----------------------------------------------------------- The Ushshaq Masri Rotations A Constant Structure Superset of the 1-3-7-9 Hexany ----------------------------------------------------------- The 1-3-7-9 hexany with an added step at 13/8 forms this Constant Structure (CS) scale suggesting a variation on the Arab Maqam Ushshaq Masri, a mode explained below: 48 54 56 63 72 78 84 96 1/1 9/8 7/6 21/16 3/2 13/8 7/4 2/1 0 203.9 266.9 470.8 702.0 840.5 968.8 1200 9:8 28:27 9:8 8:7 13:12 14:13 8:7 203.9 63.0 203.9 231.2 138.6 128.3 231.2 All steps are superparticular, and a quick overview of the rotations in Scala will confirm that we have a Constant Structure: 1/1 : 9/8 7/6 21/16 3/2 13/8 7/4 2/1 9/8 : 28/27 7/6 4/3 13/9 14/9 16/9 2/1 7/6 : 9/8 9/7 39/28 3/2 12/7 27/14 2/1 21/16: 8/7 26/21 4/3 32/21 12/7 16/9 2/1 3/2 : 13/12 7/6 4/3 3/2 14/9 7/4 2/1 13/8 : 14/13 16/13 18/13 56/39 21/13 24/13 2/1 7/4 : 8/7 9/7 4/3 3/2 12/7 13/7 2/1 2/1 This scale also has an attribute much investigated by Jacques Dudon: it is differentially coherent, with difference tones between adjacent scale steps or their octave equivalents also being present in the scale itself. A diagram of these difference tones and octave-equivalent scale steps may make this concept clearer: 3/2 2/1 7/4 9/8 3/2 3/2 3/2 6 2 7 9 6 6 12 48 54 56 63 72 78 84 96 1/1 9/8 7/6 21/16 3/2 13/8 7/4 2/1 Here is a Scala file and listing: ! ushshaq_masri-7a.scl ! Basic 7-note Ushshaq Masri as superset of 1-3-7-9 hexany 7 ! 9/8 7/6 21/16 3/2 13/8 7/4 2/1 | 0: 1/1 0.000 unison, perfect prime 1: 9/8 203.910 major whole tone 2: 7/6 266.871 septimal minor third 3: 21/16 470.781 narrow fourth 4: 3/2 701.955 perfect fifth 5: 13/8 840.528 tridecimal neutral sixth 6: 7/4 968.826 harmonic seventh 7: 2/1 1200.000 octave --------------------------------------------- 1. Maqam Ushshaq Masri and the Arab Tradition --------------------------------------------- In Arabic "Ushshaq Masri" means "Egyptian Masri," with Ushshaq as the name for a modal type and also a word meaning "lovers," as in Rawda al-Ushshaq, "the Garden of Lovers." In our Constant Structure, we have one possible septimal realization of the general Ushshaq Masri pattern, placed in modern Arab theory on the standard final or note of repose D, and with "d" showing an Arab half-flat lowering a step by approximately a quartertone, and a minus or "-" sign showing a note lowered by a comma from its Pythagorean position: Busalik Bayyati |---------------------------|------------------| D E F- G- A Bd C- D 1/1 9/8 7/6 21/16 3/2 13/8 7/4 2/1 9:8 28:27 9:8 8:7 13:12 14:13 8:7 T B T T J J T To describe the general patterns of the Busalik and Bayyati genera, we may find it helpful to borrow the medieval Near Eastern symbols of T for a _tanini_ or tone; B for a _bakkiye_ or semitone; and J for a _mujannab_ or neutral second. The pattern for Busalik is T B T T, or tone-semitone-tone-tone; while that for Bayyati is J J T or two neutral seconds plus a tone. These symbols have the advantage of leaving open the question of precise intonation. While the lower pentachord of T B T T is in Arab theory called either Nahawand or Busalik, the latter name tends to imply a lower position for the minor third step, with the usual position for Nahawand, in one interpretation, at around 32/27 (294.135 cents). A tuning at 7/6 is one possible realization of the lower position favored for the Busalik genus. Scott Lloyd Marcus, _Arab Music Theory in the Modern Period_ (PhD dissertation, University of California, Los Angeles, 1989), pp. 203-213, reports the understanding of some Arab musicians that in a mode of this type the minor third and minor seventh degrees may be lowered by a "comma" -- meaning not a fixed measurement, but a small adjustment. While the 7/6 and 7/4 degrees of our Ushshaq Masri set could agree with one interpretation of such an adjustment, the narrow 21/16 fourth may be more an alteration brought about by the hexany structure, although lowered fourths as well as minor seventh steps are sometimes associated with other modal families, thus see Marcus, ibid. at 210 (Maqam Rast). ------------------------------------ 2. A Hybrid Approach: Maqam or Raga? ------------------------------------ Although our heptatonic scale fits the general concept of Maqam Ushshaq Masri, a vital aspect of maqam practice is the routine use of inflections within the framework