Page  348 ï~~ Combination Product Sets and Other Harmonic And Melodic Structures John H. Chalmers, Jr, Department of Pathology, Baylor College of Medicine, Houston, Texas 77030, and Ervin M. Wilson, 844 N. Avenue 65, Los Angeles, California 90042. INTRODUCTION: In 1965, we began a series of theoretical studies in microtonal music using the facilities of the computer center of the University of California, San Diego, where the first author was a graduate student in Biology. At that time, hand calculators did not exist, and even the simplest theoretical investigation entailed large quantities of tedious manual computation. As a first step, therefore, we set about the compilation of extensive tables of equal temperaments, linear temperaments, modified equal temperaments, and just intervals. Having obtained these research materials, we were then in a position to study individual equal temperaments in detail with the goal of identifying optimal equal temperaments for the realization of certain harmonic and melodic concepts, derived in part from Partch (1949) and Fokker (1949, 1975). The modified equal temperament and linear temperament were closely related to these optimal systems and comprised the second phase of our studies. The construction of reference tables of just intervals was the third of our projects, and it led to the discovery, by the second author, of a new class of harmonically and melodically symmetrical pitch structures or scales, the Combination Product Sets (CPS). The Combination Product Sets have proven to be both musically useful and theoretically fascinating, and an introduction into their genesis and properties is the fourth phase of our investigations. EQUAL TEMPERAMENTS: Our first project was to examine all the equal divisions of the octave containing from 5 to 120 tones in terms of the harmonic series. This was done in two parts; the first was the calculation and tabulation of the basic intervals of each system by the following method, while the second was the analysis proper or System Interval Equivalents. The octave was divided into N tones by inserting N-1 arithmetic means between 0 and 1. The resulting numbers are the binary or base-2 logarithms of the intervals. The antilogarithms of these numbers are the corresponding roots of 2, which were then multiplied by 440hz to generate pitches for the octave A to a. The reciprocals of the roots yielded string lengths,and multiplication of the logarithms by 1200 resulted in cents. The analysis part operated quite differently. In order to find the closest approximation of a given interval (i.e., 3/2) in an N-fold equal temperament, the base-2 logarithm of the interval was multiplied by N. This product was rounded to the nearest integer, and the difference between this value and the just value was calculated (in cents). The relative error or DEFECT in fractional degrees was then computed by multiplying the error in cents by N/1200. A given interval may be considered as adequately represented if the DEFECT is 0.2 or less. An excerpt from these tables is presented in Table lA. Many copies of the complete tables were distributed to other investigators such as A.D. Fokker, and M.J. Mandelbaum. A similar 1200-tone table was published in the journal, Xenharmonikon (Chalmers, 1974). We next turned our attention to the relatively unexplored region between 120 and 1200 tones per octave. In this study, we employed our analysis program with a more compact format to save space. 348 * 1982 John H. Chalmers, Jr. and Ervin M. Wilson.

Page  349 ï~~ The conclusions we drew from this first investigation were that the low numbered systems, 9,12,17,19,22,29,31,34,41,43,46,53....., are superior to their neighbors, when judged by their degree of approximation to the intervals just intonation, as has been found by other workers (Barbour, 1953; Stoney, 1970; Mandelbaum, 1961; Fokker, 1963, 1968; Hall, 1981a, 1981b; Yunik & Swift, 1980). However, if only partial sets of harmonic intervals or even non-harmonic intervals (i.e.,, the Golden Ratio), are desired for certain melodic effects, other equal temperaments might be preferred (5,7,10,13,14,15,16,24,36). With respect to the large number divisions, the 270-tone (Van Hoerner, 1974; Mandelbaum, 1961), and 612-tone (Van Prooijen, 1978; Bosanquet, 1876; Hall, 1981a, 1981b; Mandelbaum, 1961; Smith, 1978) equal temperaments stand out. The 270 tone division embodies the harmonic series adequately through the 13th harmonic, while the 612 is acceptable throught the 11th (Table 2A,B). Although such large numbers of tones are not recommended for acoustic or electromechanical instruments, they may well have utility as (hyper) chromatic fields for digital synthesis where for technical reasons, a large but still limited number of tones is available. Bosanquet-type generalized keyboards for these two system have also been designed (Wilson, unpublished). MODIFIED AND LINEAR TEMERAMENTS: Modified equal temperaments are stretched and shrunk octave variants of normal equal temperaments, where the octave itself is tempered slightly to improve the consonance of a particular harmonic interval. A good example is Ivor Darreg's variant of 31-tone equal temperament, 49 3, which yields a just 3/1 (twelfth) by stretching each octave to 1203.278 4. Analogous divisions can be constructed for other systems such as Kolinsky's, (1959) and Novaro's (1951) 7 3/2 for 12, and for other intervals, i.e., (44 5 for 19-T.E.T., and 87 7 for 31 T.E.T.. Another approach to use a least squares' technique which, with the harmonics 4,5,6, and 7, yields an octave of 1200.609823 4 for 31-tone equal temperament. Although we considered a number of shrunk octave modifications, (i.e., Fokker's "tempered" 31-T.E.T of 1198.15 4, the experience of Martin and Ward (1961) suggests that these should be less acceptable. Linear temperaments also have a harmonic genesis in that the purity of another interval in addition to the octave is maintained. The prototypes for this class of tuning are the Meantone (5/4 pure), the Helmholtzian (5/4 pure) (Ellis, 1877) and the Pythagorean (3/2 pure). These systems are models for equal temperaments, and for instruments with more than 12 tones per octave such as the Wilson-Hackleman Clavichord-19 (Hackleman, 1976) and the Scalatron (Secor, 1975), they are not only practical but may be musically superior. We programmed cycles of altered Fifths and Fourths which converged to certain harmonic intervals. Having identified such cycles, we combined them in various ways to generate new linear temperaments with novel harmonic properties (Table 3A). Examples are meantone-like systems with perfect 7/4's, F - 696.8826 4; 7/5's, F - 697.0854 4; 11/8's, F - 697.2954 4; and 21/16's, F - 697.3437 4. Positive systems similar to Helmholtz's and the 41-tone equal temperament were also constructed. With somewhat more ingenuity, systems related to the 22-tone equal temperament were also found. The least squares' method may also be applied to linear temperaments and the negative least squares' fifth for the harmonics 3,5, and 7 has 696.8843 cents (Table 3B). These experiments are discussed in detail elsewhere by Chalmers (1974b). The implications of this phase are that the 22,31, and-41 tone equal temperaments are optimal systems, for which the linear temperaments are harmonically dynamic analogues. 349

