GÖTTERZAHLEN AND SCALE STRUCTURE

By RICHARD DUMBRILL

The Neo-Babylonian text UET VII, 126¹ , published by Professor O.R. Gurney, shows that in l.4² a fourth-string is listed bilingually as Sumerian sa.4.tur, ‘fourth, small-string’, and Akkadian a-ba-nu-[ú], ‘Ea-created’. It is the only string in the text which is qualified by the name of a deity. The pairing of Ea with this fourth string, whilst the god is usually associated with the number 40³ , attracted my attention in the light of a line in ASKT, p. 128, rev. 5f., where Ištar praises herself in the following terms: sa.a i.bí mčn sa.a a.ba mčn LI mahru anaku LI arku anaku. But there I had hoped that ‘LI’, in the Akkadian translation of the Sumerian phrase would have equated to Sum. sa = pitnu (or pidnu⁴), ‘string’, as had been thought until recently⁵ , thus translating as "I am the first string, I am the last string⁶" . However, Dr George further brought to my attention that a new reading of the controversial ‘LI’, as Akkadian še-ta⁷‘net’, compromised seriously the validity of my assumption. But this does not hinder the fact that being associated with the number 15 the goddess remains an entity comparable to Ea’s quantity and that the two deities can be expressed as the ratio of 40/15. So despite the erroneous reading of ‘LI’ we still have evidence of two gods which not only associate with numbers but also with musical quantities as I shall now demonstrate.

Now the ratio of their respective numbers, 15 and 40, should reveal some musical coherence which in turn should be echoed with other ‘Hauptgötter-numbers’ such as 60⁸ from Anu, (also 21⁹) ; 50 for Enlil, 30 for Sin, 20 for Šamaš, 10 for Bel Marduk, 14¹⁰ for Šakan and Nergal and 6 for Adad. My assumption is that, in a musico-mythographical context, god numbers would have expressed either units of frequencies or units of string lengths, or both, as shown below, from Ištar to Anu:

15 30 40 50 60

1. If these figures express units of frequencies, then the highest note would be Anu’s with 60 and the lowest, Ištar’s with 15. Their ratio, 60/15, can be reduced to 4/1; the ratio of Anu (60) to Sin (30), 60/30 = 6/3 = 2/1, and that of Sin (30) to Ištar (15). 30/15, also equals 2/1. Since the doubling of the frequency number equates to the octave interval, then the distance between Ištar and Anu is two octaves. Sin stands right in the middle, an octave higher than Ištar and one lower than Anu.

2. If these figures express units of string lengths, then Anu is, with 60 units, the longest string, the bass note. Sin is one octave below Ištar and one above Anu. The ratios of string lengths are thus in reciprocal relation to the ratios of frequencies.

It seems appropriate at this point to introduce the musical cent or centičme since it is the most tangible unit of tonometry. The conversion of ratios into musical cents consists in multiplying the log to base 10 of the quotient of the division between the denominator and numerator of the ratio by the constant 3986.314. This method produces a scale composed of 1200 units in which equal semitones measure 100 cents. Thus, 1/1 = 0 cents; 2/1= 1200 cents, the octave; 9/8 = 204 cents, the Pythagorean tone; 3/4 = 498, the just fourth; 2/3 = 702, the just fifth, etc.

From this we see that the gods’ respective numbers are contained in the span of the top octave. Anu, Enlil, Ea and Sin provide with the tonal infrastructure for the Babylonian scale as shown below:

SIN EA ENLIL ANU

0 498 884 1200

Fundamental Fourth Sixth Octave

Anu/Enlil 60/50 = 6/5 = 316 = just minor third

Enlil/Ea 50/40 = 5/4 = 386 = just major third

Ea/Sin 40/30 = 4/3 = 498 = just fourth

Sin/Šamaš 30/20 = 3/2 = 702 = just fifth

Šamaš/Bel 20/10 = 2/1 = 1200 = octave.

With regards to the four gods contained in the upper octave, it will be interesting to note that their numbers produce the same intervals as those nomeclated in CBS 10996. This is the reason behind the presence of only two intervals, thirds and fifths in that text.

I would like to point out that these numbers, in a musicological context, could only have originated from empiricism applied exclusively to a fretted instrument such as the lute. The reason for this is simple enough as any unfretted paradigm such as a harp or a lyre, would not have allowed for any comparative metrology for the reason that frequency variations on such instruments result not only from the length, but also from variation in the tension and mass of their strings. With the lute, variations of frequencies are defined only by the accurate positioning of frets or of fret marks on the neck of the instrument, providing for the basic parameters for the elaboration of ratios.

When I first drafted this paper, there was a major obstacle to my thesis as the oldest iconographic representation of a lute dated from two seals of the Akkadian period, BM 89096 and BM 28806. There was a discrepancy between the period of the origins of the theory and the first apparition of a lute. This would have implacably disposed of my thesis, but it is only recently that Dr Dominique Collon acquired on behalf of the British Museum, a seal cylinder of the Uruk period on which there is an incontestable representation of a musician playing the lute. The seal is cataloged as BM WA 1996-10-2,1, and predates the aforementioned ones by some 800 years, reconciliating the periods for theory and practice, to my greatest relief and satisfaction.

Later, the Greeks established their system around other numbers which I believe exempt of animism. Theirs were 12-9-8-6. The Babylonian system fits in as its series 6-5-4-3 precedes the Greek system in the series: 12-9-8-6-5-4-3. If the Babylonian figures are multiplied by 2 for the purpose of comparativity, then we have 12-10-8-6. It is the Babylonian value of 10 versus the Greek 9 which is essential to the distinction between both systems. Whilst the Babylonians attached more importance to minor and major thirds, 6/5 and 5/4, respectively, the Greeks looked on the fourth 12/4 and the tone 9/8.

Professor O.R. Gurney originally objected to my thesis on a purely Assyriological standpoint stating that the god-numbers were all part of a kabbalistic system that was invented in the Middle Assyrian period (c. 15th century BC) and has nothing to do with music. But in a post scriptum to one of his letters to me, Gurney states that Middle Assyrian scribes were merely adopting an older Babylonian system, which itself might have had earlier forgotten sources in the Uruk period since the principle of the lute, which is that of the principle of fretting would not have existed in the ignorance of the fundamental Götterzahlen.

Now the question as to which preceded the other remains: were the god numbers at the origins of the Uruk fret system of the lute or were the fret ratios at the origins of the god-numbers? I am willing to give preference to the second option as I am presently animated by subjectivist obnubilation.

Other Articles:

The Uruk Lute: Elements of Metrology
The Morphology of the Babylonian Scale