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Musical Consonance and Cochlear Mechanics

vonReinhart Frosch

Neuausgabe

Musical Consonance and Cochlear Mechanics – Frosch

schnell und portofrei erhältlich bei beck-shop.de DIE FACHBUCHHANDLUNG

Thematische Gliederung:

Musikalische Akustik, Tontechnik, Musikaufnahme

vdf Hochschulverlag 2012

Verlag C.H. Beck im Internet:www.beck.de

ISBN 978 3 7281 3513 1

Preface In 1792, Maria Anna von Berchtold zu Sonnenburg, i.e., W.A. Mo-zart’s sister Nannerl, answering eleven questions, wrote „Data zur Biographie des Verstorbenen Tonn-Künstlers Wolfgang Mozart“ (Da-ta for the biography of the deceased musician Wolfgang Mozart). Her answer to the second question contains the following lines:

“[…] Der Sohn war damahls drey Jahr alt, als der Vater seine sieben-jährige Tochter anfieng auf dem Clavier zu unterweisen. Der Knab zeugte gleich sein von Gott ihm zugeworfenes ausserordent-liches Talent. Er unterhielte sich oft lange Zeit bey dem Clavier mit Zusammensuchen der Terzen, welche er immer anstimte, und sein Wohlgefahlen verrieth dass es wohl klang. […]“

Rough translation: „[…] The son was then three years old, when the father started to instruct his seven-years-old daughter on the piano. The boy showed right-away his extraordinary talent thrown at him by God. Often he amused himself a long time at the piano searching to-gether the thirds, which he always played, and his pleasure betrayed that it sounded well. […]”

In his psycho-acoustic one-boy experiments, young Mozart stud-ied sensory consonance, i.e., he judged the consonance of isolated, context-free two-tones. The present text, too, is focussed onto the sen-sory consonance of two simultaneous complex tones. In Part One, psycho-acoustic experiments are described. Some of these experi-ments were informal one-man studies, while others involved fairly large groups of subjects and subsequent statistical analysis. Part Two contains selected chapters of cochlear mechanics; a more complete treatment was undertaken in my earlier book “Introduction to Coch-lear Waves” [Frosch (2010a)]. In Part Three, selected chapters on sen-sory-consonance theories are presented. That treatment, too, is far

Extract from: Reinhart Frosch, Musical Consonance and Cochlear Mechanics © vdf Hochschulverlag 2012

from being exhaustive; a broader survey of sensory-consonance theo-ries is contained, e.g., in the book “The Psychology of Music” [Deutsch (1999)].

The present volume is intended to add weight to the hypothesis that our preference for certain two-tones is not only due to education, but is based on the physiology of our hearing organs.

The readers are expected to know biology, physics, and mathe-matics at high-school level. Exercises and their solutions are included at the end of most sections.

The author wishes to thank his friends and relatives who took part in the psycho-acoustic experiments described in Chapters 9 and 10. Interesting “threads” on topics treated in the present volume are spun by the Auditory E-mail List, [email protected]; therefore I permit myself to dedicate the present volume to that group. Brugg, Switzerland, September 2012. R. F.

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3.3 Deeper Harmonic Complex Tone C4 The results of the first one-man consonance experiment are presented in Fig. 3.2 below. Here, the pitch of the deeper tone corresponded to the just-C-major-tuning-system note C4 and so had a frequency of 264 Hz. The frequency of the higher tone was varied, in steps of an equal-temperament sixteenth-tone (12.5 cents, 8 tuning units) from 264 Hz to 528 Hz, i.e., to the pitch of the just-C-major-tuning-system note C5.

Fig. 3.2. Consonance ratings of two simultaneous harmonic complex tones; deeper

tone at 264 Hz.; intervals of 0–1200 cents.

For the ratings of the 97 different two-tones of Fig. 3.2, the Swiss school-grade system was applied. Grade 6 was attributed to very con-sonant two-tones, and grade 1 to very dissonant two-tones. I restricted myself to integral and half-integral ratings (1.0, 1.5, 2.0, … , 6.0). As discussed on pages 34–36 above, the first micro-tuning of the DX11 synthesizer contained two-tones ranging from zero to 472 tuning units, i.e., from zero to 737.5 cents. For larger intervals, the tuning of keys 13–61 was changed. The tuning of keys 1–12, however, was main-tained at the original small intervals (zero to 125 cents); these intervals

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were used as standards; i.e., the consonance ratings for larger intervals were determined by comparison with the small intervals.

