Music Chem

Chemistry for Everyone

Musical Chemistry: Integrating Chemistry and Music

W

A Nine-Part Series on Generating Music from Chemical Processes Mahadev Kumbar Department of Chemistry, Nassau Community College, Garden City, NY 11530; [email protected]

We live in a world that is not only tantalizingly beautiful, it is also filled with spine-tingling mysteries, pleasant surprises, and mind-boggling intrigues. Human nature includes curiosity and the impulse to understand the world around us: experts use the tools of their fields to explore the natural world. For example, chemists investigate the world in terms of atoms, molecules, and chemical reactions. One (perhaps surprising) aspect of the natural world is that each and every process in nature—chemical or otherwise—produces some kind of sound, whether audible (20 Hz–20 kHz) or nonaudible (<0 Hz and >20 kHz), characteristic of that process. Those sounds, I believe, are the music that is the universal language of the natural world. Music and chemistry—being integral parts of the natural world—have shared commonalities: they both use fundamental units as building blocks (chemical elements in chemistry and musical notes in music); they both have time dimensions that are meaningful; and both find expression in the language of mathematics. Few attempts have been made in the past to understand the relationship between music and macroscopic and microscopic processes, including atoms and elementary particles (1–3). No studies, as far as I know, relate music to chemical reactions, although the problem is of considerable interest to both chemists and musicians. Such investigations may lead to discovery of new classes of chemical reactions not yet seen. Therefore, in a ninepart series, I have explored the musical aspects of a variety of chemical reactions and also a few other chemical processes. Their titles are shown in List 1; detailed explanations for each, as well as sound files, are provided in the Supplemental Material.w Overview Music is an art; yet it is based on the science of mathematics. Chemistry is a science that is also describable in the language of mathematics. Hence, it is possible to express chemistry in the language of music. In part I, I have tried to answer this fundamental question: do chemical reactions make music?

List 1. Titles for Each in a Nine-Part Series Developed To Integrate Chemistry and Music

I: Can Chemical Reactions Make Music?



II: Fourier Analysis of First-Order Reaction Kinetics

III: Ordinary Chemical Reactions IV: Oscillating Chemical Reactions

V: Nuclear Decay Reactions

VI: Pharmacokinetic Reactions VII: Enzyme Kinetic Reactions VIII: Musical Properties of Electron Transition IX: Musical Properties of the Periodic Table



Techniques for Using Chemistry To Produce Sound To answer this question, I designed a process that starts with a chemical reaction and progresses through several steps that ultimately lead to a musical piece playable either on a computer or with a musical instrument. The steps include: describing the nature of the sounds; establishing a correspondence between the perceptual attributes and physical attributes of the sounds; and transforming the sounds via mathematics and principles of music theory into music that humans can play on musical instruments or computers with appropriate software. The various steps involved are shown in the following flow diagram: chemical reaction Fourier transform

frequency amplitude spectrum phase music theory music (computer music or instrumental music)

sound

The main quest here is to transform the time domain (chemical reaction) to the frequency domain (music) using the Fourier transformation. Once the physical attributes are extracted through this technique, they are then converted into perpetual attributes using music theory so that music can be produced. In that respect, this article describes a strategy to transform aspects of chemical reactions into music. Mathematics under It All Before music can be derived from chemical reactions (part I), the essential mathematical techniques and the methodology need to be developed. This is achieved in part II using the firstorder reaction as a model function: (1) A B The reaction rate can be measured either in terms of rate of disappearance of reactant (A) or rate of formation of product (B). The equation for the first kind is

Ct  C0 e  k1t



(2)

where Ct is the concentration of A at time t, C0 is the concentration of A at time zero, and k1 is the first-order rate constant. Similarly, the equation for the second kind can also be written.

www.JCE.DivCHED.org  •  Vol. 84  No. 12  December 2007  •  Journal of Chemical Education 1933

