-------------------------------------------------
Zest-24: An Aaron/AKJ Matrix
Tempered variations on septimal (2-3-7) JI
-------------------------------------------------
One fascinating chapter in the development of just intonation (JI)
systems and their use to compose beautiful music is Gene Ward Smith's
designing of two 12-note tunings in septimal JI based on prime factors
of 2-3-7, named "Aaron" and "AKJ" in honor of his fellow composer
Aaron Krister Johnson. Johnson then returned this compliment in the
best way by using these tunings to create some inspiring music, for
example in the context of some acclaimed musical settings for a
production of Ibsen's play _Peer Gynt_ which draw on a variety of just
and tempered tunings.
Here we shall first briefly describe these just tunings, and then
explore a 19-note tempered variation found within the Zest-24 tuning
system inspired by a superset combining "Aaron" and "AKJ." This
"Aaron-AKJ matrix" as realized in the irregularly tempered Zest-24 has
interesting resemblances to and divergences from the more familiar
intonational patterns of just intonation and equally tempered systems.
-----------------
Table of Contents
-----------------
1. The "Aaron" and "AKJ" tunings of Gene Ward Smith
2. Complicating things: The Aaron/AKJ superset and Zest-24
2.1. A superset takes form: A 6x4 matrix
2.2. Zest-24: septimal approximations and congruity
2.2.1. Lattice mapping and alternative steps
2.2.2. Three septimal generation strategies in Zest-24
2.2.3. Accuracy and congruity
3. Complicating things further: a 19-note superset
3.1. Filling out a 6x4 framework
3.2. Neutral intervals and tempered shadings
3.3. Septimal and interseptimal colors: gamelan
4. Tempering the Aaron/AKJ lattice: A comparative view
4.1. A single-chain solution: Strategy 1 and 22-EDO
4.2. Neomedieval intervals and dieses: Strategy 2 and 63-EDO
4.3. Meantone intervals and dieses: Strategy 3 and 31-EDO
4.4. Another road: 36-EDO and the 33-cent diesis
5. Conclusion: An envoi
---------------------------------------------------
1. The "Aaron" and "AKJ" tunings of Gene Ward Smith
---------------------------------------------------
The "Aaron" and "AKJ" tunings map conveniently to a conventional
12-note keyboard, and avoid any direct occurrence of the septimal
comma or comma of Archytas at 64:63 (27.26 cents). Here is a lattice
diagram for the Aaron tuning, with 3:2 fifths shown in the horizontal
direction and 7:6 minor thirds in the vertical direction, and interval
sizes shown as just ratios and in rounded cents:
49/27 ---- 49/36
1032 534
| |
| |
28/27 ------ 14/9 ------ 7/6 ------ 7/4
63 765 267 969
| | |
| | |
4/3 ------ 1/1 ------ 3/2 ------ 9/8
498 0 702 204
| |
| |
12/7 ------ 9/7
933 435
A notable characteristic of this Aaron tuning is what might be called
its "depth" of septimal levels: thus, for example, the 7/6 step at a
pure septimal minor third above the "1/1" has its own step at a 7:6
third higher, or 49/36 in relation to the 1/1.
In the above diagram, a square or quadrangle formed by four notes
shows a neomedieval "quad" or four-voice sonority of 12:14:18:21
(0-267-702-969 cents), of which there are four in this system, in
relation to the note at the lower left-hand corner of a square. This
variety of quad is known as a "septimal minor seventh quad" with
intervals above the lowest voice of a 7:6 minor third, 3:2 fifth, and
7:4 minor seventh. Starting from the upper left-hand corner of such a
square, we have a "septimal major sixth quad" of 14:18:21:24 or a
rounded 0-435-702-933 cents, with intervals above the lowest voice of
a 9:7 major third, 3:2 fifth, and 12:7 major sixth.
One sign of the Aaron tuning's septimal depth is that two notes of the
set can serve as the lowest voice of either a minor seventh quad or a
major sixth quad: 1/1 and 14/9. Thus we have:
Lowest note of quad 1/1 14/9
Minor seventh quad 1/1-7/6-3/2-7/4 14/9-49/27-7/6-49/36
Major sixth quad 1/1-9/7-3/2-12/7 14/9-1/1-7/6-4/3
One consequence of this septimal depth is a somewhat smaller number of
3:2 fifths, eight, than result in a system such as the companion "AKJ"
tuning, shown here, which has nine:
28/27 ----14/9 ------ 7/6
63 765 267
| | |
| | |
16/9 ----- 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
996 498 0 702 204 906
| | |
| | |
9/7 ----- 27/14 ---- 81/56
435 1137 639
From a neomedieval perspective, the Aaron and AKJ tunings offer a
fascinating mixture of simple and complex septimal ratios ranging from
the 12:14:18:21 and 14:18:21:24 quads, which often invite standard
resolutions following 13th-14th century European patterns, to some
intriguing neutral or semi-neutral intervals. Thus the following
progression from Aaron lends a just septimal flavor to a routine type
of classic medieval cadence, with each voice either ascending by a
28:27 semitone (63 cents) or descending by a 9:8 tone (204 cents):
7/4 14/9
3/2 14/9
7/6 28/27
1/1 28/27
Here the 7:6 minor third and 7:4 minor seventh above the lowest voice
contract by stepwise contrary motion to the unison and fifth, while
the upper 7:6 minor third likewise contracts to a unison and the 9:7
major third between the middle voices expands to a fifth. This type of
resolution, with descending whole-tone and ascending semitone motions,
is termed "intensive."
The AKJ tuning provides a fine example of the "remissive" manner of
cadencing with descending semitone and ascending whole-tone motions,
again respectively 28:27 and 9:8.
4/3 3/2
7/6 9/8
1/1 9/8
14/9 3/2
Here the 9:7 major third and 12:7 major sixth above the lowest voice
expand to the fifth and octave, while the 8:7 or large whole-tone
between the two upper expands to a fourth, and the middle 7:6 minor
third contracts to a unison. The unstable 14:18:21:24 or major sixth
quad this expands to a stable _trine_ with outer octave, lower fifth,
and upper fourth, here at a just 2:3:4 (as in medieval Pythagorean
tuning).
