----------------------------------------------------------- The Ushshaq Masri Rotations A Constant Structure Superset of the 1-3-7-9 Hexany A Summary ----------------------------------------------------------- In the longer paper on "The Ushshaq Masri Rotations" , I concluded that it is possible to approach a tuning either by analyzing it in accord with the practice and theory of some musical tradition, or simply to show the rotations, steps, and intervals, leaving the matter of interpretations open. That paper took more the first approach, while this summary will take more of the second. If the 1-3-7-9 hexany (1/1-9/8-7/6-21/16-3/2-7/4-2/1) has a 13/8 step added, then a heptatonic Constant Structure results where all steps are superparticular (as also in the hexany itself), and the property of differential coherence much explored by Jacques Dudon obtains. That is, each difference tone between adjacent steps of the scale will have some octave equivalent represented in the scale itself. The following diagram shows these difference tones and their octave equivalents present among the scale degrees, as well as the steps and intervals as just ratios and in cents: 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 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 This paper seeks to offer an overview of the steps, intervals, and rotations, with studies by Erv Wilson and Kraig Grady of tunings such as Grady's Centaur as an inspiration. As in the longer paper, one good starting point is to include a Scala file, with much appreciation for Manuel Op de Coul and his generosity in designing this scale generation and analysis program and making it freely available on the World Wide Web. A Scala listing of the scale showing both ratios and cents follows in the next section, where it helps to explain the numbering of rotations. ! 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 ------------------------------------------------ 1. Rotations, Steps, and Intervals: An Open View ------------------------------------------------ The following table shows the seven rotations of our Constant Structure, while leaving open the question of whether or how to classify types of melodic steps or to group steps into genera such as trichords, tetrachords, or pentachords -- or to associate a given rotation with some known modality in this or that musical tradition. The advantage of such an open view is that it impartially leaves open many roads of interpretation. The numbering of rotations is based on the conventions of Scala, neatly illustrated by its listing for this scale, with the steps from the 1/1 to the 7/4 numbered 0-6, and the 2/1 as step 7. In the view that follows, rotations are numbered 0-6, with Rotation 0 on step 0 (the 1/1); Rotation 1 on step 1 (the 9/8), and so forth. | 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 Rotation 0 |__________________________________________________________________________| | | | | | | | | | 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 Rotation 1 |__________________________________________________________________________| | | | | | | | | |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 Rotation 2 |__________________________________________________________________________| | | | | | | | | | 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 Rotation 3 |__________________________________________________________________________| | | | | | | | | | 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 Rotation 4 |__________________________________________________________________________| | | | | | | | | | 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 Rotation 5 |__________________________________________________________________________| | | | | | | | | | 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 Rotation 6 |__________________________________________________________________________| | | | | | | | | | 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 A listing of these rotations showing both just ratios and cents may also be helpful. Rotation 0 1/1 9/8 7/6 21/16 3/2 13/8 7/4 2/1 0 203.910 266.871 470.781 701.955 840.518 968.826 1200 9:8 28:27 9:8 8:7 13:12 14:13 8:7 203.910 62.961 203.910 231.174 138.573 128:298 231.174 Rotation 1 1/1 28/27 7/6 4/3 13/9 14/9 16/9 2/1 0 62.961 266.871 498.045 636.618 764.916 996.090 1200 28:27 9:8 8:7 13:12 14:13 8:7 9:8 62.961 203.910 231.174 138.573 128.298 231.174 203.910 Rotation 2 1/1 9/8 9/7 39/28 3/2 12/7 27/14 2/1 0 203.910 435.084 573.657 701.955 933.129 1137.039 1200 9:8 8:7 13:12 14:13 