Brugg, Switzerland, September R. F.

Preface In 1792, Maria Anna von Berchtold zu Sonnenburg, i.e., W.A. Mozart s sister Nannerl, answering eleven questions, wrote Data zur Biographie des Verstorbenen Tonn-Künstlers Wolfgang Mozart (Data 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 siebenjährige Tochter anfieng auf dem Clavier zu unterweisen. Der Knab zeugte gleich sein von Gott ihm zugeworfenes ausserordentliches 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 together the thirds, which he always played, and his pleasure betrayed that it sounded well. [ ] In his psycho-acoustic one-boy experiments, young Mozart studied sensory consonance, i.e., he judged the consonance of isolated, context-free two-tones. The present text, too, is focussed onto the sensory consonance of two simultaneous complex tones. In Part One, psycho-acoustic experiments are described. Some of these experiments 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 Cochlear Waves [Frosch (2010a)]. In Part Three, selected chapters on sensory-consonance theories are presented. That treatment, too, is far

from being exhaustive; a broader survey of sensory-consonance theories 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 mathematics 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, auditory@lists.mcgill.ca; therefore I permit myself to dedicate the present volume to that group. Brugg, Switzerland, September 2012. R. F.

3.3 Deeper Harmonic Complex Tone C4 3 One-Man Experiments (Complex Tones) 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. RATING 7 6 5 4 3 2 1 0 0 200 400 600 800 1000 1200 INTERVAL [cents] 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 consonant 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 maintained at the original small intervals (zero to 125 cents); these intervals 37

Part 1, Sensory-Consonance Experiments 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 replaced the ear-phones by a music amplifier. No significant dependence 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 micropascals; 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 observed 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 consonance peaks, the spacing of intervals was reduced to 6.25 cents in order to estimate the s of the peaks with a precision of a few cents. The resulting experimental s 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 attempted to find a frequency ratio R(theor) = m/n, formed by two small integers m and n, such that the corresponding x(theor) [calculated via Equation (2.4), page 24] agreed with the experimental 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. 38

3 One-Man Experiments (Complex Tones) Number of peak Experimental x(exp) [cents] Frequency ratio R(theor) Theoretical 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 Table 3.2. Experimental s x(exp) of the consonance peaks shown in Fig. 3.2; deeper tone at 264 Hz.; intervals of 0 1200 cents. The theoretical s x(theor) in the fourth column correspond to the frequency ratios of the third column. RATING 7 6 5 4 3 2 1 0 1200 1400 1600 1800 2000 2200 2400 INTERVAL [cents] 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 corresponding s are listed in Table 3.3. 39

Part 1, Sensory-Consonance Experiments RATING 7 6 5 4 3 2 1 Number of peak Experimental x(exp) [cents] Frequency ratio R(theor) Table 3.3. Experimental s 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 (namely, to 3800 cents) are displayed; the peak s are presented in Table 3.4. 0 2400 2600 2800 3000 3200 3400 3600 3800 INTERVAL [cents] Theoretical 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 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 s above 2800 cents are seen to be less deep than those at smaller intervals. As before, 40

3 One-Man Experiments (Complex Tones) these consonance ratings were obtained by comparing the two-tones with those ranging from zero to 100 cents. Number of peak Experimental x(exp) [cents] Frequency ratio R(theor) Theoretical 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 Table 3.4. Experimental s 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 consonance 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.] 41

Part 1, Sensory-Consonance Experiments Exercise 3.7: Use Equation (2.4) to calculate the corresponding to a frequency ratio of 10/7, and verify that the central peak in Fig. 3.2 has a shoulder at that. [Solution: x = 617 cents.] 3.4 Deeper Harmonic Complex Tone G2 In a second series of one-man experiments, the deeper of the two simultaneous harmonic complex tones had the pitch of the just-c-majortuning-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). RATING 7 6 5 4 3 2 1 0 0 200 400 600 800 1000 1200 INTERVAL [cents] 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 42

3 One-Man Experiments (Complex Tones) 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. Number of peak Experimental x(exp) [cents] Frequency ratio R(theor) Theoretical 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 Table 3.5. Experimental s 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. RATING 7 6 5 4 3 2 1 0 1200 1400 1600 1800 2000 2200 2400 INTERVAL [cents] Fig. 3.6. Consonance ratings of two simultaneous harmonic complex tones; deeper tone at 99 Hz; intervals of 1200 2400 cents. 43

Part 1, Sensory-Consonance Experiments 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. Number of peak Experimental x(exp) [cents] Frequency ratio R(theor) Theoretical 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 Table 3.6. Experimental s 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. RATING 7 6 5 4 3 2 1 0 2400 2600 2800 3000 3200 3400 3600 3800 INTERVAL [cents] Fig. 3.7. Consonance ratings of two simultaneous harmonic complex tones; deeper tone at 99 Hz; intervals of 2400 3800 cents. 44

3 One-Man Experiments (Complex Tones) Number of peak Experimental x(exp) [cents] Frequency ratio R(theor) Theoretical 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 Table 3.7. Experimental s 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.] 45

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 harmonic 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 descriptions 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 Experiments 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 disagreeable beats. Condition B: In the excitation pattern generated by the dyad on the basilar membrane of the cochlea in the inner ear there are few or no wide gaps. The just specified condition A is the Helmholtz condition ; see Chapter 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 active peak. That place x can be assumed to obey an approximately 205

Part 3, Consonance Theories 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 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 z Fig. 16.1. Partial-tone patterns of the seven two-tones defined in Table 15.5, at female-singing pitch (f 1 = 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 adjacent 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 criticalband 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/b m.d. ranging from 0.5 to 2.0 (where b = f h f d 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. 206

16 A Modified Consonance Theory 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 f 1 has been chosen to be 352 Hz (note F4), near the centre of the range of the corresponding frequencies chosen in our Fig. 3.2 (264 Hz), in our Fig. 5.1 (Experiment 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 2 3 4 5 6 7 8 9 10 11 12 13 Fig. 16.2. Same as Fig. 16.1; bass pitch (f 1 = 99 Hz, note G2). z 207

Part 3, Consonance Theories 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 stimuli. The thirds (M3 and m3) and the perfect fourth (P4) are predicted to be dissonant because they violate condition A (the Helmholtz condition): 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/b m.d. ranging from 0.5 to 2.0, where b = f h f d is the beat rate; the most disagreeable beat rate b m.d. at the considered average frequency f avg. = (f h + f d )/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 distinctly 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 violation of condition A (two pairs of partial tones generating disagreeable beats). The perfect prime (P1) and the perfect fifth (P5) have a low pleasantness 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. 208

16 A Modified Consonance Theory 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 corresponding matches. The differences between Figs. 5.4 (Kaestner) and Fig. 10.2 (Frosch) may be due not only to the above-mentioned frequency difference, but also to the difference between dull and unpleasant. Conclusions on the modified consonance theory: The consonance curves for two simultaneous bowed-string-like harmonic complex tones (sensory consonance versus fundamental-frequency ratio R = f 2 / f 1 ) 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 understandable 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 basilar membrane in the inner ear. 209

Reinhart Frosch Introduction to Cochlear Waves 2010, 448 Seiten, zahlreiche Grafiken, Format 15 x 21,5 cm, broschiert ISBN 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-mortem experiments of von Békésy, who found that tones generate, in the cochlear channel, travelling hydrodynamic surface waves which are similar to waves propagating 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" response peak generated by them; evanescent cochlear waves; high-frequency plateaux; cochlear maps; certain forms of tinnitus; otoacoustic emissions; frequency glides. vdf Hochschulverlag AG an der ETH Zürich, VOB D, Voltastrasse 24, 8092 Zürich Tel. +41 (0)44 632 42 42, Fax +41 (0)44 632 12 32, verlag@vdf.ethz.ch, www.vdf.ethz.ch

Reinhart Frosch Four-Tensors, the Mother Tongue of Classical Physics 2006, 300 Seiten, Format 12 x 18,3 cm, broschiert ISBN 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. vdf Hochschulverlag AG an der ETH Zürich, VOB D, Voltastrasse 24, 8092 Zürich Tel. +41 (0)44 632 42 42, Fax +41 (0)44 632 12 32, verlag@vdf.ethz.ch, www.vdf.ethz.ch