements had practically the same purpose: transmission of important information for survival. Over an enormously long period of time tones and movements have been partially losing their original purposes, and have turned into more complicated forms (languages, music and dances), possessing a more ornamental characters and thus moving into the category Arts.

On the other hand, tones and movements are issues of natural science and technology. Scientific knowledges and technologies can contribute to developing arts further. Physiologists have tried and are trying to find out, how we hear and how we move. Physicists have studied and are studying, how sounds and movements are created. Mathematicians have tried and are trying to find mathematical order and beauty in arts. Developments of mechanics and materials contribute to construct better musical instruments, as well as concert halls. Modern electronics make it possible to conserve musical and dance performance in digital forms. Such relationships are schematically shown in Fig. 1.3.

Fig. 1.2

An ice age musical instrument (about 40,000 years old) found in a cave in the Swabian Jura region in 2008. 4.2 cm length.

©H. Jensen/Universität Tübingen

Fig. 1.3

Relationship of arts and sciences concerning tones and movements of human activities

1.2.2 Arts or Natural Sciences?

Does music belong to the arts? The answer is Yes.

Does music belong to the natural sciences? The answer is ?, Maybe yes, Maybe no or, I do not know. Probably the answer depends on which kind of music one thinks of. If one thinks about Das Wohltemperierte Klavier (The Well-Tempered Clavier) BWV 846–893 of Johann Sebastian Bach, the answer is certainly Yes. If one thinks about Erlkönig (The Erlking) Op. 1, D 328 of Franz Schubert, the answer could be No. Bach’s work is systematic with a pronounced internal structure, while in the case of the Erlkönig a dramatic story comes together with music. If one listens to the Concierto de Aranjuez of Joaquín Rodrigo, emotion develops in the first place, and the answer is ?... I cannot say anything.

In our history some natural scientists studied music as a scientific object, and some musicians explained music within the field of natural science. Below we show some of these cases.

In ancient Greece there have been seven liberal arts, which were divided into trivium (three verbal arts) and quadrivium (four mathematical arts). The former consists of grammar, rhetorics, and dialectics, and the latter of arithmetics, music, geometry, and astronomy. Interestingly, music is in the same group as arithmetics, geometry and astronomy [6]. Pythagoras (570–510 BC) and later Ptolemy (100–about 170 AD) tried to find a connection between the order of the entire cosmos and that of music according to harmonic numerical relations.

Vincenzo Galilei (1520–1591), the father of Galileo Galilei, was an Italian lutenist, composer, and music theorist. He studied pitch and string tension beyond the Pythagorean tuning, describing them as a subject of nonlinear mathematics [7].

The French composer and music theorist Jean-Philippe Rameau (1683–1764) is called father of harmony. He summarized various intervals with drawings of coordinates like in mathematics in his book Traité de l’harmonie réduite à ses principes naturels (1722) [8].

When the physicist and physiologist Hermann von Helmholtz published in 1877 his seminal book "Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik" [9] he wrote in the first lines of his introduction: In the present work an attempt will be made to form connection between the boundaries of two sciences, which, although drawn together by many natural relations, have hitherto remained sufficiently distinct—the boundaries of physical and physiological acoustics on the one side, and of musical science and esthetics on the other. The class of readers addressed will, consequently, have had very different cultivation, and will be affected by very different interests.

We could buy this text for our own introduction to this book.

1.2.3 Sender and Receiver

Music should be played and what is played should be heard. Theoretically, this process can be explained by the signal processing in physics (see Fig. 1.4).

Fig. 1.4

Signal transmission from sender to receiver

The source of the sound/noise (Sender in Fig. 1.4) can be anything: not only music instruments played in a European orchestra but also grass flutes, drums, rattles, and many instruments from non-European countries, such as the Jew’s harp (see Fig. 1.5).