of a single basic mode, as well as a rich range of modulations. These two factors together bring into play many steps beyond the seven often cited in the "textbook" definition of a given maqam. For example, Ushshaq Masri typically uses an ascending neutral sixth step (here our 13/8), but a descending minor sixth step, which in our tuning might be attractively realized by adding a 14/9 degree. This fluidity of the sixth degree is part of the basic mode, quite apart from the various modulations which may also call for additional steps. Treating our seven-note scale as a complete world in itself rather than a starting point for inflections and modulations drawing on a larger tuning system may therefore invite an approach combining some materials of Maqam Ushshaq Masri with a musical outlook more like that of raga, where a single heptatonic set invites hours of improvisation, exploration, and contemplation. ----------------------- 3. Some Basic Rotations ----------------------- Our Ushshaq Masri modality, as we have seen, has a lower Busalik pentachord (T B T T) and an upper Bayyati tetrachord (J J T). Following Scala's scheme for numbering scale degrees, with the 1/1 as step 0, we can consider this as "Rotation 0": Ushshaq Masri: Rotation 0 |-------------------------------------------|------------------------------| | T B T T | J J T | |___________________________________________|______________________________| | | | | | | | | | 9:8 |28:27| 9:8 | 8:7 | 13:12 | 14:13 | 8:7 | |___________|_____|___________|_____________|________|_______|_____________| 1/1 9/8 7/6 21/16 3/2 13/8 7/4 2/1 |-------------------------------------------|------------------------------| | Busalik | Bayyati | |-------------------------------------------|------------------------------| This tuning could also suggest a septimal version of the Persian Nawa or Nava, a modality likewise with a lower tetrachord of T B T T, and an upper tetrachord of J J T, the later in a Persian setting termed Shur, the counterpart of Arab Bayyati. Features of this modality as realized in our Constant Structure include the compact 28:27 semitone or thirdtone pulling up toward the 7/6 step, and also the 14:13 step of "2/3-tone" leading up to 7/4. In a performance over a drone, the resonance of 7/6 and 7/4, and also the special qualities of 13/8 as the "harmonic sixth," may add vertical interest to the flow of the melody. One basic rotation is a modulation to Arab Bayyati or Persian Shur on the 3/2, with the upper tetrachord of Ushshaq Masri or Nava becoming our new lower or root genus. In Near Eastern theory, it is often this root genus that gives a modal family its basic identity, an identity encompassing various inflections or other changes in the upper genus. Since the 3/2 is step 4 of Ushshaq Masri, this will be Rotation 4: Rotation 4: Arab Bayyati or Persian Shur |------------------------------|-----------------------------|-------------| | J J T | T B T | T | |______________________________|_____________________________|_____________| | | | | | | | | | 13:12 | 14:13 | 8:7 | 9:8 |28:27| 9:8 | 8:7 | |________|_______|_____________|___________|_____|___________|_____________| 1/1 13/12 7/6 4/3 3/2 14/9 7/4 2/1 |------------------------------|-----------------------------|-------------| | Bayyati | Nahawand | tone | |------------------------------|-----------------------------|----------=--| This diagram shows a typical Arab understanding, in which Maqam Bayyati consists of a lower Bayyati tetrachord (J J T) plus an upper Nahawand tetrachord (T B T), plus an upper tone to complete the octave. In an Arab style, the 4/3 step at the top of the lower genus is the _ghammaz_, roughly analogous to the confinal or dominant in European modal theory, a note often emphasized in the course of melody. A fine point of Arab theory is that the upper tetrachord of Bayyati is generally termed Nahawand rather than Busalik. Scott Marcus, in his study of _Arab Music Theory in the Modern Period_, at pp. 212-213, reports a remarkable demonstration on the `ud or lute by the Lebanese performer and scholar Ali Jihad Racy that the minor third step of Bayyati is often placed lower than the Pythagorean 32/27 (294.135 cents). Given the tuning of this instrument in pure fourths, the 32/27 is easy to find; but Racy preferred a somewhat lower position. Neither Racy nor Marcus attempts to measure or estimate the