ICONEA 2008 1 2 NEW LIGHT ON THE BABYLONIAN TONAL SYSTEM Leon Crickmore One of the most significant developments in recent musicology has been the transcription and interpretation of a number of musical cuneiform tablets dating from the second millennium B.C. It has been established that Old Babylonian music was diatonic and based on seven heptachords, corresponding to the first seven tones of the ancient Greek octave species. But a problem remains about the direction of these scales. This paper will suggest a resolution of the ‘dilemma’ reached by Kilmer in her pioneering research. It will also argue that the theoretical musicians of ancient Mesopotamia are likely to have quantified their scales, using sexagesimal arithmetic and numbers from their standard tables of reciprocals. The resulting tuning would therefore have been Just rather than Pythagorean. During the second half of the last century, our understanding of the history of music was significantly extended as a result of the transcription and interpretation of a number of musical cuneiform texts dating from the second millennium B.C. For musicians - and possibly for the general reader, too - the most accessible and succinct summary of this research is to be found in Kilmer’s article under the heading ‘Mesopotamia’, in the New Grove Dictionary of Music and Musicians. According to Kilmer 1 ‘from the Old Babylonian to the Seleucid periods a standard corpus of Akkadian terms was used to describe seven heptatonic diatonic tuning sets or scales.’ The archaeological evidence for our knowledge of the Mesopotamian tuning system, she continues: ‘derives from nearly 100 cuneiform tablets’. Of these, three main texts will be crucial to my argument: namely, CBS 10996, UET VII 74 and CBS 1766. However, before commenting on each of these, for the benefit of those who are familiar with modern musical notation by letter-names, Kilmer’s transcription of the Mesopotamian heptachords is presented (fig. 1). Musicians will note that Kilmer and the musicologists with whom she worked have assumed that the scales were rising and corresponded to the ancient 3 4 5 6 7 išartu E F G A B C Dorian D kitmu E F# G A B C Hypodorian D embūbu E F# G A B C# Phrygian D pītu E G A B C # Hypophrygian D G A B C # nīš GABA.RI* E F# G# A # B C # qablītu E# F# A # B C # F # nīd qabli E F# # # G # Lydian D# Hypolydian D# Mixolydian D# Fig. 1. * Read niš tuĥri.† Greek octave species, the names of which appear on the right. Moreover, to be even more technical for a moment, the scales have been notated chromatically within a single octave - that is thetically, rather than dynamically - a point to be considered further. The išartum mode is the only scale expressed exclusively by means of letters corresponding to the white keys of a piano. The orthographically trained will have noticed that Kilmer gives the string-pair or scale names without mimation. Commentary and Interpretation The aim of this paper is to complement the work of archaeologists and textual scholars, by providing, from a musicological perspective, a commentary on and interpretation of the content of three cuneiform texts in particular: CBS 10996, UET VII 74 and CBS 1766. CBS 10996 is a Neo-Babylonian text, published by Kilmer.2 UET VII 74 is Old Babylonian. It was originally published by Gurney,3 but later revised.4 CBS 1766 is a badly damaged tablet of uncertain provenance and date. It was only published as recently as 2006.5 In addition to a table of numbers, the text includes an unusual geometrical structure. The inscription above the numerical columns remains largely unintelligible, although recent work by a team at the British Museum suggests a link with the Middle-Assyrian song-list KAR 158. † The Old Babylonian equation of the pseudo ideogram GABA.RI has recently been rendered as niš tuĥrum. See Krispijn-Mirelman, Iraq (forthcoming). 11 ICONEA 2008 Basic tuning Fine tuning 1-5 7-5 Heptachordal name nīš GABA.RI* 2-6 1-6 išartu 3-7 2-7 embūbu 4-1 1-3 (nīd qabli) 5-2 2-4 (qablītu) 6-3 3-5 (kitmu) 7-4 4-6 (pītu) Fig. 2. * Read niš tuĥri. CBS 10996 lists fourteen pairs of integers between one and seven. The logogram ‘SA’, preceding the numbers, means a ‘string’, and suggests a tuning procedure for a seven-stringed instrument. If this is so, the odd-numbered lines from 11-24 refer to pairs of strings defining musical intervals of fifths and fourths. Modern string players still tune their instruments by fifths and fourths, although, unlike their Babylonian counterparts, modern musicians trained to think in terms of relationships between musical pitches rather than between named string-pairs, exclude the ‘unclear’ interval of the tritone (the diminished fifth or augmented fourth) from an integral role in the procedure. On the other hand, as will emerge later in the discussion of