In most of my tests, I listened to the two-tones to be judged in conventional ear-phones; the complete two-tone was presented to the left ear and, simultaneously, also to the right ear. In some tests I re-placed the ear-phones by a music amplifier. No significant depend-ence of the ratings on the method of presentation was observed. The sound-pressure level of each of the two simultaneous tones was about 65 decibels [on the conventional scale, i.e., on a scale such that zero decibel corresponds to a sound pressure amplitude of 28.7 micropas-cals; see Equation (3.7)].

In Fig. 3.2, there is a consonance minimum at 100 cents, i.e., at an interval of one semitone. If the test was started at an interval of 100 cents, and then proceeded in upward steps of 12.5 cents, the ob-served sensory consonance began to improve at ~137.5 cents and then formed a series of peaks, as shown in Fig. 3.2. Close to the conso-nance peaks, the spacing of intervals was reduced to 6.25 cents in or-der to estimate the interval sizes of the peaks with a precision of a few cents. The resulting “experimental” interval sizes of the consonance peaks are listed in the second column of Table 3.2.

For each of the ten consonance peaks listed in Table 3.2, it was at-tempted to find a frequency ratio R(theor) = m/n, formed by two small integers m and n, such that the corresponding interval size x(theor) [calculated via Equation (2.4), page 24] agreed with the experimental interval size x(exp) listed in the second column of the table. As shown by column four of Table 3.2, in all ten cases the resulting size x(theor) differed from the experimental size x(exp) by a few cents at most.

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Table 3.2. Experimental interval sizes x(exp) of the consonance peaks shown in

Fig. 3.2; deeper tone at 264 Hz.; intervals of 0–1200 cents. The theoretical interval sizes x(theor) in the fourth column correspond to the frequency ratios of the third column.

Fig. 3.3. Consonance ratings of two simultaneous harmonic complex tones; deeper

tone at 264 Hz.; intervals of 1200–2400 cents.

In Fig. 3.3, the test results for two-tones ranging from one octave (i.e., 1200 cents) to two octaves (2400 cents) are shown. That plot, too, exhibits a series of narrow sensory-consonance peaks. The corre-sponding interval sizes are listed in Table 3.3.

Number of

peak

Experimental interval size

x(exp) [cents]

Frequency ratio

R(theor)

Theoretical interval size

x(theor) [cents]

1 0 1/1 0 2 316 6/5 316 3 387 5/4 386 4 497 4/3 498 5 581 7/5 583 6 700 3/2 702 7 813 8/5 814 8 884 5/3 884 9 975 7/4 969 10 1194 2/1 1200

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Table 3.3. Experimental interval sizes x(exp) of the consonance peaks shown in

Fig. 3.3; deeper tone at 264 Hz.; intervals of 1200–2400 cents. Columns three and four: see Table 3.2.

Finally, in Fig. 3.4 the results for two-tones ranging from two octaves (2400 cents) to an interval somewhat larger than three octaves (name-ly, to 3800 cents) are displayed; the peak interval sizes are presented in Table 3.4. Fig. 3.4. Consonance ratings of two simultaneous harmonic complex tones; deeper

tone at 264 Hz.; intervals of 2400–3800 cents. In Fig. 3.4, the consonance minima at interval sizes above 2800 cents are seen to be less deep than those at smaller intervals. As before,

Number of

peak

Experimental interval size

x(exp) [cents]

Frequency ratio

R(theor)

Theoretical interval size

x(theor) [cents]

1 1194 2/1 1200 2 1466 7/3 1467 3 1591 5/2 1586 4 1703 8/3 1698 5 1900 3/1 1902 6 2084 10/3 2084 7 2172 7/2 2169 8 2397 4/1 2400

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these consonance ratings were obtained by comparing the two-tones with those ranging from zero to 100 cents. Table 3.4. Experimental interval sizes x(exp) of the consonance peaks shown in

Fig. 3.4; deeper tone at 264 Hz.; intervals of 2400–3800 cents. Columns three and four: see Table 3.2.