adjustment, leaving 7/6 as one possible interpretation of this lowering by an undefined "comma." As Marcus also notes, some of those favoring a lowered minor third in Bayyati extend this modification to the minor sixth also, ibid. at 209. The 14/9 step is one possible interpretation of this placement "minus a comma." A measured Persian tuning for the corresponding Shur reported by Nelly Caron and Dariouche Safvate, _Les Traditions Musicales: Iran_ (Paris, Buchet-Chastel, 1966), table following p. 35, shows a lower Shur tetrachord of 136-140-224 cents, with a minor third at 276 cents, somewhat larger than 7/6 (266.871 cents) but considerably smaller than 32/27. Looking only at the seven notes of Shur represented in our tuning, we have this comparison: 136 140 224 204 72 220 204 Caron-Safvate |------|------|---------|--------|----|--------|-------| 0 136 276 500 704 776 996 1200 13:12 14:13 8:7 9:8 28:27 9:8 8:7 139 128 231 204 63 204 231 |------|------|---------|--------|----|-------|--------| Our tuning 0 139 267 498 702 765 969 1200 1/1 13/12 7/6 4/3 3/2 14/9 7/4 2/1 Very possibly in the kind of Arab tuning recommended by musicians and scholars such as Racy, as in this type of Persian practice, minor thirds, sixths, and sevenths -- and also the vital semitone step between the fifth and minor sixth, here measured at 72 cents -- may often be a bit larger than in septimal JI, but still appreciably smaller than in Pythagorean intonation. In fact, the most notable difference between the Caron-Safvate tuning and this rotation of our Constant Structure is the placement of the minor seventh at 996 cents (a just 16/9) rather than 7/4. To approach another modulation involving new genera, let us return to the original Ushshaq Masri modality, with a diagram here repeated for convenience: |-------------------------------------------|------------------------------| | T B T T | J J T | |___________________________________________|______________________________| | | | | | | | | | 9:8 |28:27| 9:8 | 8:7 | 13:12 | 14:13 | 8:7 | |___________|_____|___________|_____________|________|_______|_____________| 1/1 9/8 7/6 21/16 3/2 13/8 7/4 2/1 |-------------------------------------------|------------------------------| | Busalik | Bayyati | |-------------------------------------------|------------------------------| As already mentioned, the 14:13 step in the upper tetrachord, a smaller form of "2/3-tone," invites an ascent from 13/8 to 7/4, with the latter becoming an attractive melodic goal. Treating this 7/4 step as the upper note of a 4:3 tetrachord, 21/16-3/2-13/8-7/4, provides the basis for a modulation to a modality featuring a beautiful genus of Ibn Sina (980-1037): Rotation 3: Septimal Rast + Septimal `Ajam |------------------------------|-------------------------------|-----------| | T J J | T T B | T | |______________________________|_______________________________|___________| | | | | | | | | | 8:7 | 13:12 | 14:13 | 8:7 | 9:8 |28:27| 9:8 | |_____________|________|_______|_____________|___________|_____|___________| 1/1 8/7 26/21 4/3 32/21 12/7 16/9 2/1 |------------------------------|-------------------------------|-----------| | Septimal Rast | Septimal `Ajam | tone | |------------------------------|-------------------------------|-----------| The lower tetrachord has one of the two versions of what Ibn Sina describes as "a very noble genus," with lengths of 104:91:84:78, and steps of 1/1-8/7-26/21-4/3 or 8:7-13:12-14:13. He also described a version where 14:13 precedes 13:12, 16:14:13:12 or 1/1-8/7-16/13-4/3. The most notable difference may be the higher position of the neutral third step in the first version, as here, at 26/21 (369.747 cents), as compared with 16/13 (359.472 cents) in the second version. The 26/21 position may be more in keeping with tastes in parts of Syria, for example, while 16/13 might not be too far from standard practice in many other parts of the Arab world. Terming 8/7-13/12-14/13 as a "Rast" tetrachord may be something of a license, because the usual definition of Rast, medieval or modern, indeed involves a root tetrachord of T J J (a tone followed by two neutral seconds), but generally assumes a tone of 9:8 rather than 8:7. Ibn Sina himself does not give this or any specific name to his "very noble genus, inviting the reader to choose one. The upper tetrachord I have similarly described as a septimal variation