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This is part of a commentary on the Tao Te Ching Interpreted
Succinctly (original
order) and (alternative
order)
See also Commentary
on Tao Te Ching Interpretation: and the possibility of higher order patterning
Patterning possibilities are presented separately in detail in 9-fold
Higher Order Patterning of Tao Te Ching Insights
Navigational implications are explored in Hyperspace
Clues to the Psychology of the Pattern that Connects
This document provides some very preliminary results on the use of sound to help articulate and comprehend the pattern of 81 insights of the Tao Te Ching -- as envisaged and briefly discussed in separate documents (see above). The document should at this stage be considered a work in progress. The focus here is on identifying numeric information that could be used to drive a sound generator which would highlight resonant relationships between the insights. Part of the experiment is to determine progressively how to increase the meaningfulness of the resultant "music" as a mnemonic support for understanding the pattern of insights.
As indicated in Hyperspace Clues to the Psychology of the Pattern that Connects, the physical effects of resonance from sound and vibration are well known (for example, Chladni patterns in two and three dimensions [more]). Can psychological analogues be set up to engender the future and exert a time-binding force? Meditation on yantras and mandalas would seem to have a related function -- traditionally linked to the magic squares discussed in the accompanying paper. Within such a context, can analogues to overtones function as vehicles for particular forms of understanding? As indicated there, interesting patterns can be generated from magic squares when the numbers of the squares are replaced by symmetric symbols. These resemble Chladni patterns. Whether the magic square (or higher dimensional) patterns can be more readily comprehended through use of auditory display techniques (see NSF The Sonification Report), as seems highly probable, remains to be discovered. This could be a valuable way to explore and navigate comprehension of the relationships between the 81 insights of the Tao Te Ching -- especially in the light of any insights concerning Indian rasas .
Briefly put, this exploration focuses initally on the question as to whether the "magic" property of the mathematical relationships within a magic square can be "heard". This calls for techniques to generate tones corresponding to the constituent numbers of such squares so that the magic properties are signalled by patterns of resonance between such tones. The question is how to associate tones with numbers to ensure such effects. It is only then that the question of whether such resonance effects are significant to the relationships between the numbered insights of the Tao Te Ching can be addressed -- notably to discover whether higher order meanings are signalled, and rendered more accessible, by such magical ordering of the insights.
In the light of the work of Ernest G McClain (for example, The Myth of Invariance: the origins of the gods, mathematics and music from the Rg Veda to Plato. 1978), there is a case for exploring the patterning of the 81 insights through the manner in which the numbers associated with their order can be factored.
In a personal communication (21 November 2003), he indicates that he can think of no better place to ground research on sonification than the 81 verses of the Tao Te Ching because the pentatonic foundation of quantification in harmonic theory has a continuous history since Shuruppak c. 2650 BC. He notes:
At that point the spiral of fifths was defined arithmetically through all thirteen pitch classes (given "mantles of radiance" in cyclic modesl)--and indubitably in two different ways, both "Just" and "Spiral Fifths" in my language. Thus you are working on a solid base, rigorously disciplined for nearly 5000 years, neatly circumscribed in "smallest integers" and harnessed to parameters the ear could verify -- but shading off in all directions into the "non-being" of the infinitely complex when viewed from a secure "center of symmetry" where the "light" of insight shines brightest. I have not studied Grogono's "pan-magic" pattern carefully enough to understand it, for my own attention is on the disciplined subsets that approximate modern 12-tone tuning, and he is viewing the broader spectrum of quantified possibilities beyond it. But clarity in acoustical theory of course disappears in all directions, and I have no desire to inhibit research of any kind into its infinite complexity.
Our traditional tuning systems constitute a small "island" of clarity with a vast continuum of possibility, and you choose to view this universe "from the middle" -- as the Sumerians did, and musicians still do. So I wish you well in this general line of extended research for none of us can anticipate where it will lead.
Magic squares, however, to the best of my own present knowledge, do not restrict content to products of 2, 3, 5, and 7, the primes within ancient "Ten-ness," but their integer products were of interest partly as approximations to the square and cube roots of 2, and thus ancient harmonical numerology remained somewhat open to the influence of other integers. But I personally remain skeptical of how early in history "magic squares" acquired a status comparable to the early "net of the gods" conceived instead as the product of Mesopotamian "god numbers" and related biblical "bow in the clouds." In older cultures several integers were deified somewhat precipitately in ways that required correction on further reflection, and my own interest is mainly in working through this more limited strata of material -- still almost wholly neglected by historians of science because it must be abstracted mainly from the mythology that mathematicians tend to despise.
Inspired by Ernest McLain's approach, the numbers of the 81 insights can be factored within the framework of the panmagic square ordering of the 81 insights discussed separately in 9-fold Higher Order Patterning of Tao Te Ching Insights (see Table 3a) based on that identified by Alan Grogono.
36 2x2x3x3 |
51 1x51 |
30 2x3x5 |
65 5x13 |
80 2x2x2x2x5 |
59 1x59 |
10 2x5 |
25 5x5 |
4 2x2 |
64 2x2x2x2x2x2 |
79 1x79 |
58 2x29 |
9 3x3 |
24 2x2x2x3 |
3 1x3 |
38 2x17 |
53 1x53 |
32 2x2x2x2x2 |
23 1x23 |
2 1x2 |
17 1x17 |
49 7x7 |
28 2x2x7 |
43 1x43 |
75 3x5x5 |
54 2x3x9 |
69 3x23 |
48 2x2x2x2x3 |
27 3x3x3 |
42 2x3x7 |
77 7x11 |
56 2x23 |
71 1x71 |
22 2x11 |
1 1x1 |
16 2x2x2x2 |
76 2x2x19 |
55 5x11 |
70 2x5x7 |
21 3x7 |
0 | 15 3x5 |
50 2x5x5 |
29 1x29 |
44 2x2x11 |
8 2x2x2 |
14 2x7 |
20 2x2x5 |
34 2x17 |
40 2x2x2x5 |
46 2x23 |
60 2x2x3x5 |
66 2x3x11 |
72 2x2x2x3x3 |
33 3x11 |
39 3x13 |
45 3x3x5 |
62 2x31 |
68 2x2x17 |
74 2x37 |
7 1x7 |
13 1x13 |
19 1x19 |
61 1x61 |
67 1x67 |
73 1x73 |
6 2x3 |
12 2x2x3 |
18 2x3x3 |
35 5x7 |
41 1x41 |
47 1x47 |
11 1x11 |
26 2x13 |
5 1x5 |
37 1x37 |
52 2x2x13 |
31 1x31 |
63 3x3x7 |
78 2x39 |
57 1x57 |
The factors could be used to drive a (musical) sound generator (as discussed under Possible refinements below).
Table 2 (below) is an effort to provide number sequences in different directions from the centre of Table 1 (above) understood as a cognitive "map" for which directions can be usefully associated with compass directions (North, south, East, West, etc). The "primary" directions are therefore the prime directions (North, South, etc). the "secondary" directions are their intermediaries (North-West, South-West, etc). A "tertiary" set of "direction" might be derived from various "knight's move" sequences (in the light of the discussion of this move in Hyperspace Clues to the Psychology of the Pattern that Connects).
Primary (Rect.) | Secondary (Diag.) | Tertiary | ||||||||||||||||||||||
North | South | East | West | N-E | N-W | S-E | S-W | N-KE | N-KW | S-KE | S-KW | E-KN | E-KS | W-KN | W-KS | N-KE | N-KW | S-KE | S-KW | E-KN | E-KS | W-KN | W-KS | |
56 | 40 | 15 | 21 | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | |
28 | 68 | 50 | 70 | 71 | 77 | 46 | 34 | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | . | |
24 | 12 | 29 | 55 | 75 | 17 | 7 | 45 | 43 | 49 | 74 | 62 | 22 | 60 | 42 | 20 | . | . | . | . | . | . | . | . | |
80 | 52 | 44 | 76 | 53 | 79 | 41 | 67 | 3 | 17 | 18 | 6 | 1 | 66 | 27 | 14 | 10 | 30 | 63 | 5 | 69 | 44 | . | . | |
. | . | . | . | 4 | 36 | 57 | 11 | 59 | 30 | 31 | 37 | 16 | 72 | 48 | 8 | 25 | 51 | 78 | 26 | 32 | 47 | . | . |
Initial work has focused on feeding data of the above kind into Lars Kindermann's downloadable MusiNum: The Music in the Numbers software, notably in the light of its experimental use in the Elenyscope.
The software allows a simple table of data (such as from a spreadsheet) to be loaded. Each column of the table is then associated with a particular "voice". Each such voice can then be edited in terms of: instrument (any of the standard Midi instruments), speed, scale, note, etc.
After tweaking the instruments, the result may be saved in an internal format or as a Midi file.
Each "direction" identified earlier may be considered metaphorically as a "wind" (Wind of the North, etc). The numbers associated with the steps along each direction towards the periphery of Table 1 can then be treated as a column of a data table. Each column then becomes one instrument or "voice".
Primary | Secondary | ||||||
N | S | E | W | N-E | S-W | S-E | N-W |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
56 | 40 | 15 | 21 | 71 | 34 | 46 | 77 |
28 | 68 | 50 | 70 | 75 | 45 | 7 | 17 |
24 | 12 | 29 | 55 | 53 | 67 | 41 | 79 |
80 | 52 | 44 | 76 | 4 | 11 | 57 | 36 |
52 | 80 | 76 | 44 | 11 | 4 | 36 | 57 |
12 | 24 | 55 | 29 | 67 | 53 | 79 | 41 |
68 | 28 | 70 | 50 | 45 | 75 | 17 | 7 |
40 | 56 | 21 | 15 | 34 | 71 | 77 | 46 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
For the purposes of the exercise, "awareness" is considered as initially at the centre of Table 1 at zero (see discussion relating to this in the separate paper). Each "step" away from that centre point evokes 8 voices, which contiuue to play through each successive step out to the boundary of the panmagic square. However, since it is a panmagic square, the voice is allowed to continue playing "around the cognitive world" back to the point of origin -- by assuming that the most distant points are contiguous.
The first trivial result -- indicating the feasibility of the procedure -- can be heard by clicking here. Minimal effort has been used to select, tweak and tune the 8 voices. The media player can of course be set to repeat. This may represent the first crude effort to "play" a "magic square" -- although it is very likely that magic tables have been used to organize classical compositions in a number of traditions, notably in Asia.
The question is how trivial is the result and whether and how it suggests that this approach may be refined given the possibilities of tweaking the MusiNum software and driving it with more interesting patterns of numbers evoking resonance effects (possibly based on factors as identified above).
Can the approach be used to build up a mnemonic soundscape that can be appropriately navigated from any point?
The question is how the factors in Table 1 might be used to drive the sound generator such as to ensure that any resonance effects are recognizable by the ear in contrast to other effects.
Anthony Judge: