Sitzungsber. Abt. II (2005) 214: 43–79 Mathematical-Physical Properties of Musical Tone Systems By Bruno J. Gruber (Vorgelegt in der Sitzung der math.-nat. Klasse am 23. Juni 2005 durch das k. M. Peter Steinhauser) Abstract In this article the fundamental properties of the physical-mathematical laws are studied upon which music is based. Thus this article is not a study of music as such, but a study of the properties of the component elements which are the ‘‘building blocks’’ of music in the hands of the composers. These ‘‘building blocks’’ are the musical sound frequencies – the musical tones. It is the intrinsic physical- mathematical structure of the musical tones, and the interrelationships among these tones, which is studied. The physical-mathematical properties of the tones, together with the physical-mathematical laws governing the relationships between the tones, that forms the musical tone system which is discussed in this article. The fundamental properties of musical tones (also referred to as intrinsic internal musical properties) rest upon the laws of physics and mathematics. As such, they are independent of cultural priorities regarding the perception of beauty in musical tones, even though these fundamental properties form the basis upon which the various musical systems are built. The cultural priorities find their way into music via the composers, the musicians and the public in general. The arguments used in this article are based mostly upon physics (acoustics) and mathematics. The language used rests upon the concept of symmetries and leads to a geometrical interpretation of the basic internal properties of music in the form of a (mathematically scaled) lattice in a three-dimensional space. Each dimension rep- resents a ‘‘rescaled octave system’’, based upon the three prime numbers 2, 3, and 5. The (scaled) lattice points represent the possible musical tones – with respect to some reference tone. The various musical scales are represented by subsets of lattice points satisfying certain conditions. This geometrical picture leads to a convenient and easy interpretation of the properties of the individual musical scales, and moreover, serves 44 B. J. Gruber to illustrate clearly the relationships between the various musical scales. As examples, early western and modern western musical scales are treated in some detail. And, for the purpose of comparison, two eastern musical scales (Japanese Noh-scales) are discussed, in addition. A map of the musical lattice tones from the 3-dimensional geometrical lattice space upon the line of musical frequencies leads back to the standard theory of musical tones. A careful distinction needs to be made between the concept of a mathematical scale transformation, as it pertains to lattice points and musical ratios, and the musical scales (staff) which are used to write down musical scores (notes). 1. Introduction This introduction serves as a summary of both standard and well es- tablished properties of acoustics, as well as a summary of the new results which are derived in this articleÀ basedÁ upon those properties [1]. 2 The frequency ratios of the 1 -octave system can be expressed in the form n; ; ¼ 2n 1 þ ; lin 2 0 2 0 0 1; n ¼ 0; Æ1; Æ2; ... 2 linð0; 0; 0Þ¼0; linðn; 1; 0Þ¼linðn þ 1; 0; 0Þð1:1Þ with 0 an arbitrarily chosen sound frequency (within the range of frequencies of human hearing). Eq. (1.1) states that the range of frequencies lin can be segmented into a set of musically equivalent intervals, called octaves, distinguished by pitch ð2nÞ only. The entire set, Eq. (1.1) will be called an octave system (Fig. 1). For a given value of n, the frequency interval 2n 1 þ ; 0 1; ð1:2Þ 2 0 2 will be called the n-th octave,withn ¼ 0 the basic octave. The reason for this segmentation of the line of frequencies into (musically Fig. 1. The standard 2-based octave system with respect to the reference tone 0 Mathematical-Physical Properties of Musical Tone Systems 45 equivalent) octaves results from the fact that the human mind perceives the ratio of two simultaneously sounded frequencies and 0 ¼ 2 ð1:3Þ 0 as pleasing, i.e. as a consonance. The arbitrarily chosen frequency 0 will be called the reference frequency for the octave system. The parameter =2 has been chosen in this form since 0 =2 1 represents a cycle (Sect.À 3).Á 2 An exponential form for the frequency ratios of the 1 -octave system is given by nþ n; ; ¼ 2 2 ; 0 1; ð1:4Þ exp 2 0 0 2 with 0; ; ¼ 22 ; 0 1; ð1:5Þ exp 2 0 0 2 the basic octave. Due to the difference in the two laws for lin and exp, the frequencies lin and exp differ for =2 ¼ =2 for all parameter values except for the values ¼ ¼ 0; 1 ð1:6Þ 2 2 for which the two frequencies agree, Fig. 2. In fact, the parameter values are related to each other by ¼ log 1 þ : ð1:7Þ 2 2 2 That is, for a given frequency lin ¼ exp the values of the two param- eters =2, =2 disagree, or for the same parameter value =2 ¼ =2 two different frequencies lin 6¼ exp are obtained, except for =2 ¼ =2 ¼ 0; 1. The octave property is thus represented by n ¼ linðn; 0; 0Þ¼expðn; 0; 0Þ¼2 0; ð1:8Þ i.e. the frequencies for which the two frequencies agree (the conso- nances of the octave system based upon the reference frequency 0). The parameterÀ Á values =2 ¼ =2 ¼ 0; 1 characterize the octave 2 tones of the 1 -octave system – namely the initial points and the endpoints of each octave on the line of frequency ratios, Eq. (1.8) and 46 B. J. Gruber Fig. 1. These octave tones can be looked upon as (scaled) lattice points on the line of frequency ratios. Eq. (1.7) is a relationship of musical frequency ratios (defining musical tones) to the intervals (distances) between musical tones. That is, for two tones given by 1=0 and 2=0, the interval between the two tones is defined as 1 2 1 2 1 À ¼ log2 À log2 ¼ log2 : 2 2 0 0 2 Thus, the intervals for the octave tones =0 of Eq. (1.8) with respect to are given by 0 2 ¼ log2 ¼ n log2 ¼ n: 2 0 1 This shows that in the lattice picture of musical tones, Fig. 1, the interval between neighboring lattice points is equal to 1 (i.e., equal to 1,200À cents),Á while the lattice point n corresponds to the frequency 2 n ratio 1 . À Á 2 It will be shown that, in addition to the 1 -octave lattice system, two ‘‘rescaled octave systems’’ can be introduced which are based 3 5 upon the frequency ratios 2 and 3. These ‘‘rescaled octave systems’’ are given as 3 m 5 r ¼ and ¼ : 0 2 0 3 The product of these three octave systems constitutes the musical tones which are discussed in this article, namely the frequency ratios 2 n 3 m 5 r ¼ ðn; m; rÞ; ð1:9Þ 0 1 2 3 n; m; r integers. Eq. (1.9) can also be interpreted as representing musical lattice points in a (mathematical) 3-dimensionalÀ Á space. 3 The interval distances for the ‘‘rescaled 2 -octave system’’ are given by 3 m 3 ¼ log ¼ m log 2 2 2 2 2 À Á and for the ‘‘rescaled 5 -octave system’’ by 3 5 r 5 ¼ log ¼ r log : 2 2 3 2 3 Mathematical-Physical Properties of Musical Tone Systems 47 The familiar logarithmic formula for the intervals between musical tones of the type given by Eq. (1.9) then becomes 3 5 ¼ log2 ¼ n þ m log2 þ r log2 ; ð1:10Þ 2 0 2 3 that is a linear expression in terms ofÀ threeÁ discreteÀ Á parametersÀ nÁ; m; r. 2 3 5 The coefficients of Eq. (1.10), log2 1 , log2 2 and logÀ2 Á3 , rep- 2 resent the intervalÀ distancesÁ À alongÁ the three lines of the 1 -octave, 3 5 and the rescaled 2 - and 3 -octaves. See Figs. 8, 9, 13 and 15. For the lattice points =2 ¼ 0 holds and thus the corresponding frequencies are well defined and can be compared, ðn; 0; Þ 0 0 2nÀn : 1:11 0 ¼ ð Þ ðn ; 0; 0Þ Since the reference tone 0 can be arbitrarily chosen it is possible to associate an octave system with any musical tone, in particular also with the tones of a musical system, like the musical system of the ancient Greek Lyre, the diatonic Pythagorean musical system, etc. Considering for the moment the natural diatonic musical system, each of the tones (the German notation for the tones is used; see Sect. 6) cd e f g a hc1 0 9 5 4 3 5 15 0 ¼ 10 ð8Þ0 ð4Þ0 ð3Þ0 ð2Þ0 ð3Þ0 ð 8 Þ0 20 ð1:12Þ serves as a reference tone for its own octave system 0 ðn; 0; 0 ¼ cÞ n n ¼ 2 ;n; ; 0 ¼ c ¼ 2 1 þ ð0 ¼ cÞ; ð0 ¼ cÞ 2 2 ðn; 0; 0 ¼ dÞ n n ¼ 2 ;n; ; 0 ¼ d ¼ 2 1 þ ð0 ¼ dÞ; ð0 ¼ dÞ 2 2 ðn; 0; 0 ¼ eÞ n n ¼ 2 ;n; ; 0 ¼ e ¼ 2 1 þ ð0 ¼ eÞ; ð0 ¼ eÞ 2 2 . ð1:13Þ n with all the tones 2 0 properly defined ð=2 ¼ 0Þ, i.e. consonances. It holds then ðn; 0; 0 Þ 0 0 ¼ 0 : ð1:14Þ ðn; 0; 0Þ 0 0 Thus, the ratio of the 0 octave system with respect to the 0 octave system exists and is given by the ratio of the reference frequencies.