Sometimes a usual major sixth or minor seventh quad calls for more
unconventional resolution, however, as with this example in Aaron:
12/7 49/27
3/2 49/36
9/7 49/36
1/1 49/27
The unstable major sixth quad and resolving trine have vertical
interval sizes identical to those of the previous example; but here
the voices either descend by 168-cent steps of 54:49, or ascend by
semitones of 343:324 or about 99 cents. The first step, slightly
larger than the 11:10 neutral second of Ptolemy at 165 cents, might
approach the transition zone between a large neutral and small major
second; while the second is a bit larger than a usual Pythagorean
semitone at 256:243 or 90 cents, and almost identical to 18:17, the
latter being wider by the tiny superparticular ratio of 5832:5831
(0.30 cents).
The overall impression of this progression with its 168-cent
"near-major" second and 99-cent middling semitone steps may tend
toward the intensive, but with a rather different color than a usual
intensive resolution with its 203-cent and 63-cent steps. For one
approach to describing the myriad interval sizes and shadings offered
by just and tempered systems alike, an approach often followd here,
see "Regions of the Interval Spectrum: Some Concepts and Names,"
.
This last resolution showcases one of the virtues of a compact system
like the 12-note Aaron or AKJ: the limited range of positions and
resolutions impels use to explore possibilities that might often be
overlooked in a larger tuning. While an identical resolution would be
available in any septimal JI system with the notes to support it, in
the Aaron system it is the _one_ logical way of expanding by stepwise
contrary motion from the major sixth quad 14/9-1/1-7/6-4/3 to a
complete 2:3:4 trine. Having had it thus compellingly brought to our
notice, we can look for it in other systems along with the usual
intensive or remissive formulae which might otherwise overshadow it.
Such an opportunity arises in the AKJ tuning, where the 12:14:18:21
quad 16/9-28/27-4/3-14/9 has a routine remissive resolution -- but
also an alternative resolution involving neutral second steps:
Remissive Equable
14/9 3/2 14/9 81/56
4/3 3/2 4/3 81/56
28/27 1/1 28/27 27/14
16/9 1/1 16/9 27/14
In the first resolution, we have the usual steps of 9:8 and 28:27,
here with ascending whole-tones and descending semitones in the
remissive manner. In the second, we have voices either ascending by
steps of 243:224 at 141 cents, or descending by steps of 784:729 at
126 cents -- the latter precisely twice the size of the 28:27 semitone
favored in the tunings of Archytas. These steps are quite close to the
superparticular ratios of 13:12 (139 cents) and 14:13 (128 cents),
differing respectively in the wide and narrow directions by the small
rational interval of 729:728 (2.376 cents).
Following George Secor, we may term this second cadence "equable," a
term inviting a bit of explanation. Here the lower two voices, for
example, contract from a 7:6 minor third (16/9-28/27) to a unison (on
27/14), a total melodic motion or contraction of 7:6 or 267 cents. In
an equable cadence, as here, this total motion of 7:6 -- or, more,
generally, some type of minor third -- is realized as two neutral
second steps, here at 141 cents (ascending) and 126 cents
(descending). Secor derives the term "equable" from the Equable
Diatonic of Ptolemy with its tetrachord of 12:11:10:9 (0-151-316-498
cents), where a larger 6:5 minor third is likewise divided into
neutral second steps of 12:11 and 11:10. The present example more
closely resembles a tetrachord described by the Persian philosopher
and music theorist Ibn Sina (known in Latin Europe as Avicenna) around
the early 11th century, 28:26:24:21 (0-128-267-498 cents), with a 7:6
minor third divided 14:13:12.
The AKJ tuning also presents opportunities to use neutral intervals
vertically, as in this resolution where a neutral seventh at 448:243
or 1059 cents and a neutral third at 896:729 or 357 cents contract
respectively to a fifth and unison:
4/3 7/6
16/9 14/9
81/56 14/9
Here the lower voice ascends by a 784:729 step of 126 cents -- also
used in the previous example -- while the upper voices descend by
large 8:7 septimal whole-tones. Depending on the context, a 126-cent
step might be heard as either a large semitone or small neutral
second. Our present examples invite the latter interpretation, in
contrast to the situation of 19-EDO used in a Renaissance meantone
fashion, where an almost identical step size of 126 cents routinely
serves as the usual diatonic semitone -- in effect, a kind of tempered
variation on the 16:15 step (112 cents) of pental or 5-limit JI.
With their mixture of simple and complex septimal ratios, the Aaron
and AKJ sets are admirably crafted JI systems. Here are scale files
for use with Manuel Op de Coul's outstanding and free Scala program
available at ; in the Scala
scale archive to accompany the program these tunings are included
under the names of pipedum_12k (Aaron) and pipedum_12i.scl (AKJ).
! gws_aaron.scl
!
Septimal scale by Gene Ward Smith for Aaron K. Johnson
12
!
28/27
9/8
7/6
9/7
4/3
49/36
3/2
14/9
12/7
7/4
49/27
2/1
! gws_akj.scl
!
64/63 and 6561/6272, Gene Ward Smith, 2004
12
!
28/27
9/8
7/6
9/7
4/3
81/56
3/2
14/9
27/16
16/9
27/14
2/1
----------------------------------------------------------
2. Complicating things: The Aaron/AKJ superset and Zest-24
----------------------------------------------------------
One aspect of exemplary tuning systems such as Aaron and AKJ which
might be deemed as possibly less than optimal is the way that they can
inspire variations exhibiting "intonational sprawl": a desire for more
notes, more intervals, more stable and unstable sonorities -- and
arguably considerably less elegance and coherence. The remainder of
this article just might serve as an apt illustration.
From time to time, I have considered how the Aaron and AKJ tunings
might be adapted to the Zest-24 temperament, a name standing for
"Zarlino Encompassing Spectrum Tuning" since the system is based on a
modified version of Gioseffo Zarlino's 2/7-comma meantone of 1558, the
earliest known European temperament to be described in mathematically
precise terms. Zest-24 consists of two 12-note circles, each with
eight fifths (F-C#) tuned as in Zarlino's meantone at 2/7 of an 81:80
or syntonic comma (21.51 cents), or about 6.14 cents narrow, and the
others equally wide by about 6.42 cents. The two circles are placed at
50.28 cents apart, the enharmonic diesis of Zarlino's 2/7-comma
meantone.
At least, this is the "theoretical" version. In practice, I have
realized the system on a Yamaha TX-802 synthesizer in 1024-EDO, which
slightly modifies the interval sizes and curiously permits some
"fine-tuning" to optimize certain intervals. Here I will often cite
the 1024-EDO values for steps and intervals, with the caution that a
range of minutely different realizations or "nanotemperaments" of
Zest-24 are possible. For example, one might use 288-EDO, or a
nanotemperament based on John Brombaugh's _temperament unit_ (TU)
equal to 1/720 of a Pythagorean comma (531441:524288 or 23.46 cents)
in which each fifth is impure in one direction or the other by either
180 TU (1/4 Pythagorean comma) or 216 TU (3/10 Pythagorean comma).
The latter solution is virtually identical to the 1024-EDO version
used here.
! zest24n.scl
!
1024-tET/EDO nanotemperament of Zest-24
24
!
50.39062
70.31250
120.70312
191.01562
241.40625
287.10938
337.50000
383.20313
433.59375
503.90625
554.29687
574.21875
624.60938
696.09375
746.48437
778.12500
828.51562
887.10938
937.50000
996.09375
1046.48438
1079.29688
1129.68750
2/1
----------------------------------------
2.1. A superset takes form: A 6x4 matrix
----------------------------------------
Getting back to Aaron and AKJ, I got the idea around the end of 2007
or beginning of 2008 that a superset combining the two tunings might
be a good place to start for a Zest-24 realization. Such a superset
maps as follows:
49/27 ---- 49/36
1032 534
| |
| |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
63 765 267 969
| | | |
| | | |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
996 498 0 702 204 906
| | | |
| | | |
12/7 ------ 9/7 ----- 27/14 ---- 81/56
933 435 1137 639
Note that the longest horizontal chain of 3:2 fifths has six notes,
including the 1/1 of the tuning, ranging from 16/9 to 27/16. We might
term this the "breadth" of the matrix or lattice. The longest vertical
chain of 7:6 minor thirds has four notes, also including the 1/1,
ranging (in an upward direction on the diagram) from 12/7 to 49/36. We
might describe this dimension as the "depth" of the matrix or lattice.
Here is a Scala file:
! aaron-akj-merger16.scl
!
Superset merging GWS aaron and akj tunings (16)
16
!
28/27
9/8
7/6
9/7
4/3
49/36
81/56
3/2
14/9
27/16
12/7
7/4
16/9
49/27
27/14
2/1
---------------------------------------------------
2.2. Zest-24: Septimal approximations and congruity
---------------------------------------------------
To arrive at a tempered version of an extended Aaron/AKJ matrix in
Zest-24, I started by mapping the 16-note superset of these two
12-note JI tunings to what looked like the best Zest-24 "fit," with F
as the 1/1. In the following notation, an asterisk (*) shows a note on
the upper keyboard, raised by the enharmonic diesis of 2/7-comma
meantone at 50.28 cents. The just sizes in cents of the ratios being
approximated are shown in parentheses, along with the actual Zest-24
sizes and keyboard locations:
(1032) (534)
49/27 ---- 49/36
Eb* Bb*
1034 543
| |
| |
(63) (765) (267) (969)
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb/F* Db/C* Ab Eb
70/50 766/746 274 983
| | | |
| | | |
(996) (498) (0) (702) (204) (906)
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
983 492 0 696 192 887
| | | |
| | | |
(933) (435) (1137) (639)
12/7 ------ 9/7 ----- 27/14 ---- 81/56
D* A* E* B*
938 434 1130 626
Here all near-3:2 fifths, of which 14 are available, are impure by
about 6 or 7 cents in either the narrow or the wide direction.
Representations of the just 7:6 minor third (267 cents) vary in size
from 254 to 287 cents. This rather wide range is worthy of further
comment, and we will examine it more closely below after considering
some related aspects of lattice mapping and interval generation.
--------------------------------------------
2.2.1. Lattice mapping and alternative steps
--------------------------------------------
While the JI lattice for the combined Aaron/AKJ matrix has 16 notes or
positions, our tempered mapping actually brings into play 17 notes of
Zest-24. The 28/27 and 14/9 steps on the lattice each may use two
alternative notes or "versions" in Zest-24: for 28/27, a choice
between Gb/F* at respectively 70/50 cents from F, the 1/1; and
likewise for 14/9, a choice between Db/C* at 766/746 cents from F.
Depending on the musical context, either "version" of these steps
might be chosen to obtain more accurate septimal approximations in a
given progression, or more generally to fine-tune the melodic and
vertical color of a passage.
Thus we have the curious situation where one note in a JI system maps
to two or more in a temperament, rather than vice versa! However, the
more familiar situation where two or more JI ratios map to a single
tempered value also applies: here Eb at 983 cents from F represents
both 16/9 (996 cents) and 7/4 (969 cents), rather as in 22-EDO.
Starting with the 16-note JI lattice, we have added two extra notes
with the alternative versions of 28/27 (Gb/F*) and 14/9 (Db/C*), but
subtracted a note by using Eb for both 16/9 and 7/4, arriving at 17
notes, or one more than in the JI version.
A few examples may help to clarify the use of the alternative steps
Gb/F* and Db/C*, and also the dual use of Eb as 16/9 and 7/4. To
explore the nuances made possible by the alternative steps, let us
begin by considering this remissive cadence to Ab where a minor
seventh sonority contracts to a fifth:
Eb* -- -50 -- Eb Eb* -- -50 -- Eb
963 709 983
Db -- +217 -- Eb C* -- +237 -- Eb
696 709 696
Gb -- +204 -- Ab F* -- +237 -- Ab
Vertically, Gb-Db-E* at 0-696-963 cents rather closely approximates a
just 4:6:7 (0-702-969 cents), with the outer 7:4 minor seventh narrow
by about 6 cents and the upper 7:6 minor third at 267 cents virtually
pure. In contrast, F*-C*-Eb* (0-696-983 cents) has a somewhat
different color, with the minor seventh wide of 7:4 by some 14 cents
(a tad more than the 13 cents of 22-EDO), and the upper minor third
at 287 cents wide of 7:6 by a full 20 cents (providing, in fact, a
fine representation of 13:11 at 289 cents).
Melodically, both progressions notably use in the highest voice a
small descending semitone Eb*-Eb at 50 cents, which could be
considered a highly tempered variation of 28:27 at 63 cents. The first
version is otherwise closer to conventional septimal JI, with the
lowest voice ascending by 204 cents, a virtually just 9:8, and the
middle voice by 217 cents, a tempered compromise which, like the
218-cent step of 22-EDO, might represent either 9:8 or 8:7 (231
cents). The second version further accentuates the melodic contrast
between generous whole-tone and compact semitone motions: the lower
voices ascend by 237-cent steps, a bit wider than a just 8:7, and more
than four times as large as the incisive 50-cent semitone motion of
the highest voice!
In this instance, from either a vertical or melodic point of view, the
choice of Gb and Db more closely models septimal JI, while that oF F*
and C* provides a stimulating variation. A different situation arises
if we desire a remissive cadence to F with a major sixth quad
expanding to a complete trine:
Eb -- +217 -- F Eb -- +217 -- F
(913) (1200) (933)
Db -- -70 -- C C* -- -50 -- C
(696) (696) (696)
Bb -- +204 -- C Bb -- +204 -- C
(422) (696) (442)
Gb -- -70 -- F F* -- -50 -- F
The first version with a major sixth quad Gb-Bb-Db-Eb at 0-422-696-913
cents has a very pleasant neomedieval color rather distinct from a
just 14:18:21:24 (0-435-702-933 cents). The 422-cent major third is
about 13 cents narrow of 9:7, and like the 424-cent third of 17-EDO
suggests harmonic complexity rather than a simple just ratio; the
913-cent major sixth, about 20 cents narrow of 12:7, is much closer to
22:13 (911 cents). While the 274-cent minor third between the middle
voices is much closer to 7:6, only about 7 cents wide, the overall
vertical effect is very characteristically neomedieval but not so
accurately "septimal."
This time the second version with F* and C* yields an unstable
vertical sonority generally closer to septimal: 0-442-696-933 cents,
with the outer major sixth a virtually just 12:7. However, there are
some notable variations elsewhere from pure septimal sizes. Thus the
442-cent major third is wide of 9:7 by about 7 cents, almost identical
to the 7-step interval of 19-EDO; while the minor third between the
middle voices at 254 cents is narrow of 7:6 by about 13 cents. While
the 442-cent major third is on the outskirts of 9:7, it verges on what
I term the "interseptimal" region with its own distinct character
found between the 9:7 major third and the 21:16 narrow fourth (471
cents); the 254-cent minor third falls squarely within the
interseptimal region between the 8:7 tone and 7:6 third. The major
second between the two highest voices at 237 cents, about 6 cents
wider than 8:7, also approaches this latter interseptimal region,
which might be said to begin by around 147:128 (240 cents).
To sum up, our first version has unstable vertical intervals tending
toward the "middle" region between Pythagorean and septimal, an
oft-favored neomedieval color, while our second version offers
generally closer septimal approximations with some interseptimal
tendencies. Melodically, the first resolution uses 70-cent semitones
which not only mathematically but aurally might be considered the
closest equivalents of 28:27 at 63 cents; in fact, George Secor has
suggested that around 70 cents may represent an "optimal" size of
melodic semitone in this type of musical style. The second version
has the narrower 50-cent semitones very common for septimal
realizations in Zest-24. One might prefer either effect, depending
on taste and context.
Thus the choices of Gb/F* and Db/C* on the lattice serve both to
facilitate the most accurate septimal approximations of certain
sonorities and, more broadly, to make available some fine shadings of
vertical and melodic color.
The dual role of Eb as 16/9 and 7/4 can be briefly demonstrated by
these two progressions:
Eb -- -217 -- Db
(983) (696)
Eb -- +217 -- F C -- +70 -- Db
(983) (1200) (696) (696)
Bb -- +204 -- C Ab -- -204 -- Gb
(492) (696) (274) (0)
F F -- +70 -- Gb
In the first resolution by oblique motion, F-Bb-Eb (0-492-983 cents)
represents a tempered version of the just 9:12:16 (0-498-996 cents)
with its outer 16:9 minor seventh formed from two pure 4:3 fourths.
The fourth then ascends to the fifth, and the unstable minor seventh
to the octave. This beautiful idiom, typical of Perotin and his era
(c. 1200), here takes on a certain septimal cast with the narrowly
tempered minor seventh about midway between 16:9 and 7:4 -- an effect
which Paul Erlich has found charming in 22-EDO and allied systems.
In the second resolution, this time by contrary motion, a tempered
12:14:18:21 minor seventh quad contracts to a fifth in an intensive
manner. The lower 274-cent minor third is not too far from 7:6, while
the 983-cent minor seventh at F-Eb now takes on the role of a somewhat
wide 7:4. This progression features the 70-cent melodic semitones,
here ascending in the intensive manner, which Secor has described as
around the optimal size -- at least in a neomedieval setting.
------------------------------------------------------
2.2.2. Three septimal generation strategies in Zest-24
------------------------------------------------------
Within the Zest-24 version of the Aaron/AKJ matrix we find three
different ways of generating septimal approximations, one of them
examplified in the last examples. As in 22-EDO, near-septimal
intervals can arise from a chain of wide fifths (or narrow fourths).
Thus F-Bb-Eb, with the first fifth at 492 cents and the second at a
narrower 491 cents, yields F-Eb at 983 cents. Adding another 491-cent
fourth to the chain, F-Bb-Eb-Ab, yields F-Ab at 274 cents. Carrying
the process a step further with another fourth at 492 cents, we get
F-Bb-Eb-Ab-Db, yielding F-Db at 766 cents, only 1.5 cents wide of 14/9.
The accuracy of this direct-chain-of-fifths method is comparable to
but slightly less than that of 22-EDO with its fifths around 709
cents; in the 1024-EDO version of Zest-24, wide fifths have sizes of
either 708 or 709 cents (more precisely, 707.812 or 708.984 cents).