Exercise 3.5:

Using Tables 3.2–3.4 and Figs. 3.2–3.4, write a list of the frequency ratios R = m / n [where m and n are small integers] of those conso-nance peaks which have obtained ratings of 6.0 or 5.5. [Solution: 1/1, 5/4, 3/2, 5/3, 2/1, 5/2, 3/1, 4/1, 5/1.] Exercise 3.6:

Consulting, if necessary, a music lexicon or a music-theory textbook, write down the names of the two-tones found in Exercise 3.5. [Solution 1/1 = prime; 5/4 = just major third; 3/2 = just fifth; 5/3 = just major sixth;

2/1 = octave; 5/2 = just major tenth; 3/1 = just twelfth; 4/1 = fifteenth; 5/1 = just major seventeenth.]

Number of

peak

Experimental interval size

x(exp) [cents]

Frequency ratio

R(theor)

Theoretical interval size

x(theor) [cents]

1 2397 4/1 2400 2 2788 5/1 2786 3 3100 6/1 3102 4 3369 7/1 3369 5 3600 8/1 3600

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Exercise 3.7:

Use Equation (2.4) to calculate the interval size corresponding to a frequency ratio of 10/7, and verify that the central peak in Fig. 3.2 has a “shoulder” at that interval size. [Solution: x = 617 cents.] 3.4 Deeper Harmonic Complex Tone G2

In a second series of one-man experiments, the deeper of the two si-multaneous harmonic complex tones had the pitch of the just-C-major-tuning-system note G2. The frequency of that note is deeper than that of C4 (264 Hz) by a just eleventh, i.e., by a factor of 3/8. Thus the frequency of the deeper tone was chosen to be 99 Hz. The timbre of the two tones was unchanged (voice “Harmonica”; see Fig. 3.1, page 31). Fig. 3.5. Consonance ratings of two simultaneous harmonic complex tones; deeper

tone at 99 Hz; intervals of 0–1200 cents. The consonance curve in Fig. 3.5 differs strongly from that in Fig. 3.2 (page 37) although the range of two-tone intervals is 0–1200 cents in both cases. In Fig. 3.5, there are fewer consonance peaks, and the

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peaks are somewhat wider. A comparison of the corresponding Tables 3.2 and 3.5 yields that the latter table contains no consonance peaks at two-tone frequency ratios of 6/5, 7/5, 8/5, and 7/4. The integers in the “theoretical” frequency ratios R(theor) of Table 3.5 are less than or equal to five. Table 3.5. Experimental interval sizes x(exp) of the consonance peaks shown in

Fig. 3.5; deeper tone at 99 Hz.; intervals of 0–1200 cents. Columns three and four: see Table 3.2.

The corresponding data for intervals ranging from one to two octaves are presented in Fig. 3.6 and Table 3.6. Fig. 3.6. Consonance ratings of two simultaneous harmonic complex tones; deeper

tone at 99 Hz; intervals of 1200–2400 cents.

Number of

peak

Experimental interval size

x(exp) [cents]

Frequency ratio

R(theor)

Theoretical interval size

x(theor) [cents]

1 0 1/1 0 2 375 5/4 386 3 500 4/3 498 4 700 3/2 702 5 881 5/3 884 6 1200 2/1 1200

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A comparison of Figs. 3.6 and 3.3 yields a result which is similar to that of the above-mentioned comparison of Figs. 3.5 and 3.2: For deeper two-tones there are fewer consonance peaks, and the peaks are wider. Table 3.6. Experimental interval sizes x(exp) of the consonance peaks shown in

Fig. 3.6; deeper tone at 99 Hz.; intervals of 1200–2400 cents. Columns three and four: see Table 3.2.

The results for intervals ranging from two to somewhat more than three octaves are presented in Fig. 3.7 and Table 3.7. Fig. 3.7. Consonance ratings of two simultaneous harmonic complex tones; deeper

tone at 99 Hz; intervals of 2400–3800 cents.

Number of

peak

Experimental interval size

x(exp) [cents]

Frequency ratio

R(theor)

Theoretical interval size

x(theor) [cents]

1 1200 2/1 1200 2 1587 5/2 1586 3 1900 3/1 1902 4 2169 7/2 2169 5 2400 4/1 2400

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Table 3.7. Experimental interval sizes x(exp) of the consonance peaks shown in

Fig. 3.7; deeper tone at 99 Hz.; intervals of 2400–3800 cents. Columns three and four: see Table 3.2.

Exercise 3.8:

Same task as in Exercise 3.5 (page 41), for Tables 3.5–3.7 and Figs. 3.5–3.7. [Solution: 1/1, 3/2, 2/1, 5/2, 3/1, 4/1.]

Number of

peak

Experimental interval size

x(exp) [cents]

Frequency ratio

R(theor)

Theoretical interval size

x(theor) [cents]

1 2400 4/1 2400 2 2788 5/1 2786 3 3100 6/1 3102 4 3369 7/1 3369 5 3600 8/1 3600

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16 A Modified Consonance Theory As shown in Section 15.3 above, the consonance theory of Hermann von Helmholtz fails to explain the high experimental consonance rankings obtained by major and minor thirds composed of two har-monic complex tones in the female-singing pitch region. That failure was corrected by a modification which I first published in Section 3.3 of my book “Mitteltönig ist schöner” [Frosch (2001)]. Additional de-scriptions of my modification appeared in Section 3.4 of the English translation of the just mentioned book, “Meantone Is Beautiful” [Frosch (2002)], and in a proceedings paper, “Psycho-Acoustic Exper-iments on the Sensory Consonance of Musical Two-Tones” [Frosch (2007)].

According to my modified theory, the sensory consonance of a dyad (i.e., a two-tone) formed by two simultaneous harmonic complex tones is high if the dyad fulfils both of the following conditions: Condition A: The dyad contains few or no pairs of partial tones which generate dis-agreeable beats. Condition B: In the excitation pattern generated by the dyad on the basilar mem-brane of the cochlea in the inner ear there are few or no wide gaps. The just specified condition A is the “Helmholtz condition”; see Chap-ter 15 above. Condition B constitutes my proposed modification of the Helmholtz theory.

As discussed in Section 14.3, a soft or medium-level sine-tone causes, in a healthy human cochlea, a strong vibration of an about one millimetre long piece of the basilar membrane, at the place of the “ac-tive peak”. That place x can be assumed to obey an approximately

Extract from: Reinhart Frosch, Musical Consonance and Cochlear Mechanics © vdf Hochschulverlag 2012

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8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

z

linear relation, given by our Eq. (14.13), to the critical band number z, which was found, by psycho-acoustic experiments, to depend on the sine-tone frequency f according to our Eq. (14.11). P8 g g g 3 gaps M6 g g 2 gaps P5 g g g 3 gaps P4 g g g 3 gaps M3 g r 1 gap m3 g r 1 gap P1 g g g 3 gaps Fig. 16.1. Partial-tone patterns of the seven two-tones defined in Table 15.5, at fe-

male-singing pitch (f1 = 352 Hz; note F4); see text. In Figs. 16.1 and 16.2, the critical-band numbers z corresponding, according to Eq. (14.11), to the partial-tone frequencies f occurring in each of the seven dyads are indicated by filled circles. Pairs of adja-cent partial tones separated by a critical-band-number difference ∆z > 2 (and thus yielding, on the basilar membrane, excitation peaks separated approximately by a distance ∆x > 1.7 mm) are conjectured to cause a disagreeable gap, and are indicated, in Figs. 16.1 and 16.2, by the symbol g. If Figs. 16.1 and 16.2 are extended to higher critical-band numbers z, no additional gaps with ∆z > 2 are found in any of the seven dyads. Pairs of partials yielding, via Eq. (15.1), a ratio b/bm.d.

ranging from 0.5 to 2.0 (where b = fh – fd is the beat rate) are thought to cause a very disagreeable roughness, and are indicated in Figs. 16.1 and 16.2 by the symbol r.

