
A QuasiWilsonian "Septendecene" in Zest24:
A septimal genus 17 in a tempered setting

One of the treasures of Erv Wilson's writings generously made
available on the Anaphoria Embassy website is a paper on _Some
Basic Patterns Underlying Genus 12 & 17_ (1980, revised 1981, 1983).
While this paper is rich with many ideas, one which captured my
imagination was generating a 17note tuning or "genus 17" by taking a
7note diatonic mode in a just intonation (JI) system such as
Ptolemy's Intensive Diatonic and then transposing it to each of the
six steps of a Pythagorean hexachord (1/1 9/8 81/64 4/3 3/2 27/16).
This concept appears at the very opening of the presentation (p. 1),
with a focus on Pythagorean (23 prime) and 5limit (235 prime)
tunings.
What would happen if I applied the same basic idea to a septimal
system based on ratios of primes 237  or, indeed, a temperament
approximating these ratios like Zest24? That was an exciting
question, and I suspected that the answers might be not only
interesting but musically inspiring.
Quickly I should note that using the very attractive term
"septendecene" for such a 17note grouping generated by hexachord
transposition may be something of a poetic license, since Wilson
himself uses it on some unnumbered pages dated and copyrighted 1977 to
describe a different method of generating a 17note tuning by
tetrachord rotation. However, just as the term "duodene" for a 12note
set can have various uses (often referring specifically to a style of
JI tuning, but sometimes applied more generally, as by Bosanquet), so
"septendecene" might freely apply to various types of 17note sets,
including those generated by the hexachord transposition method.
And now to my adventure.

1. Hexachord transposition: a JI model

To start with, I focused on this JI scale as a starting point, one of
my favorites in either pure or tempered realizations. We could call it
a septimal neomedieval Dorian mode since it follows the basic step
pattern of a medieval European Dorian mode, here colored by the
delightful 7:6 minor third and 7:4 minor seventh:
 
1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1
0 204 267 498 702 906 968 1200
9:8 28:27 8:7 9:8 28:27 8:7
204 63 231 204 63 231
As this diagram shows, there are two symmetrical tetrachords of
1:19:8:7:64:3 or 24:27:28:32. The step sizes of 9:8, 8:7, and 28:27
are identical to those of the diatonic of Archytas, called by Ptolemy
the Tonic Diatonic, albeit here with a different order or permutation
of steps (here we have 9:828:278:7, and there 8:728:279:8).
Now for a step with great medieval resonance for me: we simply
transpose this 7note septimal Dorian mode to each of the six steps of
a Pythagorean hexachord: 1/19/881/644/33/227/16. People
interested in the hexachord side of things might visit:
For our present purposes, I might just remark that the hexachord has
six notes and thus five adjacent melodic steps or intervals. Four of
these are 9:8 tones, while the centrally located semitonne has a size
of 256:243 (about 90 cents). Transposing our septimal mode to each
step gives this arrangement:
1/1: 1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1
9/8: 9/8 81/64 21/16 3/2 27/16 243/128 63/32 9/8
81/64 81/64 729/512 189/128 27/16 243/128 2187/2048 567/512 32/27
4/3: 4/3 3/2 14/9 16/9 2/1 9/8 7/6 4/3
3/2: 3/2 27/16 7/4 2/1 9/8 81/64 21/16 3/2
27/16: 27/16 243/128 63/32 9/8 81/64 729/512 189/128 27/16
Ordering this resulting set of 17 notes to an octave, we might arrange
these notes on a lattice with 3:2 for the horizontal dimension and the
septimal minor third at 7:6 for the vertical dimension. In this type
of lattice, each square or quadrangle (e.g. 1/17/63/27/4) shows a
complete 12:14:18:21 tetrad (a rounded 0267702969 cents), with the
lowest note of the tetrad at the lower left corner. Here are there are
six such sonorities:
765 267 969 471 1173 675 177
14/9  7/6  7/4  21/1663/32189/128567/512
      
16/9  4/3  1/1  3/2  9/8 27/16 81/64729/512  2187/2048
996 498 0 702 204 906 408 612 114