Fig. 1.5

Jew’s harp is an instrument consisting of a flexible metal or bamboo tongue or reed attached to a frame. Top: Slovakian jew’s harp drumb’a (CC BY 4.0 by Krzysiu), bottom: Assam jew’s harp gogona made from bamboo (CC BY-SA 3.0 by Abhi kal)

The most frequent principle of sound transmission is the following:

A sound source is excited by a linear or surface vibration. A resonance body which is attached to the sound source amplifies the vibration, radiating into the surrounding air. Human ears function as a receiver. The acoustic waves reach our ears and are transmitted into the auditory cortex.

1.3 Early History of Musical Notations

There are certainly many people who have not been familiarized to our system of musical notations, and the corresponding scripts are as difficult to decipher as a Chinese text. One has to learn musical notations like a new script from a different language. Many have done such things readily during their education, like learning the characters of our alphabet.

It took thousands of years to develop our alphabets to their current form. Just consider the character A: in the Phoenician alphabet (around 1000 BC) the first character was aleph ( $$=$$ bull head) symbolized as in the sketch in Fig. 1.6, top. This character was turned around and later took the form of the Greek Alpha A or the Latin A. Similarly, the third letter gaml symbolized by the camel’s back and turned into the Greek letter Gamma $$\Gamma $$ , precursor of our C or G (Fig. 1.6, bottom).

Fig. 1.6

Development of our alphabet. Top: from aleph to A, bottom: from gaml to $$\Gamma $$

Fig. 1.7

Old music notes of Hurrian hymn of the 14th century BC in the ancient Syrian city of Ugarit (public domain, author unknown)

1.3.1 Mesopotamia, Egypt and Greece

For the development of musical notation we also look back on many centuries. Melodies have certainly been handed down by listening and copying in early times (even now in some areas). There must have been at a point in time the need to find a way to write them down for further use. We have an impressive example from the times of ancient Mesopotamia.

In the early 1950s, archaeologists unearthed several clay tablets from the 14th century BC in the ancient Syrian city of Ugarit. As shown in Fig. 1.7, these tablets contained cuneiform signs¹ in the Hurrian language, which turned out to be the oldest known piece of music ever discovered, a 3400 year-old cult hymn [10].

Egyptian and other ancient cultures must have had some kind of musical notation for documentation, but the first completely deciphered one was created by the Greek (Seikilos epitaph, Fig. 1.8), perhaps already around the 3rd century BC. Letters and other signs were used for pitch and tone duration—supposedly derived from the first 7 characters in their alphabet, corresponding to the strings of the kithara. (A kithara is exhibited in the gallery of historical instruments, below in Sect. 1.4.) Other symbols written above the line served to indicate the tone duration [11].

Fig. 1.8

Left: the Seikilos column made with marble, engraving poetry and musical notation, right: the inscription in detail. Nationalmuseet Denmark

(CC BY-SA 3.0 by Nationalmuseet)

1.3.2 Notation Systems in the Middle Ages

In Europe this symbolism got lost with the end of the Roman Empire and only little was conserved and later deciphered. Outside Europe, though, there evolved different notation systems, in China as well as in India. Since the 13th century the Arab notations became known, based on the handed down Greek tradition. Quite generally, the music notation served then to memorize improvised music and less to conserve melodies for generations to come.

In the Middle Ages (from the 9th century AD on) European monasteries started a new type of notation for Gregorian chorals: the neumes - graphical symbols to identify certain melodic phrases, as shown in Fig. 1.9 [12]. The neumes serve to indicate by a kind of stenographic arrows the approximate course of a melody. Lines were added, as well, starting with just two for tones F and C.

Fig. 1.9

Left: cheironomic neumes (neumes without lines) describing psalm verses Jubilare deo universa terra, right bottom: neumes with 4 lines, a sample of Kýrie Eléison XI (Orbis Factor) from the Liber Usualis; some times the cheironomic neumes appear on top of the lines (public domain)

It was then Guido da Arezzo  (991/992—after 1033),² who at the beginning of the 11th century added parallel lines to the already existing lines for notes F and C in order to grasp precisely the tone intervals between the lines. He also invented the clef, at the time mainly in terms of the C-clef.

