Annexes to Patterns of N-foldness

By Anthony Judge

*© 2003 Copyleft Anthony Judge *(*tony@laetusinpraesens.org*)*.
The memes above would enjoy entering into other patterns with your assistance
– provided they are appropriately cared for in their journey. *

The concept schemes identified here are discussed in the paper
on: Patterns
of N-foldness; comparison of integrated multi-set concept schemes as forms of
presentation. This was prepared for a sub-project meeting of the Forms of
Presentation group of the Goals, Processes and Indicators of Development (GPID)
project of the United Nations University (UNU). They were published in *Patterns
of Conceptual Integration*. Brussels, UIA, 1984, pp 161-204

Each concept scheme is the subject of an Annex listed below. The items within each Annex are ordered according to the number of set elements. This number is given as that portion of each (non-zero) item number before the decimal. The number after the decimal is a sequence number of no significance.

It should be stressed that the exercise is tentative and experimental.

Following the brief introductory references to the work[s) from which the items within each Annex were obtained, the items are ordered sequentially by a numeric code structured as follows:

- when 0 signifies: general contextual extracts
- "non-zero" : number of elements in the concept set described in that extract.

- used to separate and sequence extracts with the same initial number;

**Order of items** within such groupings is not significant,
except that occasionally related items have been placed together, especially
for contextual extracts.

**Example:** 12.2 indicates, the second extract containing
information on a concept set of 12 elements. Some references to annex material
in the introductory paper are made in the form 17.12.2, meaning Annex 17, item
12.2.

The extracts within each of the following Annexes were selected in the light of the following considerations:

a. Brief indication of the general approach.

b. Remarks touching on the problem of comprehension.

c. Remarks concerning the use of number or geometry as an ordering aid.

a. Any clear identification of a complete set generally indicated by a reference to a number of elements.

b. Duplicate references to the same set were included when this might assist comprehension of the set,

c. Sufficient contextual material was given, when available, to assist comprehension of the significance of the set within the concept scheme.

d. Extracts which seemed of questionable interest were included if there seemed some possibility that they might later prove to be of signi ficance in relation to corresponding extracts in other annexes.In general, the extracts were selected in the spirit of a "data gathering exercise", namely they might (or might not) prove to be of interest in continuing, this investigation.

## Annexes

- Annex 0: UNU/GPID Project: It is appropriate to employ the same presentation method to the GPID concept scheme as it is now emerging.
- Annex 1 : Geometry of Meaning: This is a modern effort to order a complex pattern of information on change and development in the light of physical concepts of dimensionality and control.
- Annex 2 : Book of Changes: This is the 3000-year old Chinese I Ching which is conceived as encoding the complex pattern of changes in physical and social phenomena. It has been of considerable interest to Leibniz (philosopher), Jung (psycho-analyst)and western mathematicians, and its poetic expression has proved highly acceptable to a segment of western society.
- Annex 3: Catastrophe theory: This is a new controversial way of thinking about change in all kinds of phenomena in the light of the mathematics of differential topology.
- Annex 4: Tibetan Buddhism: This is a highly structured traditional scheme of concept sets which, because of both illiteracy and the absence of paper, uses powerful imagery to facilitate memorability and communicability,
- Annex 5: Genetic code: This recent fundamental breakthrough in the biological sciences groups a number of concept sets in a highly integrated pattern. ·
- Annex 6: Chinese Communist terminology: This is included because it illustrates the importance, in one major non-western political system of concept sets governed by number.
- Annex 7: Tonal patterns of Rg Veda chanted poetry: The Rg Veda is, in terms of survival over 4000 years, the most successful active communication vehicle. The concept scheme interlinks many concept sets in a very powerful way.
- Annex 8: Movement and dance notation: This concept scheme is one of the most widely accepted frameworks through which understanding of dance is ordered.
- Annex 9: Chinese art of war: This traditional scheme is even now considered basic to ordering perceptions of strategy and tactics.
- Annex 10: Art of colour: Artists achieve certain visual effects by selecting intuitively amongst a range governed by a perceptionoriented concept scheme distinct from the colour preoccupations of physicists and chemists.
- Annex 11: Islamic cosmological doctrine: As in the case of Tibetan Buddhism, this concept scheme has been of special significance to Islamic culture for an extended period.
- Annex 12: Language and transformational-generative grammars: Language itself should be rich in concept schemes which are themselves a form of language. This Annex, unlike the others, considers aspects of current thinking which have rendered superficial the traditional concept sets in this area.
- Annex 13: Thermodynamics: This fundamental discipline is concerned with the description of change in physical processes. It has been applied by analogy to social processes. Its pattern of concepts is very well integrated. Unlike the other concept schemes, the concept sets are not explicitly set out. An attempt is made in this Annex to show how they might emerge for comparison with other schemes.
- Annex 14: Periodic classification of chemical elements: This fundamental scheme is included because of the comprehensibility of the pattern governing the complexity of the information ordered.
- Annex 15 : Systematizes: This modern scheme, formulated by a philosopher-mathematician , is included because of the variety of phenomena it encompasses and the leads it offers to understanding number-governed patterning complexity.
- Annex 16: Periodic coaction coordinate system: This ambitious modern scheme is included because it Durports to order patterns of interaction in a variety of complex systems.
- Annex 17: Synergetics; geometry of thinking: This highly original and well-integrated scheme is included because of the multiplicity of concept sets it includes and the leads it provides as to how transformations between them may be accomplished.
- Annex 18: Polygons and polyhedra: This Annex indicates the sets of polygons and polyhedra. It is significant in the light of the previous Annex as indicating how set elements can be interrelated in an integrated whole.
- Annex 19: Topological features of polyhedra: Again, in the light of Annex 17, the number-governed sets associated with this material offer useful indi cations as to how such sets are interrelated in patterns.
- Annex 20: Chladni patterns: This is included as a systematic study of the range of patterns arising form the vibration of a surface area. It is significant in that it indicates how a zone is "broken up" into sub-zones.
- Annex
21: Levels of declarations of principles: This is a separate experiment
in articulating principles based on different numbers of elements. (Although
originally published with the other annexes, it formed the topic of a separate
paper)