In the second method, also illustrated by some examples in the last
section, a near-septimal interval is generated, for example, by adding
the 50-cent diesis between the two 12-note circles of fifths and
keyboards to the 217-cent major second Db-Eb to produce Db-Eb* at 267
cents, virtually identical to a just 7:6 minor third; and similarly to
the 913-cent major sixth Gb-Eb, producing Gb-Eb* at 963 cents, about 6
cents narrow of 7:4. The following cadence, discussed above, uses both
these intervals:
Eb* Eb
Db Eb
Gb Ab
In like fashion, subtracting the diesis from a fourth at 491 or 492
cents produces a wide major third at 441 or 442 cents, about 6 or 7
cents wide of 9:7 (e.g. Eb*-Ab, F*-Bb); and subtracting it from a
minor seventh at 982 or 983 cents yields a near-just 12:7 major sixth
at 932 or 933 cents (Bb-Ab*, Eb*-Db). Subtracting a diesis from a
287-cent minor third yields a 237-cent large tone about 6 cents wide
of 8:7 (e.g. C*-Eb). Another cadence cited above uses these intervals:
Eb F
C* C
Bb C
F* F
Some other septimal approximations of this type are slightly less
accurate but also very useful. Subtracting a diesis from a 274-cent
minor third (e.g. F*-Ab) yields a wide major second or tone around 224
cents, about 7 cents narrow of 8:7. Likewise, adding a diesis to a
926-cent major sixth (e.g. Ab-F*) produces a 976-cent minor seventh
equally wide of 7:4. Here are two examples with intervals of the
opening unstable sonority shown in rounded cents, the first featuring
the 976-cent seventh, and the second the 224-cent tone between the two
highest voices, with the 50-cent diesis also playing a premier role in
both progressions as a melodic interval. The reader may note that for
the sake of variety the first progression takes the liberty of
cadencing on Ab*, a step not actually included in the present 17-note
version of the Aaron/AKJ matrix, but added in the 19-note version to
be discussed below (Section 3.1):
F* Eb* Ab Bb
Eb Eb* F* F
Bb* Ab* Eb F
Ab Ab* Bb* Bb
(0-268-709-976) (0-441-708-932)
To sum up, our second strategy for obtaining septimal approximations
is to start with an interval somewhere between Pythagorean and
septimal formed within a single chain of fifths, and add or subtract a
diesis. In some contexts, this strategy and our first of generating
near-septimal intervals within a chain of fifths combine very nicely
to produce a scale such as this approximation of Dariush Anooshfar's
"Iranian Diatonic" in 125-EDO (iran_diat.scl in the Scala archive),
with roundings of Anooshfar's steps and intervals in parentheses -- an
example again calling for the step Ab* from the coming 19-note version
of the matrix:
(0) (221) (442) (490) (710) (931) (979) (1200)
Bb* Db Eb Eb* F* Ab Ab* Bb*
0 224 441 491 708 932 982 1200
224 217 50 217 224 50 218
(221) (221) (48) (221) (221) (48) (221)
Our third strategy is identical to that prevailing in a regular
2/7-comma meantone tuning, or also, for example, in 1/4-comma meantone
or 31-EDO: add a diesis to a usual major meantone interval, or
subtract it from a minor meantone interval, to generate an
approximately septimal major or minor interval _of the same
category_. Thus a 312-cent or 313-cent minor third less the diesis
(actually 50.39 cents in a 1024-EDO realization) produces a near-7:6
at a rounded 261 or 262 cents, about 4-6 cents narrow (e.g. B*-D,
E*-G); a 383-cent major third plus the diesis yields a 434-cent major
third (e.g. F-A*, G-B*) about 1.5 cents narrow of 9:7. A 1008-cent or
1009-cent minor seventh less a diesis produces a narrow minor seventh
in the range of 957-959 cents, on the outskirts of 7:4 but also near
the upper end of the interseptimal zone between 12:7 and this ratio.
The following progression, with its excellent 70-cent semitones
derived from a usual 121-cent meantone diatonic semitone less a
diesis (E*-F, B*-C), illustrates these intervals:
D C
B* C
G F
E* F
(0-262-696-957)
Likewise, adding a diesis to a meantone major second at 191 or 192
cents produces a wide major second of 241-243 cents, near 8:7 but
entering the lower interseptimal range between this ratio and 7:6; a
major sixth at 887 or 888 cents plus a diesis produces a near-12:7
sixth at 938-939 cents near the lower edge of the 12:7-7:4
interseptimal region. These intervals are featured in this resolution,
again with 70-cent semitones (this time descending):
D* E*
C B*
A* B*
F E*
(0-434-696-938)
One way to get an overview of the three strategies interact in
building the tempered Aaron/AKJ matrix is to see how notes from the
two Zest-24 chains of fifths interact in forming septimal quads, as
shown in the following diagram along with the sizes of the fifths. In
addition to the seven quads present in the JI version of Aaron/AKJ, an
extra quad is available at Ab-Bb*-Eb-F* and shown accordingly:
709 708 696 708 709
Eb* -- Bb* -- F*/Gb - C*/Db --- Ab --- Eb
| | | | | |
| | | | | |
Gb --- Db --- Ab ----- Eb ---- Bb ---- F ---- C ---- G ---- D
696 708 709 709 708 | 696 | 696 | 695 |
| | | |
D* --- A* --- E* --- B*
696 696 696
In the left portion of the diagram, corresponding to the upper part of
the lattice diagram (Section 2.2), the second strategy of adding a
diesis to an interval in the Pythagorean-septimal range prevails,
together with the first strategy of generating septimal approximations
within a single chain of fifths, which becomes especially important
around and slightly to the right of the middle portion. As we move
further to the right, the third strategy of modifying major or minor
meantone intervals by a diesis becomes the rule.
Quickly examining the fifth sizes suggests that the first two
strategies apply to regions of a Zest-24 circle where large fifths
predominate (708 or 709 cents), producing major or minor intervals
which themselves reasonably approximate septimal ratios and/or do so
when modified by a diesis. For example, a 926-cent major sixth such as
Db-Bb itself provides a nice near-12:7, and when a diesis is added
(Db-Bb*) additionally yields a pleasant near-7:4 at 976 cents.
The third strategy, not so surprisingly, applies in portions of a
circle with small meantone fifths (695 or 696 cents), where adding a
diesis to a usual meantone major interval or subtracting it from a
minor one produces some variety of septimal flavor -- with 9:7 or 14:9
most accurate (434 or 766 cents); 7:6 or 12:7 a bit less so (261-262
or 938-939 cents); and the representation of 8:7 or 7:4 intriguingly
"peripheral" or interseptimal in flavor (241-243 or 957-959 cents).
-----------------------------
2.2.3. Accuracy and congruity
-----------------------------
In a JI system such as Aaron or AKJ, or our superset of the two, a
pure ratio such as 7:4 or 9:7 is typically represented precisely by
itself -- I say "typically," because some JI systems might produce
variant sizes within a few cents of these ratios serving in effect as
"virtually tempered" equivalents.
In equal or other regular temperaments such as 22-EDO, 31-EDO, or
36-EDO with more or less close septimal approximations, we generally
assume that any given just ratio will have one consistent tempered
representation: 9:7, for example, mapping to a size of 436 cents in
22-EDO, 426 cents in 31-EDO, and 433 cents in 36-EDO.
In a typical well-temperament with fifths tempered in a single
direction (either wide or narrow of 3:2), for example George Secor's
17-tone well-temperament (17-WT), the situation becomes rather more
complex because interval sizes will vary as one moves around the
circle. Thus Secor's 17-WT has the best 9:7 approximations at 429
cents in the nearer portion of the circle, with major third sizes
getting smaller as one moves into more remote territory, and
eventually arriving at a just 14:11 (418 cents) in the furthest
portion of the circle. However, the strategy for obtaining the closest
representation of 9:7 available at any given position remains the
same: take a chain of four fifths up from the note in question.