Page  350 ï~~ JUST INTONATION TABLES: Our attention was next led to the computation of extensive tables of just intervals because scales and harmonic progressions in just intonation were the musical materials which we wished to map into the various equal temperaments studied previously. Since the set of just intervals is potentially infinite, we decided to use no prime numbers larger than 41, and, furthermore, to limit the number of prime factors which could define a composite interval. These limits were achieved by a function termed Complexity, which was defined as the sum of the prime factors times the absolute value of their exponents. For example, 3/2 has a Complexity of 3; 9/8, 6; 8/5, 5; and 15/14, 15. The Complexity limit varied between 24 and 60 in various studies. A sample of these tables is given in Table lB. The 24 limit table was published in 1974, in Xenharmonikon (Chalmers, 1974a). COMBINATION PRODUCT SETS: Surprisingly, even a Complexity limit of 60 proved not to be sufficient for defining the modulations of some intracate musical structures. In order to increase the usefulness of the just tables, it was decided to print them out in several different keys or pitches. While experimenting with different sets of pitch bases for the tables, Wilson discovered a class of harmonically symmetrical pitch structures called Combination Product Sets (CPS). Combination product sets are generated by taking products of N harmonic factors, R at a time. When R - N/2, the resulting pitch sets may be partitioned into sets of paired harmonic and subharmonic chords, or alternatively, into a spectrum of lower homologues. For example, 4 factors (i.e.,1,3,5 and 7), taken 2 at a time, produce the HEXANY, a six tone set partitionable into 4 pairs of inversionally related triads. Six factors, 3 at a time, generate the EIKOSANY of 20 tones, divisible either into 15 pairs of tetrads or 15 species of Hexany, each of which occurs twice. The next homologue is the 70-tone HEBDOMEKONTANY, ( ), which partition into eikosanies, hexanies and pentads. The generation an properties of these remarkable structures are described in Tables 4,5,6,7 and Figure 1. Higher.homologues of 252, 924........ tones, as well as CPS of lesser symmetry, where R # N/2, also exist and may be found by reference to Pascal's Triangle. It should be noted that CPS are harmonically symmetrical in that no single tone is structurally a unique tonic and each tone is included in the same number of consonant chords, both harmonic and subharmonic. Combination Product Sets may also be extended to create larger tonal aggregates. A tone may be added to each of the 8 triads of the Hexany to form a complete tetrad with all four harmonic factors. The resulting 14-tone stellate hexany is termed a "Mandala" from its appearance in certain graphic representations (Table 7B). The number of tones (S) in a stellate CPS.is S - 2P-C where P=(N+ R- 1)I C- NI RI (N-1) I and RI (N-R) I Therefore, the stellate Eikosany has 92 tones and the stellate Hebdomekontany, 590. Another means of elaborating CPS is to create clusters in which CPS share common tones, intervals, chords or sub-CPS. Thus Hexanies can be joined to form 7,12 and larger tone complexes, while Eikosanies and Hebdomekontanies to form still greater aggregates. Combination Products Sets of i-tones have characteristic symmetrical melodic patterns, Im/2' 12, Il> 12, 1m/2> tm/2+l, where I1,I2*******m/2 are not necessarily 350

Page  351 ï~~ distinct intervals and Im/2+1 completes the octave. In the case of the Hexany, this reduces to c,b,a,b,c,d where a+2b+2c+d equals the octave (N). A search program was written to find articulate melodic hexanies in many of the equal temperaments examined in Part 1 and to extract from each melodic hexany the generating tetrad which, in equal temperament, corresponds to the quartet of harmonic factors. For the general melodic Hexany, c,b,a,b,c,d, the generating tetrad is 0,b, b+c, a+2b+c. The Hexany is reconstructed by summing the notes of the tetrad, 2 at a time, (modulo 12). It may be necessary to transpose by -b or rotate the Hexany to restore the original keynote and mode. As an example, the intervals, 221223, define the 12 tone Hexany, 024579, whose tetrad is 0 2 4 7. The Hexany is also known as the Guidonian hexachord and if combined with its transposition at the fifth, generates a major scale as a complex of two' Hexanies (traditionally referred to as mutation) (Table 7D). The paired chords or facets of melodic Hexanies may be found by placing the Hexany on a hexagram or octahedral lattice as in Table 4 after one triad pair has been identified. If the intervals, c,b,a,b,c,d, are summed in order, the notes, 0, c+b, and a+2b+c, are one triad and its complement, the other. These may be placed on the hexagram so that diagonally opposite tones sum to N-d. Structurally consonant chords are those which appear as triangles and which do not contain diagonally opposite tones. The number of Hexanies in a N-tone equal temperament increases roughly as an exponential function of the N and exceeds 100 by N-21 (Table 7C). Thus Hexanies, per se, are likely to be more useful in the small numbered systems; whereas in the large systems, they would most likely be used in an eikosany or higher CPS context. In 12-tone equal temperament, 16 hexanies exist (Table 7A), where they have been listed, analysed, and their occurences in the writings of other theorists noted (Forte, 1973; Martino, 1961; Babbitt, 1955; Gamer, 1967; Winograd, 1966; Rothenberg, 1966, 1975, 1978). Balzano (1978, 1980) has also developed criteria similar to Rothenberg's and Winograd's. The melodic patterns of the CPS may be extrapolated to include certain melodically symmetrical scales which do not necessarily have also their symmetrical harmonic properties. The addition of another pair of symmetrically disposed intervals to the Hexany creates an 8-tone set (octony, octatrope, etc.) with the pattern, d,c,b,a,b,c,d, e. This scale may be partitioned into 8 pairs of inversionally related tetrads, but lacks the defined chordal generator which characterizes the CPS. For this reason the -any suffix will not be used for scales of this type unless they are also CPS (Hexany, Eikosany, Hebdomekontany, etc.). The properties of an 8-tone set are described in Table 8A which shows its partition into tetrads on the oktagram or cubic lattice. As with the Hexany, the consonant chords of these scales may be found by summing the intervals and selecting alternate tones from 0. On the cubic lattice, consonant chords are of 3 types - faces, corners and corner-diagonals which consist of a vertex tone and the three tones connected to it by face diagonals. Chords containing tones on principal diagonals are dissonant.' There are only 10 different 8-tone sets of this type in 12-tone equal temperament, and these have been tabulated and analyzed briefly in Table 8B. The best known is the String of Pearls scale (#4), although #10 has also been mentioned by Wilding-White (1961) who lists several of the Hexanies too (Table 7, #10,#13,#14). In fact, the 8-tone and higher numbered homologues (10,12,14,.. 351

Page  352 ï~~..20....etc.) are more interesting in temperaments with more than 12-tones. In f3-tone equal temperament, the 8-tone set of Table 8A is a Moment of Symmetry, generated by a cycle of 5 or 8 steps (Wilson, unpublished; Chalmers, 1975, 1979) and has been used by Easley Blackwood in his "Twelve Microtonal Etudes for Electronic Music Media" (recorded in 1980). We recommend the Combination Product Sets to the computer composer who desires an organic and non-coloristic context for microtones. Combination Product Sets offer a non-diatonic, but still tonal, in the expanded sense of Partch, Fokker and others, answer to the continual search for new musical resources. One possible compositional approach, among many, would be to circumnavigate an Eikosany, Hexany by Hexany using common triad progressions with optional stellation of each Hexany. Similarly, the more complex and subtle Hebdomekontany may be transversed by Eikosanies with common tetrad or Hexany modulations. Hexanies and Hexany-clusters have been used by the composers, Glen Prior and Gary David, in several works already. The melodic versions of Hexany, Eikosany and Hebdomekontany, as well as their harmonically less symmetrical homologues, may be employed to utilize those equal temperaments considered less than optimal with regard to the harmonic series (13,18,20,21, etc.). In these systems too, acoustic consonance may be exploited, as many non-senary triads and higher chords are reasonably in tune. Even non-harmonic intervals may be made relatively consonant or dissonant by careful control of steady-state spectra by using additive synthesis with nonintegral overtones as necessary. (Kameoka and Kuriyagawa, 1969; Plomp and Levelt, 1965; Pierce, 1966). Thus, we feel that we have developed from our early results at USCD in the in the late 1960's some very general techniques for new scale construction and harmonic usage. REFERENCES Babbitt, M. 1955. The Score and I.M.A Magazine 12:53-61. Balzano, G.J. 1978. The Structural Uniqueness of the Diatonic Order, in Cognitive Structures of Musical Pitch, Symposium of the Western Psychological Association, San Francisco. 1980. Computer Music Journal 4:66-84. Barbour, J.M. 1957. Tuning and Temperament, Michigan State University, East Lansing. Bosanquet, R.H.M. 1876. An Elementary Treatise on Musical Intervals and Temperament. McMillan and Co., London. Chalmers, Jr., J.H. 1974a. Xenharmonikon 1:26-57. 1974b. Xenharmonikon 2:36-41. 1975. Xenharmonikon 4:64-78. 1979. Xenharmonikon 7&8:156-167. Ellis, A.J. 1877. In: On the Sensations of Tone as a Physiological basis for the Theory of Music, Herman L.F. Helmholtz, A.J. Ellis, Translator, Dover Publications reprint 1954. New York. 352