The concept scheme described here is discussed in the paper on: Patterns
of N-foldness: comparison of integrated multi-set concept schemes as forms of
presentation. This was prepared for a sub-project meeting of the Forms of
Presentation group of the Goals, Processes and Indicators of Development (GPID)
project of the United Nations University (UNU). The annexes were published in
*Patterns of Conceptual Integration*. Brussels, UIA, 1984, pp 161-204

The Rg Veda is a collection of hymns which constitutes the earliest literary document of the Indian tradition, which is believed to have been composed between 2,500 and 1,500 BC. "The hymns are still chanted and there is considerable interest in exploring both their special use of language and their relationship to the mathematics of music. The following points are extracted from two complementary books:

- Ernest G.McClain. The Myth of Invariance. Boulder and London, Shambhala, 1978
- Antonio T de Nicolas. Meditations through the Rg Veda. Boulder and London, Shambhala, 1978

0.1 "In the beginning was tone. This is the most important clue to bear in mind in our effort to understand the Rg Vedic conception and use of Language and of languages. The whole of the Rg Veda is chanted....we have already pointed out the sophisticated musical-metrical structure of the hymns; and it is precisely on this model of musical tones that the meaning of the hymns is grounded....

Tone is a sound of a certain fixed pitch....man discovered that the intervals between the tones could be defined by the ratios of the lengths of pipes and strings that sounded them. It was the ear that made ratios invariant: by its vivid memory of the simpler intervals, the ear made the development of a science of pure relations possible within the theory of numbers, the tone-field now being isomorphic with the number field....The ratios of the first six integers defined the primary building blocks: the octave 1:2; the fifth 2:3; the fourth 3:4; the major third 4:5; and the minor third 5:6. That tones recur cyclically at every doubling or halving of frequency or wave-length is the 'basic miracle of music'. From this acoustical phenomenon, the number 2 acquires its 'female* status; it defines invariantly the octave matrix within which all tones come to birth. Here, in this initial identification of the octave with the ratio 1:2, is the root of all the problems which haunt the acoustical theorist, problems which the ancient theorist conceived as symbolizing the evil and disorder of the universe. The octave refuses to be subdivided into subordinate cycles by the only language ancient man knew – the language of natural number, or integers, and the rational numbers derived from them. It is blunt arithmetical fact that the higher powers of 3 and 5 which define subordinate intervals of music never agree with higher powers of 2 which define octave cycles. It is man's yearning for this impossible agreement which introduced a hierarchy of values into the number field. For our ancestors, the essence of the world and of the numbers which interpreted that world was sound, not substance, and that world was rife with disagreement among an endless number of possible structures." (de Nicolas, p. 55-56)

0.2 "Therefore, from a linguistic and cultural perspective, we have to be aware that we are dealing with a language where tonal and arithmetical relations establish the epistemological invariances....Language grounded in music is grounded thereby on context dependency; any tone can have any possible relation to other tones, and the shift from one tone to another, which alone makes melody possible, is a shift in perspective which the singer himself embodies. Any perspective (tone) must be "sacrificed" for a new one to come into being; the song is a radical activity which requires innovation while maintaining continuity, and the "world" is the creation of the singer, who shares its dimensions with the song.

In ancient times, the infinite possibilities of the number field were considered isomorphic with the infinite possibilities of tone....Today in the West, we use number to constrict all possibility to an economically convenient limit; the international pitch standard...and the limitation to 12 equal semitones within the octave are antithetical to the spirit and needs of music. Rg Vedic man, like his Greek, counterparts, knew himself to be the organizer of the scale, and he cherished the multitude of possibilities open to him too much to freeze himself into one dogmatic posture. His language keeps alive that "openness" to alternatives, yet it avoids entrapment in anarchy. It also resolves the fixity of theory by setting the body of man historically moving through the freedom of musical spaces, viewpoint transpositions, reciprocities, pluralism, and finally, an absolute radical sacrifice of all theory as a fixed invariant." (de Nicolás, p. 57)

0.3 "In a language ruled by the criteria of sound, perspectives, the change of perspectives and vision, stand for what musicologists call "modulation". Modulation in music is the ability to change keys within a composition. To focus within this language, and by its criteria, is primarily the activity of being able to run the scale backwards and forwards, up and down, with these sudden changes of perspective. Through this ability, the singer, the body, the song and the perspective became an inseparable whole. In this language, transcendence is precisely the ability to perform the sang, without any theoretical construct impeding its movement a priori, or determining the result of following such movement a priori." (de Nicolás, p. 192)

0.4 "It would be a radical misunderstanding of the Rg Veda to read it with the detached objective aloofness with which we in the West are accustomed to view whatever is presented to our speculative reason. This is the precise error of knowledge which the Rg Veda is trying to correct....The Rg Vedic understanding of man centers the activity of creation back in man himself. Man is the center of both time and eternity, each subletting a different language of the lattice (as in the mathematics of lattice theory), and the passage from one to the other is the passage up or down the lattice from one structure of expression to another within the relationship of action defined by the lattice. Man is at the center of his own activity, creating and recreating himself and his cosmos in relation to how efficiently he climbs or descends the contextual multiplicity within which he constantly operates." (de Nicolás, p. 186-73

