Jay KappraffNew Jersey Institute of Technology Department of Mathematics University Heights Newark, New Jersey 07102 USA To
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Introduction to Arithmetic
[1]. This book is one of the only surviving documentations of
Greek number theory. Little is known about the life of Nicomachus,
and the period of his life can only be estimated to lie between
the middle of the first century and the middle of the second
century AD, making him contemporary with Theon of Smyrna and
Ptolemy. I will discuss a pair of tables of integers found in
the Arithmetic and show how they lead to a general theory
of proportion. These include the system of musical proportions
developed by the neo-Platonic Renaissance architects Leon Battista
Alberti and Andrea Palladio, the Roman system of proportions
described by Theon of Smyrna, and the Modulor of Le Corbusier.
My interest in this article is to show how all of these systems
are derived naturally from the Nicomachus tables. Although the
architectural details have been discussed elsewhere, I will describe
the Roman system of proportions in some depth.This essay should not be looked at as an historical study.
M.L. D'Ooge [1] has admirably presented Nicomachus's treatise
in an historical context, and I have nothing to add. Rather I
shall take a single theme found in the
These tables follow a kind of Pascal's law in which each number is the sum of the two above it. Note that in Table 1 each column ends when the integer is no longer divisible by 2, while the second table ends when integers are no longer divisible by 3. The motif of each table is expressed by the triangle of numbers shown to the left. Successive elements in the rows of Table 1 (Table 2) are in the ratio of 1:2 (1:3) while rows successive elements in each column are in the ratio of 2:3 (3:4). Also successive numbers on right leaning diagonals are in the ratio 4:3 (9:4). Tables 1 and 2 are generalized to any real number x as shown in Table 3. Again each column ends when an element is not divisible by
x. The terms of the geometric series on the diagonal are seen
to be (1+x) x:1 ® 1:(1+x) and
x:(1+x)Nicomachus actually thought of this relationship as a transformation and says in Chapter XXIII of Book I,
( He then goes on to cite another example: From 16, 28, 49 (ratio 4:7) comes either 16, 44, 121 (ratio 4:11) or 49,77,121 (ratio 7:11). Apart from the archaic language, this transformation appears to relate to the Farey sequence shown in Table 4. Farey sequences are, in turn, related to representations of rational numbers as continued fractions with applications to dynamical systems [3,4]. They also have applications to musical tuning systems [5,6]. The numerator (N) and denominator (D) of any rational number in the Farey sequence is gotten by adding the numerators and denominators of the two terms that brace it from the rows above it, e.g., 3/5 is braced by ½ and 2/3 so 3/5 = (1+2)/(2+3). In the Farey sequence each rational number x = N/D in one row gives rise to two rationals, 1: (1+x) = D/(N+D) and x:(1+x) = N/(N+D) in the next row, e.g., 2:5 ® 2:7 and 5:7. In this sense, Nicomachus indicates an early awareness of the concepts underlying the Farey sequence which has become an important tool of modern mathematics and music.
Table 5 nicely reflects the three means that pervade mathematics, the geometric, arithmetic, and harmonic means. Given the sequence: a c b, the arithmetic mean is defined as: b-c = c-a or c = (a+b)/2 the harmonic mean is defined as : (b-c)/b = (c-a)/a or c = 2ab/(a+b). Notice in Table 5 that each element is the geometric mean of the elements to the left and right of it; an element of a given row is the arithmetic mean of the two elements that brace it from above; while a given element fits between the two numbers in the row below as the harmonic mean. What about the mean relations expressed by Table 6? Again
each row is a geometric progression and again each element of
a given row fits between the two numbers in the row above and
below. However, an element of a given row is now a kind of generalized
arithmetic mean defined by c=(a+b)/3 while a given element fits
between two numbers in the row below it as a generalized harmonic
mean c = 3ab/(a+b). These mean relations can be generalized to
Table 3 where now the generalized arithmetic mean c
If b = ax and x ³
f, then
a £ c_{a} < b and
a < c_{h} £ b,where c_{a} = (a + b)/x and c_{h}
= xab/(a + b).For the elements in Table 2, we can define another table in which each element is the common (not generalized) arithmetic mean of the two elements above it while each element is the common harmonic mean of the two elements below it as shown in Table 7. A similar table for the geometric series based on 1:x can be created, although the table for x=2 is, of course, identical with the original and the table for x = f collapses into a single row with the other rows repetitions of this row. There is one noteworthy relation tying the common and generalized means for a given x together. Place the corresponding arithmetic and harmonic means, both common and generalized, between any pair of elements from Table 1 and 2, e.g., the sequence of common and generalized means between 6 and 12 for Table 1 and 12 and 36 for Table 2 are: The ratio of successive integers in this sequence are written below, and the product of these ratios equals 1/2 for the 1:2 sequence and 1/3 for the 1:3 sequence. In a similar manner, it can be shown that the product of the ratios for a 1:x sequence is 1/x.
Table 5 was the basis of the system of musical proportions
used by Leon Battista Alberti in Renaissance Italy [10],[11].
