enumeration of synthetic musical scales by matrix algebra

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Enumeration of Synthetic Musical Scales by Matrix Algebra and a Catalogue of Busoni Scales Author(s): Robert M. Mason Source: Journal of Music Theory, Vol. 14, No. 1 (Spring, 1970), pp. 92-126Published by: on behalf of the Duke University Press Yale University Department of MusicStable URL: http://www.jstor.org/stable/843038Accessed: 21-12-2015 07:52 UTC

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92

Enumeration of

Synthetic Musical Scales

by Matrix Algebra

I have made an attempt to exhaust the possibilities of the arrangement of degrees within the seven-tone scale; and succeeded, by raising and lowering the intervals, in estab- lishing one hundred and thirteen different scales. These 113 scales (within the octave C-C) comprise the greater part of our familiar twenty-four keys, and, furthermore, a series of new keys of peculiar character. But with these the mine is not exhausted, for we are at liberty to trans- pose each of these 113, besides the blending of two such keys in harmony and melody. *1

Ferruccio Busoni (1866-1924)


93

and a Catalogue

of Busoni Scales

ROBERT M. MASON

Busoni's Problem

In 1966 the musical world celebrated the 100th anniversary of the birth of Ferruccio Busoni. Although he is better known, of course, as a composer and performing artist whose creative life spanned the transitional period between the romantic and modern styles, Busoni is recognized also for his pioneering contributions to musical theory. As a theoretician he must be given credit for posing a sparkling little problem in applied combinatorial analysis - a problem of musical scale synthesis that later intrigued J. Murray Barbour. *2


94

As almost everyone knows, the seven letters from A to G, ar- ranged in the order C, D, E, F, G, A, B, denote that sequence of tones and intervals referred to as the "one-octave, ascending scale of C major", or commonly, the "C major scale". Busoni's proposal for forming synthetic musical scales by sharping or flatting tones of the C major scale was studied by Barbour in his 1929 paper. Since a sharp (#) or a flat (6) is a musical sign telling the performer to raise or lower a tone one halfstep in pitch, indiscriminate applications of sharps and flats to the C major scale might, in the equitempered system, lead to tonal duplication (exemplified by G#Ab) or to overlapping (exemplified by ElFL), which are prohibited. Barbour circum- vented both of these complications by rephrasing the conditions of Busoni's problem in terms of a particular musical instru- ment - the harp - and the way in which its natural scale can be modified by pedal settings.

Usually the note C is marked as the starting point or entrance to the scale of C major. If some other letter is chosen instead to signify the "keynote", and the remaining letters are rotated, the resulting cyclic rearrangement of the notes of the C major scale is termed a "modal variant" of the C major scale. Having decided to exclude modal variants, Barbour counted the scale- tunings of the harp subject to this criterion by determing first the number of two-tone fragments, then the number of three- tone fragments, and so on, until he reached complete scales. He found that Busoni's count of synthetic musical scales starting on C is somewhat less than the true number (155) of such scales formed in accordance with Busoni's declared method.

In his 1949 paper, which was presented before the Acoustical Society of America, Barbour elaborated on his earlier results. He observed that "not all [sixty-six] heptatonic scales can be notated with seven letter names unless double sharps and flats are used. "

Although retaining the general pattern of Barbour's approach, the present study systematizes the enumeration by using the algebra of matrices, a method well adapted to digital compu- ters. By employing standard mathematical procedures as a framework within which to carry out the required enumeration, rather than the less-familiar harp tunings used by Barbour, the present construction readily permits an extension of the solution to five inflections (x, #, , 6, 6 , ). An examination of the catalog of Busoni scales (see Table 1) accompanying this enumeration demonstrates the correctness of Barbour's observa- tions. Yet, such an examination reveals that six inflections


95

are needed in order to write down all sixty-six heptatonic scales in every key.

Moreover, putting the matter in terms of musical intervals and their relationships avoids the introduction of Barbour's con- cepts of duplication and overlapping. It also demonstrates that an enharmonic mapping of the literal alphabets onto the twelve tones of the equitempered system is not essential to solving Busoni's problem. Until an enharmonic mapping is made in the concluding section, observe that Fb is not equal to E, and Cx (C#a) is not equal to D. This is true for the simple reason that no assumption has been made to the contrary.

Synthetic Musical Scales

It will be helpful in the beginning if certain terminology, ex- plained in detail in a recent paper by the present author *3, is stated once more in brief summary. Thus, a "musical interval" is made up of two tones whose corresponding frequencies are recognizable as an accepted musical interval relationship. Such an interval relationship always can be expressed by the traditional terms: "perfect, major, minor, augmented, diminished, first, second, third, and so on." This provides as- surance that no matter which two notes are chosen from the musical staff, the resulting interval will be a musical interval. To give several extreme examples, D#F### forms an augmented third, C6E6b6 forms a diminished third, and FbbBx forms a quintuply augmented fourth.

By definition adopted here, an interval is "disconnected" and the associated interval relationship is "partitioned" by inserting one or more note names in between the endnotes. For example, the interval CC, whose endnotes are in an octave-interval relationship, can be disconnected in many ways. One familiar way would be to split it into seven smaller intervals: CD, DE, EF, FG, GA, AB, BC. So, the C major scale CDEFGABC may be considered to arise from disconnection and to correspond to a partition of the octave into seven interval parts.

The simple idea of octave partitioning suggests a very general concept, to be called a "synthetic musical scale". A "synthetic musical scale" is a partition of the octave into smaller musical interval relationships. From this general point of view, the final interval of any given synthetic musical scale closes a cir- cuit to bring the melodic path back to the keynote, and if the octave CC momentarily is considered to be indistinguishable from the unison CC, the melodic path may be said to return to


95 its starting point. To illustrate, various musical intervals can be connected to form a chain beginning with C, as follows:

CD + DE + EF + FG + GA + AB + "BC" = CC. (1) The interval "BC" is enclosed in quotation marks to indicate that it is the final interval required to complete the synthetic musical scale. In 1929 Barbour measured these intervals in semitones, which is equivalent to restating equation 1 as:

2+2+ 1 +2+2+2+ 1 = 12. (2) Thus, equation 2 must be interpreted from the standpoint of equitemperament.

For present purposes, it is better to re-express equation 2 as:

2 + 2 + (-5)+2 + 2 + 2 + (-5) = 0, (3) in order not to introduce equitemperament. The zero on the right-hand side of equation 3 now represents the octave CC in the same fashion as a zero residue represents an integer mul- tiple of the modulus in an arithmetic congruence. Now some of the integers in equation 3 bear algebraic signs. This results from the fact that equation 3 refers to a series of tones in Pyth- agorean temperament and is therefore not the same as equation 2 in equitemperament.

Recalling that the Pythagorean system is based upon the octave and the fifth using the ratios 2:1 for the octave and 3:2 for the fifth, it is possible to express the frequency ratio f(T) of an arbitrary Pythagorean tone T as the number f(T) = 2s(3/2)t:1= 2s(3/2)t, where s and t are suitably chosen integers and the basic reference frequency is unity. The set of all tones bearing the same note name T has the set of associated frequencies:

....

2-2(3/2)t, 2-1(3/2)t, (3/2)t, 21(3/2)t , 22(3/2)t ..... This set forms an equivalence class with respect to the relation

of octave congruence, so that the whole class can be represented by a single member, which for convenience is takento be (3/2)t. In other words, the appropriate variable factor 2s is suppressed while the tones with the same value of t are lumped together. Hence, the frequency ratio of a given interval XY can be written as:

R = f(Y)/f(X) = (3/2)m:(3/2)n = (3/2)m-n:1,


97

where m and n are suitably chosen integers. The numbers that appear in equation 3, so-called "I-values" (see Reference 3), are defined to be the logarithm on the base 3/2 of this ratio, that is to say, I = m - n.

Although it has just been demonstrated that equation 3 is deeply rooted in the Pythagorean system, the physical significance attached to I-values is not vital, and in fact is restrictive to the present outlook. Rather, consider these values as numerical labels that have been attached to the various names of musical interval relationships without regard to temperament, simply to help mechanize the enumeration process. Then, the door remains open for assigning any physical interpretation to the resulting scale notations that will be useful, without mental encumbrance. Accordingly, enlarging upon ideas found in the first paragraph of Section 5 of Reference 2a, "Busoni scale" may be defined independently not only of the C major scale but also of both the Pythagorean and equitempered systems as being any synthetic musical scale of eight degrees that comprises a chain of undiminished seconds. Of course the synthetic scale restriction implicitly assumes a 2:1 octave ratio still. Since the undiminished seconds can be shown to correspond to I-values -5, 2, 9, 16, 23, 30, and so on (see Reference 3), an equivalent definition may be stated as follows: A Busoni scale is any ascending sequence of eight tones whose adjacent notes bear interval relationships with I-value labels equal to -5, 2, 9, 16, 23, 30, and so on.

Restating the conditions of this definition mathematically reveals that the Busoni scales are related to, and in fact struc- turally specified by solutions of the following partition equation:

O = N1(-5) + N2(2) + N3(9) + N4(16)+ N5(23)+ N6(30) + .... (4)

where the coefficients of the I-values (that is, the Ni) are non- negative integers that together total seven. Obviously, for the major scale, N1 = 2, N2 = 5, and for i> 2, Ni = O. But there are other solutions to this partition equation corresponding to other Busoni scales. Interest consequently centers on allpos- sible sets of coefficients Ni that will satisfy equation 4.

To find these solution sets it is best to start out by giving successive integral values to N1 beginning with the smallest ad- missible. It is soon obvious that no solution exists for N1


93

equal to either O"or 1. It follows that all Busoni scales must contain at least two minor seconds. Also evident is the fact that no solutions are possible for N1 greater than 6. Conse- quently all Busoni scales are devoid of intervals stretched be- yond the quadruply augmented mark. (See Reference 2b.)

Each row of Chart 1 contains a set of coefficients Ni that bal- ances the partition equation, equation 4. To illustrate the meaning of the entries in this chart, the fourth solution says that

4(5-) + 2(2) + 1(16) = 0.

This equality implies that many different Busoni scale structures or modes exist (the exact number being 7!/(4!2!) = 105) that are formed by arranging four minor seconds (m 2nd), two major seconds (M 2nd), and one doubly augmented second (++ 2nd) in every possible way. The factorial expressions on the right state the number of interval permutations that are possible for each combination of intervals specified by the N's. Barbour established the total to be 462 heptatonic modes. As a check, this result can be shown to agree with the coefficient of the zero-th power term in the expansion of the associated generating function: *4

-5 2 9 16 23 30 (u +u +u +u +u +u ). (5) The number of different Busoni scale structures, 462, when divided by the number of modal variants for a heptatonic scale, 7, yields the number of harmonically distinct seven-tone scales that are theoretically possible, 66.

Before attempting to count the Busoni scales themselves, it is advisable to digress briefly for a discussion of literal notation and a review of matrix algebra.

Literal Notation

Consider two sets, a set of letters A, B, C, D, E, F, G called L and a set of inflections .., x, x, . , , 6 6, ... called I. Modern musical notation is based on a new set made up of ordered pairs produced by choosing the first element of the pair from L and the second from I, in that order. This so-called "Car- tesian product" set, written L x I (read "L cross I"), is termed a literal alphabet. If the set I is restricted to an odd number of inflections centered on b, the resulting truncated literal al-


99

CHART

1

Heptatonic Scale Family

Classification I N Nmber ofBusoni (Barbour, 1949)l IScale Structures

Class I 2 I7 5= 21 5!2! Class II 3 3 1 i 140

1 1 1--1 =3! 3!

7! Class III 114 1 2 4I2--'!= 105 412! Class IV 4 2 0 1 ----1 105

Class V 5 0 1 1 1 11 = 42 5.! Class VI 5 1 0 0 1! = 42 5!

Class VII 1 6 0 0 0 0 1 7 = 7 6!

Intervalship m 2nd M 2nd + 2nd ++ 2nd +++ 2nd +++- 2nd Total: 462 Relationshi p1


100

phabet is said to be "balanced".

