Hello, William, Paul, and all. In another thread, William, you observed that JI was attractive for you, quite apart from the question of pure concords, because of the variety of melodic step sizes. Here I might add that many people who like JI or near-JI systems find the sequence of superparticular melodic steps, for example, very attractive. Paul, you replied by observing not only JI systems can offer this melodic variety, but also certain types of tempered systems. My purpose here is to introduce such a system with two 12-note circles in a modified meantone temperament at 36.328 cents apart. This modified meantone is only one example of what I term the Diversity of Gradations or DOG approach, which can also take various just or near-just forms. This temperament is hardly "near-just," since each of the 24 fifths is impure by something between 4.689 and 7.029 cents, in one direction or the other! One place to begin, before exploring some smaller sets to illustrate the DOG approach, is by documenting the full 24-note tuning and its intervals. ! ordinaire1024-24.scl ! Two chains of ordinaire1024.scl at 36.2 cents apart 24 ! 36.32812 78.51562 114.84374 193.35937 229.68749 284.76563 321.09375 387.89063 424.21875 502.73437 539.06249 581.25000 617.57813 697.26563 733.59375 775.78125 812.10938 890.62500 926.95312 993.75000 1030.07812 1085.15625 1121.48437 2/1 That's the whole tuning, with each 12-note circle involving some DOG "outcrossing" -- I don't say cross-breeding, because that's become rather a trademark of Graham Breed! The short story is that the 10 notes or 9 fifths in the range of F-G# are tuned in a meantone (or the most even 1024-ed2 approximation) quite close to 31-ed2. The remaining three fifths, to balance the circle, are tempered at 708.984 cents, almost identically to 22-ed2 (709.091 cents). So, in a sense, our tuning is a "mutt" closely related to the well-known lineages of Nicola Vicentino (1/4-comma or 31-ed2), and who else but Paul Erlich (22-ed2 and the decatonic system)! It has some features from each line of descent, and others that make it a different kind of mix. To see what the DOG approach can involve, let's look at a simple diatonic tuning, approximating the famous diatonic of Archytas, with its steps of 9:8, 8:7, and 28:27 (204, 231, and 63 cents). We'll go for the form which is one variation on the medieval European Dorian mode: 1/1-9/8-7/6-4/3-3/2-27/16-7/4-2/1. Above a drone, this has 7/6, 4/3, 3/2, 7/4, and 9/4 all as relatively simple and aurally apparent concords, as well as sonorities for three or more voices such as 4:6:7, 12:14:18:21, and the Erlichan Ninth, 4:6:7:9 (I call it that because I learned about it from you, Paul, as kind of offshoot for me of the medieval European 4:6:9). Here's our 7-note tuning: ! ordin24-archytan7.scl ! Archytas or septimal diatonic from C#* 7 ! 206.25001 273.04689 502.73439 697.26564 915.23438 970.31251 2/1 Here we're using our best approximations for 9/8, 7/6, and 7/4 -- give or take a cent, or more precisely 1/1024 octave, for the latter two. Our major sixth, a 27/16 in the JI version, is here somewhat wider at 915.234 cents, a near-just 56/33 (915.553 cents). The approximate 56/33-7/4 step at 55.078 cents happens to be almost identical the diatonic semitone of 22-ed2, at 54.545 cents, maybe lending a bit of extra "xenharmonic" quality. The 9/8-7/6 step, by comparison, is 66.797 cents, not far from the just 28/27 (62.961 cents) of Archytas. Looking at the sizes of whole tones and semitones is one way to know that we're in DOG country. Here's a listing of all the intervals found from each of the seven locations: We have tones at these sizes: 194.5 cents (meantone between 10/9 and 9/8) 206.3 cents (near 9/8, just value 203.910 cents) 218.0 cents (eventone between 9/8 and 8/7) 229.7 cents (near 8/7, just value 231.174 cents) Also, we have semitones at 55.1 and 66.8 cents, adding a bit of variety. Our minor thirds, even in this simple diatonic, illustrate the DOG approach also. At 273.0, 284.8, and 296.5 cents, they approximate 7/6 (6.2 cents wide); 33/28 (like 56/33, virtually just); and 19/16 (1.0 cents narrow) or 32/27 (2.3 cents wide). One interesting ramification of this diatonic is 4:6:7:9, here 0-697.2-970.3-1406.3 cents, so that 9/4 as well as 7/4 is quite close to just. The accuracy of 4:7:9, of course, is balanced by the greater inaccuracy of 3/2, here narrow by 4.689 cents, or almost exactly 1/5 Pythagorean comma. Now let's go to 8 notes by adding, in a medieval European fashion, a minor sixth step also, at 775.8 cents, with the lower part of a PDF file to which we already linked, and is here repeated for convenience, showing the matrix of intervals. This eight-note set, a medieval Dorian with the fluid sixth degree, includes a DOG flavor of a 6:7:9 tritriadic scale, with an approximation of this sonority featuring a 273-cent third at the 1/1, 4/3, and 3/2 steps. At the 9/8 position, we also get a tempered variation on one possible Greek version of the Archytas diatonic, approximately 1/1-28/27-32/27-4/3-3/2-14/9-16/9-2/1. Now it's some time for some real "xen" quality. At the 4/3 step, we have a tempered but quite recognizable version of Erv Wilson's 1-3-7-9 hexany, albeit with a meantone major second rather than a just or near-just 9/8: 0-194.5-273.0-467.6-697.3-970.3-1200 