Guido da Arezzo is also known for the Guidonian hand (Fig. 1.10). This was a mnemonic device for sight-singing. (This kind of device had existed before his time, though.) He used the joints of the hand for teaching his hexachord and solfège (a method of naming pitches by using syllables) [13].

Fig. 1.10

Guidonian hand from a manuscript from Mantua, last quarter of 15th century (Oxford University MS Canon. Liturg. 216. f.168 recto) (Bodleian Library) (NB: Color image is unsourced–black-and-white image in file history from MS Canon Liturg. 216) (public domain)

Since the 14th century tablatures have become common, where the number of lines corresponds to the number of strings on which the position of the gripping hand is marked, as depicted in Fig. 1.11 [14]. This approach has survived until today, e.g., with chord notations for the guitar.

Fig. 1.11

Tablatur notes: Fol. 44r from Robertsbridge Codex with the beginning of Tribum, quem non abhorruit. British Museum (public domain)

Whereas Guido da Arezzo’s invention worked with 3 and later 4 lines (still in use in some of church music), we are now accustomed to a 5-line system, which came into use in France in the 16th century and developed to be the basis of modern music writing. Five lines proved to be well adapted to the average capacity of the human voice. This modern system will be used throughout the following chapters.

1.4 Gallery of Historical Instruments

In early times many people on the continents of Africa, America, Asia and Europe have developed their own ways to sing and play pentatonic melodies. These reflect their relation to folk music and sounds for other common needs and events. We describe non-European music in Chap. 5. However, in this section we introduce some musical instruments which show a few quite remarkable cultures, where the pentatonic or a similar musical system has been cultivated.

Fig. 1.12

The woman standing to the right plays a harp: The three musicians, Tomb of Nakht, Thebes

(public domain from The Yorck Project (2002) [15])

Lyres and harps count among the oldest instruments with several strings, originating in Sumerian and Mesopotamian culture (see Fig. 1.12). From Egypt they found their way southwards to Nubia and East Africa, including Ethiopia.

If we have a look at Ethiopia, Eritrea and Somalia, we detect a historically popular instrument, the krar. As seen in Fig. 1.13 (left), this plucked instrument, belonging to the class of lyres, has 5 or 6 cords guided in a trapez-like manner across a resonating corpus made of wood. The strings are tuned along a pentatonic scale which can be adapted to the voice pitch of the singer [16]. The instrument to the right in Fig. 1.13 is a pluriarc of West Africa. Five strings are spanned from the neck to the bridge. It is played by plucking the strings either with fingers or with a plectrum [17].

Fig. 1.13

Left: krar, right: pluriarc

(Photos by KT in Volker Bley’s gallery)

In Greece, where so many rules on tones and scales were invented, a traditional instrument was also made of strings mounted on a wooden frame: the kithara (ancient root for the word guitar), played mainly for noble events, in particular in honor of the god Apollo (see Fig. 1.14). In parallel one used the lyre, a smaller instrument without a foot basis [18].

Fig. 1.14

Muse tuning two kitharas. Detail of the interior from an Attic white-ground cup from Eretria, in 465

(photo by Jastrow (2005): public domain)

The kithara had 5–12 strings and came about in the 8th/7th century BC. For playing see the person standing on the right in Fig. 1.15. The player fixed the instrument on his breast. His right hand plucked the strings with a plectrum, while the left hand damped the sound and provided higher tones by shortening the strings [19].

Wind instruments were widely used for playing pentatonic music, with flutes and reed instruments being the most popular among them. The aulos was an exotic double reed woodwind instrument, a precursor of the modern oboe. It existed already in ancient Egypt and Italy. The primary form consisted of a pair of cylindrical (conical) bore pipes with double reed mouthpieces and finger holes (see the person in the middle of Fig. 1.15). Early aulos were made from reed stems, wood or bone.