The musical symbolism required for the remainder of this discussion is customary and consists of two balanced literal alphabets that are obtained by taking two particular subsets of L x I. These subsets are based on three and five inflections and contain twenty-one and thirty-five literals, respectively. Graphically, the two literal alphabets may be depicted as sets of points on a planar lattice, as shown in Figure 1.

The following brief review of the simple mathematical elements called "matrices" is included here to facilitate the mathematically untrained musical reader's understanding of the subse- quent material.

Matrix Algebra

First of all, a matrix P is defined as a rectangular array of numbers p, which may be represented as follows:

P11 P12 * "

'Pln P21 P22 * * P2n

P ~. (6)

Pm1 Pm2 " Pmn

By convention the first subscript on p denotes the row in which that p occurs, the second subscript denotes the column. The next paragraph explains what is meant by "matrix multiplication".

Provided that the matrix Q has exactly as many rows as the matrix P has columns, the product of P and Q may be defined to be another matrix R:

R = PQ. (7)

Otherwise, the operation of matrix multiplication is undefined. The element rij appearing in the ith row and jth column of the product matrix R is obtained as follows: Multiply the first element in the ith row of P by the first element in the jth column of Q, then the second element in the ith row of P by the second element in the jth column of Q, and so on. The sum of all these products,


101

FIGURE

I Literal Alphabets

o o o o o o o 35 LITERAL ALPHABET

/ 0 0 0 0 0 0

0O 21 LITERAL ALPHABET

z -

-0--0-j--o- z0 0 0 0 0 0 0 -

"I I

A a c D F F o

LET TERS

LETTERS


102

n r. = Pil1j +Pi 2q2j ++.. +P innj=1 ikqkj ' (8)

is the element required to occupy the ith row and jth column of the product matrix R.

This definition implies that matrix multiplication is associative and distributive, but not commutative. These terms mean that for any three matrices for which multiplication is possible, say U, V, W, it is always true that:

U(VW) = (UV)W ASSOCIATIVE LAW (9a) U(V + W) = UV + UW DISTRIBUTIVE LAW (9b)

and it is often true that

UV VU FAILURE OF COMMUTATIVE LAW (9c) If P and Q denote the same matrix, then formula 8 gives its second power or square. In this case, however, the number of rows m must equal the number of columns n for the multiplication to be defined (see equation 6). The square of the matrix P is denoted by

P2 = (P)P = P(P).

Successively higher positive integral powers of a matrix can be defined recursively, so that

P3= (2)p= p(p2)

and, in general,

Pv= (Pv-1 )=p(pv-1). (10) Precedence Matrices

In general a precedence matrix" is defined to be a square array (m = n) of zeros and ones, such that the entry in the ith row and the jth column equals 1 if a corresponding precedence relation holds, and the entry equals 0 otherwise. A precedence matrix P is a mathematical way of showning, for example, how certain events (the soundings of tones) E1, E2 ....., En are ordered in time (when a scale is played). If tone Ei is struck


103

immediately before tone E in any scale, then the element in the ith row and the jth column of P, i. e., pij, is set equal to one. If tone Ei is not struck immediately before tone Ej, this element is assigned the value zero.

Now if a 1 appears in the ith row and the jth column of a given precedence matrix, then a 0 must appear in the jth row and ith column of the same matrix, since usually if something immediately precedes something else, it cannot also immediately follow. By the same token, all the elements in the ith row and the ith column (that is, the elements P11' P22

*.... Pii*

.... Pnn) must be zero, because in normal circumstances something cannot immediately precede itself.

Example

Consider the direct precedences defining the ascending C major scale (see equation 1), which constrain the scale steps to be taken in a fixed order. Thus, if "

104

"note C immediately precedes note D"; the 1 at the intersection of the F column and the E row means that in this same scale "note F immediately follows note E".

The Three-Inflectional Case

To apply the previous ideas to Busoni's problem, it is neces- sary to work with larger matrices. The matrix P for the three- inflectional case (see Figure 2) is constructed as follows. First, the twenty-one literals of the alphabet are listed both across the top and down the left-hand side of the matrix. A given literal designates the same row and the same column. For example, D6 designates row four and column four in the chosen matrix. Subject to this single restriction, the literals may be assigned arbitrarily. *5 From Barbour's point of view, these twenty-one designations are the various notes of the harp.

Second, each pair of literals having a direct precedence is en- tered into the matrix by the following rule of construction: Whenever the interval relationship between the lower note (labeling row i) and the upper note (labeling column j) is an undiminished second, the element pij occuring in the ith row and the jth column of the matrix P is assigned the value 1; otherwise this element is assigned the value zero.

Precedence matrices are useful not only because they represent the sequence in which steps must be taken, but also because their powers have an interesting interpretation. To illustrate, if Z denotes a precedence matrix, the entry in the ith row and the jth column of Z2 gives the number of second-order precedences implicit in Z. The next few paragraphs describe the formation of P2.

A precedence matrix is squared by multiplying it by itself using the row-by-column rule given before for ordinary matrix multiplication (see equation 8). As it now stands, however, the matrix P (Figure 2) is too large to handle conveniently. Luckily the calculation can be simplified by a mathematical trick. The details of the trick are as follows: Let x and y denote the following submatrices of P:

x = 1 1 and y = 0 1 1 . (13) 11Then the matrix P may be written more compactly as follows:01

Then the matrix P may be written more compactly as follows:


105

FIGURE The P matrix - a precedence matrix for Busoni scales based on a twenty-one literal alphabet. (Zero elements are suppressed.)

C 111 C II1

Db 1 1 1

D I1l

Eb,1 1 1 E 1 1

Fb,1 1 1 F 1 1 1 E 1I1 EG 1

G 1 1 1

F I 11

A 11

G l1l

A1 1

11 Ab I 1

Bb, 1 1 1 B 11 B$t 1


100

Ox00000 00x 0000 000y000

P = 0000 x 00 . (14) 00000x0 000000x yO000000

The resulting seven by seven "partitioned matrix" is a symbolic expression of the original twenty-one by twenty-one matrix (Figure 2). Note that each 0 in the compact expression really stands for a submatrix of zeros:

000 0= 0 0 0 . (15)

000

Partitioned matrices are multiplied in the normal way by treat- ing the submatrices as if they were elements. For multiplication of partitioned matrices to be possible, the first matrix must be partitioned as to columns in the same way that the second matrix is partitioned as to rows. By matrix multiplication, successive powers of P, in its partitioned form, can be determined easily.

When the precedence matrix P is squared, the elements in the resulting product matrix show the number of Barbour's three- tone fragments. For clarity,

0 x 000002 0 Ox2 0 0 0 0

00 x 000 0 0 0 Oxy 0 0 0

000 y 000 0 0 0 Oyx 0 0

0000 x 00 = 0 0 0 0x2 0, (16) 0 0 0 0 0 x 0 0 0 0 0 0 Ox2

000000 x xy 0 0 0 0 0 0

y 000000 Oyx 0 0 0 0 0

where the particular submatrices are calculated as follows:

111 111 233

x2= 111 111 = 233, (17a)

011 011 122


107

111 111 123 xy = 1 1 1 0 1 1 = 1 2 3 (17b)

011 001 012

111 111 233 yx= 011 111 = 122, (17c)

001 011 011

(Note from equations 17b and 17c that yx xy; this illustrates the behavior expressed in 9c.) As an example of the interpretation of 17a, consider the position of x2 in the top part of P2:

Eb E E# CT 2 3 3 C 2 3 3 C# 1 2 2

This portion of P2 shows that there are just two three-tone fragments connecting note C to note Eb. These are, of course, CDbEb and CDEb. (See Reference 2a; in particular, Table 1.) The corresponding entry of P3 must be zero because there are no four-tone fragments connecting note C to note Eb. The product matrix p3 is multiplied again by P and the process continues until the seventh power is formed. Discussion of these powers is postponed until a later section.

The Five-Inflectional Case

The precedence matrix Q for Busoni's problem in the five- inflectional case is shown in Figure 3. Let x and y now stand for the following submatrices of Q:

11111 11111 1 1 1 1 1 0 1 1 1

x = 0 1 1 1 1 and y = 0 0 1 1 1 (18) 0 0 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1

Then Q has the form symbolic of P and its first seven powers can be determined easily.

Technically there are inconsistencies in both P and Q, because in every Busoni scale the penultimate tone leads once again to the starting tone. *6 As an example, the leading tone B of the scale of C major returns to the tonic C, which was the starting point. These inconsistencies are permissible because such loops are legitimate and it is only in the seventh powers of P


1o08

and Q that they manifest themselves for the first time.

Complete Listing of Busoni Scales

Table 11t gives an exhaustive presentation of the 1254 Busoni scales possible with five inflections. The 363 synthetic musical scales counted by Barbour are embedded in this catalog and are marked by asterisks. Each arrowhead indicates the entrance of a new scale structure and the beginning of a family of heptatonic scales related by transposition. These 66 Busoni scale families are categorized completely by the following in- dex, which gives the line number 1 of the first scale in each family to have the designation h for every characteristic harmony of cardinality seven. *7

h 1 h 1 h 1 h 1 h 1 h 1 127 0003 379 1118 487 1029 671 0864 749 0935 875 0875 191 0370 381 0057 491 1213 687 1178 755 0990 877 0737 223 0209 415 0234 493 1154 695 0945 757 0473 885 0155 239 1086 431 1220 499 0711 699 0800 823 0627 925 1107 247 0952 439 0306 501 0111 701 1246 827 0601 939 0427 251 0759 443 0899 505 0825 719 0590 829 1069 941 1239 253 0175 445 0439 607 0325 727 1195 847 0779 949 1188 319 0129 463 0972 623 0539 731 0257 855 1143 981 0197 351 0843 471 1016 631 1047 733 0392 859 0912 1367 0092 367 0403 475 0648 635 0479 743 0674 861 0683 1371 0452 375 0564 477 0034 637 0350 747 0287 871 0513 1387 1252

Tables 2 and 3 contain numerical data about the many scales listed in Table 1. These data were found by evaluating the successive powers of matrices P and Q. Table 2 displays the sum

S21 =P+P + P + P + P + P + p7 (19) obtained by adding the corresponding elements of each of these seven matrices. Similarly, Table 3 displays the sum

S35 = Q + 02 3

Q4 + Q5 + Q6 + Q7 (20)

By summing in this way, the information contained in each power matrix is condensed without disturbance.

There are three kinds of numbers in Tables 2 and 3. Those along the main diagonal (bold) show at a glance the number of

t All Tables have been placed at the end of the article. - ed.


109

FIGURE

The Q matrix - a precedence matrix for Busoni scales based on a thirty-five literal alphabet. (Zero elements are suppressed.)

C~)1 1 1 1 1 C 1111 Cs 1 1 1 Cx 1 1 Db 1 1 1 1 1 Dfr 1 1 1 1 1 D 1111

Dx 1 1 EbC1 1 1 1 1 E 1 1 1 1 E 111

El 1 Ex 1 FbC 1 1 1 1 1 FC 1 1 1 1 1 F 1 1 1 1 Ft 1 1 1 Fx 1 1 Gb, 1 1 1 1 1 Gfr 1 1 1 1 1 G 1111 Ge 1 1 1 Gx 1 1 A~lP 1 1 1 1 1 Afr 1 1 1 1 1 Ab 1111 AW 1 1 1 Ax 1 1 B 1 1 1 1 1 Bf I 1! 1 1 B 111 Bt 1 1 Bx I


110

complete Busoni modes that correspond to the row or column label. Those in boxes along the main diagonal (italic), excluding main diagonal entries (bold), relate to two-octave Busoni scales and thus should be regarded as spurious solutions. And those situated elsewhere (roman) reveal the number of fragments that radiate from the corresponding row label as keynote or lead into the corresponding column label as endnote.