UET VII 74, the Babylonian tuning system could be construed as a cyclic procedure for the correction of tritones. Kilmer6 interprets the seven ‘dichords’ (pairs of strings) in my left-hand column as a description of a method for tuning seven strings to each of seven modes or heptachords, with the outcome I have already indicated in figure 1. Smith and Kilmer7 interpret the dichords of the even-numbered lines between 11 and 24 - that is, those in the righthand column of figure 2 - as a means of ‘fine-tuning’ the thirds and sixths in each of the seven scales, usually through the adjustment of the common string whose number is underlined in the figure. They consider the likely function of this procedure would be to make the thirds and sixths sound ‘sweeter’. This would imply bringing the basic Pythagorean tuning closer to what acousticians call Just tuning - another matter to be considered in greater detail later. The dichords in the even-numbered lines have their own textual descriptions. UET VII, 74 Kilmer8 states that it was this text (which she refers to as U. 7/80, its field number) which convinced scholars that heptatonic diatonic scales must be the correct interpretation of the tuning tablets. Unfortunately, it has also left her own pioneering research work ‘on the horns of a dilemma.’ 9 12 For in the secondary literature concerning CBS 10996 and UET VII 74, a difference of opinion emerges about whether the heptachordal scales should be interpreted as rising or falling. Musicologists have been uncertain about whether the word ‘qudmu’ (‘foremost string’) in CBS 10996, refers to the string sounding the highest or the lowest pitch. When Gurney first published UET VII 74 in 1968, everyone assumed that the scales defined in the tablet were ascending. However, some years later, the musicologist, Vitale,10 argued that the string descriptions ‘thin’ and ‘small’ in UET VII 126 must refer to higherpitched strings, and in consequence the scales in UET VII 74 ought to be descending. Then the Assyriologist, Krispijn,11 proposed an improved reading of the twelfth line of UET VII 74 which supported Vitale’s view. The relevant portion of line 12 originally read: ‘NU SU’, ‘no more’, that is, ‘end of sequence’. Krispijn considered that damaged signs were compatible with ‘ĥu-um’, and suggested ‘nusuĥ(u-um)’, the infinitive of the verb ‘nasaĥum’, ‘to tighten’. Gurney12 therefore, issued a revised transliteration, as a result of which most textual scholars and musicologists have accepted that (with regard to UET VII 74 at least) the scales defined must be falling. Such a consensus, however, created a problem for Kilmer, for while it is true that the tuning procedures she had derived from CBS 10996 can be applied in either an upward or a downward direction, the change of direction results in different names for the scales. The only scale which retains the same name whether rising or falling is embūbum. Fig. 3 indicates the anomalies in nomenclature. Vitale išartum embūbum nīd qablim qablītum kitmum pītum nīš GABA.RI* Kilmer nīd qablim embūbum išartum nīš GABA.RI* pītum kitmum qablītum Fig. 3. *Read niš tuĥrim. Kilmer frankly admitted this dilemma, but at the same time expressed her belief that ‘we have not arrived at the end of the discussions of this subject’ and ‘perhaps the answer will lie in our eventual ability to understand how ‘pitch sets’ could work either up or down’.13 A possible escape route out of this dilemma, was published earlier this year.14 15 The musicologists who assisted in the recovery of the Mesopotamian tuning system were perhaps too eager to relate its scales to the octave species of ancient Greece. Kilmer16 notes that no-one has yet identified a Sumerian or Akkadian word for ‘octave’. ICONEA 2008 The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week. Nicomachus, writing in the second century A.D., devotes the whole of the fifth chapter of his Manual of Harmonics to the thesis that ‘Pythagoras, by adding the eighth string to the seven-stringed lyre, instituted the attunement of the octave’ (for full text and commentary see Levin17). The pioneering musicologists were not comparing like with like, but seven-note scales (heptachords) with eight-note scales (octachords). Thus, for example, when defined as a series of tones (t) and semitones (s), the heptachord išartum would be stttst, corresponding to the first seven tones of the ancient Greek Dorian scale, rising. But the first seven notes of the falling Dorian octave, starting from the octave above the original note, displays a different pattern: ttsttt - the pattern of the heptachord with the alternative name in figure 3, that is ‘nīd qablim’, corresponding to the Lydian octave species and our modern major scale. Each of the heptachords forming a pair in figure 3 are in fact the mirror image of