In Zest-24, however, realizing the Aaron/AKJ matrix means routinely
using three different strategies for generating septimal flavor
intervals that may vary greatly in size and color: for example, as
noted, representations of 7:6 ranging from 254 to 287 cents (or more
specifically at 254, 261/262, 267, 274, or 287 cents). The extremes of
this range would hardly be called "septimal" in a usual sense: 254
cents is squarely in the interseptimal range between 8:7 and 7:6,
while 287 cents is a fine representation of 13:11 rather than 7:6.
From one point of view, this "inaccuracy" might be taken as a high
musical virtue, an element of "variegation" in which septimal flavor
intervals shade into other neighboring realms. From another, if a more
or less consistent septimal color is indeed our objective, these
vagaries will often be mitigated in textures for three or more voices
when "peripheral" intervals such as minor thirds at 254 or 287 cents
are heard in sonorities also featuring intervals in the more central
septimal range.
If we think in terms of 12:14:18:21 or 14:18:21:24 quads rather than
isolated intervals or dyads, the following variation on the lattice
diagram may bring this point home. Here the unstable intervals of
minor seventh quads are shown: the minor seventh, two minor thirds,
and also the major third between the middle voices. Where two
alternative "versions" of a lattice step are available in Zest-24, I
have chosen the version for a given quad which seems to yield the more
accurate septimal approximations. Here the Aaron/AKJ lattice is
modified to show, as discussed in the last section, an additional quad
which arises in the tempered version at Ab-Bb*-Eb-F*, with F* serving
in effect as a "49/48" of sorts (a virtually just 7:6 above the
approximate 7/4 at Eb).
49/27 ---- 49/36 . . 49/48
Eb* Bb* (F*)
1034 543 50
| 976 | 976 |
| 267 268 | 268 267 |
| 441 | 441 |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb/F* Db/C* Ab Eb
70/50 766/746 274 983
| 963 | 982 | 983 |
| 267 254 | 274 274 | 274 287 |
| 442 | 434 | 422 |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
983 492 0 696 192 887
| 959 | 959 | 957 |
| 262 262 | 262 262 | 262 261 |
| 434 | 434 | 434 |
12/7 ------ 9/7 ----- 27/14 ---- 81/56
D* A* E* B*
938 434 1130 626
Reading from the top of the lattice, we see that the first row of
quads have 976-cent minor sevenths and virtually just 7:6 minor
thirds, with a 441-cent major third between the middle voices (6 cents
wider than 9:7) as the "outlier" from a septimal viewpoint. This
situation may be a bit analogous to that of 31-EDO, where the 426-cent
major third is notably less accurate than the other septimal
approximations in 12:14:18:21. For example, using standard meantone
spellings:
7/6 ------ 7/4
C# G#
271 968
| 968 |
| 271 271 |
| 426 |
1/1 ------ 3/2
Bb F
0 697
In the middle row of quads, outlying intervals include minor thirds at
254 or 287 cents, and major thirds at 422 or 442 cents. From a
septimal point of view, these not-so-accurate representations are
"balanced" by others in a given quad. More specifically, above the
lowest voice of each quad we find a minor third within 7 cents or so
of 7:6 (either the virtually just 267 cents or the reasonably close
274 cents), and a minor seventh at 963 cents (about 6 cents narrow of
7:4) or 982/983 cents (comparable to 22-EDO). The outlier intervals
occur between two upper voices. Here we might consider either the
concept of Paul Erlich that the near-3:2 fifths and near-7:4 minor
sevenths of such sonorities may give them a certain "rootedness" (with
a septimal flavor, I might add); or the common contrapuntal guideline
that in some circumstances tense (or intonationally imprecise)
intervals between two upper voices may be less significant than those
above the lowest voice.
In the lowest row of quads, formed as in a regular 2/7-comma meantone
temperament by adding or subtracting a diesis from a usual meantone
interval, major and minor thirds are quite close to 9:7 (434 cents,
nearly just) and 7:6 (261-262 cents, somewhat narrow). Here the
"problematic" interval, from a septimal perspective, is the narrow
minor seventh at around 957-959 cents, 10-11 cents narrower than 7:4
and actually a bit smaller than the 960-cent interval found in 5-EDO
and more generally 5n-EDO (any equal division of the 2:1 octave with a
size divisible by five). This narrow minor seventh, around the upper
end of the interseptimal zone between 12:7 and 7:4, is a very
attractive attribute of 2/7-comma -- although not necessarily ideal if
one is seeking impeccable approximations of septimal JI.
Some of these examples and characterizations raise questions as to
varying judgments of accuracy regarding different septimal intervals,
and also the _direction_ of inaccuracy. Thus while the Zest-24
interval of 422 cents, the 424 cents of 17-EDO, or even the 426 cents
of 31-EDO would be regarded as distinct from an "accurate"
representation of 9:7 at 435 cents, the comparably inaccurate (from a
mathematical viewpoint) representation of 7:4 at 982 cents in 22-EDO,
or 982/983 cents in Zest-24 as realized in 1024-EDO, may be deemed
musically accurate enough. Paul Erlich has suggested that 9:7 may be a
"shallower valley" or region of psychoacoustical blend or "simplicity"
than 7:4.
Similarly, the 982/983-cent minor sevenths of Zest-24 (or 22-EDO) are
impure by about 13-14 cents in the wide direction, a greater
mathematical inaccuracy than that of the 957-959-cent minor seventh at
about 10-11 cents narrow. Yet from a septimal perspective, the first
might be readily accepted and the second deemed more problematic --
albeit with relish, if one also has a taste for interseptimal color,
and finds it "impure in the more intriguing direction."
Leaving these questions open, we might define a property of
"congruity" which fits the structure of our tempered 17-note
realization in Zest-24 of the Aaron/AKJ matrix:
(1) Every note in the system should be the member of at least
one 12:14:18:21 (or 14:18:21:24) quad.
(2) Every pair of vertically aligned notes in the lattice
should form a minor third within about 20 cents of 7:6
(the 287-cent third, more precisely 207.109 cents, is wide
by 20.2385 cents).
(3) Every note in the lattice should be within about 20 cents
of the septimal ratio which it represents from the 1/1.
In the setting of Zest-24, where each of the 24 notes can participate
in some flavor of "near-septimal" quad, the first rule exerts no
constraints on possible expansions of our lattice to include more
notes from the tuning, or to add new lattice positions using some of
the existing notes in dual or multiple roles.
The second and third rules, however, do exert such constraints.
Consider, for example, this "conceptual expansion" of the lattice
using only the existing 17 notes:
49/27 ---- 49/36 ---- 49/48 ---- 49/32
Eb* Bb* F* C*
1034 543 50 746
| 976 | 976 | 963 |
| 267 268 | 268 267 | 267 254 |
| 441 | 441 | 442 |
28/27 ---- 14/9 ------ 7/6 ------ 7/4 ----- 21/16
Gb/F* Db/C* Ab Eb Bb
70/50 766/746 274 983 492
| 963 | 982 | 983 | 996 ?
| 267 254 | 274 274 | 274 287 | 287 300 ?
| 442 | 434 | 422 | 409 ?
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
983 492 0 696 192 887
| 959 | 959 | 957 |
| 262 262 | 262 262 | 262 261 |
| 434 | 434 | 434 |
12/7 ------ 9/7 ----- 27/14 ---- 81/56
D* A* E* B*
938 434 1130 626
Here C* at 746 cents above the 1/1 at F represents 49/32 (738 cents)
as well as 14/9 (765 cents); and Bb at 492 cents represents 21/16 (471
cents) as well as 4/3 (498 cents). All of these representations are
within about 20 cents of just -- albeit those of 14/9 and 21/16 just
barely so, with the latter simply a tempered 4:3, as in a 22-EDO
version of this lattice -- so that the third rule is observed.
However, looking to the quad C-Eb-G-Bb at the right end of the middle
row, we note a problem under the second rule of congruity. This rule
requires that we "connect the dots" between the vertically aligned
notes G-Bb and test as to whether they form a minor third "within
about 20 cents" of 7:6. In fact, they form a 300-cent minor third wide
of this septimal interval by some 33 cents, thus creating an
"incongruity."
Accordingly, the vertical line on the lattice diagram connecting these
two notes is filled in with question mark characters, signalling the
septimal incongruity. It should be emphasized that the C-Eb-G-Bb quad
at 0-287-696-996 cents is a fine neomedieval sonority routinely mixed
with septimal ones both in a "classic" variation on 13th-14th century
European styles and in more "modernistic" idioms -- but one itself
having more of a Pythagorean than a septimal cast, an impression
reinforced by the virtually just 16:9 minor seventh at 996 cents and
the 409-cent major third, about a cent wider than the Pythagorean
81:64 and almost identical to 19:15. Thus this quad, while indeed
quintessentially neomedieval, is outside the scope of the Aaron/AKJ
matrix proper.
To put the issue another way, by using Bb to fill a 21/16 position on
the matrix, we would be advertising that it could provide an
approximation of a 7:6 minor third above G, the 9/8 -- but, in fact,
the resulting 300-cent interval is outside our "within about 20 cents"
guideline.
The third rule, that each ratio on the lattice in relation to the 1/1
should be represented by a Zest-24 note within about 20 cents of that
ratio, also exerts constraints on expansion of the system. For
example, suppose that we sought to add another quad at the lower right
corner of the lattice:
49/27 ---- 49/36 ---- 49/48
Eb* Bb* F*
1034 543 50
| 976 | 976 |
| 267 268 | 268 267 |
| 441 | 441 |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb/F* Db/C* Ab Eb
70/50 766/746 274 983
| 963 | 982 | 983 |
| 267 254 | 274 274 | 274 287 |
| 442 | 434 | 422 |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16 --- 81/64
Eb Bb F C G D A
983 492 0 696 192 887 383
| 959 | 959 | 957 | 959 |
| 262 262 | 262 262 | 262 261 | 261 262 |
| 434 | 434 | 434 | 434 |
12/7 ------ 9/7 ----- 27/14 ---- 81/56 -- 243/224
D* A* E* B* F#*
938 434 1130 626 121
Here the problem under the third rule, curiously, is not with a
septimal ratio but with the representation of the Pythagorean 81/64
(408 cents in JI) by A at 383 cents, almost 25 cents narrow of this
ratio! The rule, or at least this application of it, may seem more
technical than musical, since A is the meantone major third of
2/7-comma just as D at 887 cents is the major sixth, and the fact that
the latter is only about 19 cents rather than some 24 cents narrow of
its Pythagorean lattice ratio (27/16 at 906 cents) hardly makes a
dramatic aural or stylistic difference. The new quad B*-D-F#*-A which
would be formed by admitting A as an "81/64" is in fact virtually
identical to the others in this row likewise derived from altering
usual meantone intervals by a 50-cent diesis.
However, this concept of "congruity" where every just ratio in a
lattice with respect to the 1/1 is represented in a tempered
realization by some value within a given tolerance -- "around 20
cents" is, of course, merely one arbitrary choice! -- here serves the
practical function of setting some constraints on the expansion of our
Aaron/AKJ matrix to include more positions and Zest-24 notes.
From a theoretical perspective, the distinction between the second and
third rules of congruity is an interesting one. In realizing a
septimal JI lattice, a system such as extended 1/4-comma meantone or
31-EDO, for example, easily meets the second rule of supplying a
reasonable and indeed excellent approximation of 7:6 in every position
(271 cents in 31-EDO, and 269 or 275 cents in 1/4-comma); but moves
outside the third rule as interval sizes within a single chain of
fifths drift further and further from Pythagorean JI values.
We might perhaps speak of "strict congruity" meeting both the second
and third rules, whatever standard of accuracy one might choose in a
given context, and "relaxed congruity" where the second rule is met
but not necessarily the third. Alternatively, we could use the term
"absolute congruity" for the accurate mapping of each lattice ratio
with respect to the 1/1, and "relative congruity" for the accurate
realization of certain defining intervals (e.g. here the 3:2 fifth and
7:6 minor third) at all relevant positions.
--------------------------------------------------
3. Complicating things further: A 19-note superset
--------------------------------------------------
Having arrived at our Zest-24 lattice with its 16 positions and 17
notes, can we expand the system further while remaining within the
rules of strict congruity? In some recent examples, we have already
added a 17th position, 49/48, at the upper right corner of the
lattice, filled by F* (50 cents above F, the 1/1), an existing note
which also represents one alternative version of 28/27. At 50 cents,
F* is about midway between the smaller 49:48 at 36 cents and the
larger 28:27 at 63 cents, just ratios differing by the comma of
Archytas at 27 cents.