Fig. 1.15

Dancers and musicians, tomb of the leopards, Monterozzi necropolis, Tarquinia, Italy. UNESCO World Heritage Site. Fresco a secco. Height (of the wall): 1.70 m. (public domain)

We move to the north: The Finnish kantele or Estonian kannel (Fig. 1.16) is a box zither with 5 pentatonic strings fixed on a burnt-out and carved-out birch trunk, frequently tuned in D major or D minor. Later on the string number was increased up to as many as 23 strings. It can be played in two ways: either the musician puts the long strings directly at his front (Haapavesi style), or the more traditional player would turn the shorter strings towards himself (Perhonjoki-style) [20].

Fig. 1.16

Top: Finnish kantele with 5 strings (CC BY-SA 3.0 by TheYellowFellow), bottom: Estonian kannel with 6 strings (CC BY-SA 3.0 by Adeliine)

Fig. 1.17

Left: a Zen Buddhism monk blows shakuhachi in walking (CC BY-SA 3.0 by Corpse Reviver), right: nose flute played by a Paiwan (The Paiwan are an indigenous people of Taiwan) (CC BY 2.0 Presidental Office Taiwan)

Moving on to Asia, China founded a tonal system similar to the one that Pythagoras achieved. However, later it was developed in ways different from the European one. The shakuhachi is an ancient Chinese longitudinal, end-blown bamboo flute (chǐ bā), which was introduced to Japan in the 7th century. The instrument is tuned to the minor pentatonic scale. As shown in Fig. 1.17 left, Zen Buddhism monks used this instrument for blowing meditation [21]. There are 4 holes on the front and one hole on the back side.

A nose flute (Fig. 1.17 right) is a popular musical instrument played in South East Asia, Polynesia, the Pacific Rim and in Africa. It is made from a single bamboo section and has up to four finger holes. There are a recorder type as shown here and a transverse type.

Its provenance is the Persian area and, around the year 1500, it was transmitted through Islam to the Asian East including Malaysia and Thailand. Its tonal range is two octaves. Due to its thin or sharp penetrating sound, it is preferentially used in large orchestras. Similar instruments have been in use in countries like Northern India (tangmuri), Tibet (gyaling) or Nepal (mvali).

The Kulintang is a horizontal row of nipple gongs played on the south-Philippine island Mindanao [22]. The instrument consists of a row/set of 5 to 9 graduated pot gongs, horizontally laid upon a frame arranged in order of pitch with the lowest gong found on the player’s left, based on pentatonic principles (see 8 gongs on the table in Fig. 1.18).

Fig. 1.18

A set of gongs of Kulintang on a table, Insets are a top view of the two gongs (bottom middle) and sticks (bottom right)

(CC BY-SA 2.5 by Philip Dominguez Mercurio)

The saung (a Burmese string instrument) is an arched horizontal harp and is said to be the only surviving harp in Asia [23]. Different from the European harps, the resonator is placed horizontally, as shown in Fig. 1.19. There are thirteen or sixteen strings, which have been traditionally made of silk (now often made of nylon).

Fig. 1.19

Saung musician in 1900 (public domain)

The Indonesian islands of Java and Bali are home to the gamelan music. It consists of many gongs, sounding plates, some xylophones, drums, and perhaps flutes, strings or voices. Noteworthy is a relief panel on a wall of Borobudur: a musical ensemble, probably a gamelan ensemble is engraved (Fig. 1.20). Borobudur is the world’s largest Buddhist temple built in the 9th century in Central Java, Indonesia [24].

Fig. 1.20

Musicians performing in a musical ensemble, bas-relief of Borobudur, probably an early form of gamelan

(CC BY 3.0 by Gunawan Kartapranata)

Finally we arrive on the American continent. The wall paintings dating from 775 AD at the Bonampak ceremonial complex [25] near today’s border of Guatemala (Fig. 1.21) indicate that in the Maya period people played instruments such as trumpets, flutes, whistles, and drums.

Fig. 1.21

Maya musicians: twin trumpeters standing side by side in a 12(?)-man orchestra, blowing higher overones; in Bonampak temple (public domain, author unknown)

A horizontal log drum, the teponaztli, played an important role in the Aztecs culture (see Fig. 1.22) [26].

