Marduk and his horned dragon

Marduk is a character like Indra who was associated with a very large number and antediluvian floods, though Noah seems adapted not to mention 8,640,000,000. The reality of these floods is based upon an idea revealed in this number. The head number 864 is the earliest which can form the seven note heptatonic octave whose intervals are far from perfect. Winds can be viewed as blowing into the octave's tone circle, and the heptatonic has seven but having defined seven tones (including the tonic here to be called D) the gaps remaining between B and C and between E and F, that we call semitones, are unplanned so as to be numerically and acoustically imperfect as equal the ratio 256/243, This imperfection was called by Pythagoreans the leftovers (leimmas), and they highlight the fact that when a cycle of fifths and fourths emanate from D (clockwise and anticlockwise within in the octave's tone circle), powers of three can only be eliminated to achieve the exact 2 required of octave doubling, by a semitone fraction involving three to the power of five = 243 (figure 1).

HE sevenWinds

Figure 1 The Serpentine cycle of ascending and descending fifths leading to the Pythagorean Heptatonic using imagery generated by Harmonic Explorer.

Tiamat's seven heptatonic winds are already blowing apart the ability to play in more than one "key" or scale and there is a strange semitone lurking within the scale that will create other creatures such as diminished thirds if a player skips notes. And the problem can only get worse as powers of three accumulate to prefigure an audible disagreement in the chromatic twelve notes which instead become thirteen, hence the weapon of Marduk is that of thirteen winds which, blown into Tiamat's "mouth", the octave's tone circle, blow her (the octave) apart with an excessive g-sharp exceeding (enharmonically) a diminutive a-flat (octave based on D). Also, from the numerical perspective which Marduk represented, there has been a flood of numbers in which D required a limiting number of 724496 to resolve all the tones as integers from D. Whilst the ear hears musical tones and intervals within an implicit logarithmic world, the raw numerical view of Tiamat's mouth would look somewhat different and so figure 2 shows the inflation of Taimat's mouth.

reconstruction Tiamat 13 winds

Figure 2 Mapping out Taimat's octave "mouth" as she generates chromatic dissonance of thirteen winds G# and Ab on left, opposite D.
As with figure one but now three further fifths extend new tones either side of D-d, whose theoretical limit of integer tone numbers rises up therefore from 3^6 = 729 (Plato's Tyrant) to 2^10 = 1024 (D = 724496 on right) of that as Marduk inflates her with his "flood weapon" of powers of two.

According to the archaeomusicology of Ernest G McClain, flood myths are concerned with the innovation of the prime number five, which the seven zeros after Marduk's head number (864) give him seven powers of. The height of Marduk's mountain generates a near perfect Ab-G# with Tiamat's "head" of D and Tiamat can then be stacked up in pieces to use much smaller limiting numbers and a number of different scales in a tuning system we call Just intonation, present in the ancient near east since the Sumerians (4th millennium BCE) which preceded Marduk's Babylon. The lowest limit then required to play five scales can be mapped out for the octave 360-720, as in figure 3.

reconstruction 720

Figure 3. The familiar tone-burst of Tiamat stacked now on top of herself. Here we blow Tiamat up from the central ring of limit 45 through 90, 180, 360, to reach the limit 720.
To play different scales requires using just two rows of the darkened bricks in the new tone mountain. The number 720 is 360 "days and nights" of a schematic year, also though used by Egyptians as a secular year of 360 days, perhaps in harmonic deference. Chromatism cannot set in until 4 times 720 = 2880, but then the a-flat is higher than the g-sharp and hence not enharmonic.

Whereas Pythagorean tuning relied on only one type of successive tuning interval, the fifth (and its complement the fourth), Just intonation resolves pure tone major (5/4) and minor thirds (6/5) so to enable the improved (smaller numbered) semitone of 16/15 and introducing a new tone of 10/9, to cancel threes and fives and achieve the octave doubling of 2. Therefore, within Just intonation skipping notes in the scale arrives at thirds and fourths and fifths that are pure low numbered ratios. These days, Equal Temperament has kicked over the traces of this early musicological myth and its necessities, by making every tone irrational as the twelfth root of two and every interval within a compromise, apart from octave doubling.