0.5 "The propositional Logic of Quantum Mechanics and Classical Logic...are opposed to each other in the following sense: Classical Logic is a propostional Logic of two-valued truth-functional propositions, the logic of classes and of quantization.....On the other hand, quantum Logic is a logic appropriate to the context: language about contexts and the language appropriate to the contexts. It is a two-valued propositional logic which varies only in dropping the distributive laws for 'and and 'or' and replacing them by some weaker form of connection, like a modularity principle. This quantum logic is an ortho-complemented non-distributive lattice. Since this logic is formalizable, it becomes possible to clearly determine complementarity and exhibit the logical structure of the dialectic of context-language dependence. This is the core of Bohr's notion of complementarity." (de Nicolás, p. 35-6)

0.6 "Therefore, the possibility exists, for philosophy and for man, to continue viewing the world on the classical model as an. objective whole, and remove all subjective variations from it. A viewing of this sort gives us a public object only in its systematic aspect in an objective (scientific) domain which, on constitution is already a closed system....On the other hand, there is also the possibility of viewing according to the Modern Physics model. Within this model, the way of procedure is through a community of inseparables (subject-object, observer-observed, mind-body, etc) which were artificially created in the first place, and which indirectly indicate man's progress in transcending the knowledge of his own context-language dependence through the activity of the men of science, philosophy, and by man in general."(de Nicolás, p. 37-8)

0.7 "No embodied-vision is possible without the sacrifice of the totalitarian tendencies of partial viewpoints: this totalitarianism is primarily due to the fact that partial viewpoints demand that man be reduced to the theoretical only. The sacrifice the Rg Veda offers points out the way for man to return to his whole body through the mediation of a language of images through which i dialogue between opposing viewpoints is not only carried out. but all theoretical totalitarianism is cancelled. From the point of view of logic, this sacrifice involves a partial ordering of languages in a non-Boolean logic, the non-Boolean character of which is the mediation from growth and liberation." (de Nicolás, p. 187)

0.8 "Hindu mythology seems to have aimed at and achieved a total unity between the physical and the metaphysical, with number theory providing the ground for an absolute certainty of viewpoint." (McClain, p. 85)

0.9 "A musical cosmology which aimed to harmonize the fields of both visual and aural experience under the principle of smallest integers could scarcely avoid stumbling across or systematically developing that material we have presented...Since attention is being directed powerfully to number, we must look for some more convenient way of codifying algebraic operations than our tone-mandalas have provided; they are convenient only for displaying the metric properties of tone-numbers. In the Hindu yantras to be studied next we shall see how simple pebble patterns codify all the algebraic aspects of musical number theory in triangular arrays which look like the mountains from which Hindu and Hebrew gods alike hurl thunderbolts at their enemies." (McClain, p. 41-2)

0.10 "We are immersed in a symbolism which revelled in its power to express a cosmic unity, invented by men whose mathematical rigour matched their musical and poetic feeling, and in whom reverence was matched with humour."(McClain, p.53)

0.11 "Rgvedic man was enveloped by sound....While the other sensory media provided discontinuity, sound alone, in spite of its evanescene, gave Rgvedic man the instance of eternal presence and unity he so well used to further develop the world of rta. the well-formed instant." (de Nicolás)

0.12 "This study will develop the hypothesis that the "lattice logic" which de Nicolás perceives in the Rg Veda was grounded on a proto-science of number and tone. The numbers Rgvedic man cared about define alternate tunings for the musical scale. The hymns describe the numbers poetically, distinguish "sets" by classes of gods and demons, and portray tonal and arithmetical relations with graphic sexual and spatial metaphor. Vedic concerns were with those invariances which became the focus of attention in Greek tuning theory. Because the poets limited themselves to integers, or natural numbers, and consistently used the smallest integers possible in every tonal context, they made it possible for us to rediscover their constructions by the methods of Pythagorean mathematical harmonics." (McClain. p. 3)

0.13 "Throughout my study I shall focus on invariances,that is, on patterns which remain the same in different contexts. This theme of invariance was expressly formulated by Marius Schneider: "In view of the inconstancy of the world of form, primitive man questions the reality of static (spatial) phenomena and believes that transient (temporal) dynamic rhythms are a better guide to the substance of things." "(McClain, p. 7)

0.14 "The logic of India is profoundly geometric. Its mándalas and yantras present the observer with static forms which could only be achieved by dynamic processes. Our problem here is to learn to see these forms as Socrates yearned to see his own ideal forms, "in motion"."(McClain, p. 6)

0.15 "Mary Danielli states that 'numbers in early times are almost invariably associated with the symbolism of the mándala.' "Anthropologically, the mándala is concerned among other things with increase and growth. Keeping these within the natural pattern, so that they do not disintegrate into fragments." The mándala is 'a wordless symbol' which 'speaks for itself'. It is a model 'discovered by man in himself, and 'no one knows how early he became conscious of it'. The mándala has 'several systematisations' attached to it simultaneously; it represents 'restraint, organisation, harmony, and the cultivation of the individual as a creative person'. In brief, 'Man is the mandala'. Professor Danielli's analysis of the technical characteristics of a mándala is summarized in six points that are applicable to all of the mándalas in this book:

1. The mándala is concentric around a center [applicable both to tone circles and reciprocal yantras).

2. The mándala is symmetrical around three axes [illustrated by all tonal yantras); alternately, 2 axes may appear.

3. The mándala is self-replicative (i.e. its patterns can be "translated" arithmetically and "transposed" tonally and "rotated" and "reflected" geometrically).

4. The mándala is three dimensional in fact, although often represented as a two-dimensional base plan [i.e. most of the yantras in this book are multiplication tables for the prime numbers 3 and 5, hence "two-dimension-al", but powers of 2 constitute a "third-dimension" relevant to any tonal realization and must be supplied mentally).

5. Every part communicates with every other part (i.e. musical intervals overlap within the confines of the octave and provide alternate channels of "communication" between any two tones.)

6. Every line, every junction, and every space has a meaning (i.e. every junction is a number and a tone, every space is a tonal interval and a ratio, and every line is a route of connection.)" (McClain, p.55-6)

0.16 "The central geometric image in the Rg Veda is the mándala of the "single-wheeled chariot of the Sun", harmonizing moon months with solar years and the signs of the zodiac....Every odd number has a unique "angular value" in the tone-mandala. Since Hindu cosmic cycles are numbers divisible by 2, 3, and 5, we can examine their potential musical implications by plotting the location of all products of the prime numbers 2, 3, and 5 in relevant tone-mandalas.. ..The "divine male number 3" will generate cuts or "spokes" which lie within about half a degree of the idealized twelve spokes of equal-temperament (tuning)....The "human number 5" will generate "poorer" spokes....We shall encounter 7 not as a "tone-value", but as the most important limiting number in several sets, particularly in its role as exponent." (McClain, p. 9 and 25)

1.1 "It is a theme of much ancient mythology that the Divine Unity is a hermaphrodite, producing a daughter, "2", by a process of division without benefit of a mother. God is "1", but he cannot procreate except via his daughter, "2", the female principle and mother of all. Numerical and musical relationships provide the metaphor: ratio theory is "music" for the ancients". (McClain, p. 21

1.2 "But from a central perspective, god " 1 - geometric mean has been transformed to least common denominator for the tones represented in the diagram here, 720, and sends his rays through the table in all directions. All perspectives are equally relevant." (McClain, p. 53)

1.3 ."In the Rg Veda, Asat, the non-existent, is:

- the whole differentiated primordial chaos
- a place of silence" self-destruction, disappearance
- not letting exist that which longs to exist – a field condition out of which all differentiation in human experience emerges for the poets
- the original receptacle, foundation, of the waters (McClain, p. 21)

2.1 "A vibrating string of any reference length can be halved to sound the octave higher or doubled to sound the octave lower." (McClain, p. 19)

2.2 "The number 2 is "female" in the sense that it creates the matrix, the octave, in which all other tones are born. By itself, however, it can only create "cycles of barrenness", in Socrates metaphor, for multiplication and division by 2 can never introduce new tones into our tone mándala," (McClain, p. 19-20)

2.3 "The great expansion of the number sets in later diagrams is motivated, I believe, by the effort to approximate as exactly as possible the irrational square root of 2 which is needed to locate a tone symmetrically opposite the mean on D, that is, precisely in the middle of our octave." (McClain, p. 37)

2.4 "The model for all existence (Sat) – hence of everything which can be named or numbered – is Indra...The continuum of the circle (Vrtra) embraces all possible differentiations (Indra). The conflict between Indra and Vrtra can never end: it is the conflict between the field of rational numbers and the continuum of real numbers. Integers which introduce new "cuts" in the tone-mandala demonstrate "Indra-power" over Vrtra; Vrtra is "cut to pieces" in every battle with the Gods, but his death would be their own. 'Without the Asat or its equivalent Vrtra, the Dragon, there would be no Indra, nor even the gods for he is their container." (McClain, p. 21)

3.1 "The Greeks spoke of "three genera" – diatonic, chromatic, and enharmonic – so that the apt metaphor of musical "generation" has always been part of the Western tradition....Unless one has the experience of actually tuning an instrument, behaving as "midwife" to successive tones, the metaphor of generation may have little meaning." (McClain, p. 19)

3.2 "Three tones provide the framework on which Western theory developed from its earliest Greek foundations up through the nineteenth century. They are... defined by the "musical proportion" 6:8 :: 9:12....In ratio theory the number 9 is the arithmetic mean" within the octave module C6:12 = 1:2} and 8 is the "sub-contrary" or "harmonic" mean....The Greeks conceived these two "means" as being the fixed limits of their tetrachords within each of which the two interior strings were "movable" in pitch. Western musicians think of these tones as "dominant" and "subdominant" in the harmonic vocabulary of recent centuries. In our tone mándala, with D as reference tone, G and A are simply the "opposite" meanings of 3, as 3 and 3 ." (McClain, p. 25-7)

3.3 "These three tones play interchangeable roles as arithmetic, harmonic, and geometric means...In Vedic metaphor they are apparently symbolized by 'Indra and the Asvin Pair". (McClain, p. 26-7)

3.4 "It is by drinking whole lakes of Soma juice that Indra's powers expand. Those lakes, I suggest, are yantras, which lead to larger yantras. Increased insight comes from multiplication to larger sets....The "expanded consciousness" which Soma brings is first of all insight into musical experience = number theory. We are looking in particular for sets which remain invariant under reciprocation....In a sense, this essay will be finished when we understand how the Vedic poets arrived at the twelve "spokes" for the Sun's chariot within the number field generated by our yantras. The Soma we care about is indeed a "triply-mingled sraught" which "flows round into worlds", for it is developed from three prime numbers, 2, 3, and 5, or from the "Pythagorean triple" 3:4:5. These numbers are our "filters" for studying Hindu cosmological numbers. Numbers of the form 2p3q5r are clearly a "triply-twisted thread". (McClain, p.49

4.1 "Three tones play interchangeable roles as arithmetic, harmonic and geometric mean and thus define the limits of two tetrachords within each of which the Greeks conceived the two interior strings as "movable" in pitch." (McClain, p. 26-27)

4.2 "The two fundamental tetrachord frames can be filled with "movable sounds" generated by the prime number 5 according to exactly four patterns, giving four pentatonic (five tone) sequences containing all the basic diatonic-heptatonic tones." (McClain, p. 28-9)

