Science and Music – The Impact of Music

Music and Mathematics

Manfred R� s^cHroeder (Göttingen)

With 5 Figures

Abstract

In September 1962, in the presence of Jacqueline k^ennedy and Maestro b^ernstein, Philharmonic Hall at the new Lincoln Center for the Performing Arts in New York was ceremoniously inaugurated� But its much touted acoustics soon revealed severe limitations� There were disturbing echoes, the celli could not be heard during tutti passages and, most seriously, the listeners felt “detached” from the music emanating from the stage; there was no sensation of being “enveloped” by the music�

The author, then at nearby Bell Laboratories, was one of a committee of four acoustical experts charged by Lin­

coln Center with the analysis of these problems� Extensive testing revealed that the lack of low­frequencies (the

“celli problem”) could be attributed to an insufficient reflection of low-frequency energy by the overhead acoustic panels� The echoes were easily localized and removed by acoustic absorbers� But the main problem, the “lack of envelopment”, had to wait several years before its find clarification. In 1969, after having assumed the directorship of the Dritte Physikalische Institut of the University of Göttingen, the author petitioned the German Science Foundation (DFG) to support a large­scale investigation of concert hall acoustics, involving 22 halls in Europe and America� The main conclusion from this study was that good acoustics for music requires energetic sound waves striking a listen­

er’s head from the sides – something that most modern halls with low ceilings and wide seating areas could not pro­

vide� To remedy this defect, the author suggested the use of sound diffusing surfaces on the ceiling and side walls of concert halls based on number­theoretic principles, such as quadratic­residues� Quadratic­residues diffusers are now used world­wide, not only in concert halls, but also in recording studios, radio stations and even private homes�

Another field were mathematics has had a considerable impact, going in fact back to the Pythagoreans, are musical scales� New scales have been constructed based on the frequency interval 3:1 replacing the familiar octave interval 2:1� Number theory guarantees excellent harmonic properties if the 3:1 interval is subdivided into 13 equal steps�

Number theory has also yielded interesting recursive algorithms for the generation of simple melodies reminiscent of baroque music�

Zusammenfassung

Im September 1962 wurde die Philharmonic Hall des neuen Lincoln Center for the Performing Arts in New York im Beisein von Jacqueline k^ennedy und Maestro b^ernstein feierlich eröffnet� Aber die im Voraus viel gepriesene Akustik zeigte bald ernstliche Mängel� Es gab störende Echos� Die Celli waren während tutti­Passagen unhörbar, und, besonders gravierend, die Hörer hatten das Gefühl von der Musik auf dem Podium „abgekoppelt“ zu sein; es fehlte das Gefühl von der Musik „umgeben“ zu sein�

Der Autor, der damals bei den nahegelegenen Bell Laboratories arbeitete, wurde Mitglied eines Komitees von vier akustischen Experten, die vom Lincoln Center gebeten wurden, die Probleme der Philharmonic Hall zu analy­

sieren� Ausgiebige Messungen zeigten, daß der Mangel an tiefen Frequenzen (das „Celli­Problem“) auf die ungenü­

gende Reflektion von niedrig frequenter Energie durch die akustischen Panele an der Decke zurückgeführt werden konnte. Die Echos waren leicht zu lokalisieren; sie wurden durch akustische Absorber entfernt. Aber das Haupt­

problem, der „Mangel an Eingehülltsein“ mußte mehrere Jahre auf seine Lösung warten� Im Jahre 1969, nachdem der Autor die Leitung des Dritten Physikalischen Instituts der Universität Göttingen übernommen hatte, bat er die Deutsche Forschungsgemeinschaft (DFG), eine groß angelegte Studie der Konzertsaal­Akustik von 22 Sälen in Europa und Amerika zu unterstützen� Das Hauptergebnis dieser Studie war, daß eine gute Akustik für Musik ener­

giereiche seitlich einfallende Schallwellen am Kopf des Zuhörers erfordert – eine Bedingung, die in den meisten modernen Sälen mit niedriger Decke und großer Breite nicht erfüllt ist� Um diesem Manko zu begegnen, empfahl

Manfred R. Schroeder

10 Nova Acta Leopoldina NF 92, Nr� 341, 9 –15 (2005)

der Autor die Benutzung von schallsteuernden Oberflächen an der Decke und den Seitenwänden von Konzertsälen, die auf zahlentheoretischen Prinzipien, wie quadratischen Resten, basieren� Quadratische­Reste­Diffusoren werden heute weltweit angewendet, nicht nur in Konzertsälen, sondern auch in Aufnahme­Sudios, Rundfunksendern und selbst privat zuhause�

Ein anderes Gebiet, in dem die Mathematik erfolgreich angewendet wurde, ist die schon auf die Pythagoreaner zurückzuführende Konstruktion von Tonleitern, zum Beispiel solche, in denen das bekannte Frequenzverhältnis 2:1 (Oktave) durch ein Frequenzverhältnis 3:1 ersetzt wurde� Die Zahlentheorie garantiert exzellent harmonische Eigen­

schaften, wenn man das 3:1 Intervall in 13 gleichgroße Schritte einteilt�

Zahlentheoretische Prinzipien führen auch auf interessante rekursive Algorithmen zur Erzeugung einfacher Me­

lodien, die an Barockmusik erinnern�

This paper considers three of the many interactions between Music and Mathematics:

(i) Concert Hall Acoustics, (ii) New Musical Scales,

(iii) Algorithms for Generating Melodies�

1. Concert Hall Acoustics

In September 1962, in the presence of Jacqueline kennedy and Maestro bernstein, Philhar­

monic Hall at the new Lincoln Center for the Performing Arts in New York was ceremoni­

ously inaugurated�

Figure 1 shows Philharmonic Hall in its original configuration in 1962. Note the overhead hexagonal acoustic reflection panels.

But its much touted acoustics soon revealed severe limitations� There were disturbing echoes, the celli could not be heard during tutti passages and, most seriously, the listeners felt “detached” from the music emanating from the stage; there was no sensation of being

“enveloped” by the music�

The author, then at nearby Bell Laboratories, was one of a committee of four acoustical experts charged by Lincoln Center with the analysis of these problems� Extensive testing revealed that the lack of low­frequencies (the “celli problem”) could be attributed to an insuf­

ficient reflection of low-frequency energy by the overhead acoustic panels.

Figure 2 shows the acoustic energy along the middle aisle on the main floor. Note the dramatic attenuation of the low frequencies around 125 Hz compared to 750 Hz�

Before beginning the acoustic measurements, the author asked the ushers, students of the Juillard School of Music, whether there wasn’t a good location� Their answer was: yes, Seat A­15, which was therefore included in the measurements� And, indeed, instead of a 30­deci­

bel (dB) attenuation, there was only a 3­dB drop between 750 Hz and 125 Hz�

The echoes were easily localized and removed by acoustic absorbers� But the main prob­

lem, the “lack of envelopment”, had to wait several years before its find clarification.

In 1969, after having assumed the directorship of the Dritte Physikalische Institut of the University of Göttingen, the author petitioned the German Science Foundation (DFG) to support a large­scale investigation of concert hall acoustics, involving 22 halls in Europe and America� The main conclusion from this study was that good acoustics for music requires energetic sound waves striking a listener’s head from the sides – something that most modern halls with low ceilings and wide seating areas could not provide�

Fig. 1 Philharmonic Hall, New York, in its original state in 1962. Note the overhead acoustic reflection panels.

To remedy this defect, the author suggested the use of sound diffusing surfaces on the ceiling and side walls of concert halls based on number­theoretic principles, such as quadratic­resi­

dues�

Figure 3 shows a quadratic residue diffuser in cross­section based on the prime number p = 17� For p = 17, the quadratic residues are a_n = 0, 1, 4, 9, 16, 8, 2, 15, 13, 13, 15, ��� (Note that 5² = 25 which is congruent 8 modulo 17 and 6² = 36 which is congruent 2 modulo 17 etc�)�

The depth of the channels are proportional to these residues a_n creating local reflection coef­

ficients r_n = exp (2π i a_n/17)� It can be shown that the Fourier Transform of the (periodic, with period 17) r_n has constant magnitude resulting in a uniform diffusing pattern at 16 different frequencies, see Figure 4�

Quadratic­residue diffusers are now used world­wide, not only in concert halls, but also in recording studies, radio stations and even private homes�

Manfred R. Schroeder

12 Nova Acta Leopoldina NF 92, Nr� 341, 9 –15 (2005)

Fig. 2 Philharmonic Hall, New York: The received acoustic energy from the stage along the center of the main floor.