8:7 9:8 28:27 203.910 231.174 138.573 128.298 231.174 203.910 62.961 Rotation 3 1/1 8/7 26/21 4/3 32/21 12/7 16/9 2/1 0 231.174 369.717 498.045 729.219 933.129 996.090 1200 8:7 13:12 14:13 8:7 9:8 28:27 9:8 231.174 138.573 128.298 231.174 203.910 62.961 203.910 Rotation 4 1/1 13/12 7/6 4/3 3/2 14/9 7/4 2/1 0 138.573 266.871 498.045 701.955 764.916 968.826 1200 13:12 14:13 8:7 9:8 28:27 9:8 8:7 138.573 128.298 231.174 203.910 62.961 203.910 231.174 Rotation 5 1/1 14/13 16/13 18/13 56/39 21/13 24/13 2/1 0 128.298 359.472 563.382 626.343 830.253 1061.427 1200 14:13 8:7 9:8 28:27 9:8 8:7 13:12 128.298 231.174 203.910 62.961 203.910 231.174 138.573 Rotation 6 1/1 8/7 9/7 4/3 3/2 12/7 13/7 2/1 0 231.174 435.084 498.045 701.955 933.129 1071.702 1200 8:7 9:8 28:27 9:8 8:7 13:12 14:13 231.174 203.910 62.961 203.910 231.174 138.573 128.298 ----------------------- 2. A Table of Intervals ----------------------- A table of intervals, suggested by Kraig Grady's table for Centaur, may also be helpful: Constant Structure: 1-3-7-9 hexany plus 13/8 The 28 intervals (1/1-2/1) Unison | Octave 1/1 0.000 | 2/1 1200.000 _________________________|__________________________ Semitones | Major sevenths 28/27 62.961 | 27/14 1137.039 _________________________|__________________________ Neutral seconds | Neutral sevenths 14/13 128.298 | 13/7 1071.702 13/12 138.573 | 24/13 1061.427 _________________________|__________________________ Major seconds | Minor sevenths 9/8 203.910 | 16/9 996.090 8/7 231.174 | 7/4 968.826 _________________________|__________________________ Minor thirds | Major sixths 7/6 266.871 | 12/7 933.129 _________________________|__________________________ Neutral thirds | Neutral sixths 16/13 359.472 | 13/8 840.528 26/21 369.747 | 21/13 830.253 _________________________|__________________________ Major thirds | Minor sixths 9/7 435.084 | 14/9 764.916 _________________________|__________________________ Fourths | Fifths 21/16 470.781 | 32/21 729.219 4/3 498.045 | 3/2 701.955 _________________________|__________________________ Tritones (smaller) | Tritones (larger) 18/13 563.382 | 13/9 636.618 39/28 573.657 | 56/39 626.343 _________________________|__________________________ ------------------------- 3. Some Commas and Dieses ------------------------- Kraig Grady's table of intervals for Centaur also shows the "span" or range of interval sizes within each category, and the "gap" between two adjacent categories (e.g. whole tones and minor thirds). Here this information is presented for our Constant Structure in the form of a diagram showing differences between adjacent interval sizes, and thus highlighting some of the main commas and dieses found in the tuning. 28:27 27:26 169:168 27:26 62.961 65.337 10.274 65.337 0 62.961 128.298 138.573 203.910 1/1 28/27 14/13 13/12 9/8 |-----------------|-------------------|---|---------------------| 2/1 27/14 13/7 24/13 16/9 1200 1137.039 1071.702 1061.427 996.090 64:63 49:48 96:91 169:168 27.264 35.697 92.601 10.274 203.910 231.174 266.871 359.472 369.747 9/8 8/7 7/6 16/13 26/21 |-------|---------|-----------------------------|---| 16/9 7/4 12/7 13/8 21/13 996.090 968.826 933.129 840.528 830.253 27:26 49:48 64:63 65.337 35.697 27.264 369.747 435.084 470.781 498.045 563.382 573.657 26/21 9/7 21/16 4/3 18/13 39/28 |---------------------|----------|-------|------------------|---| 21/13 14/9 32/21 3/2 13/9 56/39 830.253 764.916 729.219 701.955 636.618 626.343 ----------------------------- 4. Interval grid a la Centaur ----------------------------- Kraig Grady's interval grid designed for the Centaur tuning offers another way of getting an overview of intervals and rotations: Interval grid (another view of rotations) 0 203.910 266.871 470.781 701.955 840.528 968.826 1/1 9/8 7/6 21/16 3/2 13/8 7/4 _________________________________________________________________ | 1/1 | = 9/8 7/6 21/16 3/2 13/8 7/4 | 203.910 266.871 470.781 701.955 840.528 968.826 | 9/8 | 16/9 = 28/27 7/6 4/3 13/9 14/9 203.910 | 996.090 62.961 266.871 498.045 636.618 128.298 | 7/6 | 14/9 27/14 = 9/8 9/7 39/28 3/2 266.871 | 764.916 1137.039 203.910 435.084 573.657 701.955 | 21/16 | 32/21 12/7 16/9 = 8/7 26/21 4/3 470.781 | 729.219 933.129 996.090 231.174 369.747 498.045 | 3/2 | 4/3 3/2 14/9 7/4 = 13/12 7/6 701.955 | 498.045 701.955 764.916 968.826 138.573 266.871 | 13/8 | 16/13 18/13 56/39 21/13 24/13 = 14/13 840.528 | 359.472 563.382 626.343 830.253 1061.427 128.298 | 7/4 | 8/7 9/7 4/3 3/2 12/7 13/7 = 968.826 | 231.174 435.084 498.045 701.955 933.129 1071.702 In peace and love, Margo mschulter@calweb.com