The critical text for understanding the musical notation system is the so-called 021“mathematical text” which comes from Nippur and dates to 500 B.C.
The “mathematical text” includes in each line a pair of numbers and the name of the pair of numbers. Each line in the list begins or contains the Sumerian sign SA. In Sumerian, SA can mean sinew or gut and is also one of the signs in the word for cat. SA can also mean string of a musical instrument. It seems that gut was used to string musical instruments thousands of years ago. Each line of the tablet that begins with SA names two strings. The names of strings are known from the second of the theory texts, a lexical text from Ur. In the “mathematical text,” the word SA is followed by a string name and then SA is repeated with another string name. In the same line two numbers are given, and SA is again repeated with a special word following. The two numbers in the line, in conjunction with the SA, must stand for the distance between the two strings identified by the first two SA’s in the line; that is, the two numbers identify the musical interval between the two strings. This—the name of the interval—seemed the only logical meaning for the last appearance of SA in the line. This meaning was later confirmed by another tablet which we have not yet discussed.
Thus, in the “mathematical text,” the SA lines take the following form:
SA [meaning “string”] fore [the name of a string] and SA [string] fifth, 1, 5, SA [interval] ni
µs ûgabarî or (“rise of the duplicate”) [the name of the interval]
SA [string] next and SA [string] fourth-behind, 2, 6 SA [interval] ishartu (“normal”)
The names of the strings, as we have said, were also found in the lexical text from Ur. In addition, the “mathematical text” contains a section in which the lines contain only two numbers and the name of the interval preceded by SA (meaning “interval”). These lines take the following form:
1, 5 SA ni
Thus, this confirms that SA in conjunction with the special word following it identifies the interval and its name. For example the numbers 1, 5 indicate an interval of a fourth, say from C to F. The numbers 1, 3 could indicate an interval of a third, say C to E.
The “lexical text” names nine different strings but the “mathematical text” refers to only the first seven strings listed in the “lexical text.” This suggests that the musical intervals listed in the “mathematical text” (and consisting of two notes or strings) are to be played on a seven-note scale, with the eighth note being the same as the first. This reasoning was later confirmed by the so-called “tuning text,” which is the third theory text.
Thus from the mathematical and lexical texts, we learned the names of the strings, the names of 14 musical intervals (7 fourths and fifths, 7 thirds and sixths) and the fact that the scale consists of seven notes before repeating the octave.
Additional information comes from the last of the theory texts, which is a catalog of cult songs. A portion of this text simply lists love songs in particular tunings.
What was meant by the different tunings is revealed in the already mentioned “tuning text.”
The “tuning text” contains instructions for tuning—or more precisely, for changing the tuning of—a lyre.
The “tuning text” takes the following form:
If the lyre was in X tuning, but the Y interval was unclear, you alter two particular strings and then the lyre is in Z tuning.
The key to understanding these instructions is to understand what is meant by “unclear.” Oxford musicologist, David Wulstan, first suggested that “unclear” referred to a particular interval (known to musicians as a “tritone,” an augmented fourth or diminished fifth) which sounds a bit discordant, unstable and, if you will, unclear. By changing one of two strings on the lyre by a half-step, this interval would be “clear,” but the instrument would then be in another tuning (or “key”) and another interval in the new scale would then be “unclear.” This interval could then be made “clear” by the same retuning process; the lyre would then be in yet another tuning.
Although this sounds somewhat technical, it can be understood by anyone who can play a scale on a piano.
As we have seen, the ancient scale had seven notes, just as we do: for example, C, D, E, F, G, A, B. Then the octave C is repeated. Unlike a piano, however, the ancient lyre had no “black” notes. Imagine a piano on which you cannot play the black notes. It would be possible to play seven different scales on such a piano, one beginning on each of the white notes. Each of these scales would be different because, without black notes, the half steps would come in different places within the scale. (The half steps on a piano are between E and F, and B and C; there are no black notes between these intervals.) With black notes we can play a half step (or half tone) above or below any other note. But without the black notes, we can’t do this. Using only white notes, we have seven different scales on the piano, one beginning on each of the notes. These are the seven tunings identified in the “mathematical text.”a
We can play each of these seven different scales on the piano without retuning the piano simply by beginning on the next higher white note each time. But this takes 14 successive notes. On the lyre, it would take 14 different strings. To obtain these seven different scales on a lyre of only 9 or 11 strings, it was necessary to re-tune the instrument so that the half steps would occur in the right places depending on which of the seven scales we wanted.