cents. And 0-467.6-697.3-970.3 cents gives a 16:21:24:28 effect, something else again! We can resolve this to 3/2-9/8, with the outer voices of the minor seventh, here a near-just 7:4, contracting a fifth in the best medieval European tradition. That is to say, the minor seventh to fifth progression is classic, not necessarily having the seventh tuned at 7:4 rather than 16:9, and much less the 16:21:24:28, a trademark of LaMonte Young, as I learned from Kyle Gann. Other things are starting to happen in this 8-note set, but let's bring them to the fore with a 12-note set, which could serve as an interesting chromatic set on its own, but has extra possibilities as part of the complete 24-note set (like a perfect fourth and fifth available for every location). A new PDF file documents all the intervals of this 12-note subset: ! ordin24-archytan12.scl ! Archytas diatonic (medieval Dorian) as basis for 12-note set 12 ! 78.51563 206.25001 273.04689 424.21875 502.73439 618.75001 697.26564 775.78126 915.23438 970.31251 1121.48438 2/1 Here the idea is a chromatic set with an emphasize on major and minor intervals somewhere in the range from around Pythagorean to septimal -- plus some middle or Zalzalian intervals in the bargain! This is where a DOG approach can offer a greater variety of melodic steps than either 22-ed2 or 31-ed2, although each has its own unique advantages! Before getting to the Zalzalian intervals, let's note how a DOG system can have more than one type of path to intervals approximating a ratio such as 7/4, or even to a single interval size, such as our near-7/6 minor third at 273 cents. From our large major third step at 424.2 cents, a near-just 23/18, we have a chain of fourths 0-491.0-982.0-273.0 cents, with three of these narrow tempered fourths (each almost identical to 22-ed2, or also the harmonic fourth at 85/64, 491.269 cents) forming our near-7/6 third. This is a path essentially the same as in 22-ed2. However, we can also get a 273-cent third from a chain of two meantone fourths (at 502.7 cents) plus a near-21:16 fourth (here at 467.6 cents, close to the representation of 21:16 in 36-ed2 or 72-ed2). We see these chains at the 1/1 step (0-502.7-970.3-273.0 cents), where the middle fourth at 4/3-7/4 is the near-21:16; and at the 4/3 step (0-467.6-970.3-273.0 cents), where it is the first fourth in our chain -- very notable in our tempered 1-3-7-9 hexany above! Now for the Zalzalian intervals, which run a gamut from small to large, and happen as melodic steps to approximate certain superparticular ratios -- pretty much by "dumb luck," as George Secor once said of the happy range of step sizes in his 17-tone well-temperament or 17-WT. The Zalzalian seconds illustrate this point: 127.7 cents (near 14/13, 128.298 cents) 139.5 cents (near 13/12, 138.573 cents) 151.2 cents (near 12/11, 150.637 cents) 157.0 cents (near 23/21, 157.493 cents) Of course, these ratios may be less elegant than in a near-just system with fifths closer to 3/2, where lots of related ratios like 13/8 will also be close to just. Nevertheless, they provide the melodic variety we seek for Near Eastern styles, letting us know that we're barking up the right tree -- if not for the ultimate maqam system, at least for some interesting variations. To give a quick example, our 915-cent or 56/33 step has an ascending Rast at 0-206.3-363.3-491.0-709.0-903.5-1060.0-1200 cents. This might be a reasonable Syrian Rast -- apart from the question of how fourths or fifths tempered by 7 cents might go over with Syrian musicians (Ozan Yarman does sometimes use this degree of tempering in tunings like his Yarman24 series) -- with a bright third at 363.3 cents, not too for from 21/17 (365.825 cents), and a Zalzalian seventh at 1060.5 cents, close to 24/13. The classic steps at 9/8 and 27/16 are quite close to just. Descending, the seventh step is often lowered to minor, and here we get a shading of 0-206.3-363.3-491.0-709.0-903.5-982.0-1200 cents. This tempered septimal minor seventh at 982.0 cents, almost identical to the 22-ed2 approximation of 7/4, may fit the style suggested by at least one Turkish cura with a fret at around 7/4. In the ascending form of Rast, the lower Rast tetrachord has steps of 206.3-157.0-127.7 cents; and the upper tetrachord at the 3/2 has steps of 194.5-157.0-139.5 cents. In the descending Rast, the tetrachord from the 3/2 is 194.5-78.5-218.0 cents. The result is great melodic variety, and also a considerable contrast between the larger and smaller Zalzalian steps (at 157 cents and 128 or 139 cents). Finally, these Archytan tunings of 7, 8, and 12 notes focus on the Pythagorean-to-septimal range of major and minor intervals, and also on the Zalzalian intervals -- but not on prime 5, the main attraction of a modified meantone, of course! In the complete 24-note system, of course, prime 5 abounds; but the idea here was to explore some of the other aspects of this DOG system. As more of an approach than a paradigm, DOG emphasizes such themes as irregular temperament, multiple chains or circles of fifths, and an interest both in diatonic (including maqamic) and more "unconventional" types of structures. And as a proven for getting melodic and general variety, JI is definitely included! Margo Schulter mschulter@calweb.com 4 November 2013