2. In a tempered context: Zest24

My idea was now to apply this kind of scale generation to the crafting
of a 17note subset or "genus 17" in Zest24. Briefly I should explain
that Zest24, the Zarlino Extraordinaire Spectrum Tuning, consists of
two irregularly tempered 12note circles each based on Gioseffo Zarlino's
renowned 2/7comma meantone temperament of 1558. Each circle has eight
fifths (FC#) in this regular temperament, with the other four fifths
tuned equally wide to close the circle.
The two circles are placed at a distance of about 50.28 cents, the
enharmonic diesis of Zarlino's regular tuning. While we may find
within each circle some intervals in the Pythagoreanseptimal portion
of the spectrum, the interweaving of the circles provides septimal
approximations in many more locations. At the same time, the element
of irregular temperament causes intervals and sonorities to shift more
or less subtly in color as we move around the system: sonorities often
mix distinctively "septimal" intervals with others shading into
neighboring regions of the spectrum.
My "genus 17" project may illustrate these points as well as some
other quirks of the temperament. As a starting point, I sought out a
reasonably close approximation of the neomedieval septimal Dorian mode
discussed above. Here an asterisk (*) shows a note in the upper
12note circle of fifths raised by the 50.28cent diesis:
Db Eb Eb* Gb Ab Bb Bb* Db
0 217 267 504 708 925 975 1200
217 50 237 204 217 50 225
Here, as in the 237 JI version, the septimal minor third and seventh
come from a different chain of fifths than the other notes of the
mode  respectively the upper and lower 12note circles. The near7:6
(or ~7:6 for short) step at Eb* is very slightly wide of pure, and the
~7:4 step at Bb* is wide by this tiny amount plus the temperament of
the large fifth Eb*Bb*, about 6.42 cents, or a bit less than seven
cents in all.
While the melodic step GbAb at a rounnded 204 cents is virtually
identical to the 9:8 step of the JI version, we find two steps of 217
cents: DbEb and AbBb. These tempered steps are almost equal to the
mean or average of 9:8 (204 cents) and 8:7 (231 cents); we might call
them "septimal eventones." As we'll see, two of these eventones yield
a nearjust 9:7 major third, a feature useful in our 17note system.
The major sixth DbBb, from a chain of three large fifths, is around
925 cents  and yields a fair approximation of 12:7 (933 cents).
The scale locations in our 237 JI version of the mode where 8:7
steps or large tones appear also have ratios fairly close to 8:7 in
this tempered version: Eb*Gb at 237 cents and Bb*D at 225 cents.
These intervals, as in the just version, involve two notes from
different chains of fifths. These sizes of 225 and 237 cents, or more
precisely about 224.59 and 237.16 cents, illustrate a characteristic
feature of this irregular temperament: interval sizes often graduated
in steps of 12.57 cents, or a quarter of the 50.28cent diesis.
Melodically notable are the two 50cent semitones EbEb* and BbBb*,
actually diesis steps considerably smaller than the JI step of 28/27,
the semitone or thirdtone of Archytas at 63 cents. These 50cent
semitones are a routine feature of Zest24 in septimal contexts, for
example, and rather narrower even than the diatonic semitone of 22EDO
at 54.54 cents or 1/22 octave.