In a C-major scale, a half step occurs between E and F, and between B and C. Otherwise we proceed by whole steps or whole tones (that is, with a black note between). If we begin a scale on D without using the black notes, the half steps will occur at a different place within the scale (between the second and third, and sixth and seventh notes) and this is why a white-note scale beginning on C is different from a scale beginning on D.
If we understand how stringed instruments are tuned, we will be able to understand better what is meant by “unclear” in the tuning text. People who tune stringed instruments (including pianos) commonly do so by tuning a cycle of fifths, that is by starting on a particular note and tuning a fifth interval above it. Thus if we start on C, we could tune G above it. This interval is easily accessible to the ear, it is reliable, and is repeatable; in short, it is “clear.” After tuning G in relation to C, the tuner can then proceed to tune D in relation to just-tuned G. To do this, he descends from the G to a fourth below it; this interval is also clear and easily recognizable because, although the interval is a fourth, if inverted, it is a fifth; that is, D above G is a fifth rather than a fourth. So the tuner begins on the particular strings and tunes by ascending fifths alternating with descending fourths. When he has tuned all the strings in this relationship, the instrument is “in tune.”
Without the black notes, all scales will include a tritone (an augmented fourth or—the same thing—a diminished fifth). In a C-scale, this tritone or interval occurs between F and B. This interval is discordant or “unclear.” To “clear” it one string must be raised or lowered to change it into a pure fourth or fifth. This will alter the place where the half-step occurs within the sequence, and will change the “all-white note” scale.
At one point in the tuning tablet, the instructions are to lower the bottom string of the augmented fourth, thus producing a pure fifth.b If this were done seven times to each augmented fourth, the entire instrument would be tuned one-half step lower, and if we followed this procedure seven times, we would be, in terms of note relationships, precisely where we were at the beginning. That is why, in the tuning text, there is the instruction “no more” after the instructions for seven tunings.
This understanding of making an “unclear” interval clear and thereby creating a different scale or tuning, is consistent with everything else known about texts and no other hypothesis explains all the data.
At this point, the scholars working on the tablets knew from the mathematical and lexical texts the names of the tunings (or scales) and the names of the intervals. The names of the seven tunings were identical to the names of seven intervals. But which tuning applied to which interval? Professor Hans Kummel of the University of Hamburg put forward the technical but ingenious hypothesis that each tuning was named for the interval of a fourth or fifth with which the tuner began to tune the instrument. The tuning text instructed the tuner to “clear” the “unclear” interval by lowering a particular augmented fourth (or diminished fifth) one half step, thereby creating either a “clear” fourth or a “clear” fifth. The tuning then 022proceeded by tuning ascending fifths and descending fourths. The interval from which the tuning began (the “clear” interval created by lowering one of the strings of the “unclear” interval) became the name of the particular tuning.
In this way, scholars were able to identify, the seven scales that were named after the seven intervals of fourths and fifths. The remaining intervals—thirds, sixths, and the tritone—could then be identified exactly by their place in any particular scale.
Now it is time to return to the song text tablet from Ugarit which contains the words to our Hurrian hymn. Below two parallel lines on this tablet the text seems to contain the musical notation to the song.
The notation text takes the form “word + number, word + number, word + number, etc.” The words were easily identifiable as Hurrianized forms of the Akkadian intervals we had already deciphered and understood. These intervals could be played! (We didn’t and still don’t know what note they start on—i.e. what pitch—but we do know the interval relationship and where the interval occurs within the scale.)
Another way of expressing the form of the musical notation written below the parallel lines is:
interval name + number, interval name + number, interval name + number, etc.
interval name + 3, interval name 1, interval name 3, etc.
(using the actual Akkadian name of the interval)
qablite 3, irbute 1, qablite 3
(using the Hurrian form of the interval name, which is actually what is used in the notations below the parallel lines)
qabitu 3, rebutu 1, qablih 3
What the interval name meant was known but what was the significance of the number following it? Several possibilities were tried but the only one that “worked” (in the sense of matching the lyrics) was taking the number to indicate the number of times the interval, like a chord, was repeated. Thus, qablite 3 meant to repeat the qablite interval three times. This hypothesis fit well with the total number of syllables in the song. Certain minor philological assumptions had to be made about the song text, but with these assumptions (relating to the number of syllables), the text fit the music quite closely.
The resulting music is reprinted in the article. The particular tuning is indicated in the label or colophon on the reverse of the song tablet. This song is to be played in nid-qabli tuning.
In this particular tuning, the half tones fall in the same places they do in our modern major scale. Thus our song was written in the equivalent of a modern major scale.