Note the steep drop for the low notes (125 Hz)�

Fig� 3 Quadratic­residue diffuser based on the prime number 17�

Nova Acta Leopoldina NF 92, Nr� 341, 9 –15 (2005) 13 2. New Musical Scales

Many musical scales are based on the octave, i�e� a frequency ratio 2 : 1� By subdividing the octave into 12 equal ratios or “semitones”, 2^1/12,the mostimportant intervals, can be well approximated. For example, the perfect fifth, having a frequency ratio of 3 : 2 = 1.5 is well represented by seven semitones: 2^7/12 = 1�498 instead of 1�5, where the difference is called

“Pythagorean comma”� The scales based on 2^1/12 are called “well­tempered” scales�

In the 1970s Heinz b^oHlen and John R� P^ierce introduced musical scales based on the frequency interval 3 : 1� They found that by subdividing the “tritave”, as he called it into 13 equal ratios, they could approximate several ratios of small integers very well� Thus, for ex­

ample, the ratio 5 : 3 = 1�666��� is well represented by 6 “tritave semitones”:

3^6/13 = 1�660 which is very close to ₃⁵ [1]

and, amazingly,

3^4/13 = 1�402 close to 5

7, [2]

with an error of less than 2 parts in a thousand� boHlen and Pierce proceeded to construct both mayor and minor musical scales based on the frequency ratio 3 : 1 in close analogy to the scales based on the octave (2 : 1)�

Fig. 4 Back scatter from quadratic-residue diffuser: The diffuse reflection from the diffuser shown in Fig. 3. Note the preponderance of desirable lateral reflections.

Manfred R. Schroeder

14 Nova Acta Leopoldina NF 92, Nr� 341, 9 –15 (2005)

What is the mathematical reason behind these uncanny coincidences? The answer comes from number theory, namely continued fractions, as Kees ^van P^rooijen, a Dutch music theo­

rist, first discovered. For example, if we desire to approximate the integer 5 by a power of 3, we have to consider the ratio

3

6 2 1 1 1

1 5 log

3 log

r

���

[3]

and approximate it by a rationed number:

4

19

13� [4]

Thus, 5 equals about 3^19/13 and similarly 7 is close to 3^23/13� As a consequence well­tempered tritave scales can be based on the “semitone” 3^1/13, which equals about 1�463 customary semi­

tones� Music composed using the tritave scales needs time to be appreciated�

3. Algorithms for Generating Melodies

Consider the integers 0, 1, 2, 3, 4, ��� written in binary notation 0, 1, 10, 11, 100, ��� and count the number of 1s: 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2 ���

Fig� 5 Every 63^rd term of the number­theoretic sequence 01121223122322334 derived from the binary representation of the integers� Note the “fractal” nature of the notes�

The result is a self-similar sequence because taking every second term reproduces the (infi­

nite) sequence: 0, 1, 1, 2, 1, ��� � Converting these numbers to a musical scale, say C major, with C = 6, D = 7, etc� results in an attractive self­similar melody� However, taking every 3^rd or 7^th or, say, every 63^rd term, see Figure 5, results in a distinctly “baroque” sounding tune�

Many musical pieces have been composed involving this and similar algorithms, see (and listen to !) www�reglos�de/musinum�

Further Reading

boHlen, H�: 13 Tonstufen in der Duodezime� Acustica Vol� 39, pp� 76–86� Stuttgart: S� Hirzel (1978) P^ierce, J. R.: The Science of Musical Sound. New York: Scientific American Library, W. H. Freeman 1983 scHroeder, M.: Fractals, Chaos, Power, Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman 1991 scHroeder, M� R�: Number Theory in Science and Communication� Berlin: Springer 1999 (Third Edition)

Prof� Dr� Manfred R� s^cHroeder Rieswartenweg 8

37077 Goettingen Germany

Phone: +49 551 21232