Page  353 ï~~ Fokker, A.D. 1949. Just Intonation. Martijnus Nijhoff, The Hague. 1975. New Music with 31 Notes. Verlag fur Systematische Musikwissenschaft GmbH. Bonn, Translated by Leigh Gerdine. 1963. Proc. Kon. Nederl. Akad. Wetensch. A66(1):1-6. 1968. Proc. Kon. Nederl. Akad. Wetensch. B71:251-266. 1969. Proc. Kon. Nederl. Akad. Wetensch. B72:154-168. Forte, A. 1973. The Structure of Atonal Music, Yale University Press, New Haven. Gamer, C. 1967. J. Music Theory 1:32-59. Hackleman, J.S. 1976. Xenharmonikon 5:46-48. Hall, D.E. 1981a. Acoustical Numerology and Lucky Equal Temperaments (in preparation). 1981b. A Systemic Evaluation of Equal Temperaments Through N-612. (in preparation). Kameoka, A. and M. Kuriyagawa. 1969. J. Acoust. Soc. Amer. 45:1460-1469. Kolinsky, M. 1959. J. Amer. Musicol. Soc. 12:210-214. Mandelbaum, M.J. 1961. Multiple Division of the Octave and the Tonal Resources of 19-tone Temperament. Indiana University (diss.). Martin, D.W. and W.D. Ward. J. Acoust. Soc. Amer. 33:582-585. Martino, D. 1961. J. Music Theory 5:225-273. Novaro, A. 1951. Sistema Natural de la Musica. Mexico. D.F. Partch, H. 1949. Genesis of a Music, University of Wisconsin Press, Madison, reprinted 1974, 1979, by Da Capo, New York. Pierce, J.R. 1966. J. Acoust. Soc. of Amer. 40:249. Plomp, R and W.J.M. Levelt. 1965. J. Acoust. Soc. Amer. 38:548-560. Rothenberg, D. 1966. Technical Report, AF-AFOSR 881-65 and 987-66 (NR-348-012). 1975. Lecture Notes in Computer Science 22:126-141. 1978. Math. Systems Theory 11:199-234. Secor, G. 1975. Xenharmonikon 4:36-40. Smith, R.K. 1978. Unpublished Letter to J. Chalmers. Stoney, W. 1970. Theoretical Possibilities for Equally Tempered Musical Systems. In: The Computer and Music, Harry B. Lincoln editor, Cornell University Press, Ithaca p. 163-171. Van Prooijen, K. 1978. Interface 7:45-56. Von Hoerner, 5. 1974. Psychology of Music 4:18-28. Wilding-White, R. 1961. J. Music Theory 5:275-286. 353

Page  354 ï~~ F Wilson, E.M. 1975. Kenharmonikon 4:61. Winograd, T. 1966. An Analysis of the Properties of "Deep" Scales in a T--Tone Temperament. Unpublished, cited by Gear, 1967. Tunik, K. and G.D. Swift. 1980. Computer Music Journal 4:60-65. TABLE I A. EQUAL TEMPERAMENT TABLE. 12-TONE TEMPERAMENT.doNo.-Momrr rrrrr rrrrr.rrrrrrrr rrw rrrrsrirr DEGREE L06S2 CENTS ROOTS PITCH STRING LENGTH +..r-m t-.~._owr~w.r"..r.rurwwr apo, 001.083333 100.00000 1.059463 466.1638.943874 --.w.".....""4-------.."M~ ". --wr"I...1 ZL46. Â~"rAtiAai.1..--.. aszaat " 3.250000 300.0000 1.189207 523.2511.840896 5.416667 500.0000 1.334840 517.3295.749134 1.583333 700.0000 1.496307 659.2551.667420 *....-...L------""rw~w a 6 "w6i ACQ I~i0 ww0A LA&QL S w /iC At "t.........abZlL--".. 9.750000 900.0000 1.681793 739.988.594634 ".wwr w",,~..Akil 3 -.".rA..QQ tQ.QQ- - wr.1a7AL12..."..71.ThQLw,..........dll -._ 11.916667 1100.000 1.881749 630.6094.529732 --- rf~wwry- - - ---- -S""rr.w"" flI tlWIL&XAJ. QL1YALLMLL..r "----.""Â~"" DEEC.-.0196 DEFECT..1369 CMTS.13'559L....Lhu..JkA~.d8. -_ DEFECT..3117 rsua. IL DEFECT. -.0391 V aaat -2 L w 1s.M 1 -t -St J DEFECT..4868 DEFECT. -.4053 DEFECTu.1173 DEFECT. -.0496 DEFECTS.0249 DEFECT. 0 DEFECT. -.2x27 DEFECT. -.2956. -..-LU- LtL.S~~---- ~~utU 'l....t r.rn ~ ~at"u B. JUST INTONATION, 24-LIMIT TABLE. * Up 0 lI it I 1S 354 34 11 14 I "1 3.3 4.4 " 0.0 1 "7 0 I "t S -a 4.a S 0 0 I 0 S 0 S 0 0 0 0 " 0 I 7 0 0 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 S 0 0 0 0 0 a 01 13 * 0 o o * 0 O" i 0 " o o * 0 * 0 o " * 0 * 0 * p * 0 o 0 o a * 0 * 0 " 0 * 0 * 0 o 0 I 0 " 0 * 0 a I lACloqas i" 1" O 0 " 0 r, 0 0 O 0 O 0 O 0 O 0 * 0 * 0 O 0 * 0 0 0 * 0 * 0 * 0 0 0 * 0 * 0 * 0 * P * 0 O t O t * 0 Â~ I U3 S B 0 0 S S 0 0 0 0 0 0 0 0 0 0 " S S 39 31 * 0 O 0 O 0 O 0 " 0 o " O 0 * 0 O 0 0 0 O 0 O 0 " 0 " 0 O 0 * 0 O 0 " 0 " 0 * 0 * 0 * 0 O 0 * 0 O 0 " I i, S 0 p 0 0 0 0 S 0 0 0 t 0 0 0 0 0 0 " I LAB~a CO15S aclooL 11 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I 0. 4111 1,4 14It9.4J+1 ito * 9s100.P'o~ttO.441401 0 701 9156 04S. 01) cO0. OigO 19e. 1060 A180. 1110 t0o.??2A lt3. ta 4*4. JtJ* iof. aIj 646.440, 194.11I4i 1. 'SOhuOS 1.9119614 1.Pd 1S3 l 1.9041') 1. 604003V &.boflhal 1.r53614 1. Si1tvOS 1.131111 1.0.1 4011 & * tenet so0bo6S 161.bwPoo 043.4.1 aO 150. 019.0 00..lo~ee Oit.3111%0 664.900 0g. 4044O 0106.60!?1 40.Uo6'd0 * 60Oo,0i 1w4. 10e.2 549. 1t10 1h0. Octs ids. S0r 11.. 1535 010. 1)*v@ 400. 004,0 aoa. hS?) ")1.40145 ere.0it +. 6oo. 1 ib.. 015110. 1SS JWBO.04)'...10w 'oAF * o150151.bs0 1405.)Is1is} ""?s)Jkw.0?)t1.0s.oO.. Fk