Focusing attention on Table 2, the sequence of entries reading from left to right along the bottom row of S21 states that there is a unique two-tone fragment from B# to C#. Building upon that, there is a single three-tone fragment ending on D and another ending on D#. Continuing in the same direction, there is just one four-tone fragment from B# to Eb, although there are two of them from B# to E and from B# to E#. Moreover, there are a total of nine five-tone fragments with keynote B#. (Ac- tually 1 + 3 + 5 is a row sum of p4; column sums have a mean- ingful interpretation too. ) Likewise, there are 22 six-tone fragments starting on B#, of which four terminate on G6, and there are 57 seven-tone fragments starting on B#, of which 13 terminate on Ab. Again, there are 92 attempts (35 + 57) to form Busoni scales starting on B# that failed because the endnotes (B6 and B, respectively) do not match the keynote. Final- ly, there are 57 successful attempts to form Busoni scales starting on B#.

The numbers (bold) generated along the main diagonals of Tables 2 and 3 - that is, along a straight line extending from the top left to the bottom right corner of S21 and S35 - comprise an enumeration of complete Busoni modes originating and termi- nating at notes labeling the row or column in which they appear. If these diagonal elements are summed the result is called a "trace". In mathematical notation the trace of either S21 or S35 may be represented by the same formula,

n n tr S

= Skk, or tr S = s ij. (21) k=1 i= 1,

j=i Since tr S21 = 2541 = 7 X 363, there are 2541 Busoni modes and 363 Busoni scales based on the twenty-one literal alphabet. Similarly, since tr S35= 8778= 7 X 1254, there are 8778 Busoni modes and 1254 Busoni scales based on the thirty-five literal alphabet.


111

Busoni Mode Names

The first family of scales set off by arrowheads in Table 1 oc- cupies lines 0001 to 0029 inclusive. Undoubtedly the scales in this first section are harmonically equivalent because they are modal variants of the major scale. To speak in precise terms, when the tone in the r-th column of the first section is taken to be the keynote of the scale, the scales in this equivalence class are named as follows:

r mode 1 Lydian, 2 Mixolydian, 3 Aeolian, 4 Locrian, 5 Major, 6 Dorian; and 7 Phrygian.

Convenient mode names for the remaining scales of Table 1 (that is, lines 0030 to 1254) can be created simply by attaching the prefix "co-" to the mode name for section one corresponding to the column containing the keynote. For example, the C har- monic scale (line 0488, r = 3) would be in the co-Aeolian mode.

Conclusion

The Basoni scales within the first section are enharmonically identical in accordance with the collowing chart.

0001 = 0013 = 0025 0002 = 0014 = 0026 0003 = 0015 = 0027 0004 = 0016 = 0028 0005 = 0017 = 0029 0006 = 0018 0007 = 0019 0008 = 0020 0009 = 0021 0010 = 0022 0011 = 0023 0012 = 0024


112

Referring again to Table 1, it is apparent that within any section listing more than twelve Busoni scales those scales in lines N and N + 12 are enharmonically identical. A close look, however, shows that not all sections list thirteen or more scales. Some Busoni scales remain to be found.

As previously indicated, several conclusions can be drawn from an inspection of Table 1.

(a) An alphabet having only three inflections (#, 0, b) - as implicit in Barbour's earlier study - is insufficient to yield a Busoni scale to represent each of the sixty-six possible harmonically distinct classes of heptatonic scales.

(b) An alphabet having five inflections (x, #, b6, 6 ), although sufficient to yield at least five representatives, nevertheless is insufficient to ensure that all twelve enharmonically distinct Busoni scales in each such class appear at least once in the list.

(c) An extension of Table 1 to alphabets having six or more inflections is trivial.

Enlarging upon conclusion (c), the shortest section in Table 1 extends from line 1250 to line 1254 inclusive. As this section is incomplete, it readily suggests seven harmonically equivalent heptatonic scales that cannot be written as Busoni scales using only a thirty-five literal alphabet. An example of such a scale, which is of interest because it cannot be written with just the five inflections x, #, 4, 6, bb without having to use some one of the letters A, B, C, D, E, F, G more than once, is to be found by taking, say, the seven leftmost tones of Figure 4. Obviously the letter A cannot appear in any trial spelling of this scale, and therefore, of necessity, one of the remaining letters must be repeated. By introducing a sixth inflection, say a triple sharp, * = x# = ###, a definitive solutionto Busoni's problem can be obtained. This small luxury permits the last section of Table 1 to be continued in the manner shown in Chart 2. *8 Other short sections can be continued in the same way.

The remarks of the preceding paragraph lead to the following two theorems.

THEOREM 1: An alphabet having six inflections is both neces- sary and sufficient to ensure that all twelve enharmonically distinct Busoni scales in each of the sixty-six possible harmonically distinct classes of heptatonic scales appear at least


113

FIGURE 4 Enharmonic mapping of the thirty-five literal alphabet onto the twelve-tone

equitempered system.

Tones 0 1_ 2 f3 4 5 6 7 8 9] 10111 - Bx Cx - Dx - Ex Fx - Gx - Ax

B C B$ C - D - E$ F$ - G$ - A - C - D

FE F - G

-

A - B - D - E F - G - A - B C

D - Ebb F? - - Ab - B C -

CHART

2

1250 FO Gx A$ B C Di EOO 1251 COO Dx E$ F$ G AO B p 1252 G b Ax SB C$ D EO FO 1253 DI0 Ex Fx G$ A BS CO 1254 AO Bx Cx D$ E F GO 1254+i FD F* Gx A$ B C DW 1254+2i BO C* Dx E$ F$ G AO 1254+3i FO GO Ax SB C$ D EO 1254+4i CO D*. Ex Fx G$ A Bs 1254+5i GO A. Bx Cx D$ E F 1254+6i Df E* F. Gx AS B C 1254+7i AO B3 C* Dx E$ F$ G


114

once in the list analogous to Table 1.

THEOREM 2: Every scale that has seven or fewer tones can be written without using any letter more than once, although to do so may require as many as six inflections.

Thus, it may be said that harmonies containing seven or fewer distinct tones are "weakly diatonic'.


115

TABLE 1

BUSONI SCALES

W0001 FOO GOO AOC BOO COO DD E EOO 0068" BO C D E F$ G A 0002 COO DOO EOO FO GOO AOO BOO 00690 F G A B C$ D E 0003 GOO A3 S BOO CO DOO EOO FO 00700 C D E F$ G$ A B 0004 DO EOt FO GO At, BOO CO 0071* G A B C$ D$ E F$ 0005 A, BOO CO DO EOO FO GO 0072* D E F$ G$ A$ B C$ 0006 EO FO GO AS BO CO DO 00730 A B C$ D$ E$ F$ G$ 0007 BOO CO DO ES FO GO AS 00740 E F$ G$ A$ Bt C$ D$ 0008* FO GO AS BO CS DO ES 0075 B C$ D$ E$ Fx G$ A$ 0O09 CS DO ES F GO AS BO 0076 F$ G$ A$ B$ Cx D$ E$ 00100 GO AS BO C DO ES F 0077 C$ D$ E$ Fx Gx A$ B3 00110 DO ES F G AS BO C 0078 G$ A$ B$ Cx Dx E$ Fx 00120 AS BO C D ES F G 0079 D$ E$ Fx Gx Ax B$ Cx 00130 ES F G A B3 C D 0080 A$ B$ Cx Dx Ex Fx Gx 0014* BO C D E F G A 0081 E$ Fx Gx Ax Bx Cx Dx 0015* F G A B C D E W0082 FOO GOO A3S BOO CO DO EOO 0016* C D E F$ G A B 0083 COO DSS ESS FO GO AS BOO 00170 G A B C$ D E F$ 0084 GO A S BO CO DO ES FO 0018* D E F$ G$ A B C$ 0085 DSS ESS FO GO AS BO CS 0019 A B C$ D$ E F$ G$ 0086 AS B3O CS DO ES F GO 0020 E F$ G$ A$ B C$ D$ 0087 EO FO GO AS BO C DO 0021 3 B C$ D$ E$ F$ G$ A$ 0088 BOO CO DO ES F G AS 00220 F$ G$ A$ B$ C$ D$ E$ 0089* FO GO AS B3 C D ES 0023 C$ D$ E$ Fx G$ A$ B$ 00900 CS DO ES F G A B3 0024 G$ A$ B$ Cx D$ E$ Fx 0091* GO AS B3 C D E F 0025 D$ E$ Fx Gx A$ B$ Cx 00092 DO ES F G A B C 0026 A$ B$ Cx Dx E$ Fx Gx 0093* AS B3 C D E F$ G 0027 E$ Fx Gx Ax B$ Cx Dx 0094* ES F G A B C$ D 0028 B$ Cx Dx Ex Fx Gx Ax 0095* BO C D E F$ G$ A 0029 Fx Gx Ax Bx Cx Dx Ex 00960 F G A B C$ D$ E

W0030 FO GOO A3C BOO COO DO EDO 00970 C D E F$ G$ A$ B 0031 COO DO EOO FO GOO A 3 BOO 00980 G A B C$ D$ E$ F$ 0032 GOO A3S BOO CO DOO EO FO 0099* D E F$ G$ A$ B$ C$ 0033 DO ES F0 GF AGO BA CS 0100 A B C$ D$ E$ Fx G$ 0054 A3S BOO CO DO EOO F GO 0101 E F$ G$ At Bt Cx D$ 0035 EO FO GO AS BO C DO 0102 B C$ D$ E$ Fx Gx A$ 0036 BOO CO DO ES F c G AS 0103 F$ G$ A$ B$ Cx Dx E$ 007* FO GO AS B3 CS D ES 0104 C$ D$ E$ Fx Gx Ax B$ 00380 CS DO ES F GO A B3 0105 G$ A$ B$ Cx Dx Ex Fx 0039 GO AS BO C DO E F 0106 D$ E$ Fx Gx Ax Bx Cx 00400 DO ES F G AS B C W0107 FOO GOO AOO BOO C DD ESS 00410 AS BO C D ES F$ G 0108 COO DOO EOO FO G AS B3O 00420 ES F G A B3 C$ D 0109 GOO A3S BOO CO D ES FO 0043* BO C D E F G$ A 0110 DOO EOO FO GO A B3 CS 0044* F G A B C D$ E 0111 ASS B3O CS DO E F GO 00450 C D E F$ G A$ B 0112 E O FO GO AS B C DO 0046* G A B C$ D E$ F$ 0113 BOO CO DO ES F$ G AS 00470 D E F$ G$ A B$ C$ 01140 FO GO Ap BO C$ D ES 0048 A B C$ D$ E Fx G$ 0115* CS DO ES F G$ A BO 0049 E F$ G$ A$ B Cx D$ 0116* GO AS BO C D$ E F 0050 B C$ D$ E$ F$ Gx A$ 01170 DO ES F G A$ B C 0051 F$ G$ A$ B$ C$ Dx E$ 0118* At BO C D E$ F$ G 0052 C$ D$ E$ Fx G$ Ax B$ 011O ES F G A B$ C$ D 0053 G$ A$ B$ Cx D$ Ex Fx 0120 Bt C D E Fx G$ A 0054 D$ E$ Fx Gx A$ Bx Cx 0121 F G A B Cx D$ E

W0055 FSS GOS ASS BOS CS DSS ESS 0122 C D E F$ Gx A$ B 0050 COO DO S ES S FO GO AS BOS 0123 G A B C$ Dx E$ F$ 0057 GOS ASS BOS CS DO ESO FO 0124 D E F$ G$ Ax B$ C$ 0058 DOS ESS FO GO AS BOO CS 0125 A B C$ D$ Ex Fx G$ 0059 ASS BOO CO DO ES FO Gk 01206 E F$ G$ At Bx Cx D$ 0060 DE FO GO AS BO CS DO W0127 FSS GOS ASS BO CS DSS ESS 0061 BSS CS DO ES F GO AS 0128 COO DOS ESS F GO AS BOS 00620 FO GO AS BS C DS ES 0129 GoS AS B3P C DO ESS FO 00630 CS DO ES F G AS BS 0130 DO S ESS FO G AS BOS CS 00640 Gk AS BO C D ES F 0131 ASS BOS CS D ES FO GO 0065' DO ES F G A BS C 0132 D FO GO A BO CS DO 0066o AS BS C D E F G 0133 BOO C5 DO E F GO AS 00670 Eb F G A B C D 0134' FO GO AS B C DO ES

(Table Continues)


116

01355 Cb DO Eb F$ G Ab Bb 0295 F G A Bx Cx D$ E 01360 Gi Ab Bb C$ D Eb F F 0206 Fbb Gbb Ab BDb COi Dib Ebb 01S70 D L EL F G$ A SB C 0207 COO DOi Ei FO GOO AuO BOO 0138 AO Bi C D$ E F G 0208 GOO A S BOi CO Di ELi Fi 01Si9 LE F G A$ B C D 0209 DOO EOO F Gk AOO BOi CO 0140 S Bi C D E$ F$ G A 0210 AOO BOi C Di ELi FO Gk 01410 F G A B$ Cl D E 0211 ELi FO G Ai BOi CO DO 0142 C D E Fx G$ A B 0212 BOi C D DE Fi Gk Ai 0143 G A B Cx D$ E F$ 02135 Fi Gk A Bi Ci D L Ei 0144 D E F$ Gx A$ B C$ 0214* CO DO E F Gk Ai Bi 0145 A B C$ Dx E$ F$ G$ 02150 Gk Ai B C Di EL F 0146 E F$ G$ Ax B$ C$ D$ 0216* Di Ei F2 G Ai 9i C 0147 B C$ D$ Ex Fx G$ A$ 02170 Ai Bi C$ D Ei F G 0148 F$ G$ A$ Bx Cx D$ E$ 0218* Ei F G$ A SB C D

w0149 Fii Gii AiS Bi C- Di Eu 02190 BS C D$ E F G A 0150 Cii Dii EiO F Gk A Bi o02200 F G A$ B C D E 0151 Gii AiD BOi C Di Ei Fi 0221* C D E$ F$ G A B 0152 Di Ei i Fi G Ai i Ci 0222* G A B$ C$ D E F$ 0153 Aii BOi Ci D Ei F Gk 0223 D E Fx G$ A B C$ 0154 Ei Fi Gk A SB C Di 0224 A B Cx D$ E F$ G$ 0155 BOi C Di E F G Ai 0225 E F$ Gx A$ B C$ D$ 0156 Fi Gk Ai B C D EL 0220 B C$ Dx EL F$ G$ A$ 01570 Ci Di Ei F$ G A Bi 0227 F$ G$ Ax B$ CQ D$ EL 015 Gk Ai Bi C$ D E F 0223 C$ D$ Ex Fx G$ A$ B$ 0159 DO Ei F G$ A B C 0229 G$ A$ Bx Cx D$ El Fx 016o Ai Di C D$ E F$ G W0230 Fii Gii Ai BOi Cii DO EuO 01l10 Ei F G A$ B CQ D 0231 Cii Dii Ei Fi Gii Ai BDi 012 S Bi C D El F$ G$ A 0232 Gii Aii Bi Ci Dii Ei Fi 01630 F G A B$ C$ D$ E 0233 Di i Ei i F Gk A i BS Ci 0104 C D E Fx G$ A$ B 0234 AuO BOu C DO EuO F Gk 0165 G A B Cx D$ E$ F$ 0235 Eii Fi G Ai BOi C Di 010 D E F$ Gx A$t $ C$ 0250 BOi C D Ei Fi G Ai 0107 A B C$ Dx E$ Fx G$ 02357 F Gk A Bi Ci D Ei 0108 E F$ G$ Ax B$ Cx D$ 02358 Ci Di E F Gk A Bi 0169 B C$ D$ Ex Fx Gx A$ 0239 Gi Ai B C Di E F 0170 F$ G$ A$ Bx Cx Dx EL 02400 D EK F$ G Ai B C

w0171 Fii Gii Au S Bi C Di EuO 0241* Ai Bi C$ D E F2$ G 0172 Ci DWI Ei Fi G Ai BDi 02420 Ei F G$ A: Bi C$ D 0172 Gii Aii BOi C D Ei Fi 0243* Bi C D$ E F G$ A 0174 Dii EOi Fi G A Bi Ci 02440 F G A$ B C D$ E 0175 AiO BOi Ci D E F Gi 0245* C D E$ F$ G A$ B 0176 Ei Fi Gi A B C Di 02460 G A B$ C$ D E2 F$ 0177 Bii C DO E F$ G Ai 0247 D E Fx G$ A B$ C$ 0178* Fi Gi Ai B C$ D Ei 0248 A B Cx D$ E Fx G$ 01790 Ci D Ei F$ G$ A Si 0240 E F$ Gx A$ B Cx DO 01W8 G Ai Bi C$ D$ E F 0250 B C$ Dx E$ F$ Gx A$ 01810 DO Ei F G$ A$ 3 C 0251 F$ G$ Ax Bt C$ Dx E$ 018o Ai Di C D$ E$ F$ G 0252 C$ D$ Ex Fx G$ Ax t$ 0185* Ei F G A$ B$ C$ D 0253 G$ A$ Bx Cx D# Ex Fx 0184 Bi C D E$ Fx G$ A 0254 Fi Gi Ai BOi C Dib EOi 015 F G A B$ Cx D$ E 0255 Cii DOb Ei Fi Gi Aii BOi 0136 C D E Fx Gx A$t 0256 Gii Ai Bi Ci Di ELi Fi 0187 G A B Cx Dx E$ F$ 0257 Dii EDi F Gk Ai Dii Ci 0188 D E F$ Gx Ax B$ C$ 0258 Ai BOi C Di Ei Fi Gk 0189 A B C$ Dx Ex Fx G$ 0259 ELi Fi G Ai DB Ci Di 0190 E F$ G$ Ax Bx Cx D$ 0200 BOi Ci D Ei F Gi Ai

W0191 Fii Gii AiO i C Di Eu 0261e Fi Gi A Bi C Di Ei 0192 Cii DOi Ei $F G Ai BOi 022, Ci Di E F G Ai iS 0193 Gk Ai i B C$ D Ei Fi 023* Gk Ai B C D EK F 0194 DiB Ei i Fi G$ A Bi Ci 0 24 Di E F$t G A Bi C 0195 AuO mu Ci DI E F Gk 0257 Ai Di C$ D E F G 0196 Ei ' F Gk At B C Di 02M0 Ei F G$ A B C D 0197 Bii Ci Di E$ F$ G Ai 070 S Bi C D$ E F$ G A 01960 Fi Gi Ai B$ C$ D Ei 0N8e F G A$ B C$ D E 0199 Ci Di Ei Fx G$ A Bi O0209 C D El F$ G$ A B 0200 Gi Ai Bi Cx D$ E F 0270* G A 3$ C$ D$ E F$ 0201 DO Ei F Gx A$ B C 0271 D E Fx G$ A$t C$ s5 A SOi C Dx Et F$ G 0272 A B Cx D$ EL F$ G$ 020S Ei F G Ax Bg C$ D 0273 E F$ Gx At Bg C$ D$

04 BS C D Ex Fx G$ A 0274 3 C$ Dx El Fx G$ A$ (Tabk Continues)


117

0275 F$ G$ Ax B$ Cx D$ E$ 0345 COO DOt E,

F GO AO BOO 0276 C$ D$ Ex Fx Gx At B$ 0346 GOO

AO, tS C DO

E, FO

0277 G$ A$ Bx Cx Dx E$ Fx 0347 DOt

EOO F G AO SO CO ,0278 Ft,

GOO AO ?BO

CO DO E,

0348 At, ?BO

C D E,

F GO 02709 CO

DOt E, F, GO AO

?BO 0349

E, FO G A

?t C DO

0280 GOt Att ?t

C DO E Ft,

0350 ?B

CO D E F G AO 0281

Dfrt E, F GO

At Bt CO 0351*

F, GO A B C D

E, 0282 AOt Btt C DO EO F GO 0352 CO DO E F$ G A sB

02t3 E, FO G AO BS C DO 0353

GO, AO B C$ D E F 0284

t3 CO D EO F G AO 0354* DO EO F$ G$ A B C 0285*

Ft GO A SI C D Et 0355"

AO SO C$ D$ E F$ G 02t0

CO DO E F G A so 03560 EO F G$ At B C$ D 0287* Gk AO a C D E F 0357* BS C DI E$ F$ G$ A 02t88 DO E0 F$ G A B C 0358M F G A$ B$ C$ D$ E 0289* AO Sk C$ D E F$ G 0359 C D E$ Fx G$ A$ B 020 E0 F G$ A B C$ D 0360 G A B$ C x D$ E$ F$ S0291* O C D$ E F$ G$ A 0361 D E Fx Gx A$t B C$ 0292* F G A$ B C$ D$ E 0362 A B Cx Dx E$ Fx G$ 033* C D El F$ G$ A$t 0363 E F$ Gx Ax B$ Cx Do 0294* G A B$ C$ D$ Et F$ 0364 B C$ Dx Ex Fx Gx A$ 0295 D E Fx G$ At B$ C$ 0365 F$ G$ Ax Bx Cx Dx Et 020 A B Cx D$ E$ Fx GI 0366

Ft Gt0 AO S C DO E 02 E F$ Gx A$ A$ Cx D$ 0367 COO DOb E,

F G AO tOO 0296 B C$ Dx E$ Fx Gx A$ 0368

G, At C D EG

Ft 020 F$ G$ Ax t$ Cx Dx Et 030 D C0 E DF G A St

C 0300 C$ D$ Ex Fx Gx Ax 0$ 0370 At, BOO C D E F GO 0301 G$ A$ Ox Cx Dx Ex Fx 0371 EOt FO G A 0 C DO W0302 Ftt GOt AO BOO C DO Ett 0372 ?tO CO D E F$ G AO 0303 CO Dkt E FO,

G AO BOt 03730F Ft

G A B C$ D E0 0304 GOO At, t ,

CO D E,

FO 0374* CO DO E F$ G$ A 0S 0305 D EOt

F GO A SO C 0375* GO AO B C$ D$ E F 030 At, BOO C DO E F GO 0370* DO EO F$ G$ A$ B C 0307 E tFO G AO B C DO 03770 AO St C$ D$ Et F$ G 0300 tOCO D E0 F$ G AO 03780 0E F G$ A$ 0$ C$ D 0309 FO GO A SO C$ D E 0379 St C D$ E$t Fx G$ A 03100 CO Dt

E F G$ A O 10380

F G A$ B$ Cx D$ E 03110 GO AO B C D$ E F 0301 C D E$ Fx Gx A$ B 0312 DO E4 F$ G A$ B C 0382 G A B$ Cx Dx E$ F$ 0313*

At B0 C$ D E$ F$ G 0383 D E Fx Gx Ax B$ C$ 03140 E0 F G$ A B$ C$ D 0304 A Cx Dx Ex Fx G$ 0315* S C D$ E Fx G$ A 0385 E F$ Gx Ax Bx Cx D$ 0310 F G A$ B Cx D$ E 038

FP, Gt AO B C

D, E, 0317 C D E$ F$ Gx A$ 0307 CO Dtt

ED F$ G AO Bt, 0318 G A B$ C$ Dx E$ F$ 0388 GO

At, SO C$ D EO FO

0310 D E Fx G$ Ax B$ C$ 0380 Dt EOt

F G$ A SO CO 0320 A ? Cx D$ Ex Fx G$ 0390 At, BOO C D$ E F GO 0321 E F$ Gx A$ Bx Cx Dt 0301 Et FO G A$ ? C DO

93222 Ft, GOO AO St CO D, Ett 03 Bt CO D E$ F$ G AO 0323 COO D t0 E F GO A, BOO 00330 FO GO A B$ C$ D Et 0324 GO

At 0 C DO E t FO 0394 CO D E Fx G$ A 0S 0325 Dt El F G AO BOO CO 0305 GO AO Cx DI E F 0326 At, BOO C D E0 FO GO 030 DO EO F$ Gx A$ ? C 0327 EOO

tF G A SO CO Di 0397 AO

St C$ Dx Et F$ G 0328 BO CO D E F GO AO 038 EO F G$ Ax B$ C$ D 0329 FO, GO A B C DO EO 0300 0t C D$ Ex Fx G$ A 03300 CO DO E F$ G AO S 0400 F G A$ Bx Cx D$ E 03310 GO AO C$ D ED F F 1041 FPO GtO A SO CO Dbt EOt 0332 DO EO F$ G$ A SO C 0402 COO DOt E F GO At, tBO 0333" AO ?t Ct Dt E F G 04013 GO, At, B C DO EO FO 03340 E4 F G$ A$ ? C D 0404 D0, E tF$ G AO SOt CO 03350 Bt C D$ Et F$ G A 0405 At, tBO C$ D E0 FO, Gt 03360 F G A$ B$ C$ D E 0400 EOt Ft G$ A SO CO DO 0337 C D Et Fx G$ A B 0407 BOOC 0 D$ E F GO At 0330 G A B$ Cx D$ E F$ 0400 FO, GO A$ 3 C Dt0 E4 0330 D E Fx Gx At ? C$ 0409 C Ct D Et F$t G At 0S

0340 A 0 Cx Dx Et F$ G$ 04100 GO AO B$ C$ D E0 F 0341 E F$ Gx Ax B$ C$ D$ 0411 DO EO Fx G$ A BS C 0342 ? C$ Dx Ex Fx G$ A$ 0412 Ab Bt Cx D$ E F G 0343 F$ G$ Ax Bx Cx D$ Et 0413 E0 F Gx A$ ? C D W0344 Fb0 GOO AO St CO DO E, 0414 0S C Dx E$ F$ G A

( Table Continues)


11i8

0415 F G Ax B$ C$ D E 0485* CO D ES F Gb AS B3

0416 C D Ex Fx G$ A B 0486* GO A B3 C Db ES F 0417 G A Bx Cx D$ E F$ 0487* DO E F G AS B3 C

W0418 FO GOb A Bb CS Db EbO 0488* AS B C D ES F G 0419

Cbb DOb E F GO AS B3O 04890 ES F$ G A 3B C D

0420 GbO ASb

B C DO ES FO 0490* Bb C1 D E F G A

0421 DOb Ebb F$ G AS B3 CO 04910 F G$ A B C D E 0422

Ab BOO C$ D ES F GO 04920 C D$ E F$ G A B

0423 Ebb

FO G$ A B3 C DO 04930 G A$ B C$ D E F$ 0424 BOb CO D$ E F G AS 04940 D E$ F$ Gj A B C$ 04250

Fb GO A$ B C D ES 04950 A B CQ D$ E F$ G$

04260 CO DO E$ F$ G A 3B 0496 E Fx G$ A$ B C$ D$ 04270 GO AS B$ C$ D E F 0497 B Cx D$ E$ F$ G$ A$ 0428 DO ES Fx G$ A B C 0498 F$ Gx A$ B$ C$ D$ E$ 0429 AS Bb Cx D# E F$ G 0499 C$ Dx E$ Fx G$ A$ B$ 0430 ES F Gx A$ B C$ D 0500 G$ Ax B$ Cx D$ E$ Fx 0431 B0 C Dx E$ F$ G$ A 0501 D$ Ex Fx Gx A$ B# Cx 0432 F G Ax B$ C$ D$ E 0502 A$ Bx Cx Dx E$ Fx Gx 0433 C D Ex Fx G$ A$ B ,0503 Fbb Gb ASS B3b Cbb Db ESb 0434 G A Bx Cx D$ E$ F$ 0504 COO DO ESb FO GOO AO 3BO

,0435 Fbb GOO A Bb C Db Ebt 0505 Gbb AS B3b CS Dbb ES FO 0436 Cbb Dbb E F G AS BOb 0506 DbS ES Fb Gb Ab) Bb CO 0437 GOb Abb B C D ES FO 0507 'ASb Bb CO DO Ebb F GO 0438 Dbb Ebb F$ G A Bb CO 0508 Ebb F Gb AS B3b C DO 0439 AOO BOO C$ D E F GC 0509 BO C DO ES FO G AS 0440 EOO FO G$ A B C DO 05100 FO G AS B3 CS D ES 0441 BOO C0 D$ E F$ G AS 0511* CS D ES F GS A BD 04420 F0 GO A$ B C$ D ES 0512* GO A B3 C DO E F 04435 CS DO E$ F$ G$ A BD 05135 DO E F G AS0 B C 04440 GO AS B$ C$ D$ E F 05140 AS B C D ES F$ G 0445 DO ES Fx G$ A$ B C 05150 ES F$ G A BD C$ D 0446 AS B3 Cx D$ E$ F$ G 05160 B3 C$ D E F G$ A 0447 ES F Gx A$ B3$ C D 05170 F G$ A B C Di E 0448 B3 C Dx E$ Fx G$ A 05180 C D$ E F$ G A$ B 0449 F G Ax B$ Cx D$ E 05190 G At B C$ D E$ F$ 0450 C D Ex Fx Gx A$ B 05200 D E$ F$ G$ A B$ C$ 0451 G A Bx Cx Dx E$ F$ 0521 A B$ C$ D$ E Fx G$

W0452 FOO GOO A B C DO EOO 0522 E Fx G$ A$ B Cx D$ 0453 COO Dtt E F$ G AS BOO 0525 3 Cx D$ E$ F$ Gx A$ 0454 GOO AOO B C$ D ES FO 0524 F$ Gx A$ B$ C$ Dx E$ 0455 DOO EOO F$ G$ A BD CS 0525 C$ Dx E$ Fx G$ Ax B$ 056 AOO BOO C$ D$ E F GO 0526 G$ Ax B$ Cx D$ Ex Fx 04,7 EOO FO G$ A$ B C DO 0527 D$ Ex Fx Gx A$ Bx Cx 0458 B3O CS D$ ES F$ G AS 0 0528 Ftt GO Att B3 CS Dtt Ett 0459" FO GO A$ B$ C$ D ES 0529 COO DO ESS FO GO AS BOO 0460 CS DO E$ Fx G$ A BD 0530 GOO AO BOO CO DO EOO FO 0461 GO AS B$ Cx D$ E F 0551 DOO ES FO GO AS BOO CO 0462 DO ES Fx Gx A$ B C 05352 ASS BE CS DO ES FO GO 0463 AS BD Cx Dx E$ F$ G 053355 ES F GO AS BD CS D 0464 ES F Gx Ax B$ C$ D 0534 BOO C DO ES F GO AS 0465 BD C Dx Ex Fx G$ A 05355 FO G AS BD C DO ES 0466 F G Ax Bx Cx D$ E 05360 CS D ES F G AS BO

W0467 FSS GOO A$ B C DS EOO 0537* GO A BD C D ES F 0468 COO DOO E$ F$ G AS B O 0538* DO E F G A BD C 0469 GOO ASS B$ C$ D ES FS 0539* AS B C D E F G 0470 DSO ESS Fx G$ A BD C 0540* ES F$ G A B C D 0471 ASS BOS Cx D$ E F G 054M1 BS C$ D E F$ G A 0472 ED FO Gx A$ B C DO 05420 F G$ A B C$ D E 0473 BOO C Dx E$ F$ G AS 0543 C D$ E F$ G$ A B 0474 FO GO Ax B$ C$ D ES 0544" G A$ B C$ D$ E F$ 0475 CS DO Ex Fx G$ A B 09545* D E$ F$ G$ A$ B C$ 0476 GO AS Bx Cx D$ E F 0546" A R$ C$ D$ E$ F$ G$

W0477 FSS GO ASS BOS COO DSS ESS 0547 E Fx G$ A$ B$ C$ D$ 0478 COO DO ESS FO GOO ASS BSS 0548 B Cx D$ E$ Fx G$ A$ 0479 GOO AS BDO CS DSS ESS FO 0549 F$ Gx At B$ Cx D$ E$ 0480 DSS ES FO GO ASS BOO CO 0550 C$ Dx E$ Fx Gx A$ B$ 0481 A BDO CS DO ED, FO GO 0551 G$ Ax B$ Cx Dx E$ Fx 0482 EP. F GO AS SSS CO DO 0552 D$ Ex Fx Gx Ax B$ Cx 0483 BOO C DO ES FO GO AS 0553 At Bx Cx Dx Ex Fx Gx 0484* FO G AS BD CS DO ES 30554 FSS GO ASS BOO CS DO ESS

(Table (ontinues)


119

0555 COO DO ESS FO GO AS BOO 0625 ASS B3 CO D ES F GO 0556 GOO AO BOO CO DO ES FO 0626 EOO F GO A B3 C DO 0557 DSS ES FO GO AS BO CO 0627 B3O C DO E F G AS 0558 AOO BO CO DO ES F GO 06289 FO G AS B C D ES 0559 EOO F GO AS B3 C DO 0629* C k D ES F$ G A 3B 0560 BO C DO ES F G AS 06s0* GC A B3 C$ D E F 05610 FO G AS B3 C D ES 06351 DO E F GC A B C 05620 CS D ES F G A B3 06352 AS 8 C D$ E F$ G 0563 GO A BO C D E F 06533 ES F$ G A$ B C$ D 0564* DO E F G A B C 0634* O C$ D E$ F$ GO A 05650 AS B C D E F$ G 0650 F G$ A SB C$ DO E 0566 ES F$ G A B C$ D 0656 C Dt E Fx G$ At B 0567 3 BO C$ D E F$ Gt A 0657 G At B Cx D $ Et Ft 0568 F Gt A ct$ Dt E 06538 D E$ F$ Gx A$t $ C 0569* C Dt E F$ Gt A B 0659 A B$ C$ Dx E$ Fx G$ 0570 G A$ B C$ Dt E$ F$ 0640 E Fx Gt Ax B$ Cx D$ 05710 D Et F$t G At Bt C$ 0641 B Cx Dt Ex Fx Gx At 0572 A Bt Ct Dt Et Fx Gt 0642 F$t Gx At Dx Cx Dx Et 0575 E Fx Gt At Bt Cx DI P0645 FO GO AS BO C DO EOO 0574 B Cx Dt E$ Fx Gx At 0644 COO D.5 EOO F G AS BOO 0575 Ft Gx At Bi Cx Dx Et 0645 GOO AO 3B C D ES FO 0576 C$ Dx E$ Fx Gx Ax B$ 0644 DF EO FO G A B3 CO 0577 Ct Ax Bt Cx Dx Ex Fx 0647 ASS B3 CO D E F GO 0578 Dt Ex Fx Gx Ax Bx

Cx 0648 EOO F GO A B C DO W0579 FSS GO ASS 3BO C DO EOO 0649 B3O C DO E F$ G AS 05880 CO DO ESS FO G AS 3BO 050

F G AS B Ct D ES 0581 GOAOBC B B CO D ES FC 06510 CO D ES F$ G$ A BO 0582 D11 E D FO GC A BO CO 06520 GO A BO Ct D E F 05815 AS BO CS D E F GO 06250 DO E F G$ A$ B C 0584 EO F G AS B C DO 06540 AS B C Dt E$ F$ G 0565 0s C DO EB F$ G AO 06550 E F$ G At Bt C$ D

0584 FC G AS BS C$ D ES 0656 BO Ct D Et Fx Gt A 05870 CS D EC F G$ A BO 0657 F G$ A B$ Cx D$ E 0588 G CG A B C D$ E F 0658 C D$ E Fx Gx A$ B

05891 DB E F G A$ B C 0630 G A$ B Cx Dx E$ F$ 05900 AS B C D Et F$ G 0660 D E$ F$ Gx Ax B$ C$ 05110 ES F$ G A B$ C$ D 0661 A B$ C$ Dx Ex Fx G$ 0592 BS C$ D E Fx G$ A 0662 E Fx G$ Ax Bx

Cx Dt 0595 F G$ A B Cx D$t E 0665 FO CSG AS 3B C DO EOs 0594 C D$ E F$ Gx A$ 0 0664 COO DO ESS F$ G AS BP3 0595 G At B C$ Dx E$ F$ 0665 GOO AS 3BO C$ D ES FO 0596 D E$ F$ G$ Ax 3O CI 0666 DOO ES FO G$ A BO CS 0597 A 3$ C$ I$ Ex Fx G$ 0667 AO B O CS D$ E F GO 0598 E Fx GO A$ x Cx DO 0668 ESSF G A$ B C 90 W0599 Fbb Gb

