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A twin 1-3-7-9 hexany set and MET-24
Comparing JI and tempered structures
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In a 1-3-7-9-11-13 eikosany, we find two 1-3-7-9 hexanies at a
distance of 13/11 (or 22/13) apart. Here the horizontal dimension
represents 3:2 and the vertical dimension represents 7:6.
1.7.13 3.7.13 7.9.13 1.7.11 3.7.11 7.9.11
|---------|---------|.........|---------|---------|
91/66 91/88 273/176 7/6 7/4 21/16
556.1 58.0 760.0 266.9 968.8 470.7
/ / / / / /
/ / / / / /
289.2 991.2 493.1 0 702.0 203.9
13/11 39/22 117/88 1/1 3/2 9/8
|---------|---------|.........|---------|---------|
1.3.13 1.9.13 3.9.13 1.3.11 1.9.11 3.9.11
We can view each hexany as two three-note chains of 3:2 fifths at
a 7:6 apart. Coming between and connecting the two hexanies are
wide fifths at 117/88-1/1 on the lower chain and 273/176-7/6 on
the upper chain. These "virtually tempered" fifths have a size of
176:117 or 706.880 cents, wider than 3:2 by 352:351 (4.925 cents).
In MET-24, the 352:351 comma is tempered out, so that three
fifths down or fourths up is very close to 13:11; but the
all-important septimal or Archytan comma at 64:63, and various
others, are observed. Our "twin 1-3-7-9 hexanies at 13:11 apart"
take on this structure.
-3.0 -0.6 +0.6 -2.0 -0.9 +1.5
553.1 57.4 760.5 264.8 968.0 472.3
|--------|---------|.........|---------|---------|
91/66 91/88 273/176 7/6 7/4 21/16
/ / / / / /
/ / / / / /
13/11 39/22 117/88 1/1 3/2 9/8
|---------|---------|.........|---------|---------|
288,3 992.6 495.7 0 703.1 207.4
-0.9 +1.4 +2.6 +1.2 +3.5
Generally the tempered values are quite close to the just ones,
with one notable difference in structure which could provide a
motive for tempering in certain stylistic contexts. In the JI
version, we have eight pure 3:2 fifths, and two wide ones at
176:117, or not quite 5 cents larger. In MET-24, all fifths are
impure by either 1.170 or 2.342 cents, a more subtle contrast.
In certain styles such as medieval European polyphony where just
3:2 fifths and 4:3 fourths are generally presumed, one _might_
argue that having all usual fifths slightly wide could be less
noticeable than the greater contrast in the JI version. In styles
of tuning such as those used for Persian tar or santur, however,
where fifths a few cents wider than pure are common, the JI
version might present no questions of this kind at all.
Some quirks of the tempered version might lead us to alternative
interpretations of certain ratios, for example:
+1.8 -0.6 +0.6 -2.0 -0.9 +1.5
553.1 57.4 760.5 264.8 968.0 472.3
|---------|---------|---------|---------|---------|
11/8 91/88 273/176 7/6 7/4 21/16
/ / / / / /
/ / / / / /
13/11 39/22 4/3 1/1 3/2 9/8
|---------|---------|---------|---------|---------|
288,3 992.6 495.7 0 703.1 207.4
-0.9 +1.4 -2.3 J +1.2 +3.5
We can view the 91/66 step of the twin hexanies in JI form as
also in this tempered form an 11/8; at 553.1 cents it's rather
closer to the latter, simpler, ratio (+1.8 cents), although not
far from the former (-3.0 cents). We can likewise view the 117/88
as also a 4/3: it's 2.3 cents narrow of the latter, simpler
ratio; and 2.6 cents wide of the former.
This "simplifying" of certain ratio interpretations also has its
complications. In reality, each of the near-7:6 thirds shown by
the vertical dimension is 264.844 cents, or about 2.0 cents
narrow of 7:6. However, the thirds at 13/11-11/8 and 4/3-273/176
suggested by this diagram would be narrow by a full 352:351 or
almost 5 cents. Of course, applying approximate JI ratios to a
temperament will always involve some compromises and distortions,
much like mapping the Earth's near-spherical surface by some
two-dimensional projection.
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1. A 7-note Constant Structure, just or tempered
------------------------------------------------
Now let's consider a 7-note Constant Structure found within our
twin hexany structure.
1.7.13 3.7.13 7.9.13
|---------|---------|
91/66 91/88 273/176
556.1 58.0 760.0
/ / /
/ / /
289.2 991.2 493.1 0
13/11 39/22 117/88 1/1
|---------|---------|.........|
1.3.13 1.9.13 3.9.13 1.3.11
Let's remap the 13/11 step to a new 1/1 step:
1.7.13 3.7.13 7.9.13
|---------|---------|
7/6 7/4 21/16
266.9 968.8 470.7
/ / /
/ / /
0 702.0 203.9 910.8
1/1 3/2 9/8 22/13
|---------|---------|.........|
1.3.13 1.9.13 3.9.13 1.3.11
This is a usual 1-3-7-9 hexany plus a 22/13 step; with a 27/16,
we'd have a standard permutation of the Archytas diatonic with
steps of 9:8, 28:27, and 8:7. Instead we have a lower pentachord
9:8-28:27-9:8-8:7 (203.9-63.0-203-9-231.2 cents), plus an upper
variation of 44:39-91:88-8:7 (208.8-58.0-231.2 cents). As with
the usual Archytan diatonic, this is a Constant Structure.
Here are the melodic steps and intervals of this mode:
203.9 63.0 203.9 231.2 208.8 58.0 231.2
9:8 28:27 9:8 8:7 44:39 91:88 8:7
|--------|-----|--------|---------|-------|-----|---------|
1/1 9/8 7/6 21/16 3/2 22/13 7/4 2/1
0 203.9 266.9 470.7 702.0 910.8 968.8 1200
In this JI version, the two small steps are unequal, at 28:27
and the narrower 91:88, a difference of 352:351. While the
classic 28:27 step of Archytas at 62.961 cents is an outstanding
example of a thirdtone, the smaller 91:88 at 58.036 cents could
be taken as either a small thirdtone or a large quartertone
(compare 33:32 at 53.273 cents).
The tempered form is rather close to this JI version, and
likewise forms a Constant Structure:
-3.0 -0.6 +0.6
553.1 57.4 760.5
|--------|---------|
91/66 91/88 273/176
/ / /
/ / /
13/11 39/22 117/88 1/1
|---------|---------|.........|
288,3 992.6 495.7 0
-0.9 +1.4 +2.6
Again making the 13/11 step the new 1/1, we have:
-2.0 +0.3 +1.5
264.8 969.1 472.3
|--------|---------|
7/6 7/4 21/16
/ / /
/ / /
1/1 3/2 9/8 22/13
|---------|---------|........|
0 704.3 207.4 911.7
+2,3 +3.5 +0.9
We might quickly take note that changing the 1/1 produces
slightly different tempered sizes for some of the intervals than
in the previous MET-24 lattices. For example, the 3/2 at this
location is 704.3 cents, or 2.3 cents wide -- as compared to
703.1 cents or 1.2 cents wide at the previous 1/1 (now 22/13).
The 7/4 approximation is now 969.1 cents, or 0.3 cents wide,
rather than 968.0 cents, or 0.9 cents narrow. However, the
approximations for 9/8, 7/6, and 21/16 remain the same: these
intervals do not change in size as we move around the tuning.
Let's compare the JI and tempered versions of our 7-note mode,
repeating the diagram for the former and adding one for the
latter:
JI version
203.9 63.0 203.9 231.2 208.8 58.0 231.2
9:8 28:27 9:8 8:7 44:39 91:88 8:7
|--------|-----|--------|---------|-------|-----|---------|
1/1 9/8 7/6 21/16 3/2 22/13 7/4 2/1
0 203.9 266.9 470.7 702.0 910.8 968.8 1200
tempered version
+3.5 -5.5 +3.5 +0.9 -1.4 -0.6 -0.3
207.4 57.4 207.4 232.0 207.4 57.4 230.9
9:8 28:27 9:8 8:7 44:39 91:88 8:7
|--------|-----|--------|---------|-------|-----|---------|
1/1 9/8 7/6 21/16 3/2 22/13 7/4 2/1
0 207.4 264.8 472.3 704.3 911.7 969.1 1200
J +3.5 -2.0 +1.5 +2.3 +0.9 +0.3 J
In the tempered version, a just ratio of either 9/8 or 44/39 is
always represented by an interval of 207.4 cents, actually rather
closer to 44/39, but still closer to 9/8 than the 200 cents of
12n-EDO. Similarly, either 28/27 or 91/88 is represented by the
same step of 57.4 cents -- very slightly narrow of 91/88, but
narrow of 28/27 by a full 5.5 cents!
In fact, the same step of 57.4 cents represents 33/32, 91/88,
and 28/27. This is one compromise of a system attempting to
represent primes 2-3-7-11-13 using only two 12-note chains of
fifths, with 57.4 cents as the distance between the chains.
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2. Rotating our Constant Structure: A 6:7:9 tritriadic scale
------------------------------------------------------------
Another rotation of our Constant Structure, just or tempered,
produces a variation on what is often termed a "tritriadic"
scale, with three triads of 6:7:9. We take the 3/2 step of the
last example as our new 1/1.
1.7.13 3.7.13 7.9.13
|---------|---------|
14/9 7/6 7/4
764.9 266.9 968.8
/ / /
/ / /
498,0 0 702,0 208.8
4/3 1/1 3/2 44/39
|---------|---------|.........|
1.3.13 1.9.13 3.9.13 1.3.11
This JI variation of the 6:7:9 tritriadic differs from the
"standard" version only in having a 44/39 rather than a 9/8
step. As in the standard version, we have pure 6:7:9 triads on
the 1/1, 4/3, and 3/2 steps.
In a lattice for the tempered version, we might use the same
ratios as in this JI version to show the intervals being
approximated:
-4.4 -2.0 -0.9
760.5 264.8 968.0
|---------|---------|
14/9 7/6 7/4
/ / /
/ / /
4/3 1/1 3/2 44/39
|---------|---------|........|
495.7 0 703.1 207.4
-2,3 J +1.2 -1.4
Here we have three approximate 6:7:9 triads, with the 7:6 minor
thirds consistently at 264.8 cents, or about 2.0 cents narrow.
As in the JI version, one of these 7:6 thirds is located above
the 4/3 step, so that this step at 760.5 cents serves as a 14/9
(4:3 + 7:6).
At the same time, this step could also be taken as a 273/176, to
which it is indeed much close: only 0.6 cents wide, as opposed to
4.4 cents narrow of 14/9. Thus the fluidity, and element of free
choice, in choosing just ratios with which to associate tempered
intervals, whether in lattice diagrams or elsewhere.
The JI version of our 6:7:9 tritriadic scale has three pure 7:6
thirds plus a pure 13:11 minor third (289.2 cents) at 44/39-4/3.
The tempered version has three 7:6 approximations at 2.0 cents
narrow, plus a 13:11 approximation at 288.3 cents, or 0.9 cents
narrow. The fifths are impure by either 1.2 or 2.3 cents (by
comparison to the JI version with four 3:2 fifths, plus a larger
one at 176:117 (3/2-44/39), wide by 352:351 or 4.9 cents).
Some other tempered intervals, however, are less accurate. These
include 9/8 at 3.5 cents wide; 14/9 at 4.4 cents narrow; and the
melodically vital 28:27 at 5.5 cents narrow. Also, the 9:7 major
thirds (435.084 cents in JI) are realized at 438.3 or 439.5
cents, respectively 3.2 or 4.4 cents wide. We can find the
smaller variety at 7/6-3/2, and the larger one at 14/9-2/1.
Beyond these in the tuning of intervals such as fifths and
thirds, the tempered version loses a melodic nuance of the JI
version: the distinction between the usual 28/27 septimal
thirdtone at 3/2-14/9, and the smaller 91/88 step at 44/39-7/6.
As we have seen, both are an identical 57.4 cents. There is
accordingly less variety, as well as considerable inaccuracy in
representing the just 28:27 step.
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3. Conclusion
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The "twin 1-3-7-9 hexanies" found at 13:11 apart in the
1-3-7-9-11-13 eikosany are emulated, albeit imperfectly and in a
somewhat simplified form, in the MET-24 temperament.
In either the just or tempered version of this 12-note twin
hexany concept, it is possible to select a 7-note Constant
Structure subset rather like a septimal or Archytas Diatonic
version of the medieval European Dorian mode, and also to find a
rotation of this subset on its 3/2 step which produces a 6:7:9
tritriadic scale.
Above all, these are beautiful and engaging structures whether
presented in their integral JI format, or in tempered variations
such as those of MET-24 or George Secor's High Tolerance
Temperament (HTT) family of tunings. Exploring a range of
variations may demonstrate the musical value of the themes
themselves.
Peace and love,
Margo Schulter
mschulter@calweb.com
October 8, 2012
.