Page  355 ï~~ TABLE 2 A. SYSTEM INTERVAL EQUIVALENTS FOR 270-TONE EQUAL TBMPERAMENT 3/2 Defect Cents 9/8 Defect Cents 15/8 Defect Cents 23/16 Defect Cents - 158 -.0601 -.2672 - 46 -.1202 -.5344 - 245 -.1395 -.6202 - 141 - -.3617 - -1.6077 5/4 Defect Cents 11/8 Defect Cents 17/16 Defect Cents 29/16 Defect Cents a 87 ".0794 -.3530 - 124 - -.0465 - -.2068 - 24.3850 - 1.7113 - 232 -.3451 - 1.5339 7/4 - 218 Defect -.0142 Cents -.0630 13/8 - 189 Defect - -.1187 Cents - -.5277 19/16 - 67 Defect -.0596 Cents -.2648 0,r w - 187 Defect - -.4453 Cents - -1.9792 B.e SYSTEM INTERVAL EQUIVALENTS FOR 612 - TONE EQUAL TEMPERAMENT 3/2 - 358 Defect -.0029 Cents -.0058 9/8 - 104 Defect -.0059 Cents -.0116 15/8 - 555 Defect - -.0170 Cents - -.0334 23/16 - 320 Defect - -.4199 Cents - -.8234 5/4 - 197 Defect - -.0200 Cents - -.0392 11/8 - 281 Defect - -.1722 Cents - -.3376 17/16 - 54 Defect -.4727 Cents -.9269 29/16 - 525 Defect - -.0844 Cents - -.1654 7/4. 494 Defect - -.1012 Cents -"-.1985 13/8 - 429 Defect -.3309 Cents -.6488 19/16 - 152 Defect -.2684 Cents -.5262 q6, 7' - 425 Defect -.1239 Cents -.2430 355

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Page  357 ï~~ TABLE 4 GENERATION OF THE HEXANY IN JUST INTONATION 1. CHOOSE 4 HARMONIC FACTORS: A,B,C,D, i.e., 1.3.5.7 2. FORM BINARY PRODUCTS: AB, AC, AD, BC, BD, CD; 1.3 1.5 1.7 3.5 3.7 5.7 3. PLACE PRODUCTS ON HEXAGRAM SO THAT OPPOSING PAIRS CONTAIN ALL FOUR FACTORS: 1-7 A-D I-3 P-S AQ' AC 4. CONSONANT TRIADS FORM TRIANGLES; THERE ARE 4 PAIRS OF INVERSIONALLY RELATED CHORDS OR FACETS: CHORD FACETS FACETS CHORD 1.5.7 1.3 3.5 3.7 1.5 1.7 5.7 1.5.7 1.3.5 1.3 3.5 1.5 3.7 1.7 5.7 1.3.5 3.5.7 1.3 1.5 1.7 3.5 3.7 5.7 3.5.7 1.3.7 1.3 3.7 1.7 3.5 1.5 5.7 1.3.7 5. EACH TONE IS CONNECTED TO 4 OTHERS BY SIMPLE HARMONIC RATIOS. THERE ARE 6 DYADS, RAC! OCCURRING TWICE. THERE ARE 4 SPECIES OF HARMONIC AND SLJBHARW)NIC TRIAD, EACH OCCURRING ONCE. THERE ARE 8 TRIADS IN ALL AND 6 POSSIBLE TONICS. HEXANIES HAY BE LINKED BY CONW)N-TONE, COMhVN-EDGE (DYAD), OR COMMON-TRIAD MODULATIONS. 357

Page  358 ï~~ TABLE 5 A. ORNERATION OF THE ELKOSANY IN JUST INTONATION (1) CHOOSE 6 HARMONIC FACTORS, A S C D E F, I.E., 1 3 7 5 9 11. (2) FORM TERNARY PRODUCTS, A ISC, S C F, ETC., 1.E, 1.3.5, 3.7.9 ETC. (3) ARRANGE TONES ON OUTER DEKAGON SO THAT OPPOSITE TONES ARE COMPLEMENTS, AND THAT ANT THREE SUCCESSIVE TOMES IOM A TRIAD, HARMONIC OR SURNARMONIC FOLLO PATTERN IN PART C, MELON. (4) ARRANGE THE REMAINING TONES ON THE INNER DELAO0N S0 THAT EACH TONE FORMS A TETRAD WITU ITS NEAREST THREE OUTER DEKADON NEIONSORS, WITU THE SAN ERULES AS (3) ABOVE. (5) TUE EJEOSANT HAS THE MELODIC PATTERN, jchgtedcbabcd.fghljk, AND THEREFORE, MELWDIC RIKOSANIES EXIST. S. PARTITIONS OF THE EIKOSANY (1) EACH TONE IS CONNECTD T0T9 OTHERS BY SIMPLE NARNiNIC RATIOS. TMEE ARE IS DYADS, EACH OCCURRING 6 TIMES. (2) THERE ARE 20 SPECIES OF HARNONIC AND SUSHAUWNIC TRIAD, EACH OCCURRIM3 3 TICS. (3) THERE ARE 15 SPECIES OF HARMNIC AND SUSHARISNIC TETRAD, EACH OCCURAING ONCE. (4) THERE ARE 15 SPECIES OF HEXANY, EACH OCQRRING TWICE. (5) THE ENTIRE EIKOSANY CAN SE TRAVERSED SY HEEANIES USING CODON-TRIAD MODULATIONS. C. ABC DEF EIKOSANY BCF ABF ABAC BEF F C /EBC YE AI0 CEF COr AE F ACAC I -- co ' 'A. -( AI E Ia / acceec 11 " 1 I " 35EisS G 1 v546d " 358