ASS BS CS Dbb

ESS 0669 B3O C DO Ej F$ G AS 0600 CbS Db

EOO F GO ASS 3BO 06700 FO G AS 3B C$ D ES 0601 CGO AO BS O C DO EOO FO 0671 CS D ES Fx GI A BO 0602 DOO ES FO G AS BOO CO 0672 GC A B3 Cx Dt E F 0605 AS BOS CO D ES FO CS 0675 DO E F Gx A$ B C 0604 EOO F GO A Bs CS DO 0674 ASB C Dx E$ F$ G 0605 BOO C DO E F GO AS 0675 ES F$ G Ax 3B C$ D 0606 FO G A D C DO ES 0676 B3 C$ D Ex Fx GO A 0607* CS D ES FO G AS 3B 0677 F GO A 3x Cx DO E 0608* GO A 3B CO D ES F ,0678 FOO CG AS 3BO COO DSS EOO 06090 DO E F GL A B3 C 0679 COO DO ES FO GOO AOO BOO 06100 AS 3 C DO E F G 0680 CGO AS 35 CS DSS ESS FO 0611* ES F$ G A$ B C D 0681 DOO ES F GO A3S BOO CO 06120 sB C$ D E$ FO G A 0682 AOO sB C 1DO EO FO GO 06153 F GO A B$ CI D E 0683 EOO F G AS B3O CS DO 0614 C DO E Fx G$ A 3 0684 BOO C D ES FO GO AS 0615 G A$ B Cx DO E F$ 06850 FO G A BO CS DO ES 0616 D E$ FO Gx AM B CI 0686O CS D E F GO AS 3B 0617 A B$ CO Dx E$ F$ G$ 0687* GS A C DO ES F 0618 E Fx G$ Ax 3B Co D$ 0688* DO E F$ G AS BS C 0619 a Cx D$ Ex Fx G$ A$ 0689*

AS B C$ D EK F G 0620 F$ Gx Al Bx Cx D$ E$ 06900 ES F$ GO A BO C D ,0621

FSS GS AO s3 CS DS ESS 0691"

BO C$ D$ E F G A 0622 COO DS ESS F GS AS 3BS 06920 F GO A$ B C D E 0625 GOO AO sB O C DO ES FS 0693* C DO El FO G A B 0624 DSS ES FO G A BO CS 0690"

G AS B3 CO D E F$ ( Table Conhtiues)


120

0695 D E$ Fx GO A B CO 0765* C DO E$ FO GI At B 0696 A B$ Cx DB E FB GB 0760 G At BO CO Dt ES FO 0697 E Fx Gx A B CO DO 0767 D ES Fx GB A$ BO CS 06 B Cx Dx E$ FO GO At 0768 A B$ Cx D5 El Fx GO 0699 F$ Gx Ax BO CQ DO El 0769 E Fx Gx At B$ Cx D1 0700 CS Dx Ex Fx GB At B$ 0770 B Cx Dx ES Fx Gx At 0701 GB Ax Bx Cx DO Et Fx 0771 F$ Gx Ax B$ Cx Dx ES

,0702 FOP Gk A) BO) CO) D) E) $ 0772 CB Dx Ex Fx Gx Ax B) 0703 CO) DO E) F) Gk) A) BS 0773 GB Ax Bx Cx Dx Ex Fx 0704 G1# A) BO C) D), E) FO) ,774 F)) Gk A) BO) C DO EOO 0705 DO, EO F Gk AP) B) CO 0775 CO) D) E) F) G A) BO) 0706 AP) B) C DO EOO F Gk 0776 Gk) A) BO C) D E) F) 0707 EOO F G A) BOO C DO 0777 DOO E) F Gk A B) C) 0708 BOO C D E) F) G A) 0778 AP) B) C DO E F Gk 0709* F) G A B) CO D E 06779 EO F G A) B C D) 0710 C) D E F Gk A Bk 0790 B) C D E) F$ G Ak 0711* Gk A B C DO E F 0781* F) G A BO C$ D E) 07120 DO E FO G A) B C 0782 CO D E F GO A B) 0713* A) B CO D E) F$ G 0783* Gk A B C DO E F 0714* E) F$ GO A B) CQ D 0784* DO E F$ G At B C 0715* B) CO DO E F GB A 0780 AP B CQ D E$ F$ G 07160 F GO At B C DO E 0760* E) F$ G A B$ C$ D 0717* C DO E5 F$ G AS B 0787 B) CS DO E Fx Gt A 07189 G At BO CS D ES FO 0788 F GO At B Cx DI E 0719 D ES Fx GO A B$ CS 0789 C DO ES F$ Gx At B 0720 A BO Cx DO E Fx GO 0790 G At B$ CS Dx ES F$ 0721 E Fx Gx A B Cx D1 6791 D ES Fx GO Ax B$ C$ 0722 B Cx Dx E$ F$ Gx At 0792 A B$ Cx D1 Ex Fx GO 0723 FO Gx Ax Bs CS Dx ES 0793 E Fx Gx At Bx Cx DO 0724 CS Dx Ex Fx GO Ax BO W0794 F) Gk A SB C DC 1 EPP 0725 GB Ax Bx Cx DO Ex Fx 0795 CO) DO E) F Gk AP) BO)

,0726 F)) Gk AO) B) CO Dl, EPP 07096 Gk A) BO C DO EP) F) 0727 CO) DO E) F) Gk AP) BOO 0797 DO) E) F G A) BOO C) 0728 GOO A) B) C) DO EPP FO 0708 AP) B) C D E) F) Gk 0729 D1) E) F Gk A) BOO C) 0799 EPP F G A BO CO DO 0720 AP) BO C DO E) F) Gk 0900 BOO C D E F Gk A) 0751 EPP F G A) B) CO DO 0901* F) G A B C DO E) 0752 BOO C D EO F Gk A 0mT02* CO D E F$ G A) B) 0733* FO G A B) C DO E 0905* Gk A B C$ D E) F 0734* C) D E F G A) BO 0w04* DO E F$ GO A BO C 0735* Gk A B C D E) F 050s* A) B C$ D) E F G 07360 DO E F$ G A B) C 0o06* E F$ Gt At B C D 0737* A) B Cl D E F G 0807* BO C$ D) Et F$ G A 0739* E) F$ GO A B C D 08*' F GO At B$ CS D E 073 B) C$ D1 E F$ G A 06m C D1 Et Fx GO A B 07400 F GO At B C$ D E 0810 G At B$ Cx D1 E F$ 07410 C D1 Et F$ GO A B 0811 D Et Fx Gx At B C$ 0742 G At B$ C 1 D) E F$ 0812 A B$ Cx Dx Et F$ GB 0743 D Et Fx GO At B C$ 0815 E Fx Gx Ax B$ C$ D) 0744 A B$ Cx DO E$ F$ GO 0814 B Cx Dx Ex Fx GO At 0745 E Fx Gx At B$ C$ D) 0815 F$ Gx Ax Bx Cx D1 Et 0746 B Cx Dx Et Fx G9 At ,0816 FPO GO AW BO C) DW EOO 0747 F$ Gx Ax B$ Cx D1 Et 0817 COO DW EO F GO A) BOO 0748 C$ Dx Ex Fx Gx At B$ 0818 GOO AO BO C DW E) F) 0746 GO Ax Bx Cx Dx Et Fx 0819 DO) E) F G AW BO C)

,07.5 FO0 GO A) BOO C) DC0 EDO 0620 AP) B) C D E0 F GO 0751 COO DO E) F0 Gk A) BOO 0821 EO F G A B) C DO 0752 GOO A) BO C) DO E) FO 0622 BOO C D E F G AO 0753 D)) E) F GO A) B) C) 026m* F) G A B C D E) 0754 AP) BS C DO EO F Gk 0824* CO D E F$ G A BS 0755 EP F G A) BS C DO 082P GO A B C$ D E F 0756 S B C D E) F G A) 0826* DO E F$ GO A B C 0757* F) G A B C D E 027* A B C$ D1 E F$ G 0758* C) D E F G A B) M092* E) F$ GO At B C$ D 07W5 GO A B C D E F 0820 B C$ Dt Et F$ Gt A 07600 D) E F$ G A B C 06W F GO At B$ CS DO E 0761* A) B C$ D E F$ G 0831 C DO Et Fx GO At B 07627 E) F$ Gt A B C$ D 0832 G At Bt Cx Dt Et Ft 0765" B) Ct Dt E F5 Gt A 0833 D Et Fx Gx At 59 Ct 0764* F GB Al B CS D1 E 0834 A B5 Cx Dx ES Fx G9

(Tabl (Contrmxs)


121

OiS5 E Fx Gx Ax B$t Cx DI 05 C Dt Ex Fx Gt At B 0836 Cx Dx Ex Fx Gx At 006 G At Bx Cx Dt Et Ft 067 Ft Gx Ax Ix Cx Dx Et $ 0!07

F GC A Bi C DO EO 56 D F EG CO AO AB C DC EO 0908 COO DI E F G AO BOO 3956 COO DB EO F G A Bi 0960 GOO AO B C D Ei FO 64 GOO AO Di C D Ei FO 610 Dii Ei F$ G A Bi CO 0041 DO E F G A Bi CO 011 Au iB Ct D E F GO 042 AO Bi C D E F GC n12 EO 0F Gt A B C DI 083 ECE F G A B C DI 0915 CBOC Dt E F CG AB 664 B b C D E FI G A 09140 FG C At B C$ D EF 0845 FC G A B CS D EB 0t15* CO D El FI GCI A D W4e* CO D E FI GC A BO 09160 GC A BI CI DI E F 0470 GC A 3 Ct DI E F 0917 D E Fx CG At B C 088 DO E FI GC At B C 016s A DB Cx D El FI G aw AO B C$ DI El FI G 0919 E F GCx At Bt CI D ew E FI GC At IB C$ D 062 Bi CI Dx El Fx CG A 51sSl D C Dt E Fx GCt A 0921 F GCt Ax s Cx DI E 6052 F G A$t B Cx DI E 0N22 C DI Ex Fx Cx AI $ 0865 C Dt E$ Fx Gx A B 0926 C At Sx Cx Dx El FI 0854 G A SB$ Cx Dx E FI 06924 F D GC A C DO EA O s65 D E Fx GCx Ax Bt CI 0925 Ci Di E FI G Ai BOi 66 A DI Cx Dx Ex Fx CGI 20 G Ci A B C$ D Ei Fi 6"57 E Fx Gx Ax Bx Cx Dt 6027 Di Ei Ft Gt A DB Ci w656 Fi

CGi Ai B C Di Eu 6626 Au bi CI DI E F CG 6656 Cii Di Ei FI C Ai DOiB 02 Eu F CGI A B C Di 060 CG AO DB CI D Ei Fi 650 DiiC DI El Ft G Ai 0661 Dii Ei F CI A BD Ci 0310 Fi C At sl C$ D Ei 0862 Au Bi C DI E F CG 0632 Ci D El Fx CG A BD 086s Ei F C At B C Di 6635 C A D1 Cx Dt E F

0664 Di C D Et Fi C Ai 0654 Di E Fx Cx At B C 6m65 Fi C A BD CI D Ei O6S5 AiB Cx Dx Et Ft C 0"6 CO D E Fx CI A BD 0936 Ei Ft Cx Ax DB CI D 0867 Ci A 6 Cx DS E F 0967 DB CI Dx Ex Fx Ct A 068 Di E FI CGx At! C 0C38 F Ct Ax Dx Cx D E 0666 Ai Ct Dx El FI C G 6539 F GiC AIt C D Eu 0670 E FI CGI Ax Dt CI D 09640 Ci Di El FI C Ai Dii 0871 BD CI DI Ex Fx CGI A 0941 GC Ai DI CI D Ei FO 0672 F GI

1At x Cx DI E 0942 Dii E Fx CGI A BD Ci 60875 Fii CG A D1 Ci Dii Eu 0964 Au BD Cx DI E F CG 0874 Cii Di E F CG Au DBO 0644 ED F Cx AI B C Di 0875 CG Ai C Di Eu Fi 0945 BDi C Dx El FI CG A 0876 Di Ei FI C Ai Bi Ci 0464 Fi C Ax BI CI D Ei 0877 AuO BO CI D Ei FC G 6 0947 Ci D Ex Fx CGI A BD 0878 EuO F CG A BD Ci Di 096 CG A Bx Cx Dt E F 869 BiiOC Dt E F GC

A 0 6461 F GC Ai Dii CO uDi Eii 880 Fi C At B C Di Ei o~o Cu D Ei Fi CiG Ai Bi i 061 Ci D Et Ft C Ai Di 651 CiG A BD Ci Dii Eu Fi 0686' iG A SB C D Ei F 0652 Dii E F CG Au DBO Ci 06 Di E Fx CG A BD C 6953 A DB C Di Eu Fi CG 06 AiB Cx Dt E F C 954 Eu Ft C Ai Dii Ci Di

665 Ei FI Cx At S C D 0655 B D CI D Ei Fi CG Ai 66 Di CI Dx Et FI C A 0656' F CGI A BD Ci Di Ei 06887 F CGI Ax BS CI D E 06570 Ci DI E F Ci Ai BD 0888 C DI Ex Fx CGI $A B 06 CG At B C Di Ei F 0889 C At Bx Cx DI E FI 6959 Di El Ft C Ai BD C 6o66 FiO GC A BD Ci DO EuO 09600 Ai BI CI D Ei F C 0861 Ci Di E F CG Ai BOi 0961 E Fx CGI A BD C D 0862 CiG Ai B C Di Ei Fi 0962 BD Cx DI E F CG A 06893 Dii E Ft C Ai BD C 60963 F GCx A B C D E 0684 AuO BO CI D Ei F Ci 096 C Dx El FI CG A B 0865 Ei F CGt A BD C Di G65 C Ax BS CI D E FI 0896 ii C DI E F C Ai 066 D Ex Fx CGI A B CI 0867 Fi C AI B C D Ei 0967 A Bx Cx DI E F CGI 066 Ci D El FI C A BD W0968 Fi GC Ai BOi CPi Di Eu 086 CG A BI CI D E F 0966 Ci D Ei Fi CiG Ai BDi 60 Di E Fx CGI A B C 670 G Ci A Bi Ci Di Ei Fi 661 AiB Cx DI E FI C 6071 Dup E F CG Au BO Ci om Ei FI Cx AI S Ct D 6072 Au B C Di EuO F CG 0om B CI Dx El FI GIC A 6073 Eu FI C Ai BOi C Di As"4 F C Ax I CI DI E 674 BD CI D Ei Fi C Ai

(T~bI (xonxxxs)


122

0975* FO G$ A BO CO D E, 1045 Cbb D EO F GO Abb Bbb 0976* CO D$ E F GO A BO 1046 Gbb A BO C DO Ebb FO 0977* Gb A$ B C Db E F 1047 Db E F G Ab BOb Cb 0978* Db E9 F$ G Ab B C 1048 A O B C D Eb Fb Gb 0979* Ab B q C D Eb F$ G 1049 Eb FS G A BO Cb Db 0980 Eb Fx G$ A BO C$ D 1050 Bbb C$ D E F Gb Ab 0981 Bb Cx D$ E F G$ A 1051* Fb G$ A B C Db Eb 0982 F Gx A9 8 C D$ E 1052* Cb D 9 E F$ G Ab BO 0983 C Dx E9 F$ G At B 1053* Gb Al B C$ D Eb F 0984 G Ax Bl Cl D El FI 1054* Db El F$I G A BO C 0985 D Ex Fx G$ A Bl CQ 1055* Ab B CI D$ E F G 0986 A Bx Cx D$ E Fx G$ 1056 Eb Fx G$ Al B C D

W0987 Fbb G Ab BOb Cb Dbb Ebb 1057 BO Cx D$ El F$ G A 0988 COb D Eb Fb Gb Abb Bbb 1058 F Gx Al Bl CQ D E 0989 Gbb A B0 Cb Db Ebb Fb 1059 C Dx El Fx G$ A B 0990 Db E F Gb Ab BOb Cb 1060 G Ax BI Cx D9 E FI 0991 AO B C Db Eb Fb Gb 1061 D Ex Fx Gx Al B C$ 0992 EOb F$ G Ab BO Cb Db 1062 A Bx Cx Dx El FS GI 0993 Bbb C D Eb F Gb Ab W1063 Fbb G Ab B0 Cb Db Ebb 0994* Fb G9 A sb C Db Eb 1064 Cbb D Eb F Gb Ab BOb 0995* Cb D$ E F G Ab B0 1065 G b A B0 C Db Eb Fb 0996* Gb A$ B C D Eb F 10966 Db E F G Ab Bb Cb 0997* Db El F$ G A BO C 1067 A O B C D Eb F Gb 0998* Ab B3 C$ D E F G 1068 Ebb F$ G A k B C Db 0999 Eb Fx G$ A B C D 1069 Bbb CQ D E F G Ab 1000 bO Cx DW E F9 G A 1070* Fb G$ A B C D Eb 1001 F Gx Al B C$ D E 1071* Cb D$ E F$ G A BO 1002 C Dx El F$ G$ A B 1072* Gb A$ B CI D E F 1003 G Ax BI CI DI E FI 1073* Db El FI G$ A a C 1004 D Ex Fx G$ Al B C$ 1074* Ab Bl CQ D1 E F$ G 1005 A Bx Cx DW El F$I G$ 1075 Eb Fx .G$ Al B C D

91006 Fbb G Ab BOb C Db Ebb 1076 B0 Cx DS El F$ G$ A 1007 Cbb D Eb Fb Gb Ab BOb 1077 F Gx Al B$ C$ D 9 E 1008 G b A BO Cb Db Eb Fb 1078 C Dx El Fx G$ Al B 1009 Dbb E F Gb Ab B0 Cb 1079 G Ax Bl Cx D$ E9 F$ 1010 AOb B C Db Eb F Gb 1080 D Ex Fx Gx A$l E CQ 1011 Ebb F$ G Ab B0 C Db 1081 A Bx Cx Dx E$ Fx G$ 1012 Bbb C D Eb F G Ab 0 1082 Fbb G Ab BO C Db Ebb 1013* Fb G$ A B C D Eb 1083 Cbb D Eb F G Ab BOb 1014* Cb D 9 E F G A BO 1084 GO b A B C D Eb Fb 1015* Gb A$ B C D E F 1085 Db E F G A BO Cb 1016* Db El F$ G A B C 1086 A O B C D E F Gb 1017* Ab Bl C$ D E F G 1087 Eb b F G A B C Db 1018 Eb Fx G$ A B C$ D 1088 Bbb C$ D E F$ G A 1019 B0 Cx D9 9 E F$ G$ A 1089* Fb G$ A B C$ D Eb 1020 F Gx AA B CI DI E 1090* Cb D$ E FI G$ A BO 1021 C Dx El F$ G$ Al B 1091* Gb Al B C9 D$ E F 1022 G Ax B$ C$ D$ El F$ 10920 Db El F$ G$ Al B C 1023 D Ex Fx G$ Al B3 C$ 10935 A Bt C$ D$ El Ft G 1024 A Bx Cx DI El Fx G$ 1094 Eb Fx GS A$ BI CI D

w.1025 Fbb G AO BO bC Db Ebb 1095 3B Cx D 9 El Fx G$ A 1026 Cbb D Eb Fb G A BO b 1096 F Gx Al B$ Cx DL E 1027 Gbb A BO Cb D Eb FO 1097 C Dx El Fx Gx Al B 1028 DOb E F Gb A B0 Cb 1098 G Ax B$ Cx Dx El F$ 1029 AOb B C Db E F Gb 1099 D Ex Fx Gx Ax 3B C$ 1030 Ebb F$ G Ab B C Db 1100 A Bx Cx Dx Ex Fx G$ 1031 Bbb CQ D Eb F$ G Ab Il01 F?b G Ab B C DO EbP 1032* Fb G$ A BO CQ D Eb 1102 Cbb D Eb F$ G Ab Bob 1033* Cb D$ E F G9 A Bb 1103 Gbb A BO C$ D EZ Fb 1034l* G Al B C D$ E F 1104 DOB E F G$ A BO Cb 1035* Db El F$ G Al B C 1105 AMb B C D$ E F Gb 1036* Ab B9 C$ D El F$ G 1106 Ebb F$ G Al B C Db 1037 Eb Fx G$ A B$ CQ D 1107 Bbb C D E$ F$ G Ab 1038 BO Cx D$ E Fx G$ A 1108* Fb G$ A 3B C$ D Eb 1039 F Gx A$ B Cx DI E 1109 Cb DI E Fx G$I A B 1040 C Dx El FS Gx A$ B 1110 Gb At B Cx DW E F 1041 G Ax B3 Cl Dx El F$ 1111 Db El F$ Gx Al B C 1042 D Ex Ex G$ Ax b$ C$ 1112 Ab E9 C$ Dx E$ F$ G 1043 A Bx Cx 1D Ex Fx G9 1113 E9 Fx G4 Ax g9 C$ D

"1044 Fob G Ab B Cb Db Ebb 1114 Lb Cx -i9 Ex Fx G| A

( Table )(ontiues)


123

1115 F Gx A$ Bx Cx D$ E 1185 DOO E Fx G$ A BD CO 1116 FO G A BD CO DOO EOO 1186 AOO B Cx DO E F Gk 1117 COO D E F Gk Au BOO 1187 EOi F9 Gx A9 B C DO 1118 GOi A B C DO EOi Fi 1188 BOO C$ Dx E$ F9 G AO 1119 Dii E F9 G AO BDO CO 1189 Fi G$ Ax B$ C$ D Ei 1120 A B C$ D EL Fi Gk 1190 CO D$ Ex Fx G$ A BD 1121 EOO F$ G$ A BD CO DO 1191 Gk A9 Bx Cx D$ E F 1122 BDO C9 D$ E F Gk Ai 0 1192 Fui G$ A BD CO Dii Eu 1123* Fi G$ A9 B C D L Ei 1193 COO D$ E F Gk AOO BOO 1124* CO D$ El F9 G A D BO 1194 GO A$ B C DO EL FO 1125* Gk A9 B3 C9 D EL F 1195 Dii El F$ C- AO BDO CO 1126 DO El Fx G$ A BD C 1196 Aub Bi C$ D EL Fi Gk 1127 AO B$ Cx D9 E F G 1197 EuO Fx G$ A BD CO DO 1128 EL Fx Gx A9 B C D 1198 BOO Cx D$ E F Gk AO 1129 BD Cx Dx El F$ G A 1199 Fi Gx A9 B C D L Ei 1130 F Gx Ax B9 C9 D E 1200 CO Dx E9 F$ G A D BO 1131 C Dx Ex Fx G$ A B 1201 Gk Ax B$ C9 D EL F 1132 G Ax Bx Cx D9 E F$ 1202 DO Ex Fx G$ A BD C W1133 Fii G A BD Ci DO ELi 1203 AO Bx Cx D$ E F G 1134 COO D E F Gk A D BDO O 1204 Fui G$ A BD Ci DO ELi 1135 GOO A B C DO EL FO 1205 COO D$ E F Gk AO BOi 1136 DOO E F$ G AO BD CO 1206 GO A$ B C DO EL FO 1137 Au B C$ D EL F Gk 1207 Dii El F$ G AO BD CO 1138 EOO F$ G$ A BD C Di 1208 AO Bi$ C$ D Ei F Gk 1139 BOO C Db$ E F G Ai 1209 EOO Fx G9 A BD C DO 1140 Fi G$ A9 B C D EL 1210 BOO Cx DO E F G Ai 1141* Ci D$ El F$ G A BD 1211 Fi Gx A9 B C D EL 1142* Gk A$ B$ CQ D E F 1212 Ci Dx El F$ G A BD 1143 Di El Fx G$ A B C 1213 Gk Ax B$ CQ D E F 1144 AO B3 Cx D$ E F9 G 1214 DO Ex Fx G$ A B C 1145 EL Fx Gx A$ B C$ D 1215 AO Bx Cx D$ E F9 G 1146 BD Cx Dx El F9 G$ A 81216 Fii G$ A BD C Di ELi 1147 F Gx Ax B$ C9 D$ E 1217 COO D$ E F G Ai BDO 1148 C Dx Ex Fx G9 A9 B 1218 GO A$ B C D EL FO 1149 G Ax Bx Cx D$ El F9 1219 DOO El F9 G A BD CO 81150 Fii G A BD C DO ELP 1220 AuO B3 C$ D E F Gk 1151 COO D E F G Ai BDO 1221 ELi Fx G$ A B C Di 1152 GOi A B C D EL FO 1222 BDO Cx DO E F$ G Ai 1153 DOO E F$ G A BD Co 1223 Fi Gx A$ B C$ D EL 1154 AD B C$ D E F Gk 1224 Ci Dx El F G$ A BD 1155 EOO F$ G$ A B C DO 1225 GO Ax B$ C$ D$ E F 1156 BOO C$ D$ E F$ G AO 1226 DO Ex Fx G$ Al B C 1157 Fi G$ Al B CQ D Ei 1227 AO Bx Cx D$ El F$ G 11580 CO D$ El F$ G$ A Bi 81228 Fui G$ A B C DD EL 11590 Gk A B3$ C9 DL E F 1229 COO D$ E F$ G Ai BOO 1160 Di El Fx G$ Al B C 1230 GO At B Cl D EL FO 1161 AO B$ Cx D$ El F$ G 1231 Dii EL F$ G$ A BD Ci 1162 EL Fx Gx Al B3 CQ D 1232 Aii B$ CS DL E F Gk 1163 BD Cx Dx EL Fx G$ A 1233 EO Fx G$ Al B C Di 1164 F Gx Ax B$ Cx D$ E 1234 BO Cx