3. Generating a 17tone genus

To generate a 17tone set, I somewhat modified Wilson's method of
transposing a mode to the steps of a Pythagorean hexachord. Rather I
sought a Zest24 subset that would provide approximate transpositions
to each of the first six steps of the tempered version: that is, to
the steps of 0217267504708925 cents. I knew that this was
altering the parameters for Wilson's method, since these steps are at
once tempered and taken from two chains of fifths. However, with
assistance from Manuel Op de Coul's free software Scala available on
the Internet , I arrived at
this solution, shown in rounded cents:
0: 0 217 267 504 708 925 975 1200
217: 217 434 484 708 925 1129 1179 217
267: 267 484 554 759 975 1180 50 267
504: 504 708 759 1008 1200 217 267 504
708: 708 925 975 1200 217 434 484 708
925: 925 1129 1200 217 434 625 708 925
As I found, this 17note set provided septimally flavored
transpositions not only to these six steps of the original mode, but
also to the seventh step at 975 cents, thus covering all of its
degrees  and also to the step of the 17note system at 484 cents, a
narrow fourth which could be considerd a tempered equivalent of 21:16
(about 471 cents). Here are mappings and interval sizes for these
eight transpositions:
Db Eb Eb* Gb Ab Bb Bb* Db
0 217 267 504 708 925 975 1200
217 50 237 204 217 50 225
Eb F F* Ab Bb C C* Eb
0 217 267 492 708 913 963 1200
217 50 225 217 204 71 237
Eb* F* Gb* Ab* Bb* C* Db* Eb*
0 217 287 492 708 913 983 1200
217 71 204 217 204 71 217
F* G* Ab* Bb* C* Eb Eb* Eb*
0 192 274 492 696 933 983 1200
217 83 217 217 204 50 217
Gb Ab Ab* Cb Db Eb Eb* Gb
0 204 254 504 708 925 975 1200
217 50 250 204 217 50 225
Ab Bb Bb* Db Eb F F* Ab
0 217 267 492 708 925 975 1200
217 50 225 217 217 50 225
Bb C Db Eb F G Db Eb
0 204 275 492 708 900 983 1200
204 71 217 217 192 83 217
Bb* C* Db* Eb* F* G* Ab* Bb*
0 204 275 492 708 900 983 1200
204 71 217 217 192 83 217
Quickly surveying these interval sizes may suggest why due emphasis
should be given to the concept of _approximate_ transpositions, with a
dramatic degree of what be called, depending on one's editorial
viewpoint, either variegation or inaccuracy! Here a "septimal" flavor
of our Dorian is considered to be one with a minor seventh at a
rounded size of 963, 975, or 983 cents. While minor thirds
representing 7:6 are often virtually just (267 cents) or reasonably
close (275 cents), we also find sizes of 287 cents (over 20 cents
wide, and much closer to 13:11 at about 289 cents), and also 254 cents
(over 12 cents narrow, and very close to 22:19 at a rounded 254 cents).