on the Arab genus of `Ajam (T B B); it is one permutation of the Diatonic of Archytas with steps of 8:7, 9:8, and 28:27, known by Ptolemy as the Tonic Diatonic. An upper 9:8 tone completes the octave. These conjunct tetrachords, both beginning with an interval of 8:7, result in a step at 32/21 (4/3 + 8/7). A 32/21 step likewise occurs in a mode described and diagrammed by Safi al-Din al-Urmawi in the 13th century, based on conjunct symmetrical tetrachords of Ibn Sina's "very noble genus" in its 16:14:13:12 form, or 8:7-14:13-13:12. See Fazli Arslan, _Safi al-Din al-Urmawi and the Theory of Music: Al-Risala al-Sharafiyya fi al-nisab al-talifiyya; Content, Analysis, and Influences (FSTC Limited, 2007), pp. 13-15. |--- lower tetrachord ---|-- upper tetrachord --| tone | |----------|------|------|---------|------|-----|-------| 64 56 52 48 42 39 36 32 1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1 8:7 14:13 13:12 8:7 14:13 13:12 9:8 Returning to our Rast-`Ajam mode, we may note that Arab theory sometimes refers to this pairing (T J J T T B T) as Suz-i Dilara, a term more properly referring to a complex maqam with many transformations of which this modal pattern is but one face. Of these rotations, we might say that Ushshaq Masri (Rotation 0) and Bayyati (Rotation 4) present different perspectives on their common genera of Nahawand (or Busalik) and Bayyati. In contrast, our Rast-`Ajam modality (Rotation 3) with its bright 26/21 third in the lower tetrachord and its upper `Ajam genus with a semitonal ascent to the fourth of this tetrachord (16/9), introduces a different mood. ------------------ 4. Other Rotations ------------------ In addition to the rotations above, which fit an Arab view of Ushshaq Masri inviting modulations to Bayyati on the 3/2 or sometimes Rast on the fourth step (here 21/16 rather than a standard 4/3), other possibilities are open, some also with support in Arab theory. Often the root tetrachord for a standard Arab form will be available, but with a less conventional upper genus. Rotation 1, on the 9/8 step of Ushshaq Masri, produces a variation on Maqam Kurdi with a characteristic root tetrachord of B T T, with a semitone and two tones: Rotation 1: Kurdi + Bayyati |-------------------------------|------------------------------|-----------| | B T T | J J T | T | |_______________________________|______________________________|___________| | | | | | | | | |28:27| 9:8 | 8:7 | 13:12 | 14:13 | 8:7 | 9:8 | |_____|___________|_____________|________|_______|_____________|___________| 1/1 28/27 7/6 4/3 13/9 14/9 16/9 2/1 |-------------------------------|------------------------------|-----------| | Kurdi | Bayyati | tone | |-------------------------------|------------------------------|-----------| The conjunct upper tetrachord, 4/3-13/9-14/9-16/9, is our familiar Bayyati at 13:12-14:13-8:7, another permutation of Ibn Sina's noble genus. An upper 9:8 tone completes the octave. Although I am not aware of this tetrachord combination in modern Arab theory, Safi al-Din in his _Kitab al-Adwar_ or "Book of Cycles" offers a catalogue of 84 cycles or octave species. His scheme is based on a set of seven lower tetrachords and twelve upper pentachords, thus yielding 84 combinations in all. He devotes a section of his catalogue to each type of lower tetrachord, showing the 12 modes it forms when matched with each of the pentachords. As it happens, B T T (modern Kurdi) is his third tetrachord, thus forming Cycles 25-36; the upper pentachord J J T T, a modern Bayyati plus an upper tone, is his fifth pentachord, so that our basic pattern appears as Cycle 29. This concept of cycles or modes leaves open details of intonation. We can say that both the lower Kurdi, based on the Diatonic of Archytas, and Ibn Sina's division based on 13:12-14:13-8:7, were standard genera for Safi al-Din, so that their use to realize some of his cycles might not at any rate overly surprise him. The root tetrachord, at any rate, does fit an Arab view that Ushshaq Masri may modulate to Kurdi on its second step, although the upper Bayyati tetrachord may be an opportunity to explore a modal combination documented in theory by Safi al-Din but novel from the view of current Arab practice. Our next modality, Rotation 2 on the 7/6 step of Ushshaq Masri, leads us to a septimal variation on Maqam `Ajam, but with another novel element: Rotation 2: Maqam `Ajam Murassa |------------------------------------------|-------------------------------| | T T J J | T T B | |__________________________________________|_______________________________| | | | | | | | | | 9:8 | 8:7 | 13:12 | 14:13 | 8:7 | 9:8 |28:27| |___________|_____________|________|_______|_____________|___________|_____| 1/1 9/8 9/7 39/28 3/2 12/7 27/14 2/1 |------------------------------------------|-------------------------------| | Najdi | `Ajam | |------------------------------------------|-------------------------------| This rotation offers a rare modality known as Maqam `Ajam Murassa, with a lower pentachord of T T J J, two tones followed by two neutral seconds, here 9:8-8:7-13:12-14:13. The upper tetrachord is a usual `Ajam or T T B, a genus we met in Rotation 3 (Rast-`Ajam). The lower pentachord is known in Arab theory either as Najdi or by the same name as the maqam it defines, `Ajam Murassa. A usual Maqam `Ajam Ushayran, in contrast, has disjunct `Ajam tetrachords: T T B | T | T T B. One analysis views the Najdi or `Ajam Murassa pentachord as a Rast tetrachord (T J J) with a tone added below, thus T T J J. Whatever the analysis, this beautiful form also occurs in Turkish music as one interpretation of the genus Penchgah (either T T J J or T J T J). However, the Turkish Maqam (or Makam) Penchgah uses this lower pentachord as part of a modal and musical structure distinct from that of the Arab `Ajam Murassa. What these forms share is a pentachord with a smallish augmented fourth, here at 39/28, and more generally about a 9:8 tone higher than the usual placement of the neutral third of Rast, as here at 26/21 in Rotation 3. It is a great pleasure to encounter a rare Arab maqam of the `Ajam family as we explore this Constant Structure. The location of `Ajam Murassa on the step a tone below the final of our Rast family mode (Rotation 3) also parallels the standard Arab scheme placing untransposed Rast on C and `Ajam Murassa on lower Bb, although the septimal intonation may be a less conventional element. Having already surveyed the common modulations of Rotations 3 and 4, we move to Rotation 5 on the 13/8 step of Ushshaq Masri. Rotation 5: A Buzurg Variant |---------------------|-----------------------------|----------------------| | J T | T B T | T J | |_____________________|_____________________________|______________________| | | | | | | | | | 14:13 | 8:7 | 9:8 |28:27| 9:8 | 8:7 | 13:12 | |_______|_____________|___________|_____|___________|_____________|________| 1/1 14/13 16/13 18/13 56/39 21/13 24/13 2/1 |---------------------|-----------------------------|----------------------| | Buzurg | Nahawand | Rast | |---------------------|-----------------------------|----------------------| The lower three notes of this mode are identical to those of a genus of the epoch around 1300 known as Buzurg, defined in one form as 1/1-14/13-16/13-4/3-56/39-3/2. However, we have neither a 4/3 nor a 3/2 degree, nor any close variations (e.g. 21/16 or 32/21). While our mode is quite different from Buzurg or other medieval cycles based on combinations of a lower tetrachord and upper pentachord, a different kind of modern Arab model can better accommodate this step structure. Modal patterns in the Sikah (or Segah) family typically consist of a lower J T trichord, plus a central tetrachord of some kind, plus an upper trichord. This general template seems helpful here, and nicely provides a guide to modes of the Sikah family which may not have a 4/3 or 3/2 step. Although the medieval concept of Buzurg refers to a pentachord, to speak of a "Buzurg trichord" is useful in distinguishing our 1/1-14/13-16/13 from a modern Sikah trichord, also beginning with a smallish neutral second, but favoring a usual 9:8 tone and a lower neutral third step, perhaps somewhere in the range of 63/52 to 11/9. The middle tetrachord, 16/13-18/13-28:27-21/13, has a Nahawand or Busalik structure (T B T), but here in the compressed space of 21:16 rather than 4:3. The upper trichord may be described as Rast, since it has T J, a tone plus a neutral second adding up to a largish neutral third (here 26/21). This mode might share one of the tendencies of modern Sikah family modes: an inclination for the upper Rast trichord to seek completion by cadencing on the next step above the 2/1, here the 14/13 (or, in this octave, the 28/13). The musical focus thus becomes a Rast tetrachord (here a septimal variation) of 21/13-24/13-2/1-28/13 -- or, 21/13 as the 1/1, 1/1-8/7-26/21-4/3! Indeed the name Sikah (an Arabic rendering of Persian Segah) refers to the "third" step -- more specifically, the third step of a Rast family Maqam. From this point of view, the 2/1 step of our Buzurg variant is the 26/21 step of the septimal Rast modality we encountered in Rotation 3 -- or, more precisely, that step in the upper octave. As with Rast and the Sikah family modalities of modern Arab music, we may thus modulate between a Rastlike structure based on tetrachords, and the distinctive trichord-tetrachord-trichord pattern of our Buzurg variant. Our last modal form found in this Constant Structure, Rotation 6 on the 7/4 step of Ushshaq Masri, offers something else again: Rotation 6: A Harmonic Maqam Jaharkah |-------------------------------|-----------|------------------------------| | T T B | T | T J J | |_______________________________|___________|______________________________| | | | | | | | | | 8:7 | 9:8 |28:27| 9:8 | 8:7 | 13:12 | 14:13 | |_____________|___________|_____|___________|_____________|________|_______| 1/1 8/7 9/7 4/3 3/2 12/7 13/7 2/1 |-------------------------------|-----------|------------------------------| | `Ajam | tone | Rast | |-------------------------------|-----------|------------------------------| While the lower septimal `Ajam tetrachord at 8:7-9:8-28:27 and the upper septimal Rast at 8:7-13:12-14:13 are familiar from other rotations, this arrangement yields a scale often following the series of harmonics 7-14: 1/1-8/7-9/7/4/3-3/2-12/7-13/7-2/1. Expressed as an integer series, this would be 42:48:54:56:63:72:78:84. This modal pattern with lower `Ajam (T T B), a middle tone, and upper Rast (T J J) is known in Arab theory as Jaharkah (Persian Chahargah), referring to "the fourth step" of Rast, or here the Rast family variation of Rotation 3. Maqam Jaharkah, like the rarer `Ajam Murassa of Rotation 2, belongs to the `Ajam family; and both of these modalities feature neutral steps making them memorable variations on the `Ajam theme of T T B. For Jaharkah, the neutral seventh step serves as a modal trademark. ------------- 5. Conclusion ------------- Adding a 13/8 step to the 1-3-7-9 hexany results in a differentially coherent Constant Structure where each rotation realizes a septimal form of some enticing modal pattern. We find more or less close fits with currently established Near Eastern patterns (Rotations 0, 4, 6); rare but recognized modalities (Rotation 2, Maqam `Ajam Murassa); modalities combining medieval genera in ways possibly novel then as well as now (Rotation 3), or recognized in medieval theory but, at least today, opening new possibilities (Rotation 1, Safi al-Din's Cycle 29); or creating innovative patterns inviting some mixture of both medieval and modern approaches (Rotation 5). In one sense, these rotations may be seen as a kind of aleatory music: the 1-3-7-9 hexany was likely not designed specifically for Near Eastern styles, and adding a 13/8 step has many welcome but unanticipated consequences I did not consider when drawn to this solution for a heptatonic Constant Structure. My approach in this paper of canvassing some Near Eastern practice and theory as a basis for approaching these septimal rotations has the advantage of placing the steps and intervals in a musical context; but also the possible disadvantage of beginning with certain preconceptions which are certainly in the eyes and ears of the beholder or listener. The boundaries defining trichords, tetrachords, and pentachords, for example, are ways of organizing musical materials, but not necessarily inherent in the materials themselves. Varying Near Eastern views on how best to parse the genera making up a given maqam provide one illustration of this reality. A different and equally viable approach is simply to show the steps and intervals of these rotations, and leave the interpretations to musical experience. Indeed, just as medieval and Renaissance polyphony of Europe may be transcribed (if and when transcription is useful, as opposed to reading from the original notation) with or without barlines, so listings of these rotations with and without possible divisions into genera may together give a better perspective on the creative possibilities. In peace and love, Margo Schulter mschulter@calweb.com