DS El F$ G Ai 1165 C Dx Ex Fx Gx Al B 1235 Fi Gx Al B$ C$ D EL 1166 G Ax x Cx Dx E FI 1236 Ci Dx El Fx BG A DB

"1167 Fu G A B C Di ELi 1237 GO Ax B$ Cx D$ E F 1168 COO D E F$ G A BOO 1238 Di Ex Fx Gx Al B C 1186 Gif A B C$ D EL FO 1239 AO Bx Cx Dx EL F$ G 1170 DO E F$ G$ A DB C

--1240 FiG $ Al B C Di E 1171 AO B C$ D$ E F Gi 1241 COO D$ El F$ G Ai BDO 1172 EOO F$ G$ Al B C Di 1242 GOO Al Bl C$ D Ze FO 1173 BOO C$ D$ El F$ G Ai 1243 DO El Fx G$ A BD CO 11740 Fi G$ Al B3 C$ D EL 1244 Aki BI Cx DL E F Gi 1175 Ci D$ EL Fx G$ A BD 1245 EOi Fx Gx Al B C Di 1176 Gi Al B3 Cx D$ E F 1246 BOO Cx Dx El F$ G Ai 1177 DO EL Fx Gx Al B C 1247 Fi Gx Ax B3 C$ D EL 1178 Ai 3B Cx Dx EL Ft G 1248 Ci Dx Ex Fx G9 A BD 1179 EL Fx Gx Ax B3 C$ D 1249 Gi Ax Bx Cx DL E F 1180 BD Cx Dx Ex Fx G$ A

-1250 Fii Gx A$ B C Di Eu 1181 F Gx Ax Bx Cx DS E 1251 Cii Dx El F$ G Ai BDi 81182 Fii G A$ B C Di EuO 1252 Gii Ax B3 C$ D EL Fi 1183 Cii D El F% G Ai ii 1153 Dii Ex Fx G9 A BD Ci

1184 Gik A 3l C9 D EL Fi 1254 Ai ix Cx D9 E F Gi


124

TABLE

2

THE MATRIX S%

C0 C C$ DW D DI Ek E E4 Fk F Ft Gk G G# Af A A# Bk B B$ Ck 59 155 251 1 1 1 2 3 3 2 5 8 7 15 15 22 37 37 59 96 96 C 59 155 251 1 1 1 2 3 3 2 5 8 7 15 15 22 37 37 59 96 96 C 35 92 149 0 1 1 1 2 2 1 3 5 4 9 9 13 22 22 35 57 57 Dk 24 63 102 87 189 189 1 1 1 1 2 3 3 6 6 9 15 15 24 39 39 D 24 63 102 87 1N 189 1 1 1 1 2 3 3 6 6 9 15 15 24 39 39 DM 11 29 47 40 87 87 0 1 1 0 1 2 1 3 3 4 7 7 11 18 18

E, 13 34 55 47 102 102 14 251 251 1 1 1 2 3 3 5 8 8 13 21 21

E 8 21 34 29 63 63 92 155 155 0 1 1 1 2 2 3 5 5 8 13 13 E1 3 8 13 11 24 24 35 59 59 0 0 1 0 1 1 1 2 2 3 5 5 F0 5 13 21 18 39 39 57 96 966 57 153 249 1 1 1 2 3 3 5 8 8 F 5 13 21 18 39 39 57 96 966 57 153 249 1 1 1 2 3 3 5 8 8 Ff 3 8 13 11 24 24 35 59 59 35 94 153 0 1 1 1 2 2 3 5 5 Gk 2 5 8 7 15 15 22 3 37 22 59 96 81 177 177 1 1 1 2 3 3 G 2 5 8 7 15 22 37 37 22 59 96 81 177 177 1 1 1 2 3 3 G# 1 3 5 4 9 9 13 22 22 13 35 57 48 105 105 0 1 1 1 2 2 AO 1 2 3 3 6 6 9 15 15 9 24 39 33 72 72 105 177 177 1 1 1 A 1 2 3 3 6 6 9 15 15 9 24 39 33 72 72 105 177 177 1 1 1 A$ 0 1 2 1 3 3 4 7 7 4 11 18 15 33 33 48 81 81 0 1 1 B 1 1 12 3 3 5 8 8 5 13 21 18 39 39 57 96 96 153 249 249 3 0 1 I 1 2 2 3 5 5 3 8 13 11 24 24 35 59 59 94 153 153 B5 0 0 1 0 1 1 1 2 2 1 3 5 4 9 9 13 22 22 35 57 57


CA)

THE MATRIX S33 COO CO C C CxDWO DO D D D Dx ErO EO E

El Ex FO Ff F F

F Fx GOO GO G G$ Gx AOO AO A At Ax Br BO B B$

Bx COO 76 254 583 1063 1543 1 1 1 1 1 2 3 4 5 5 2 5 9 14 19 7 1630 49 49 23 53-102 151 151 76178 329480 480 C 76 254 83 1063 1543 1 1 1 2 3 4 5 5 2 5 9 14 19 7 16 30 49 49 23 53 102 151 151 76 178 329 480 480 C 48 164382 702 1022 0 1 1 1 1 1 2 3 4 4 1 3 6 10 14 1 20 34 349 49 234 6853 102 1 251 1548116 2178 3 20 480 Cq 20 74181 341 501 0 0 1 1 1 0 1 2 3 3 0 1 3 6 9 1 4 10 9 1934 515 34 568 510 20 48 16 07 18 320 160 Cx 6 26 69 135 201 0 0 0 1 1 0 0 1 2 2 0 1 3 5 0 1 4 9 9 1 5 14 23 23 620 43 66 66 D0 28 90201 361 521

118 3196801201

1201 1 1 1 1 1 1 2 3 4 5 3 6 10 15 15 9 19 34 49 49 28 62 111 160 160

DO 28 90201 361 521 118 319 680 1201 1201 1 1 1 1 1 1 2 3 4 5 3 6 10 15 15 9 19 34 49 49 28 62 111 160 160 D 14 48 112 206 300 62174380

680680 0 1 1 1 1 0 1 2 3 4 1 3 6 10 10 4 10 20 30 30 14 34 64 94 94 D$ 5 20 51 98 145 25 76 174 319 319 0 0 1 1 1 0 0 1 2 3 0 1 3 6 6 1 4 10 16 16 5 15 31 47 47 Dx 1 6 18 37 56 7 25 62 118

118 0 0 0 1 1 0 0 0 1 2 0 1 3 0 1 4 7 7 1 5 12 19 19 Ef 14 42 89 155 221 56 145 300 521 521 201 501 1022 1543 1543 1 1 1 1 1 2 3 4 5 5 5 914 19 191428 47 66 66 E) 9 28 61 108 155 37 98 206 361 361 135 341 702 1063 1063 0 1 1 1 1 2 3 4 4 6 10 14 14 919 33 47 47 E 4 1433 61 89 18 51 112 201201 69181 382 583 583 0 0 1 1 1 1 2 3 4 4 3 6 9 9 4 10 19 28 28

Ex 0

!

4 9 141 1 5 14 28 28 6 20 48 76 76 0 0 0

0

1 0 0

01

1

0 0

1 2 2 0

! ,

5 55 Fx 5 14 28 47 660 19 47 9 1 6 16 66 6 6 20 48 480 " 226 746102681506 1 01 1 11 2 3 4 5 5 6 014 19419 F 5 14 28 47 66 19 47 94 160 160 66 160 320 480 480 66 226

546 1026 1506 1 1

2 4 5 5 5 9 14 19 19 FI, 3 9 129 33 47 1

3 4 3 109

64 111 2 111 43 107 218 329 32943 150 3 86 697

1026 14 14

F$ 0 02 1 3 69 410 19 28 5 15 34 62 62 20 54 116 178 17820 74 190 368 546 1 16 1 0 1 1 1 36 9 9 S 0 0 1 3 5 0 4 914

1 5 14 28 28 6 20 48 76 76 6 26 74 150 226 0 0 0

0 0 2 2

0 1 5 5 S 2 5

9 14

19 7 16 30 149 49 23 53 102 151 151 23 76 178 329 48099 277 606

1086 1086 1 1 1 1 1 2 4 5 5 GA

1 3 6

10 14

41 0

4 10 20 34 34 14 34 68 102 102 14 48

16 218 320 62 178

3 716

716 0

1 1

4 4 G$ 0

!1

26 9 1 3 6

6 4 10 19 19 5 15 34 53 53

5 20

54 107 160 25 79 186

S 346

0 0

!

6 0

!

1 Gx

0 0 1 3 5

0 1 4 9 9

1 5 14 23 23

1 6 20 43 66 7 27 70 136 136 0 0 0

1 1 0 0 1 2 2

A 1 2

3 4 5

3 6 10

15 15 9

1 9

34 49 49

9 28 62 1111

160 7 99 210 370 37016 716

1086

1086 1 1 1 1

1!

3 4 1 6 1 1 4 1 2 3 30 4

14 34 64 94 18

52 116 210 210 70 186 396 606

606 0 1 1 1 1 A$ 0 0

1 2

3 0

1 3 6 6 1 4 10 16 16 1 5 15 31

4752 99 99 27 79 178 277 277 0 0

1

! ! B 0 0 1

! !

0 1 2 3 3 3 1 4 10 19 28 5 15 34 62 62 20 54 116 178 178 74 190 368 546 546 Bt 0 0 0 1

!

0 0 1 2 2

0

!1 3 5

0 4 9 14 1 5 14 28 28 6 20 48 76 76 26 74 150 226 226 Bx 0 0 0 0 0 0 0 0 0 2 0 0

3 5 0 1 4 9 9

!

5 14 23 23 6 20 43 66 K) Cr


126

RE F E R E N C ES

1 Busoni, Ferruccio. Entwurf einer neuen Aesthetik der Tonkunst (1907). Eng- lish trans. by Th. Baker (New York: G. Schirmer, Inc., 1911), pp. 29-30.

2 Barbour, J.M. (a) "Synthetic Musical Scales"' The American Mathematical Monthly, 36

(March 1929). pp. 155-160. (b) "Musical Scales and their Classification", The Journal of the Acoustical

Society of America, 21/6 (Nov. 1949), pp. 586-589.

3 Mason, R. M. "A Formula, Monogram, and Tables for Determining Musical Interval Relationships", Journal of Research in Music Education, 15/2 (Summer 1967), pp. 110-119.

4 By substituting x = u7 and using the identity:

1 n2 n-1 1 +x+ x2 +... +xn- 1-x

the generating function may be put in the form:

x-5(1 - x6)7 (1- x)7

Now by using the identity:

(1 -

x)= 1 + (n)x + n+ 2 )x2 + ..., Ixl

enumeration of synthetic musical scales by matrix algebra

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