4. Shadings and alternatives

I might add that while septimal flavors of the mode occur at eight
locations of the 17note genus, there are actually two versions
available at Eb (225 cents above our "1/1" at Db), the first already
listed above, and the second an alternative form similar to that
found at Eb* (267 cents above Db).
Eb F F* Ab Bb C C* Eb
0 217 267 492 708 913 963 1200
217 50 225 217 204 50 237
Eb F Gb Ab Bb C Db Eb
0 217 287 492 708 913 983 1200
217 71 204 217 217 71 217
These two forms differ in the tuning of the third and seventh
degrees. The first approach, closer to JI, derives the nearpure minor
third from the 217cent "eventone" EbF plus a 50cent diesis or small
semitone, thus EbF*. Likewise the minor seventh EbC* at 963 cents,
narrow of 7:4 by a bit less than six cents, is equal to the major
sixth EbC at 913 cents plus a 50cent step, EbC*.
The second approach generates all modal steps from a single chain of
fifths, with two large fifths (around 708.38 cents) or small fourths
(around 491.62 cents) producing a 983cent minor seventh (EbAbDb).
A chain of three of these large fifths down or small fourths up would
produce a minor third such as BbDb or Bb*Db* at 275 cents, or about
eight cents wide of 7:6, as can be seen in the modes on Bb and Bb*.
Here, however, the chain EbAbDbGb has a large fourth at DbGb (the
usual meantone size of 504.19 cents), so that the third is 287 cents
rather than 275 cents. This need not be seen as a flaw, but does
result in a different cast or shading than the nearjust version of
the first approach.
The first version also has the JIlike feature of the comma: a
distinction between a usual fourth (here 492 or 504 cents, and in JI
the just 4:3 at ~498 cents) and the special narrow fourth at AbC*,
almost identical to the just 21:16 (~471 cents) which would occur
in a pure tuning between the corresponding steps 4/3 and 7/4. In
contrast, the second version has the usual fourth AbDb (492 cents) at
this same location.
Thus the nearJI version can be chosen if one seeks the most accurate
septimal representations, or wants to use the beautiful 21:16 (for
example in a 16:21:24:28 sonority, AbC*EbF*, which could resolve to
BbF, all within the notes of the mode). The alternative version
offers a less "eventful" realization which tempers out the comma,
rather as in George Secor's 17note welltemperament.
While the first version more closely models JI, in fact both versions
show the "variegation" we have emphasized in representing an interval
such as the 7:6 minor third. This becomes clear when we compare the
sizes of each of the four minor thirds in a JI version and in these
tempered approaches:
JI: 1/1  7/6 9/8  4/3 3/27/4 27/162/1
7:6 32:27 7:6 32:27
267 294 267 294
Eb  F* F  Ab Bb  C* C  Eb
Tempered 1: 0  267 217  492 708  963 913  1200
267 275 254 287
Eb  Gb F  Ab Bb  Db C  Eb
Tempered 2: 0  287 217  492 708  983 913  1200
287 275 275 287
The JI version has two 7:6 minor thirds and also two Pythagorean minor
thirds at 32:27 or ~294 cents  with our 287cent or near13:11
thirds not too far from this ratio.
The second tempered version with two minor thirds at 275 cents and the
other two at 287 cents thus has a pattern somewhat comparable to that
of the JI version, but with these large and small thirds in different
positions. In JI, we find small 7:6 thirds at 1/17/6 and 3/27/4,
steps 13 and 57 of the scale; and larger 32:27 thirds at 9/84/3 and
27/162/1, steps 24 and 68. Here, however, we have small 275cent
thirds at 24 and 57, and larger 287cent thirds at 13 and 68.
The first tempered version has no fewer than four minor third sizes,
one for each of these intervals: a nearpure 267 cents, the 275cent
and 287cent sizes we have noted in the other tempered version, and
also a narrow 254 cents at BbC*. While temperament is often presented
as a strategy for "regularizing" or "evening out" some of the
variations in step and interval sizes occur in JI based on multiple
primes (235, 237, etc.), here our irregular temperament increases
the range of interval diversity  or unpredictability, again
depending on one's viewpoint.
Melodically, the second tempered version offers closer approximations
of the 28:27 thirdtone of Archytas, using 71cent semitones (actually
a just 25:24, ~70.67 cents) to represent this 63cent JI step. The
first version, in contrast, offers striking 50cent semitones.

5. A lattice overview

The choice of two tempered versions of our Dorian mode at Eb might
serve as an introduction to a general overview of the genus17
system. We can map it as lattice with the horizontal dimension showing
regular fifths at 492 or 504 cents, and vertical lines (here slanted
from left to right) showing minor thirds. When these slanted vertical
lines are dashed, they show thirds belonging to what are regarded as
"septimal flavor" tetrads approximating 12:14:18:21, with outer minor
sevenths at 963, 975, or 983 cents. Dotted lines show minor thirds
belonging to other types of tetrads moving into neighboring regions
with 996cent minor sevenths almost identical to a just Pythagorean
16:9; or 950cent minor sevenths in the fascinating "interseptimal"
region between the 7based ratios of 12:7 (933 cents) and 7:4 (969
cents). The numbers within each square or quadrangle show the sizes of
a tetrad's minor seventh, two minor thirds, and major third in cents:
Gb* Db* Ab* Eb* Bb*
554  50  759  267  975
/ 983 / 983 / 983 / 996 .
/ 287 275 / 275 275 / 275 287 / 287 300 .
/ 421 / 434 / 421 / 408 .
Db*  Ab*  Eb*  Bb*  F*  C*  G*
50  759  267  975  483  1179  675
. 950 / 963 / 975 / 975 / 963 / 950 .
. 242 250 / 254 267 / 267 267 / 267 267 / 267 254 / 250 242 .
. 454 / 441 / 441 / 441 / 441 / 454 .
Cb  Gb  Db  Ab  Eb  Bb  F
1008  504  0  708  217  925  434
. 996 / 983 / 983 / 983 / 996 .
. 300 287 / 287 275 / 275 275 / 275 287 / 287 300 .
. 408 / 421 / 434 / 421 / 408 .
Ab  Eb  Bb  F  C  G
708  217  925  434  1129  625
Within the set of "septimal" tetrads as here defined, we find minor
thirds ranging from 254287 cents, and major thirds at 421441 cents;
in a just 12:14:18:21, these intervals would have pure sizes of 7:6
and 9:7 (267 and 435 cents).
The 254cent minor third moves well into the interseptimal region
intervening between the neighborhoods of 8:7 (231 cents) and 7:6,
while the 287cent third, as we have seen, is very close to 13:11 and
approaches the Pythagorean region of 32:27. As our diagram shows,
septimal tetrads featuring these small or large sizes also have minor
thirds at the middle sizes of 267 or 275 cents closest to 7:6.
Among the major thirds of these tetrads, 421 cents is rather larger
than 14:11 (~418 cents) and approaches 17EDO (423.53 cents), while
441 cents can be regarded as a temperament of 9:7 (about 6 cents
larger than pure), but verges on the interseptimal region between this
just major third and the narrow fourth at 21:16 (471 cents). The
middle size of 434 cents is very close to a just 9:7. Interestingly,
this best approximation of 9:7 occurs within a single fifth circle,
where it results from a chain of all four large fifths at 708.38
cents (DbAbEbBbF or Db*Ab*Eb*Bb*F*), and is formed from two
equal "eventones" of 217 cents each (DbEbF or Db*Eb*F*).
Septimal and other tetrads on the lattice are generated by two basic
methods: using fifths within a single circle, or mixing notes from
both circles. The diagram thus has three rows of tetrads. In the
lowest row, tetrads are generated within the lower circle of fifths;
in the middle row, by combining notes from both circles; and in the
upper row, within the upper circle.
In the lower row, tetrads shade symmetrically from the closest
approximation of 12:14:18:21 in the center position at BbDbFAb
(0275708983 cents) to a Pythagorean flavor nicely represented by
AbCbEbGb and CEbGBb (0300708996 or 0287696996 cents). The
intermediate EbGbBbDb and FAbCEb (0287708983 cents or
0275696983 cents) are considered within a "septimal" flavor, but
have some 287cent and 421cent thirds shading somewhat in the
direction of the Pythagorean region and typical of a 17note
circulating temperament, for example.
In the middle row, the center two tetrads have 975cent minor sevenths
and pairs of nearjust 267cent minor thirds (0267708975 cents);
this symmetrically shades towards the fine interseptimal tetrads at
CbDb*GbAb* and BbC*FG* (0242696950 or 0242708950 cents).
The intermediate GbAb*DbEb* and EbF*BbC* (0254696963 or
0267708963 cents) are regarded as "septimal" and have the closest
approximations of 7:4 available in Zest24 (trivially closer on the
narrow side than a 975cent seventh almost equally impure on the wide
side), but also have a 254cent minor third shading them toward the
interseptimal region.
The highest row is much like the lowest, except that there is an
asymmetry because our 17note genus includes a chain of nine fifths
from the lower circle (CbG, with Cb at 1008 cents serving as an
equivalent to 16/9 to Db in the quartal sonority DbGbCb, for
example), but only eight from the upper circle (GbG). Thus there are
four tetrads on this row as compared to five on the lowest row, with
Bb*Db*F*Ab* in the central septimal region. Moving to the right, we
have the same transition as on the lower row to the intermediate
F*A*C*Eb* and the Pythagoreanlike CEbGBb. To the left, however,
we have only the intermediate Eb*Gb*Bb*Db*; adding to the system
the note Cb* would produce a Pythagoreanlike Ab*Cb*Eb*Gb* and make
this row identical to the lower one.
The reader may have noticed that there are actually two tetrads shown
with the lowest note of Eb: EbGBbDb in the lowest row from the
single lower circle of fifths, or EbF*BbC* in the middle row mixing
notes from the two circles. These alternative forms may be found in
the two septimal Dorian modal transpositions to Eb which we discussed
above.
There are thus 10 "septimal" tetrads in our 17note set available at
nine distinct locations (two at the location of Eb), shown on the
lattice with dashed lines on all sides. Additionally we see two fine
Pythagorean region tetrads on either end of the lower row; two
interseptimal tetrads at either end of the middle row; and another
Pythagorean region tetrad at the right end of the upper row. These
five tetrads have a slanting dotted line on one side.
We might ask how an equivalent JI lattice might look for a tuning
based on pure ratios of 237 supplying tetrads corresponding to our
tempered septimal ones. Here is one such possible system, with dotted
lines indicating tetrads that in our tempered version move clearly
into the Pythagorean or interseptimal regions, but in this just tuning
would, of course, also be pure 12:14:18:21 sonorities:
534 36 738 240 942
49/36  49/48  49/32 147/128441/256
    .
    .
63 765 267 969 471 1173 675
28/27  14/9  7/6  7/4  21/16  63/32  189/128
.      .
.      .
16/9  4/3  1/1  3/2  9/8  27/16  81/64
996 498 0 702 204 906 408
.     .
.     .
32/21  8/7  12/7  9/7  27/14  81/56
729 231 933 435 1137 639
This 25note system, in addition to the indicated 15 pure 12:14:18:21
tetrads, would also offer a wealth of Pythagorean sonorities, for
example, as a more intricate diagram of tetrads would show.