Chemistry for Everyone

Our aim here is to extract the frequency content of eq 2. If a given function satisfies Dirichlet’s condition (4), it is guaranteed that it will have frequency content. However, except for oscillating chemical reactions, all chemical reactions are not periodic in nature and go to completion in a finite time. Therefore, a methodology is developed to make the first-order reaction equation periodic using Fourier series techniques as well as a protracting method. Furthermore, discrete Fourier transformation (DFT) is applied to selected model functions to extract frequency, amplitude, and phase values. The results are summarized in the form of spectral analysis. Transforming Chemical Characteristics To Produce Specific Aspects of Sound It is important to understand how C0 and k1 influence the music. It appears that C0 simply manipulates the amplitude (loudness) without altering the quality of music (timbre), although k1 seems to influence the timbre. The frequencies produced by these model functions fall in the mHz range, which is well below the threshold frequency of human hearing (20 Hz). In addition, the amplitudes are smaller in magnitudes than audible ones. For that reason, the frequency and amplitude are magnified to hear the sounds. Sound files generated using the Csound (5) computer program are included in the Supplemental Material.w Magnified frequencies are further transformed into musical notes using principles of music theory (6, 7) based on an equal-tempered scale. Observe that the derived musical notes are dependent upon the way the magnification is carried out; magnification using the factor of 10n (where n is an integer) is applied in such a way as to bring the lowest frequency just above the frequency of the lowest musical note (27.5 Hz) in an equal-tempered scale. Then the same magnification factor is applied to the remaining frequencies. The grand staff playable 250

Amplitude

200 150 100 50 0 0

5

10

15

Frequency Index, k Figure 1. Plot of amplitude values versus the frequency index for decomposition of N2O5 (g).

on the piano is generated with the help of Mozart software (8). (This software application and the Csound computer program are used, respectively, to build musical staves and sound files in all parts of these tutorials. See the Supplemental Material for further details.w) From Reaction to Spectra to Sound In part III of this series, 25 real ordinary first-order reactions (12 reactions for the rate of disappearance of reactants and 13 for the rate of formation of products) are examined using the methodology presented in part II. The experimental data are first fitted to first-order rate equations, and subsequently used in carrying out the DFT analysis. For example, the spectral analysis for the decomposition of N2O5 (g) is shown in Figure 1. The deduced frequencies for the reactions considered here also fall in the infrasound region (<20Hz) and hence are magnified along with amplitudes to produce sound files. The musical staves are also constructed to play the music on an instrument. For example, the grand staff for decomposition of N2O5 ( g) is presented in Figure 2. Overall, the difference in temperature, type of experimental methods, and mode of chemical reactions appear to have some influence on the type and the quality of the music produced. There is a great distinction between oscillating chemical reactions and ordinary chemical reactions in terms of their lifespan; ordinary chemical reactions go to completion in finite times whereas oscillating chemical reactions either never go to completion or take a long time. Besides, oscillating reactions satisfy Dirichlet’s condition while ordinary chemical reactions do not. The study of these reactions is undertaken in part IV of the series using oscillating curves simulated by transcendental functions. The results of DFT analysis indicate that these reactions, like ordinary chemical reactions, also produce frequencies in the infrasound region. Because the simulated curves resemble the oscillating curves produced by the real oscillating chemical reactions, it is logical to assume that real oscillating reactions also have the ability to generate the music. Nuclear Decay with Pleasing Harmonics The musical apects of nuclear decay reactions are considered in part V, which also obey the first-order rate and have half-lives ranging from microseconds to years. A total of seven nuclides are considered and DFT analysis is carried out as outlined in part II. The results indicate that frequencies produced by all decay reactions (except 217Rn decay) also lie in the infrasound region. However, those of  217Rn fall within the range of human hearing. Each decay reaction produces a string of harmonics with distinct frequencies that influences the timbre. The range of frequencies for nuclear decay and chemical reactions are compared in Figure 3.

#

#

Figure 2. Grand musical staff for decomposition of N2O5 (g).

1934 Journal of Chemical Education  •  Vol. 84  No. 12  December 2007  •  www.JCE.DivCHED.org

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Chemistry for Everyone

Deriving BioBeats

and are then transformed into musical notes using music theory. The music produced by each atom is not similar and appears to be quite distinctive. The musical transition (musical–energy) diagram is developed for the H atom (Figure 4). The last part of this series, part IX, explores musical properties of the periodic table based on atomic spectra. When atoms are clustered together like groups, periods, metals, and so forth, they tend to generate unique and distinct music. Figure 5 shows

Biologically oriented systems are considered in the next two parts of this tutorial series. Part VI explores ­pharmacokinetic systems with one-, two-, and three-compartment models using specific examples. The frequencies produced by these examples are in the infrasound region, ranging from mHz to μHz. Pharmacokinetic reactions basically involve the flow of a drug in the body as time progresses and are quite different from chemical or nuclear reactions. As such, they appear to play dissimilar and distinct music compared with chemical or nuclear systems. Part VII studies the musical aspects of a few enzyme-kinetic reactions. Spectral analysis suggests that amplitude and phase values depend upon the nature of the system as well as whether the inhibitor is present or not. The deduced frequencies also fall in the infrasound region.

n6 n5

n4

Atomic Theory and Music Theory Energy

Part VIII of this series presents musical properties of atomic spectra of selected atoms of groups IA, VIIIA, and period 2. The literature wavelength values are first converted to frequencies

m

h

Brackett series

D#5

G7 n3

A#4

G7

nuclear decay reactions s

D2

A#3

F5

G7

Paschen series

n2

d

Balmer series

F#7

y

D#7 6 t 101 s

1s

Ľ2

1.667 t 10

1 Hz

Hz

3.6 t 103 s

8.64 t 104 s

3.156 t 107 s

2.778 t 10Ľ4 Hz

1.157 t 10Ľ5 Hz

3.169 t 10Ľ8 Hz

n1

Lyman series

chemical reactions

Figure 4. A musical energy (transition) diagram depicting musical notes associated with each energy level of a hydrogen atom.

Figure 3. Time spectrum for nuclear decay and chemical reactions. 1 IA

2 VIIIA

1H G#6 (Ab6)

2 IIA

3Li G5

2He

2 IIIA

2 IVA

2 VA

2 VIA

2 VIIA

4Be

5B

6C

7N

8O

9F

10Ne

D#8 (Eb8)

B5

F6

D7

C#7 (Db7)

C7

C8

13Al

14Si

15P

17S

17Cl

18Ar

D6

E6

G6

B6

33As

34Se

35Br

36Kr

E6

G#6 (Ab6)

52Te

53I

54Xe

B5

D#6 (Eb6)

D6

84Po

85At

86Rn

G5

F6

C5

Musical Periodic Table

11Na

12Mg

E5

A#5 (Bb5)

3 IIIB

4 IVB

5 VB

6 VIB

7 VIIB

8 VIIIB

9 VIIIB

10 VIIIB

11 IB

12 IIB

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28 Ni

29Cu

30Zn

31Ga

32Ge

C5

C#5 (Db5)

F5

G#5 (Ab5)

A5

G5

G5

D#5 (Eb5)

B5

37Rb

38Sr

39Y

40Zr

41Nb

C5

E5

G#5 (Ab6)

G5

G5

55Cs

56Ba

57La

72Hf

73Ta

42Mo

A#5 (Bb5) G#5 (Ab5) C#6 (Db6) G#5 (Ab5) C#6 (Db6)

43Tc

F#5 (Gb5) F#5 (Gb5)

74W

75Re

44Ru

45Rh

46Pd

47Ag

48Cd

49In

A5

G#5 (Ab5)

G5

C6

F5

E5

76Os

77Ir

78Pt

79Au

80Hg

E6

E6

G#4 (Ab4) D#5 (Eb5) A#4 (Bb4) F#5 (Gb5) G#5 (Ab5) G#5 (Ab5) A#5 (Bb5) A#5 (Bb5) A#5 (Bb5) A#5 (Bb5)

87Fr

88Ra

89Ac

G#4 (Ab4)

C5

D5

Lanthanide Series Actinide Series

58Ce

59Pr

B4

B4

60Nd

61Pm

90Th

91Pa

92U

93Np

94Pu

D#5 (Eb5)

C5

C5

C#5 (Db5)

B4

F#5 (Gb5) C#5 (Db5)

A#5 (Bb5) D#6 (Eb6)

81Ti

50Sn

C#6 (Db6) C#6 (Db6)

51Sb

A#5 (Bb5) C#6 (Db6)

82Pb

83Bi

C#6 (Db6) G#5 (Ab5) G#5 (Ab5)

62Sm

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tm

70Yb

70Lu

D5

F#5 (Gb5)

D5

D5

G5

G5

D#5 (Eb5)

G5

G5

F#5 (Gb5)

D8

Figure 5. The musical properties of the periodic table as a whole are summarized in the form of a musical periodic table.



www.JCE.DivCHED.org  •  Vol. 84  No. 12  December 2007  •  Journal of Chemical Education 1935

Chemistry for Everyone Table 1. Summary of Physical Attributes and Types of Sound Associated with the Chemical Processes Investigated Order of Frequency Range, Hz a

Chemical Process   First-order reaction

    1

  Oscillating reaction

    1

    1

  Pharmacokinetics reaction

 

  Enzyme kinetic reaction

–1×

10¯1

  Electron transition

× 10¯6 – 1 × 10¯3

1 × 10¯3 – 1 × 100

    1

Approximate values.  

×

10¯3

1 × 10¯12 – 1 × 103

  Nuclear decay reaction

a

× 10¯6 – 1 × 10¯2

×

1013

–1×

1014

Type of Sound

Type of Music b

Infrasound

Micromusic

Infrasound

Micromusic

× 100 – 1 × 101

Infrasound, very low audible

Micromusic, very faint music

    1

× 10¯2 – 1 × 101

Infrasound

Micromusic

    1

× 10¯3 – 5 × 102

Infrasound

Micromusic

Ultrasound

Macromusic

    1     1

× 10¯4 – 5 × 101 ×

   3.0

    1

×

10¯3

10¯4

–4×

–5×

101

101

b

Music in the infrasound region is termed micromusic; music in the ultrasound region is termed macromusic.

the musical properties of the periodic table as a whole, summarized in the form of the musical periodic table. Considering the musical properties of the entire periodic table on a macro level, it is intuitive to conclude that the pitch of the musical notes increases from left to right in any period and decreases from top to bottom in any group similar to firstionization energy. The summary of physical attributes and the types of sound associated with each chemical process is provided in Table 1. In conclusion, I have demonstrated that various chemical, nuclear, and biological systems indeed possess the ability to produce some kind of music that is characteristic of their nature. Acknowledgements I would like to thank F. Richard Moore of the University of California, San Diego and George C. Schatz of Northwestern University for valuable suggestions. I would also like to thank Journal staff for their diligence, skill, and support. wSupplemental

Order of Amplitude Range a

Material

and music)? Sound and music (or music and sound)? Musichemistry? In parallel with other branches of chemistry—such as physical chemistry, biological chemistry, and so forth—I have chosen to call this musical chemistry.

Literature Cited 1. Murchie, G. Music of the Spheres, Vols. I–II; Dover Publications, Inc.: New York, 1967. 2. Maruni, J; Lefebvre, R; Rantanen, M. Science and Music: From the Music of the Depths to the Music of the Spheres. In Advanced Topics in Theoretical Chemical Physics; Maruani, J., Lefebvre, R., Brandas, E. J., Eds.; Progress in Theoretical Chemistry and Physics 12; Kluwer Academic Publishers: Norwell, MA, 2003; pp 478–514. 3. de Lozanne, A. Science 2004, 305, 348–349. 4. Tolstov, G.P. Fourier Series; Dover Publications, Inc.: New York, 1962. 5. Vercoe, B. Csound, version 4.23. http://www.csounds.com/ (accessed Sep 2007).

Detailed explanatory texts and sound files for each of the nine tutorials are available in this issue of JCE Online.

6. Henry, E. Fundamentals of Music, 3rd ed.; Prentice Hall: Upper Saddle River, NJ, 1999.

Note

7. Miller, M. The Complete Idiot’s Guide to Music Theory; Alpha Books: Indianapolis, IN, 2002.

1. I had debated about what to name this series of tutorials: musical chemistry or chemical music? Music and chemistry (or chemistry

8. Webber, D. Mozart Virtuoso, version 8.01. http://www.mozart. co.uk/index.htm (accessed Sep 2007).

1936 Journal of Chemical Education  •  Vol. 84  No. 12  December 2007  •  www.JCE.DivCHED.org