The ratio of numbers in the rows of Table 5 are in proportion
1:2 which represents the musical octave. Depressing the string
of a monochord at its midpoint and sounding it with a bow yields
a tone one octave above the fundamental tone characteristic of
the whole string (see A "hexagon" of numbers was chosen within Table 5 (that is, numbers chosen so that they appear to form the vertices of a hexagon, such as 8-16-24-18-9-6, with 12 in the middle) in which adjacent numbers represented the ratio of length, width, and heights of architectural spaces such as walls, ceilings, doors, etc. That this system of proportions exhibits harmony is illustrated by taking a pair of integers from a given row, e.g., 6 and 12 and interspersing it with its arithmetic and harmonic means. Notice that the arithmetic and harmonic means insure the repetition of the ratios 2:3 and 3:4 to which they correspond . Also 2/3 x 3/4 = ½, i.e., octave is obtained by summing the intervals of a fifth and a fourth. Similar harmonic relationships hold for Tables 6 and 7. For example, place the generalized arithmetic and harmonic means between 12 and 36 in Table 6 and the common arithmetic and harmonic means between 18 and 54 in Table 7.
Beginning with the 1: q geometric series on the central row of this table, each element below the center is the generalized arithmetic mean of the pair above it ; each element above the center is the ordinary arithmetic mean of the pair below it, e.g., (q + qwhile^{2})/q = qÖ2
(q + q^{2})/2 = q^{2}/Ö2.Also, each element below the center is the generalized harmonic mean of the pair below it; each element above the center is the ordinary harmonic mean of the pair above it, e.g., q(q)(2q)/(2+2q ) = qÖ2 while 2(q/Ö2 )(q.^{2}/Ö2 )/( q/Ö2+q^{2}/Ö2)
= qThe algebra to carry out these operations can be seen more clearly by comparing the q-sequence with the discrete version of this sequence known as Pell's sequence. Both of these triples of sequences have the Pell's sequence
property : a i) Each Pell sequence has the defining property,
ii,iii ) Other additive properties are,
iv) Property 4: Any element is the double of the element two
rows above it. The algebraic properties of the Roman system can be made palpable
by considering the equivalent geometric properties based on the
interrelation of the proportions: 1, Ö2,
and q. These
relationships are shown in The architectural system based on these proportions has many
additive properties in which a small number of modules are able
to fit together in countless ways. 3
into nine subrectangles with lengths and widths from Table 8
satisfying,^{}
q ^{3} = qÖ2
+ 2q + q + qÖ2It is a good exercise to verify Equations 2 using Properties
1-4 of the Roman system. Because of the additive properties,
these rectangles can be rearranged in many ways to tile the same
rectangle. This versatility is illustrated in the design in All the tones of the Pythagorean scale are based on the ratio of string lengths in which the numerator and denominator are divisible by primes 2 and 3, e.g., the musical fifth, fourth, and octave: 2/3, ¾, and ½. The SR rectangle is particularly suited for creating the musical proportions of the Pythagorean scale, since it has a natural bisecting and trisecting property. By the construction of Figure 1 an oblique line drawn to the midpoint of the side of a rectangle cuts the diagonal of at the trisection point and the bisection of the SR rectangle divides it into two SR rectangles according to Figure 6c. Kim Williams has shown that the Medici Chapel was created with numerous SR rectangles enabling its design to be embedded with many musical ratios [11,14].
where f
Each sequence is a geometric series with ratio 1: f. Each element of one sequence fills the gap between two elements of the other sequence. Just as for Table 5 based on the ratio 1:2, each element of the Red sequence is the arithmetic mean of the pair of elements of the Blue series that brace it, while each element of the Blue series is the harmonic mean of the pair of elements from the Red series that brace it. Since the sum of a pair of integers from either the Red or Blue sequence results in another integer in that same sequence, only these two sequences are necessary to assure that the sequences have additive properties unlike the Roman system which requires an infinity of sequences. This system also has many additive properties, which enables
a rectangle to be tiled in countless ways by a small repertoire
of modules.
Let c _{a} = a(1+1/x) = b(1/x +1/x^{2}).
(A1)But, ^{2} = 1 where f = (1+Ö5)/2. (A2)Therefore it follows from Equations A1 and A2 that, _{a} £ b for
x ³
fLet c _{h} = a[1/(1/x+1/x^{2})]
= b[1/(1+1/x)]. (A3)It follows from Equations A2 and A3 that, _{h} < b for x ³ f
2. Nicomachus. 3. Anatole Beck, Michael Bleicher and Donald
Crowe. 4. Jay Kappraff. 5. R.A. Rasch. "Farey systems of musical
intonation." 6. Jay Kappraff. "The structure of Ancient Musical Scales with a Modern Twist." In preparation. 7. Tons Brunés. 8. Jay Kappraff. "A Secret of Ancient
Geometry." 9. J. Kapusta. Private communication 10. Rudolph Wittkower. 11. Jay Kappraff. "Musical Proportions
at the Basis of Systems of Architectural Proportion both Ancient
and Modern." 12. Theon of Smyrna. 13. Donald J. and Carol Martin Watts. "The
Garden Houses of Ostia." 14. Kim Williams. "Michelangelo's Medici
Chapel: The Cube, the Square, and the Root-2 Rectangle."
15. Jay Kappraff. 16. Kappraff, Jay. "Linking the Musical
Proportions of Renaissance, the Modulor, and Roman Systems of
Proportions." 17. Le Corbusier. 18. Joseph Rykwert, Neil Leach, Robert Tavernor.
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