each other. ‘embūbum’ is the only scale which keeps the same name in both columns. This is because the pattern of tones and semitones in the octave to which it belongs (the Phrygian) is palindromic: tstttst. If one were to quantify the Babylonian heptachords mathematically, using tone-numbers to express ratios of string-length, the pairs of scales carrying the same name in both columns of figure 3 would be the inverse or reciprocal scales of each other. The Greek octave species and our modern scales consist of ladders of musical pitches. It is these pitches which remain unchanged when the direction of the scale is reversed. The names of the Babylonian scales, however, may be taken to represent specific modal patterns of tones and semitones, and it is these patterns which remain identical whether the heptachord is rising or falling. If my proposed solution to the problem of nomenclature is correct, it seems likely that a remnant of the Babylonian system may have survived in our modern melodic minor scales. The upper tetrachord of such scales rises and falls in an identical modal pattern: tts, and although the pitches of the scale-ladder change when its direction is reversed, 1 2 s t t s t t t 3 t s t t s t t 4 the name of the scale does not. Figure 4 displays the modal patterns of the seven Babylonian heptachords by name. By focussing on the direction of the scales - a perennial problem in musicology - the musicological significance of UET VII 74 has not yet been explained. The tablet as a whole comprises a cyclical method of tuning and re-tuning a nine-stringed instrument through seven modes in an upward and a downward series. Each of the quatrains of the text follow a similar pattern along the following lines: (1) when the instrument is tuned to scale A, (2) the ‘unclear interval’ (assumed to be the tritone) falls between strings x and y, (3) tighten string x by a semitone (or, in part 2, tune down string y by a semitone) and (4) the instrument will be tuned to scale B. The names (‘išartum’, ‘qablītum’ and so on) refer initially to pairs of strings (the dichords in CBS 10996). The heptachords are called after the dichord which in the previous scale of the series sounded a tritone, but which by the sharpening or flattening of one of its members has now become a perfect fifth. Dumbrill,18 has elucidated the text succinctly. Figures. 5 and 6 tabulate the tuning procedure. For the construction of these figures. I have used ‘išartum’ in its descending form. Figure 5 demonstrates the cycle of tuning by ‘tightening’, as described in the first part of UET VII 74. In the ‘išartum’ heptachord the tritone lies between the fifth and the second string. The player is instructed to tighten the fifth string in order to tune the instrument to the heptachord ‘qablītum’. Subsequently, in turn, the c, g, a and e are similarly sharpened until the heptachord ‘kitmum’ is reached. If, finally, the b in ‘kitmum’ is sharpened, the instrumental tuning returns to the original ‘išartum’ tuning, but now transposed up a semitone. Figure 6 shows the tuning procedure by ‘loosening’, explained in the second part of the text. I have notated this tuning-cycle, beginning from the white-key version of ‘išartum’ used in figure 5. It could just as well have started with the transposed version of the scale with which figure 5 ends. This would simply have reversed the tuning procedure in figure. 5, until it returned to the initial white-key scale of ‘išartum’. In figure 6, however, the b, e, a, d, g and c of 5 Modal Pattern (string intervals) t t s t t t s t t t s t t t s s t t t s t 6 7 t s t t t s t String number Name išartum embūbum nīd qablim qablītum kitmum pītum nīš GABA.RI* Fig. 4. * Read nīš tuĥrim. 13 ICONEA 2008 No 1 2 3 4 Name 5 6 7 Tritone Retuning f’ e’ d’ 5-2 5G d’ 1-5 1G,8G d’ 4-1 4G d’ 7-4 7G dG’ 3-7 3G dG’ 6-3 6G dG’ 2-6 2G,9G išartum c’’ b’ s a’ t g’ t Name t s t qablītum c’’ b’ s a’ t g’ t Name e’ fG’ s t t nīš GABA.RI* b’ cG’’ t a’ t g’ t Name e’ fG’ s t t nīd qablim b’ cG’’ t a’ t gG’ s Name e’ fG’ t t t pītum b’ cG’’ t a’ t gG’ s Name e’ fG’ t t t embūbum b’ cG’’ t aG’ s gG’ t Name e’ fG’ t t s kitmum b’ cG’’ t aG’ s gG’ t fG’ t eG’ s t Fig. 5. *Read niš tuĥrim. ‘išartum’ (the twin partners of the member of the tritone sharpened in figure 5) are each, in turn, flattened, until the heptachord ‘qablītum’ is reached. The loosening of the fifth string (f) in this scale would return the tuning of the instrument to ‘išartum’, but this time tuned a semitone lower than at the start. 14 Mespotamian Music Theory Assyriologists accept that the Mesopotamians must have had their own system of music theory. The interpretation of the relevant evidence is a matter for musicologists. Before, therefore, dealing with the third cuneiform text (CBS 1766), two further questions need to be considered: