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STRUCTURAL PROPERTIES OF MUSICAL SCALES

Profile image of Norman CareyNorman CareyProfile image of David ClampittDavid Clampitt

Abstract
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This research explores the structural properties of musical scales through a mathematical lens, revealing connections between various significant music scales and particular mathematical structures. The analysis highlights the Pythagorean nature of these scales, including pentatonic, diatonic, and chromatic scales, by employing mathematical concepts like continued fractions and Diophantine equations. Additionally, the study addresses how the mind perceives musical scales and the intuitive sense of completeness associated with the diatonic scale, while proposing a framework that can be generalized to other tuning systems.

Figures (12)
Here the elements of P are arranged linearly, each element together with its corresponding note and pair of exponents: Similarly we can consider P in this two-dimensional setting, represented by the positive quadrant of the lattice Z x Z. This two-dimensional version of P was known in antiquity and is referred to as the Nicomachus Triangle. Of course, P can also be ordered in a linear sequence. The interplay between the linear (one-dimensional) sequence of P and the two-dimensional display of P provides the basis for much of what follows. The accompanying diagrams show P in a two-dimensional array and in a linear sequence. The reader will note that every pure power of 3 presents a new tone, while every pure power of 2 re-presents the origin Fp in different registers.
Here the elements of P are arranged linearly, each element together with its corresponding note and pair of exponents: Similarly we can consider P in this two-dimensional setting, represented by the positive quadrant of the lattice Z x Z. This two-dimensional version of P was known in antiquity and is referred to as the Nicomachus Triangle. Of course, P can also be ordered in a linear sequence. The interplay between the linear (one-dimensional) sequence of P and the two-dimensional display of P provides the basis for much of what follows. The accompanying diagrams show P in a two-dimensional array and in a linear sequence. The reader will note that every pure power of 3 presents a new tone, while every pure power of 2 re-presents the origin Fp in different registers.
2.4. Definition of the Region R(4 3). At this point the framework for the theory of well-formed Pythagorean scales is in lace. We proceed to offer a definition of the region Ria4,p), an operational definition in the sense that it gives a procedure for constructing all regions, making use of the set P and two properties, connectedness and complementarity.
2.4. Definition of the Region R(4 3). At this point the framework for the theory of well-formed Pythagorean scales is in lace. We proceed to offer a definition of the region Ria4,p), an operational definition in the sense that it gives a procedure for constructing all regions, making use of the set P and two properties, connectedness and complementarity.
Assuming 2“ and 3” are connected in P, the region R:,,p) is defined to be the largest non-empty connected set of complements in G,4,g). Table 2.4 gives the situation for the first four prime ratios: In the case (A, B) = (1,0), 2! = 2 and 3° = 1 are connected in P, but Ria,p) = S(4,B) = G(a,B) = 0. To avoid trivial complications later we will add the further restriction that R(4,g) be non-empty. As long as B > 1, Rya.p) # 0: Certainly s = 2938-1 and s = 24~13° are in S(4,,) as noted above. Then at least these elements (or this element, if A = B = 1) belong to R:,,g). If B > 1, then there are elements in G(4,g) that are not in S(4,g), hence elements in $4.) not in Ri 4,2). Rawpntns, OA and SP! ava anancsatad fh DP. tha san Bex xe a dabwad ta ha tha
Assuming 2“ and 3” are connected in P, the region R:,,p) is defined to be the largest non-empty connected set of complements in G,4,g). Table 2.4 gives the situation for the first four prime ratios: In the case (A, B) = (1,0), 2! = 2 and 3° = 1 are connected in P, but Ria,p) = S(4,B) = G(a,B) = 0. To avoid trivial complications later we will add the further restriction that R(4,g) be non-empty. As long as B > 1, Rya.p) # 0: Certainly s = 2938-1 and s = 24~13° are in S(4,,) as noted above. Then at least these elements (or this element, if A = B = 1) belong to R:,,g). If B > 1, then there are elements in G(4,g) that are not in S(4,g), hence elements in $4.) not in Ri 4,2). Rawpntns, OA and SP! ava anancsatad fh DP. tha san Bex xe a dabwad ta ha tha
Gis,3) consists of the 15 factors of 144 = 23° and contains the 11 elements of S(5,3) plus the four elements 144 (243*), 72 (2337), 48 (2431), and 36 (273°). The complements of these four are the first four elements of (5,3), 1 (= 2°3°), 2 (= 213°), 3 (= 2°3'), and 4 (= 273°). The remaining 7 elements form Ri5,3), the largest connected subset of complements in G(s 3). In order to show the musical significance of regions we develop the next two examples, the Structural Region and the Pentatonic Region.
Gis,3) consists of the 15 factors of 144 = 23° and contains the 11 elements of S(5,3) plus the four elements 144 (243*), 72 (2337), 48 (2431), and 36 (273°). The complements of these four are the first four elements of (5,3), 1 (= 2°3°), 2 (= 213°), 3 (= 2°3'), and 4 (= 273°). The remaining 7 elements form Ri5,3), the largest connected subset of complements in G(s 3). In order to show the musical significance of regions we develop the next two examples, the Structural Region and the Pentatonic Region.
This region embraces the key proportion of Greek musical theory 6: 8 :: 9: 12 taken over two octaves. This proportion has the property that 8 and 9 are the har- monic and arithmetic means, respectively, between 6 and 12. Musically, of course, this proportion corresponds to the structure, Tonic, Sub-Dominant, Dominant, and Tonic, which forms the framework of fixed tones for many scales throughout the world. All of this leads us to call Ris5.3) the Structural Region and the elements of Ris5,3) the structurals.
This region embraces the key proportion of Greek musical theory 6: 8 :: 9: 12 taken over two octaves. This proportion has the property that 8 and 9 are the har- monic and arithmetic means, respectively, between 6 and 12. Musically, of course, this proportion corresponds to the structure, Tonic, Sub-Dominant, Dominant, and Tonic, which forms the framework of fixed tones for many scales throughout the world. All of this leads us to call Ris5.3) the Structural Region and the elements of Ris5,3) the structurals.
A finite continued fraction is a number of the form 10G.D. Olds Continued Fractions Random House 1963; Hardy and Wright An Introduction to the Theory of Numbers Oxford University Press 1979. 3.2. Continued Fractions. The region R,,4,) is defined in terms of the connected pair of elements 24 and 3%. So far the only information we have about these numbers is that A and B must be relatively prime. We do not know anything about the ordered pairs as such, how to compute them, even whether or not the number of such pairs is infinite. The branch of mathematics which provides the tools for answering these questions is the branch of number theory referred to as the theory of continued fractions. Therefore, we will give a brief treatment of continued fractions, referring the reader unfamiliar with the topic to introductory texts in number theory for proofs and more detailed discussion.1° Y an, an oF 7PfP Y a 7 re 7
A finite continued fraction is a number of the form 10G.D. Olds Continued Fractions Random House 1963; Hardy and Wright An Introduction to the Theory of Numbers Oxford University Press 1979. 3.2. Continued Fractions. The region R,,4,) is defined in terms of the connected pair of elements 24 and 3%. So far the only information we have about these numbers is that A and B must be relatively prime. We do not know anything about the ordered pairs as such, how to compute them, even whether or not the number of such pairs is infinite. The branch of mathematics which provides the tools for answering these questions is the branch of number theory referred to as the theory of continued fractions. Therefore, we will give a brief treatment of continued fractions, referring the reader unfamiliar with the topic to introductory texts in number theory for proofs and more detailed discussion.1° Y an, an oF 7PfP Y a 7 re 7
The continued fraction algorithm illustrated above may be applied to an irra- tional number, but in this case the process continues indefinitely, generating an infinite sequence of integers. In fact, though again we do not prove it, the con- tinued fraction algorithm sets up a one-to-one correspondence between the set of all irrationals and the set of all infinite sequences of integers, to, t1,...t,... where t1,te,... are positive. We do show that any infinite sequence of integers to,t1,...t,... with t1,to,... positive defines an infinite continued fraction [to;t1,...]: If cx is defined as before. the question is whether cy, approaches a limit as k increases without bound, and we appeal to the fundamental theorem of analysis that a bounded, monotonically increasing or decreasing sequences converges to a limit. Theorems 1 through 5 may be applied to the cz, so we immediately have the result that the even convergents are strictly increasing and the odd convergents are strictly decreasing. Moreover,