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2 3 4 5 6 7 8 9 10 11 12 13

z

In Fig. 16.1, the major third (M3) and the minor third (m3) are seen to fulfil condition B especially well (only one gap), so that their good experimental consonance rankings (see Table 15.5) is understandable.

The minor sixth (m6, R = 8/5, not shown) would yield two gaps in Fig. 16.1, as does the major sixth (M6, 5/3).

In Fig. 16.1, the deeper complex-tone frequency f1 has been chosen to be 352 Hz (note F4), near the centre of the range of the correspond-ing frequencies chosen in our Fig. 3.2 (264 Hz), in our Fig. 5.1 (Ex-periment of Kaestner, 256 Hz), in our Fig. 15.1 (Theory of Helmholtz, 264 Hz), in Fig. 12.15 of Terhardt (1998, theoretical curve, 440 Hz(, and in Fig. 11 of Rasch and Plomp (1999, theoretical curve, 250 Hz). If in our Fig. 16.1 the deeper complex tone F4 is replaced by C4 (264 Hz), then the only different number of gaps is that for the perfect fourth (P4; 4/3; 2 gaps). If, on the other hand, F4 is replaced by C5 (528 Hz), then the only different number of gaps is that of the perfect fifth (P5; 3/2; 4 gaps). P8 g 1 gap M6 r no gap P5 no gap P4 r r no gap M3 r r no gap m3 r r r no gap P1 g 1 gap Fig. 16.2. Same as Fig. 16.1; bass pitch (f1 = 99 Hz, note G2).

Extract from: Reinhart Frosch, Musical Consonance and Cochlear Mechanics © vdf Hochschulverlag 2012

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According to this modified theory, sensory consonance is subtle: An increase of dissonance is avoided if the frequency difference of two neighbouring partial tones is large enough (no disagreeable beats) but also small enough (no disagreeable gap in the excitation pattern on the basilar membrane).

Condition B (i.e., consonance is high if there are no or few wide gaps in the excitation pattern on the basilar membrane) becomes fairly plausible if one considers dyads of deep tones (i.e., tones at bass pitch; deeper of the two simultaneous complex tones at ~100 Hz; see Fig. 16.2).

In the case of Fig. 16.2, perfect primes (P1), perfect fifths (P5), and octaves (P8) have no or one gap and no very disagreeable beats. In this case, condition B is about equally well fulfilled by all seven stim-uli. The thirds (M3 and m3) and the perfect fourth (P4) are predicted to be dissonant because they violate condition A (the Helmholtz condi-tion): each of them includes at least two pairs of partial tones (marked by the symbol r in Fig. 16.2) which generate very disagreeable beats [ratio b/bm.d. ranging from 0.5 to 2.0, where b = fh – fd is the beat rate; the most disagreeable beat rate bm.d. at the considered average fre-quency favg. = (fh + fd)/2 is defined by Eq. (15.4)]. Thus the modified consonance theory explains, e.g., that the pleasantness of the thirds (M3 and m3) is high in Fig. 11.1 (female-singing pitch), but is dis-tinctly lower in Fig. 11.2 (bass pitch).

The perfect fourth (P4) has a fairly low pleasantness both in Fig. 11.1 (female-singing) and in Fig. 11.2 (bass). According to Fig. 16.1, the low pleasantness of P4 at female-singing pitch is due to the violation of condition B (3 disagreeable gaps), whereas according to Fig. 16.2 the low pleasantness of P4 at bass pitch is due to the vio-lation of condition A (two pairs of partial tones generating disagree-able beats).

The perfect prime (P1) and the perfect fifth (P5) have a low pleas-antness in Fig. 11.1 (female-singing) and a high pleasantness in Fig. 11.2 (bass). Both these findings agree with the modified theory: Fig. 16.1 contains three gaps for both P1 and P5; Fig. 16.2 yields one gap only for P1, no gap for P5, and no disagreeable beats for both P1 and P5.

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The low pleasantness of the octave (P8) in Fig. 11.1 (female-singing) agrees with Fig. 16.1 (three gaps); the low pleasantness of P8 in Fig. 11.2 (bass), however, disagrees with Fig. 16.2 (one gap only, and no disagreeable beats). Possible reasons for the failure of the modified theory in the case of deep-tone octaves:

1) The low experimental rating of the octave in Fig. 11.2 is due to the experiment of Kaestner, in which the deeper tone had a frequency of 128 Hz (Fig. 5.4), higher than in my own experiment (99 Hz; Fig. 10.2; according to that latter diagram, the pleasantness of P8 is about equal to those of M3, P4, P5, and M6).

2) Kaestner’s subjects found the octaves “dull”, and therefore tended to prefer the stimuli competing against the octaves in the correspond-ing matches. The differences between Figs. 5.4 (Kaestner) and Fig. 10.2 (Frosch) may be due not only to the above-mentioned fre-quency difference, but also to the difference between “dull” and “un-pleasant”. Conclusions on the modified consonance theory:

The consonance curves for two simultaneous bowed-string-like har-monic complex tones (sensory consonance versus fundamental-fre-quency ratio R = f2 / f1) form narrow peaks.

The R-values of the peaks (R = m / n ; m and n are small integers) agree with the Helmholtz consonance theory: if the rate of the beats generated by two partial tones is in a certain range, then these beats cause dissonance.

The relative heights of the consonance peaks become more under-standable if one considers, in addition to the Helmholtz theory, also “condition B”: sensory consonance is high if there are no or few wide gaps in the excitation pattern generated by the partial tones on the bas-ilar membrane in the inner ear.

Extract from: Reinhart Frosch, Musical Consonance and Cochlear Mechanics © vdf Hochschulverlag 2012

vdf Hochschulverlag AG an der ETH Zürich, VOB D, Voltastrasse 24, 8092 ZürichTel. +41 (0)44 632 42 42, Fax +41 (0)44 632 12 32, [email protected], www.vdf.ethz.ch

Reinhart Frosch

Introduction to Cochlear Waves

2010, 448 Seiten, zahlreiche Grafiken, Format 15 x 21,5 cm, broschiertISBN 978-3-7281-3298-7

The first parts of the present text are devoted to a "passive" cochlea, i.e., to cases in which the mechanical energy generated by "active" outer hair cells is absent or negligibly small. Passive human cochleae were studied, e.g., in the post-mor-tem experiments of von Békésy, who found that tones generate, in the cochlear channel, travelling hydrodynamic surface waves which are similar to waves pro-pagating on the ocean. In spite of the fact that the travelling-wave energy starts to be transformed into frictional heat at the cochlear base already, the velocity amplitude of the basilar-membrane oscillation increases with increasing distance from base. At some place, namely at the "passive peak", that increase stops, and at greater distance from base the amplitude quickly drops to small values. At high [low] tone frequency, the distance from base of the passive peak is short [long].

Additional topics treated in this book: the outer hair cells and the "active" re-sponse peak generated by them; evanescent cochlear waves; high-frequency pla-teaux; cochlear maps; certain forms of tinnitus; otoacoustic emissions; frequency glides.

vdf Hochschulverlag AG an der ETH Zürich, VOB D, Voltastrasse 24, 8092 ZürichTel. +41 (0)44 632 42 42, Fax +41 (0)44 632 12 32, [email protected], www.vdf.ethz.ch

Reinhart Frosch

Four-Tensors, the Mother Tongue of Classical Physics

2006, 300 Seiten, Format 12 x 18,3 cm, broschiertISBN 978-3-7281-3069-3

In this monograph, based on a course that Reinhart Frosch taught at ETH, it is shown that a spectacular formal simplification of the equations representing the basic laws of classical physics (e.g. the Maxwell equations of the electromagnetic field) is achieved if the accustomed three-vectors are replaced by four-tensors.

As an introduction into the challenging subject of four-tensors, the first part of the book treats basic chapters of special relativity theory. The text is aimed at all persons interested in physics; the readers are expected to know high-school mathematics and physics. Exercises and their solutions are included at the end of most chapters.

Die Stiftung Kreatives Alter hat Reinhart Frosch für sein Buch einen Hauptpreis verliehen.