6. Keyboard layout: a regularized mapping

The ideal synthesizer implementation of our tempered 17tone genus
might be on a generalized keyboard of the kind for which Erv Wilson
has offered many mapping of just or tempered tunings. More modestly,
however, my realization uses a 24note "regularized keyboard" with two
12note manuals, one for each circle of fifths.
The term "regularized" means that each 12note manual has the same
sequence of steps and intervals. Here we might say that the two
manuals share the same variations and irregularities of the 12note
circle. The mapping for our 17genus is often rather like that of a
237 JI lattice, but with some important distinctions also:
50 267 483 554 675 759 975 1180 1250
Db* Eb* F* Gb* G* Ab* Bb* C* Db*
0 217 434 504 625 708 925 1008 1129 1200
Db Eb F Gb G Ab Bb Cb C Db
Unlike a _generalized_ keyboard, of course, there is the usual
Halberstadt distinction between tones and semitones on either manual.
There is further the element of irregular temperament: thus DbF or
Db*F* forms a nearjust 9:7, while EbG or Eb*G* forms a nearjust
81:64 at some 25 cents smaller. Certain subsets do have a certain
latticelike structure, for example a tempered version this fine
tuning by Carter Scholz identified in the Scala scale archive
(scholz.scl) as from "Simple Tune #1":
28/27 14/9 7/6 7/4
63  765  267  969
  
4/3  1/1  3/2
498 0 702

8/7
231
and realized as follows:
50 267 759 975
Db* Eb* Ab* Bb*
Db Eb Gb Ab Db
0 217 504 708 1200
Here the steps at 50, 267, 759, and 975 cents on the upper keyboard
all represent septimal ratios (28/27, 7/6, 14/9, 7/4), and 504, 708,
and 1200 cents on the lower keyboard the Pythagorean 4/3, 3/2 and
2/1. The complicating factor is 217 cents on the lower keyboard, which
might stand for either 9/8 or, as here, 8/7. Mathematically it is very
slightly closer to 9/8, and acts musically rather like 9/8 (or more
precisely 44/39 at around 209 cents) in forming a minor third with Gb
not too far from the Pythagorean 32:27 (and very close to 13:11).
However, it acts rather like 8/7 in forming a "septimal flavor" minor
seventh with Db at 983.24 cents, numerically very slightly closer to 16:9
but leaning in color toward 7:4, and approaching the 981.82 cents of
22EDO.
The net result might be kind of moderate variation on Carter Scholz's
JI tuning  with the usual inaccuracies and compromises of
temperament, of course.
I might close by briefly mentioning another kind of just tuning which
this 17genus can emulate: the 1379 hexany, with six notes forming
two pure 12:14:18:21 tetrads:
267 969 471
7/6 7/4 21/16
1.7  3.7  7.9
  
1.3  1.9  3.9
1/1 3/2 9/8
0 702 204
In our 17genus, one tempered version is:
Ab Bb Bb* C* Eb F* Ab
0 217 267 471 708 975 1200
Following the above lattice arrangement for the JI version, this could
be written:
267 975 471
Bb* F* C*
1.7  3.7  7.9
  
1.3  1.9  3.9
Ab Eb Bb
0 708 217
While the tempered version often corresponds closely with the just
tuning, there are notable variations: BbC* is an interseptimal
interval of 254 cents representing a 7:6 third; and BbAb,
representing 16:9, is a 983cent minor seventh with a 7:4like flavor.
Both of these touches somewhat alter the balance and color of the
hexany, hopefully as a stimulating ramification of the original JI
version with its own unique appeal.
Another variation on the hexany involves similar variations or
compromises:
Gb Ab Ab* Bb* Db Eb* Gb
0 204 254 471 696 963 1200
254 963 471
Ab* Eb* Bb*
1.7  3.7  7.9
  
1.3  1.9  3.9
Gb Db Ab
0 696 204
This time we have a small 254cent interseptimal third representing
7:6 above the lowest step of the hexany, and a 983cent minor seventh
at Bb*Ab* representing the 16:9 interval of 21/167/3 in the just
hexany. A subtle distinction of color is that here the 7/4 step is
represented as a narrow 963 cents (GbEb*) rather than a wide 975
cents (AbF* in the last example)  these being the closest
approximations in Zest24, about equally accurate.
These two tempered versions both feature a nearjust 21:16. Another
version involves more of a compromise involving this interval:
Db Eb Eb* F* Ab Bb* Db
0 217 267 484 708 975 1200
267 975 484
Eb* Bb* F*
1.7  3.7  7.9
  
1.3  1.9  3.9
Db Ab Eb
0 708 217
If seeking a hexany with the most accurate representations of 7:6, we
could hardly do much better in a tempered system: all three locations
for this interval on the lattice are at a virtually just 267.034
cents. Further, if one wants to maximize the number of minor sevenths
close to 7:4, we have not only the two 975cent realizations where
this ratio is expected on the lattice, but also two 983cent intervals
where the JI version has 16:9. Of course, if one relishes a just 16:9
color  or a contrast between 7:4 and 16:9 sevenths like that of the
JI version  then the other transpositions can oblige.
However, this plethora of 7:6 and 7:4 approximations does have a
consequence for the 21:16 approximation. Combining a nearpure
267cent minor third (DbEb*) with a large 217cent eventone (the
octave complement or inversion of a 983cent minor seventh) results in
an interval of about 484 cents, or 13 cents wide of a just 21:16. This
variation could be regarded as another facet of unequal temperament,
nicely illustrating the 12.57cent distinction in interval sizes that
often occurs (here from 471.22 cents in the previous examples to
483.79 cents, as compared with 21/16 at 470.71 cents).
The tuning of the near21:16 might be of special interest when using
the tempered equivalent of the hexany's 16:21:24:28 sonority found at
1/121/163/27/4. Here we compare the just version and our three
tempered emulations:
JI: 1/1 21/16 3/2 7/4
0 471 702 969
21:16 8:7 7:6
471 231 267
Ab: Ab C* Eb F*
0 471 708 975
471 237 267
Gb: Gb Bb* Db Eb*
0 471 696 963
471 225 267
Db: Db F* Ab Bb*
0 484 708 975
484 225 267
The transpositions on Ab and Gb give a close rendition of the just
21:16, while the Db transposition offers a different shading at 484
cents, interestingly close to the complex JI ratio of 189:143 that
occurs in a 24note superset of the 13791113 eikosany that I
use (482.85 cents). Each subtle gradation has its own appeal, not to
mention the JI original.
Most appreciatively,
Margo Schulter
mschulter@calweb.com