Uruk Lute

THE URUK LUTE: ELEMENTS OF METROLOGY
By RICHARD DUMBRILL

Earlier this year I examined a cylinder seal acquired by Dr Dominique
Collon on behalf of the British Museum. The piece is now listed as BM WA
1996-10-2,1, and depicts, among others, the figure of a crouched female
lutanist. The seal, which I shall not discuss here, has been identified
by Dr Collon as an Uruk example and thus predates the previously oldest
known iconographic representations by about 800 years. Little can be
said about the instrument except that it would have measured about 80
centimetres long and that some protuberances at the top of its neck
might be the representation of some device for the tuning of its
strings. Otherwise, the angle of the neck in its playing position as
well as the position of the musician's arms and hands is consistant
with one of the aforementioned Akkadian seals, namely BM 89096. This
shows that the instrument evolved very little for the best part of one
millennium, for the probable reason that it already had completed its
development, as early as the Uruk period.

The existence of the lute among the instrumentarium of the late fourth
millennium is of paramount importance as it is consequential to the
understanding and usage of ratios at that period. I am further willing
to hypothesize that the lute might have been at the origins of the
proportional system. This is what I shall now demonstrate.

The lute differs from the two other types of stringed instuments, namely
harps and lyres, in that each one of their strings produces more than
one sound. This peculiarity qualifies the lute as a fretted instrument,
not on the basis that it is provided with frets as we know them on the
modern guitar, for instance, but in that each of the different notes
generated from each of its strings is determined by accurate positions
marked on the neck of the instrument. These are defined from the
principle of ratios, and it is the principle of the stopping of the
strings along the neck of the instrument that was at the origins of the
understanding of such ratios.

DEMONSTRATION

Let us take a string that we stretch along the neck of a lute. Its
length is measured from the bridge to the nut as this is the part of the
string which vibrates freely to produce music. If a finger presses the
string onto the neck of the lute at precisely half of its length, and if
the half between the bridge and the finger is then plucked, the note
heard will be one octave higher than it sounded when it was free. The
ratio of the length of the free string to that of its half is therefore
of 2:1. Babylonian musical theory defines two intervals, thirds and
fifths. The position at which fingers should stop the strings to produce
these intervals would be 6:5 for the minor third; 5:4 for the major
third; and 4:3 for the fifth, and of course 6:3 = 2:1 for the octave. It
goes without saying that we have there a series of ratios - 6:5:4:3 -
which coincides with the god numbers of 60 for Anu, 50 for Enlil, 40 for
Ea, and 30 for Sin. This is the basic infrastructure for the Babylonian
scale. However, there is a problem since it is well attested that during
the Old Babylonian period, the scales were incontestably descending, and
not ascending as we have it on our Uruk lute. However, it can be argued
that since ratios of string lengths behave in reciprocal relation to
ratios of frequencies, that it all comes to the same. Furthermore,
ratios can be inversed: minor thirds become major sixths; major thirds
become minor sixths; fourths become fifths and fifths become fourths
etc. It is not improbable that despite of the fact that the
approportionation of its strings was made on an ascending pattern, that
the music played on the lute was composed in the characteristic style of
a descending paradigm.

With regards to units of measurements in relation to the lute, it is
possible to speculate that if the speaking length of a string equated to
2 Kي, let us say 60 centimetres, since the average forearm is 30
cms, the octave fret would have been placed at 30 cms. If the other
frets were then placed accordingly to this principle, the fret for the
minor third, 6:5, would have been placed 10 centimetres away from the
nut; the fret for the fifth, 4:3, would have been placed a further 10
centimetres down. If we take arbitrarily the free string as sounding an
unspecified `c', then the frets as placed above would produce
c-eb-g-c, a chord of c minor. Now this numeric system is unable to
provide with smaller intervals and it is possible that this is what
decided theoreticians to increase the values of the gods to 60, 50, 40,
30 and so forth as it allowed for a more complex ratio system. This is
well demonstrated with the ratio of Iëtar to Šakkan, 15:14, which
is situated outside the pantheon of the higher gods (60-50-40-30). But
if we make it fit in the pantheon, 15:14 becomes 60:56, and thus the
fret for Šakkan would be placed 4 cms down from the nut and would
produce 119,44 cents, a large semitone. In early Sumerian metrology,
this would equate to 1 ŠU.SI. Now a ŠU.SI of 4 cms is large. It
is commonly agreed that it would have measured 1.6 cms, at some point,
and if we make it fit to our hypothetical system by means of
approportionation, then the speaking length of the string would be 48
cms. If we allow for, let us say, 10 centimetres before the nut and 10
more after the bridge for the purpose of fixation, then the length of
the whole instrument would be around 70 centimetres. This agrees with
most of the iconographia.

Two other gods pair to produce the ideal complement to the ratio of
Ištar to Šakkan (15:14): these are the second number for Anu, 21
and the number for Šamaš, 20. Their ratio 21:20 gives 84.47
cents, which added to the ratio of 15:14 = 119,44 cents, adds up exactly
to 203.91 cents. This is the so-called Pythagorean tone which equates to
the Greek ratio of 9:8. I am willing to emphasize the argument that if
up to now, the principal god numbers might have been coincidental with
the fundaments of ancient musical theory, such sophisticated figures as
the ones just debated cannot really confirm this coincidency. Thus, the
figure of 119,44 cents would have been the Babylonian equivalent of the
Greek apotome, and 84,47, the Babylonian equivalent of the Greek limma,
but not any longer using the terms in their Greek definition because
these Babylonian figures do not arise from the Greek methodology. The
prevalence of the Babylonian apotome and limma, is futher made obvious
when compared to their Greek equation which produce 114 cents for their
apotome, corresponding to the ratio of 2048/2187, and 90 cents for their
limma, corresponding to the ratio of 243/256.

It is clear, therefore that in spite of the fact that the Greek
Pythagorean tone was a fundamental ratio in their harmonic series of
12:9:8:6, the structure of their semitones was not as simple as the
Babylonian system.

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, "J.Smith" <jsmith9624@...> wrote:
>
> THE URUK LUTE: ELEMENTS OF METROLOGY
> By RICHARD DUMBRILL
>
> <snip>
>
> ... Two other gods pair to produce the ideal complement to
> the ratio of Ištar to Šakkan (15:14): these are the second
> number for Anu, 21 and the number for Šamaš, 20. Their
> ratio 21:20 gives 84.47 cents, which added to the ratio
> of 15:14 = 119,44 cents, adds up exactly to 203.91 cents.
> This is the so-called Pythagorean tone which equates to
> the Greek ratio of 9:8.
>
> <snip>
>
> It is clear, therefore that in spite of the fact that the
> Greek Pythagorean tone was a fundamental ratio in their
> harmonic series of 12:9:8:6, the structure of their semitones
> was not as simple as the Babylonian system.

I am extremely interested in all this, having done a lot of
research into (and speculation on) Sumerian and Babylonian
tuning theory. (The archives of the main tuning list can be
searched for a number of my contributions on this topic.
I had some correspondence with Dumbrill years ago too.)

What you are describing, assuming that the Babylonain limma
is the lower interval and the apotome the higher, is what the
Greeks later called "katapyknosis". In this case, the whole-tone
of ratio 9:8 has its terms multiplied by 5 so that the ratio
becomes 45:40, and a division represented by 42 is inserted
between them, to produce an upper ratio of 45:42 = 15:14
and a lower ratio of 42:40 = 21:20.

My hypothesis is that the Greek division of semitones was
indeed originally as simple as the Babylonian, and in fact
may have actually *been* the Babylonian. It was the later
theories of Pythagoras (or the pythagorians, if in fact
Pythagoras is a mythical person, which he may have been)
which mandated that all ratios be based on factors of only
2 and 3, which limited their construction of semitone division
to the now well-known pythagorean ratios.

For my own insight into the earliest known Greek
writings on tuning division, see my paper on Philolaus
in _Xenharmonikon 18_, portions of which appear in this
webpage:

http://tonalsoft.com/enc/p/philolaus.aspx

BTW, it's probably better to migrate this whole discussion
to the main tuning list, as this one is supposed to be focused
on the actual making of microtonal music. So i've cross-posted
this post to the main list, and encourages response to go there.

-monz
http://tonalsoft.com
Tonescape microtonal music software