The continued fraction algorithm illustrated above may be applied to an irra- tional number, but in this case the process continues indefinitely, generating an infinite sequence of integers. In fact, though again we do not prove it, the con- tinued fraction algorithm sets up a one-to-one correspondence between the set of all irrationals and the set of all infinite sequences of integers, to, t1,...t,... where t1,te,... are positive. We do show that any infinite sequence of integers to,t1,...t,... with t1,to,... positive defines an infinite continued fraction [to;t1,...]: If cx is defined as before. the question is whether cy, approaches a limit as k increases without bound, and we appeal to the fundamental theorem of analysis that a bounded, monotonically increasing or decreasing sequences converges to a limit. Theorems 1 through 5 may be applied to the cz, so we immediately have the result that the even convergents are strictly increasing and the odd convergents are strictly decreasing. Moreover,
Continuing in this way it is possible to compute the continued fraction expansion of logs 3 = [1;1,1, 2, 2,3,1,5, 2, 23, 2,...].
Continuing in this way it is possible to compute the continued fraction expansion of logs 3 = [1;1,1, 2, 2,3,1,5, 2, 23, 2,...].
Table 3.2.4 points to the importance of the semi-convergents to logs 3. We will show that the correspondence conjectured above holds true, but to do this we will need one more result to characterize semi-convergents in general. This motivates a closer study of Continued Fraction Theorem 8, to reinforce this result and to attempt to prove a converse theorem. onvergents “” for even. After each full convergent the value of t, is written.’
Table 3.2.4 points to the importance of the semi-convergents to logs 3. We will show that the correspondence conjectured above holds true, but to do this we will need one more result to characterize semi-convergents in general. This motivates a closer study of Continued Fraction Theorem 8, to reinforce this result and to attempt to prove a converse theorem. onvergents “” for even. After each full convergent the value of t, is written.’
From a musical point of view the most interesting family is the family whose final region is the Diatonic Region R(11,7). The interval which relates these regions is the whole step, 37/2°. The Structural, Pentatonic, and Diatonic Regions all contain whole steps, which fill in the initial octave of the Octave Region. This family is depicted below: That the diatonic region R(11,7), is the final member of this family is related to the coincidence up unti the convergents to the are [1] = +, [1; 1] = 2 and the associated wel to the golden number. 11/7 of the convergents and semi-convergents of log, 3 with golden number, ¢. Since the continued fraction for log, 3 begins [1;1,1,2,...], the first five elements in the sequence of (semi-)convergents 1; 1,1) = 3,151, 1,1] = §, and [1;1,1,2] = [1;1,1,1,1) = 8, coinciding with the first five convergents to 6 = [1;1,1,1,1,...].. The Diatonic Region is then the last region where both of the intervals which make up the region -formed Pythagorean scale are determined by convergents
From a musical point of view the most interesting family is the family whose final region is the Diatonic Region R(11,7). The interval which relates these regions is the whole step, 37/2°. The Structural, Pentatonic, and Diatonic Regions all contain whole steps, which fill in the initial octave of the Octave Region. This family is depicted below: That the diatonic region R(11,7), is the final member of this family is related to the coincidence up unti the convergents to the are [1] = +, [1; 1] = 2 and the associated wel to the golden number. 11/7 of the convergents and semi-convergents of log, 3 with golden number, ¢. Since the continued fraction for log, 3 begins [1;1,1,2,...], the first five elements in the sequence of (semi-)convergents 1; 1,1) = 3,151, 1,1] = §, and [1;1,1,2] = [1;1,1,1,1) = 8, coinciding with the first five convergents to 6 = [1;1,1,1,1,...].. The Diatonic Region is then the last region where both of the intervals which make up the region -formed Pythagorean scale are determined by convergents

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