4.3 "The four Rgvedlc "languages" de Nicolas defines have their counterparts in the foundation of all theories of music. His "language of Non-Existence" (Asat) is exemplified by the pitch continuum within each musical interval as well as by the whole undifferentiated gamut – chaos – from low to high. His "language of Existence (Sat) is exemplified by every tone, by every distinction of pitch, thus ultimately by every number which defines an interval, a scale, a tuning system, or the associated metric schemes of the poets, which are quite elaborate in the Rg Veda. The "language of Images and Sacrifice" (Yajna) is exemplified by the multitude of alternate tone-sets and the conflict of alternate values which always results in some accuracy being "sacrificed" to keep the system within manageable limits. The "language of Embodied Vision" is required to protect the validity of alternate tuning systems and alternate metric schemes by refusing to grant dominion to any one of them." (McClain, P. 3)

5.1 (See 4.2 concerning four generated patterns from prime number 5) "Notice the following structural considerations: a) every pattern is coupled with its reciprocal, b) every tatrachord is replicated in a second tetrachord which completes the octave, c) every octave is defined by a sequence of smallest integers, d) every integer set is "friends" with another integer set defining the same material from an opposite point of view (rising vs falling), and e) these four pentatonic [five-tone) sequences contain all of the diatonic- heptatonic material...Later it will Beshown that all of these new tones are 'Asvin twins', in the sense that they function as arithmetic and harmonic means in various derivative frames..(The patterns) are generated by the 'human number' 5, and they are alternative pentatonic [five-tone) 'octaves' i.e. six-tone progressions spanning the ratio 1 : 2, so that first and last tones coincide." (McClain, p. 28-9)

5.2 The 10,552 verses of the Rg Veda are divided into lessons which are themselves composed of groups of five verses (varga). [de Nicolás, p. 273)

5.3 Indra, "the Dancer, is the Lord of men," and he rules "the fivefold race of those who dwell upon the earth"

Note how his "horses" are harnessed: "Sixfold they bear him, or by fives are harnessed."

Dawn discloses "the pathways of the people" in "the lands where men's Five Tribes are settled"."

Five Bulls which stand on high full in the midst of mighty heaven",.. There are "five regions" in the world "under thy Law"

(McClain, p. 29 citing Rg Veda)

5.4 "This protopythagorean attitude toward "5" seems to have survived in later Hinduism..."

man is said to have been born of aritual having five stages; hence in every ritual the fifth offering is called 'man'"

Siva became the dancing god, "ruler of the five directions of space, of the five elements, of the five human races, of the five senses, and all that is ruled by the number 5" and he is represented as "five-faced" (McClain, p. 29)

6.1 "The Vedic calendar divided the year into six seasons, and the poets sing of "six divine Expanses," of "six directions," and of "the six expanses from which no single creature is excluded." The star-hexagon establishes exactly six directions, three pairs of reciprocals." (McClain, p. 46)

6.2 "It is the relevance of reciprocals which the interlocked triangles of the star-hexagon symbolize so beautifully....If we study the musical implications of the numbers in these sexually differentiated triangles, we can see the "drum of Siva" containing only tones arranged by perfect fourths and fifths – i.e. in "Pythagorean tuning" – emerging as a "subset" of the "Just tuning" being generated in the star-hexagon....By themselves the star-hexagon, pyramid, and drum do not produce scales: they produce the equal-interval progressions which tone-mandalas "bend round into circles" (Plato's Timaeus metaphor), a context in which a new set of appropriate "smallest integers" must be found." (McClain, p. 45)

6.3 "The star-hexagon answers Dirghatamas question: "...What is the Unborn One, Who propped the six regions apart ?" The Unborn One is the unit 1, which can mean either "whole" or "part", and which props "the six regions apart" by acting as the geometric mean for the reciprocal triangles of the star-hexagon." (McClain, p. 46)

7.1 "What about the prime number 7? In ancient times the ration 7:5 was a valued simplification of the square root of 2....The approximation is not good enough for the Rg Veda's poets. We shall encounter 7 not as a "tone-value", but as the most important limiting number in several sets, particularly in its role as exponent." (McClain, p. 25Î

7.2 Seven tones: "Tones of the "highest caste" (sa, ma, and pa) contain four srutis [quarter-tones); tones of the second highest caste (ri and dha) contain three srutis; tones of the third caste (ga and ni] contain two srutis. Tones of the fourth class would contain only micro-intervals....Tones of different musical "castes" perform different musical functions, each essential to the system as a whole. Only the highest caste can frame tetrachords; only the second caste can produce thirds; the third caste "semitones" are the "left-over" intervals within the tetrachords; the fourth caste "commas" are the micro-intervals which arise whenever the material of the preceding three castes exceeds the eleven-tone limits...The musical castes are the consequences of the structural superiority of fifths and fourths over thirds within an invariant octave frame. It is the aim of attaining purity among both fifths and thirds which produces the "low caste" commas. The absolute "democracy" of equal-temperament had to abandon the conceptual purity of its two "highest castes". Indian musicians decline to follow that example....Since they use only seven tones in any one modal set, they do not need to define the exact differences which would arise, say, if all twenty-two srutis were to be directly compared with each other." (McClain, p. 39-40)

7.3 "The seven-day week was established for Hindus and Hebrews long before it was introduced into Egypt and Greece. In the poems the seven tones of the diatonic scale – the scale which elevates 30 and 60 from mere metric convenience to the status of a necessary base – become "seven holy singers" guarding the "beloved firmly-settled station" of the Sun....The seven together, however, are a "team of Seven", "seven Bay Steeds" for the Sun's chariot, seven rays shining through "Earth and Heaven", "seven-priests, the brother-hood, filling the stations of the One." Later we shall see that these seven tone-numbers are "seven male children" generated by the odd numbers 3 and 5 within the womb of Usas symbolized by the female number 2 which defines the octave matrix as an arithmetic "double". Here we see these seven yoked in two different ways "to the one-wheeled chariot...bearing seven names"."(McClain, p. 15)

7.4 "It is the poets' idea – not mine – that the Asvins (see 3.3 above) are linked to "the Seven Mother Streams", and these life-giving streams or rivers to "the seven tones" and "seven holy singers"..." (McClain, p. 28)

7.5 "It is when the reciprocal yentras of the star-hexagon integrate 5-+7with 3+-6 that we reach the Kalpa-Brahma limits and the lessons they encode...we do need 7 as exponent to put "Visnu" (i.e. 5) in the heavens as "Sun" where he can look down on his reciprocal (5-7) in the deep. The abstract pebble notation for ratio theory makes every pebble an exponent for whatever set of ratios is involved." (McClain, p. 87)

8.1 "Since all tones recur cyclically at the octave – as the "Same" tone in one sense, but a "Different" tone in another – any octave can serve as the model for all possible octaves, at least for the general purposes of tuning theory. The cyclic structure of the octave is the invariant common to all systems of tuning." (McClain, p. 19)

8.2 "The Rg Veda...is a collection of 1028 poems...divided into ten 'circles' (monadala) books....the total of verses being 10,552...divided into eight parts...themselves divided into lessons...and these in turn into groups of five verses." [de Nicolas, p. 273)

10.1 (See 8.2 above) "The Rg Veda...is a collection of 1028 poems...divided into ten 'circles' (mándala) books..." (de Nicolás, p. 273)

10.2 "Rgvedic poets likewise never tire of celebrating the significance of ten.... Ten-ness dominates the arithmetic of the Rg Veda in ways never suspected until the appropriate triangular yantras (algebraic arrays of integers) are developed for each cosmic cycle..." (McClain, p. 6)

10.3 "Commentatots have assumed that the extravagant powers of 10 attributed to Indra's forces were simply generous tributes to the god, but numbers – factors – of the form 10n are actually part of the essential arithmetic. Again and again Indra is addressed as "Lord of a Hundred Powers"....It is by such extravagant multiples of 432 and 864 – the bounding numbers of our basic "Pythagorean" scale – that we have arrived at the Yuga numbers themselves (within an appropriate yantra), and a similar multiplication by 10 carries us to the still larger Kalpa and Brahma numbers to be studied next." (McClain, p. 75)

10.4 "The limits of the yantra, however, taken together with its reciprocal, show that the "world-egg" has developed from our central D both right and left to the limit of 3 and up and down to the limit of 5+-10. What more handsome culmination to Rgvedic "ten-ness" could we find ?" (McClain. p. 89)

11.1 There are only eleven tonal cuts in the tone-mandala defined by the rising and falling scales of the octave double 30 : 60. (McClain, p. 13)

11.2 "We cannot exceed eleven tones even by the very best integer methods without meeting a "fourth caste" comma which must be "sacrificed" in one way or another, being worth only about an eighth of a tone, subliminal to the ear in melodic context although quite audible during a monochord demonstration." (McClain, p. 76-8)

11.3 "Notice that the eleven tones...are grouped in symmetric pairs about the reference tone, 0. In Plato's "Atlantis" myth these eleven elements represent "Poseidon and his five pairs of twin sons". In the Rg Veda, Usas in her role as universal "bride" is offered the following wedding prayer: Vouchsafe to her ten sons, and make her husband the eleventh man. A Upanisad describes man's body as "a city with eleven doors"....the number of the Rudras – "the working class of heaven" – is usually given as eleven." (McClain, p. 15)

11.4 "Why the "duration of the universe" should be exactly...2^{16}
x 3^{5}x 5^{10} "years" can probably be deduced from
the yantra...The limits of invariance under reciprocity are the eleven tones
we began with (see 11.1 above) – a "father and ten sons" in "Just
tuning", now "improved" as a "father and ten sons"
in "Pythagorean tuning", that is generated exclusively by the "divine
male number 3"....The eleven invariances can be arranged as reciprocal
diatonic scales in the basic Hindu-Greek mode within the octave-double 384 :
766. familiar to Plato scholars as the Timaeus model of the "World Soul"."
(McClain, p. 87-9)

12.1 "Ancient cosmology required just enough number theory and just enough musical theory to harmonize the heavens with the (musical) scale and the calendar. "The Moon is that which shapes the years" by dividing the Sun cycle into approximately twelve sub-cycles, and so it is likewise the Moon that arouses interest in dividing the octave cycle into twelve parts even though musicians need " maximum of only seven at a time." (McClain, p. 14)

12.2 "The central geometrical image in the Rg Veda is the mándala of the "single- wheeled chariot of the Sun," harmonizing moon months with solar years and the signs of the zodiac: "Formed with twelve spokes...one wheel, navels three..." If the "twelve spokes" are the twelve tones of an octave tonal-zodiac, then the three "navels" may be powers of three prime numbers 2,3, and 5, each rotating in a sense at its own speed, correlated by any terminating number which includes all three among its factors....In a sense, this essay will be finished when we understand how the Vedic poets arrived at the twelve "spokes" for the Sun's chariot within the number field generated by our yantras....We need not wait...to sense the relevance of our yantra. Notice that...the only tonal meanings which remain invariant under reciprocation are those along the central horizontal axis (of the yantra)...."The Gods are later than this world's production" in the sense that our number field must grow systematically to some larger limit in order to produce twelve tones along this axis." (McClain, p. 9, 35, 49, and 52)

