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