Fig. 1.22

Left top: Two teponaztli, Aztec, found in Colima with a length of 59 cm (exhibited in the American Museum of Natural History, New York) (CC BY-SA 3.0 by Madman2001), left bottom: section through a drum, right: a drawing from the 16th century Florentine Codex showing a One Flower ceremony with two percussion instruments, a teponaztli (foreground) and a huehuetl (a big drum in the background) (public domain)

Appendix: Answers to the Physics & Music Quiz

This is a snapshot of a concert of a piano quartet with additional bongos.

Piano: Werner Heisenberg

1st violin: Ludwig van Beethoven

2nd violin: Albert Einstein

Cello: Jacqueline du Pré

Viola: Antonín Dvořák

Percussion: Richard Feynman

Equation above the piano: Heisenberg’s uncertainty principle [27]

$$ \Delta x\Delta p \ge \hbar $$

where $$\Delta x$$ is the standard deviation of position, $$\Delta p$$ the standard deviation of momentum and $$\hbar $$ = $$h/(2\pi )$$ the reduced Planck constant.

Fig. 1.23

Feynman diagram of electron/positron annihilation.

Details see the review article of Kaiser [29]

Mathematical symbol between Einstein and du Pré: Einstein’s field equation (general relativity) [28]

$$ \frac{8\pi G}{c^4}T_{\mu \nu } = G_{\mu \nu } + \Lambda g_{\mu \nu } $$

where $$T_{\mu \nu }$$ is the Einstein tensor, $$\Lambda $$ the cosmological constant, $$g_{\mu \nu }$$ the metric tensor, c the speed of light in vacuum and G the gravitational constant.

Diagram between Feynman and Heisenberg: a Feynman diagram (Fig. 1.23).

References

1.

J. Nichols, The origin and dispersal of languages: Linguistic evidence, in The Origin and Diversification of Language, ed. by N. Jablonski, L.C. Aiello (California Academy of Sciences, San Francisco, 1998), pp. 127–170

2.

C. Perreault, S. Mathew, Dating the origin of language using phonemic diversity. PLoS One 7, e35289 (2012)ADSCrossref

3.

D. Bickerton, Adam’s Tongue: How Humans Made Language, How Language Made Human (Farrar, Sraus & Giroux, New York, 2009)

4.

N.D. Cook, Harmony, Perspective and Triadic Cognition (Cambridge University Press, Cambridge, 2012), pp. 216–219

5.

N.J. Conard, M. Malina, S. Münzel, New flutes document the earliest musical tradition in southwestern Germany. Nature 460, 737 (2009)ADSCrossref

6.

E.B. Castle, Ancient Education and Today (Penguin Books, Baltimore, 1969)

7.

H.F. Cohen, Quantifying Music: The Science of Music (Springer Netherlands, Dordrecht, 1984), pp. 78–84Crossref

8.

J.-P. Rameau, Traité de l’Harmonie Réduite à ses Principes Naturels (Dr. Jean-Baptiste. Christophe Ballard, Paris, 1722)

9.

H. Helmholtz, Die Lehre von des tonempfindungen, als physiologische Grundlage für die Theorie der Musik (Friedrich Vieweg und Sohn, Braunschweig, 1863)

10.

M.L. West, The Babylonian musical notation and the Hurrian melodic texts. Music Lett. 75, 161–79 (1994)Crossref

11.

T.J. Mathiesen, Apollo’s Lyre: Greek Music and Music Theory in Antiquity and the Middle Ages (University of Nebraska Press, Lincoln, 1999)

12.

D.G. Sunol, G.M. Durnford, Textbook of Gregorian Chant According to the Solesmes Method (Kessinger Pub. Co., Whitefish, 2003)

13.

W.G. McNaught, The history and uses of the solfège syllables. PMRA 19, 35–51 (1893)

14.

J. Caldwell, English Keyboard Music Before the Nineteenth Century (Dover Publication, Mineola, 1985)

15.

10.000 Meisterwerke der Malerei (DVD-ROM), distributed by DIRECTMEDIA Publishing GmbH. ISBN: 3936122202

16.

A. Kebede, The bowl-lyre of Northeast Africa. Krar: the devil’s instrument. Ethnomusicology 21, 379–395 (1977)

17.

U. Wegner, Afrikanische Saiteninstrumente (Museum für Völkerkunde, Berlin, 1984), pp. 82–92, 153f

18.

J.G. Landels, Music in Ancient Greece and Rome (Routledge, London, 1999)Crossref

19.

O.J. Brendel, Etruscan Art (Yale University Press, New Haven, 1995)

20.

R. Apanaviǒius, Ancient Lithuanian Kanklės (Institute of Ethnomusic, Vilnius, 1996)

21.

J, Keister, The Shakuhachi as a spiritual tool: a Japanese buddhist instrument in the west. Asian Music 35, 104–105 (2004)

22.

K. Benitez, The Maguindanaon Kulintang: Musical Innovation, Transformation and the Concept of Binalig (University of Michigan, Ann Harbor, 2005)

23.

T. Miller, S. Williams, The Garland Handbook of Southeast Asian Music (Routledge, London, 2008)

24.

https://​en.​wikipedia.​org/​wiki/​Borobudur#cite_​ref-Raffles1814_​13-1

25.

M.D. Coe, The Maya (Thames & Hudson, London, 1999)

26.

M.D. Coe, Mexico: From the Olmecs to the Aztecs (Thames & Hudson, London, 2002)

27.

W. Heisenberg, Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927)ADSCrossref

28.

A. Einstein, Relativity: The Special and General Theory, translated in 1920 (H. Holt and Company, New York, 1961)

29.

D. Kaiser, Physics and Feynman’s diagrams. Am. Sci. 93, 156–165 (2005)Crossref

Footnotes

1

Cuneiform is a writing system with wedge-shaped marks. It is one of the earliest systems of writing, invented by Sumerians in ancient Mesopotamia.

2

His exact years of birth and death are unclear. The year of death could be around 1050.

Part IIOur Modern Tonal System

Music is a science which should have definite rules;

these rules should be drawn from an evident principle;

and this principle cannot really be known to us

without the aid of mathematics

—Rameau

© Springer Nature Switzerland AG 2021

K. Tsuji, S. C. MüllerPhysics and Musichttps://doi.org/10.1007/978-3-030-68676-5_2

2. Notation and Tonal Systems

Kinko Tsuji¹   and Stefan C. Müller²

(1)

Shimadzu Europa GmbH (retired), Duisburg, Germany

(2)

Professor emeritus, Institute of Physics, Otto von Guericke University Magdeburg, Magdeburg, Germany

Kinko Tsuji

Email: [e-mail]

2.1 What Is .....?

The notion of art refers to creativity found in human societies, both in theory and in physical expression. In history there have been many ways for classifying art depending on the cultural environments and viewing some fields as the main arts, others as derivatives of them. In modern times major constituents include literature (drama, poetry, prose), performing arts (among them dance, music, and theatre) and visual arts (comprising painting, sculpting, architecture, filmmaking, and others).

2.1.1 Music

We will talk about music first.

Music is an art form and cultural activity whose medium is sound organized in time. It gathers its material from the realm of tones. It thus belongs to the category of performing arts, different from the visual and/or fine arts, which are mainly static in nature.

The common elements which characterize any piece of music are melody, rhythm and harmony. Melody and rhythm proceed in time, while harmony acts at any point of the proceeding line of melody. Later we will devote more time on these characteristics (in Chap. 3), but will first mention a few other important properties of music.

2.1.2 Sound

Sound is a complex system formed from single tones: a basic tone plus several resonating single tones give rise to a series of overtones as analyzed in acoustic spectra. This creates the sonic qualities of timbre of a musical sound resulting in an intuitive tone color: glaring, soft, full, muffled, dark, dull, etc. Much more about that in Chap. 4.

2.1.3 Tone

The tone is a sound signal, as generated by periodic vibrations of an elastic body. Its pitch (tone height) is determined by vibrations of the air measured in Hertz (1 Hz $$=$$ 1/s) for frequency.

Our perception tells us that low frequency tends to sound heavy, dull, earthy and that high frequency sounds light, floating, piercing.

The intensity (loudness) of the tone is determined by the amplitude of vibrations, the dynamics of a melody by variations of amplitude. Its sound has a color as described in Sect. 4.​4. Its duration is indicated by a system of different symbols in connection with rules for tempo.

2.1.4 Noise

If a sound has a continuous frequency spectrum comprising many kinds of sounds emitted from different sources, we call it noise. That could be: the winds and waves, usual background in daily life, percussion instruments, explosions, ... Many types of noise have band-limited spectra—one hears a dominant tone (as with timpani) or they are white not permitting to perceive any tonality (rattles).

2.1.5 Spectral Characteristics of Sounds

Sounds and tones, the elements of music, have some basic properties, which we will show now in a qualitative way. A tone has a well-defined pitch, that can be measured by its frequency. We mean frequencies between 20 Hz and 20,000 Hz, which can be heard with our ears [1]. As a useful characterization of sound, that has a color, we need to determine a whole spectrum of frequencies representing all the overtones (basic tone plus series of harmonics - see Chap. 3).

In Fig. 2.1 we assemble a qualitative list of tones, sounds and their frequency spectra.

Fig. 2.1

Time courses (left column) and frequency spectra (right column): a sinusoidal sound and its single line spectrum; b sound and multiple overtone spectrum; c white noise and atonal spectrum; d band-limited noise and continuous (tonal) spectrum; e bang and its broad, line-shaped spectrum

Example: The Tuning Fork

It is not so easy to produce a pure sinusoidal sound with any musical instrument, there will be always some overtones of the fundamental frequency involved.

It is the tuning fork, (invented in 1711 by the trumpeter John Shore, see Fig. 2.2a, which emits a dominantly sinusoidal signal (as shown in Fig. 2.1a), adjusted to the standard pitch A4 $$=$$ 440 Hz, as defined in 1939 as a reference tone for the musical instruments of an orchestra [2]. Beyond the basic tone there are just very weak harmonics due to different vibrations of the fork as shown in Fig. 2.2b, c and d. Here we find sine sound vibrations (with very small deviations due to low amplitude overtones) and a more complex vibration, if the harmonics are activated by (too) strong stimulation.

Fig. 2.2

a Tuning fork; vibration of a tuning fork: b pure sine, c, d overlay of several harmonics

2.1.6 Standard Pitch

The standard pitch of today, A4 (in German notation a’) $$=$$ 440 Hz, has a long history [3]. Until the beginning of the 19th century different versions were in use, depending on the place of performance (church, opera, chamber music, ...) or the type of music played. Renaissance music used pitches up to 4 half tones higher or 3 half tones lower. In 1788 a pitch of 409 Hz was suggested in Paris, later the French Academy raised this pitch to 435 Hz.

An interesting suggestion was made by Sauveur/Chladni. They set a period of 1 s equal to a deep C ( $$=$$ 1 Hz). Eight octaves higher one would find the C4 with a frequency of 2 $$^8$$ $$=$$ 256 Hz, leading to A4 $$=$$ 426.666... Hz (in just intonation).

Nowadays there are several norms for the standard pitch issued by standard organizations of many countries. These regulations are taken with a maybe or must attitude. In Austrian and German orchestras one has introduced 443 Hz for standard pitch, because then the string instruments sound louder and fuller. In other countries the standard pitch may vary between 440 and 445 Hz.

The tone with standard pitch is played, before a rehearsal starts, by the oboe, then taken over by the concert master. His playing this note is the signal for all other members of the orchestra to tune their instruments correctly.

As mentioned above, there does not exist a unique determination of the standard pitch. Soprano singers have complained that this tone height should be reduced to be adapted to the capacities of their voices.

Different standard pitches are not appreciated by woodwind players: Whereas string instruments can be easily