13.1 "The "seventh son born singly" I take to allude to the reference mean on D, so that there are literally six sets of "twins" plus a thirteenth tone." (McClain, p. 36)

14.1 "The upright triangle 3^{p}5^{q} appears to be the Vedic "mountain
of god," and the inverted triangle (completing the yantra) is apparently
the Vedic "rain cloud" The more it "rains" from above, forcing
us to ever larger common denominators to integrate reciprocals in integer sequences,
the deeper will be the waters below our "transevering axis" which
functions as an "earth" separating the two waters. Curiously, it is
the acoustical facts which require us to pursue such increases through a triangle
of fourteen steps, and the moon which "increases" or waxes for fourteen
days is linked in the Vedic mind with our "press-stones" whose male-female
implications produce our insight: Soma the drink and Soma the Moon owe their
powers to "the Singers" who create this whole universe "with
their minds"." (McClain, p. 53)

15.1 "Our Brahma yantra is fifteen steps high. The Rg Veda tells us "the fifteen lauds are in a thousand places: that is as vast as heaven and earth in measure" ...."Fifteen-fold strong juices " are prepared for Indra....There are hints that this yantra is not peculiar to the Hindus. A Hebrew myth describes the digging of the temple foundation to a depth of fifteen "cubits"...In Egyptian mythology fourteen steps lead upwards to the throne of Osiris..." (McClain, p. 81-2)

16.1 "A reference to sixteen priests employed in sacrifices makes sense out of the sixteen elements here in the "transevering axis" (of the yantra)." (McClain, p. 89)

18.1 "The smallest integers which can define the eleven tones of the diatonic
scale (and its reciprocal) in chromatic order lie within the octave double 720
: 360 ....The limiting number, factorial 6 =720...provides, however, for a total
of eighteen tone numbers of the form 2^{p}3^{q}5^{r}...These eighteen can be regarded
as the total "indra tonal power" over Vrta (the chaos of the pitch
continuum) up to the limit of 720. each tone number being a well-defined cut
in the tone mándala." (McClain, p. 33-4)

18.2 "The eighteen chapters of the Bhagavad Gita display an internal organization which parallels our eighteen tones, suggesting that this basic tonal "18" construction played a continuing role in Hindu imagery." (McCalin, p. 37)

21.1 "...very many other verses seem to allude to the Kalpa and Brahma
yantras, specifically to the number of "pebbles" outlining the depth
and height and the extent of the "transevering axes". Across the base
of both yantras, twenty-one pebbles symbolize the powers of 3 from 3^{0}to
3^{20}....At Indra's moment of triumph over Vrta we hear that his thunderbolt
pierced "thrice-seven (=21) close-pressed ridges of the mountains".
Indra brings about the douwnfall of twenty-one "tribes"....The "Three
Times Seven pour out the milky flow" of Soma, the sacred juice, "pressed
out with stones." " (McClain, p. 81)

21.2 "In Egyptian mythology fourteen steps lead upwards to the throne of Osiris, and in his judgment hall the deceased's heart is weighed in the balance – against "the feather of the law" – while forty-two judges look on from the side: they are seated in two rows of twenty-one each, like the top and bottom rows of our reciprocal yantras..." (McClain, p. 82)

21.3 "We would seem to have reached a logical stopping place....the poet addresses "venomous" creatures...with confidence that "the three-times-seven bright sparks of fire have swallowed up the poison's strength". What has actually been "swaalowed up" is the slight discrepancy between the real number we need, the square root of 2, and its nearest rational approximations via... the most severe limitation to the products of only three prime numbers. That hymn ends , so I would like to believe, with the singer's evaluation of the slight inaccuracies introduced by this sacrificial Rgvedic "tempering": "Scorpion, thy venom is but weak."" (McClain, p. 87)

21.4 "But we are not finished. There are always more numbers....All of the quotations cited earlier concerning the twenty-one elements in the base of the Kalpa-Brahma yantras apply now to the twenty-one layers in the vertical dimensions of this yantra. A reference to Indra's lightning striking "obliquely" makes better sense if the twenty-one rows of this yantra are the "twenty-one mountains" it pierces." (McClain, p. 87-9)

22.1 "Hindu tuning theory assigns each tone from two to four possible loci, dividing the octave into twenty-two srutis which are not defined mathematically." (McClain, p. 38) (See also 7.2)

30.1 "The smallest integers which can define a diatonic scale with two similar tetrachords – a fundamental concept in both Hindu and Greek tunings – occupy a "space" of thiry units in the "octave double" 30 : 60....The numbers 30, 32, 36, 40, 45, 48, 50, 54, and 60 exhaust the tonal implications of integers 2P3q5r" 60." (McClain. p. 12)

30.2 "That the Rgvedic poets knew this tuning is an inference from the way they stressed the role of 30 and 60 in defining cycles, and the roles of 7 and 11, the number of tone-values in diatonic and chromatic sets, and from their insistence on linking these numbers to tone. The Vedic year consisted of twelve months of thirty days each...and the Vedic day contained thirty hours of sixty minutes." (McClain, p. 14)

30.3 "A cycle of thirty units harmonizes the month with the diatonic scale at the arithmetic level; later we shall see how the reciprocal scales harmonize the chromatic scale with the 360-day year." (McClain, p. 14)

32.1 The tone E flat in a rising scale (or C sharp in a falling scale), in
an octave defined by the octave double 30 : 60, is derived from 2^{5}.3^{0}.5^{0}= 32
(McClain, p. 13)

33.1 " "Thirty-three gods" play a central role in the Rg Veda....Why
the thirty- three "great" gods of the Prajapati cycle are so important
to acoustical theory can be understood best if we project them into a tone-mandala...There
we can see at a glance that the metric properties of numbers 2^{p}3^{q}5^{r} = 60^{3} are
becoming visually and aurally indistinguishable..." (McCalin, p. 66)

36.1 The tone F in a rising scale (or C sharp in a falling scale, in an octave
defined by the octave double 30 : 60, is derived from 2^{2} .3^{2} .5^{0}= 36 (McClain,
p. 13)

40.1 The tone G in a rising scale (or A in a falling scale), in an octave defined by the octave double 30 : 60. is derived from 2 .3 .5 = 40 (McClain, p. 13)

45.1 The tone A in a rising scale (or G in a falling scale),in an octave defined
by the octave double 30 : 60, is derived from 2^{0}.3^{2} .5^{1} = 45 (McCalin, p. 13)48.1
The tone B flat in a rising scale (or F sharp in a falling scale), in an octave
defined by the octave double 30 : 60, is derived from 24 .31 .50= 48 (McCalin,
p. 13)

46.2 "The ancient Indian grammarians of the Sanskrit language have identified fourty-eight sounds as worthy of notation, and in the script that was developed over the centuries each character represents that one sound unalterably.... Sanskrit notation is extremely precise." (de Nicolas, p. xi)

50.1 The tone B in a rising scale (F in a falling scale), in an octave defined
by the octave double 30 : 60, is derived from 2^{1} .3^{0} .5^{2}= 50 (McClain, p. 13)

54.1 The tone C in a rising scale (E in a falling scale), in an octave defined
by the octave double 30 : 60. is derived from 2^{1} .3^{3} .5^{6}= 54 (McClain, p. 13)

58.1 Maximum number of verses in a Rg Veda poem is 58 (de Nicolás, p. 273)

60.1 The numbers 30, 32, 36, 40, 45, 48, 50, 54 and 60 exhaust the tonal implication
of integers 2^{p}.3^{q}.5^{r}= 60, which thus defines the diatonic scale (McClain, p.12)

60.2 The tone D in a rising scale (D in a falling scale), in an octave defined
by the octave double 30 : 60, is derived from 2^{2} .3^{1} .5^{1} =60 (McCalin, p. 13)

360.1 360 (octave double 360 : 720, chromatic). D rising (D falling, see 720) (p.33)

375.1 375 (360 : 720, chromatic) D sharp rising (no falling equivalent) (p.33)

384.1 384 (360 : 720. chromatic) E flat rising (C sharp falling) (p.33)

384.2 384 (384 : 768, harmonic) D rising (D falling) (p.89)

400.1 400 (360 : 720, chromatic) E rising (C falling, see 648) (p.33)405.1

405 (360 : 720. chromatic) E rising (C falling, see 640) (p.33)

432.1 432 (360 : 720, chromatic) F rising (B falling, see 600) (p. 33)

432.2 432 (432 : 864, harmonic) D rising (D falling, see 864) (p. 61)

432.3 432 (384 : 768, harmonic) C rising (E falling) (p.89)

450.1 450 (360 : 720, chromatic) F sharp rising ( B flat falling, see576) (p. 33)

480.1 480 (360 : 720; chromatic) G rising (A falling, see 540) (p. 33)

486.1 466 (360 : 720, chromatic) G rising (no falling equivalent) (p. 33)

486.2 486 (432 : 864, harmonic) E rising (C falling) (p. 61)

486.3 486 (384 : 768, harmonic) B flat rising (F sharp falling) (p. 89)

500.1 500 (360 : 720, chromatic) G sharp rising (no falling équivalant) (p. 33)

512.1 512 (384 : 768, harmonic) A rising (G falling) (p. 89)

512.2 512 (432 : 864; harmonic) F rising (B falling) (p. 61)

512.3 512 (360 : 720, chromatic) A flat rising (no falling equivalent) (p. 33)

540.1 540 [360 : 720, chromatic) A rising (G falling, see 460) (p. 33)

576.1 576 (432 : 864, harmonic) G rising (A falling) (p. 61)

576.2 576 (384 : 768; harmonic) G rising (A falling) (p. 89)

576.3 576 (360 : 720, chromatic) B flat rising (F sharp falling, see 450) (p. 33)

500.1 600 (360 : 720, chromatic) B rising (F falling, see 432) (p. 33)

625.1 625 (360 : 720, chromatic) B sharp rising (no falling equivalent) (p. 33)

640.1 640 (360 : 720, chromatic) C rising (E falling, see 405) (p. 33)

646.1 648 (360 : 720. chromatic) C rising (E falling, see 400) (p. 33)

648.2 648 (432 : 864, harmonic) A rising (G falling) (p. 61)

648.3 648 (384 : 768, harmonic) F rising (B falling) (p. 89)

675.1 675 (360 : 720, chromat.) C sharp rising (E flat falling, see 384) (p. 33)

720.1 720 (360 : 720, chromatic) D rising (D falling, see 360) (p. 33)

720.2 "The smallest integers which can define the eleven tones of the diatonic scale (and its reciprocal) in chromatic order lie within the octave double 360 : 720 ....The limiting number, factorial 6 = 720...provides, however, for a total of eighteen tone numbers of the form 2p.3q.5F...." (McClain, p. 33)

729.1 729 (432 : 864, harmonic) B rising (F falling) (p. 61)

729.2 729 (384 : 768, harmonic) E flat rising (C sharp falling) (p. 89)

766.1 768 (432 : 864 ¡ harmonic) C rising (E falling) (p. 61)

768.2 768 (384: 768, harmonic) D rising (D falling) (p. 89)

864.1 864 (432 : 864, harmonic) D rising (D falling) (p. 61)

1028.1 1028 poems in the Rg Veda (de Nicolas, p. 273)10,552.1 10,552 verses in the Rg Veda (de Nicolas, p. 273)