Page  359 ï~~ TABLE 6 A. OENERATION OF THE KESDOEEKONTANY IN JUST INTONATION (1) C00IE S 1HARMONIC FACTORS OR GENERATORS, A C D E F,C N I.E., 1.3.5.7.9. 11.13.15. (2) FOH PRODUCTS 4 AT A TINE: A IC D, 1.3.5.7; () - 70 TONES. (3) ARRANGE TONES ON THE 70-ANY DIAGRAM ACCORDING TO DIAGRAN IN C BELOW. (4) THE 70-ANY IS PARTITIONABLE INTO PENTADS, MEZANIES AND EIKOSANIES. I. PARTITIONS OF THE HEIDOUKDNTANY (1) SACK TONE 15 CONNECTED TO 16 OTHERS BY SIFLE HARMONIC RATIOS. TBEI ARE 28 DYADS, EACH OCCURRING 20 TINES. (2) THERE ARE 56 SPECIES OF HARNONIC AND SUIHAMNIC TRIAD, EA0 OCCURING 10 TIES. (3) THERE ARE 70 SPECIES OF HARMONIC AND BUSNARNONIC TETRAD, EACH OCCURRING 4 TIMES. (4) THERE ARE 56 SPECIES OF HARMNIC AND SUINAUCMIC PENTAD, EAC OCCUAIMG ONCE. (5) THERE ARE 70 SPECIES OF HEEANT, EACH OCCURRING 6 TIES. (6) THERE ARE 28 SPECIES OF EIKOSANY.EACH OCCURRING 'NICE. (7) THE ENTIRE 70-ANY MAY E TRAVERSED BY EIKOSANIES WITH COSO0N-TETAD AND COWCN-KEANY MODULATIONS. C. 359

Page  360 ï~~ TABLE 7 A. ImMiasaIs ta-na wiQa minussa I. 3. 4. 1. 6. 9. 10. 11. 12. 13. IS. 14. 03M$79 034110 0345"9 03011 013436 0235"* 03*4410 013 *1 033679 034349 0141 031610 o3amat 012* 0135 0151 024) 0356 Di35 0154 03"6 4124 0431 0144 0121 013 01$* Isssrea 111111 211123 31113 4111 312132 221223 331231 113114 212124 222322 113113 21312) 3131 31 Â~1114 14122 21311 'set' $43310 343230 314222 "21242 224*32 143250 224232 432321 334222 0400 "21322 234232 303630 "20243 232341 32432 4-1 6-143 -32 6-110 '-323 4-30 671 '-U'4 4-3 3 13 -137 6-4 13 10 1 3 P"eriser a s I I Z p P 14 is1 I 9 Ip 10 I * r 3 1r 13 I 1" t z 1 I 3 1 11 32 wsS 01306./.4 ail, LIT amges AM 9 I 3.4 i * 7 Ustisberg. 1944 1115, 1915. its are Learlt rsr M3 (2.5,9.10,13,1*), aS five aea prust(1-6.)--12t 4-11, 71e. -I). Babbit,*1955, All Cebiseterlal Betsljt. Ail jistetwal 1*a~t wa..rrs, 11944g Or, 11, peasjets. busies, 1.3,6.1091,14I*easelt-awles srp,mass, 3-14, 4.110 1-13, sue psied a~busie.. 56,9, sad IS sa-be..au p slsts. Ports, 1013. udMe, 3,4,5S am71ee sat - Nertias's (1941) liste.hshssrds. IljC. IEXANIES PER OCTAVE B. APOD BCD MANDALA I.' to&,VbOw A ry u 5 1""1 'SC *4u * 54 46 S Inst e "sVa4". D. MAJOR SCALE AS 2 HEXANIES o0 c" r P NEXANY A' 360D AC D 0 a 4 7q

Page  361 ï~~ A. TABLE 8 GNIATION OF TNE S-TON SET IN EQUAL TEMPERANRNT 1. THESE 8-TONE SETS NAVE THE NELODIC SEQUENCE, d c b a b c da. 2. CHOOSE SEQUENCE AND PARTITION SCALE INTO TTERADS; EXAMPLE: 12212212 IN 13-TONE EQUAL TIPEARNT. (a) 0135681011, A 5+3 NOS, 0=513, SWMI0NG FROM U. (b) 0 1 3- 5 6 8 10 11 13, INITIAL TETRADS. 3. PLACE SCALE ON OKTAGRAN SO THAT OPPOSITES SUN TO N-. 4. THE 8 PAIRS OF INVERSIONALLY RELATED CONSONANT CORDS ARE OF 3 TYPES: 1) PACES, 2) CORNERS, 3) a)ENER-DIAGONALS, AND E1CLUDE CORDS CONTAINING TONES ON PRINCIPAL DIAGONALS. B. 8-TONE SETS IN 12-TONE EQUAL TWECARNT 1. 2. 3. 4. S. 6. 7. S. 9. 10. Scale 01234567 02345679 012468910 013467910 01345689 01245789 034567811 024567911 01235678 01236789 Intervale 11111115 21111123 11222112 12121212 12111213 11212113 31111131 22111221 11121114 11131113 Propriety1 Interval Vectora2 I 765442 1 566452 1 464743 SP 448444 I 546652 I 545662 I 645652 I 465472 I 654463 I 644464 forts' Na2 a-1 a-10 6-24 6-26 6-17 8-20 6-7 6-23 5-6 8-9 I Rotbanberg, 1966, 1975, 1976. 2 Porte, 1973. 361

Page  362 ï~~ FIGURE 1 A. EIKOSANY, COMPLETE LATTICE B. HEBDOMEKONTANY, COMPLETE LATICE 362