--------------------------------
3.1. Filling out a 6x4 framework
--------------------------------
In addition to applying the rules of strict congruity, we might ask
that our lattice keep within what I have termed the "6x4 matrix" of
the Aaron/AKJ superset as realized in JI with its "breadth" of up to
six notes in a horizontal row or chain of fifths, and its "depth" of
up to four notes in a vertical column or chain of 7:6 minor thirds.
49/27 ---- 49/36
1032 534
| |
| |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
63 765 267 969
| | | |
| | | |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
996 498 0 702 204 906
| | | |
| | | |
12/7 ------ 9/7 ----- 27/14 ---- 81/56
933 435 1137 639
Within this 6x4 framework, there are potentially 24 positions to be
filled -- subject in our Zest-24 realization to the rules of
congruity. Letting our expansion be shaped by these rules, we arrive
at this system:
98/81 ---- 49/27 ---- 49/36 ---- 49/48
Ab* Eb* Bb* F*
325 1034 543 50
| 963 | 976 | 976 |
| 254 267 | 267 268 | 268 267 |
| 442 | 441 | 441 |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb/F* Db/C* Ab Eb
70/50 766/746 274 983
| 963 | 982 | 983 |
| 267 254 | 274 274 | 274 287 |
| 442 | 434 | 422 |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
983 492 0 696 192 887
| 957 | 959 | 959 | 957 |
| 250 262 | 262 262 | 262 262 | 262 261 |
| 445 | 434 | 434 | 434 |
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
G* D* A* E* B*
243 938 434 1130 626
As it happens, the two instances of "alternative steps" (28/27, 14/9)
with a single JI position represented by two Zest-24 notes are
balanced out by the "dual roles" of the Zest-24 notes Eb (16/9, 7/4)
and F* (28/27, 49/48), so that we have 19 lattice positions filled by
the same number of Zest-24 notes.
Adding 98/81 at the upper left corner of the lattice raises no new
problems: Ab* at 325 cents is reasonably close to the just value of
330 cents, and either the quad Gb-Ab*-Db-Eb* at 0-254-696-963 cents
indicated in the diagram, or the alternative F*-Ab*-C*-Eb* formed
within a single chain of fifths (0-274-696-983 cents), remains within
the scope of the matrix as shaped by our congruity rules. The first
choice with its lower 254-cent minor third (Gb-Ab*) and 442-cent major
third between the middle voices (Ab*-Db) has some interseptimal
leanings, while the second with its upper 287-cent minor third and
422-cent major third leans toward the middle region between
Pythagorean and septimal.
The addition of an 8/7 at the lower left corner, represented by G* at
243 cents, is a bit more problematic, although within the letter of
our rules and the spirit of Zest-24. At 243 cents from F, the 1/1,
this note nicely exemplifies the lower interseptimal range between 8:7
and 7:6 rather than a near-just septimal realization, but is only a
bit more than 11 cents wide of the former ratio, well within our
guideline of "about 20 cents."
Here the more questionable aspect might be the minor third G*-Bb at a
rounded 250 cents, 17 cents narrow of the purported 7:6, and smaller
than any other representation of this ratio on the lattice (254 cents
being the next smallest value). This interval, however, does remain
within the "about 20 cents" rule -- and indeed is closer to 7:6 than
the 287-cent thirds appearing elsewhere. One might playfully add that
our 250-cent third is still _slightly_ closer to 7:6 than to 8:7, from
which it differs by some 18 cents -- thus duly acknowledging that our
Zest-24 variation on Aaron/AKJ indeed delves deep into interseptimal
territory. The 445-cent major third of this sonority between the two
middle voices (Bb-D*), rather close to 22:17 (446 cents), also
underscores this point.
The pleasant "outlier" qualities of this last quad may reflect the
special strategy by which some of its septimal -- or interseptimal --
intervals are generated. Two unstable intervals, the minor seventh at
957 cents (G*-F) and upper minor third at 262 cents (D*-F), arise from
the standard "meantone" strategy in this portion of the lattice which
subtracts a 50-cent diesis from a usual meantone minor interval, here
respectively G-F at 1008 cents and D-F at 313 cents.
The lower minor third G*-Bb at 250 cents results when the strategy of
diesis subtraction is applied to the 300-cent G-Bb, transitional in
the tempering of a 12-note circle between minor thirds in the meantone
realm (312/313 cents) and those in the neomedieval realm between
Pythagorean and septimal (274 or 287 cents). Just as the versatile
300-cent third may suggest an imprecise 6:5 (316 cents) in a meantone
style or a rather more accurate Pythagorean 32:27 (294 cents) in a
neomedieval style, so its diesis-reduced form at 250 cents is
intermediate between 7:6 (267 cents) and 8:7 (231 cents).
The middle major third Bb-D* at 445 cents likewise results when the
strategy of diesis addition, which with a usual meantone major third
at 383 cents yields a near-pure 9:7 (434 cents), is applied to the
somewhat larger Bb-D at 395 cents, still in the general meantone range
at a rounded 9 cents wide of 5:4. At about 10 cents wide of 9:7, Bb-D*
makes a fine neomedieval major third, but one with an interseptimal
rather than "near-just septimal" character. Thus including it in a
lattice realizing the Aaron/AKJ matrix is something of a stretch,
albeit one fitting the Zest-24 ideal of a "variegation of colors."
The G*-Bb-D*-F quad lends itself to a beautiful resolution in the
remissive manner with melodically interesting aspects:
F -- -70 -- E*
(957) (696)
D* -- +192 -- E*
(695) (696)
Bb -- -59 -- A*
(250) (0)
G* -- +191 -- A*
Two voices ascend by usual meantone major seconds of 191/192 cents
(G*-A*, D*-E*); while the others descend by small semitones of 59
cents (Bb-A*), the closest approximation in this system of the
septimal 28:27 step at 63 cents, and 70 cents (F-E*), the next closest
equivalent.
With this quad, we have stretched congruity to its limits -- and
arrived at a colorful system of 19 lattice positions and notes.
! zest24-aaron_akj19.scl
!
Zest-24 (1024-EDO version), tempered superset of GWS aaron and akj lattices
19
!
50.39062
70.31250
192.18750
242.57812
274.21875
324.60937
433.59375
492.18750
542.57813
625.78125
696.09375
746.48437
766.40625
887.10937
937.50000
983.20313
1033.59375
1129.68750
2/1
We can also analyse this lattice as derived from two contiguous chains
of fifths in Zest-24, nine notes from the lower circle (Gb-D) and ten
from the upper circle (Ab-B), as discussed above in Section 2.2.2:
709 709 708 696 708 709
Ab* --- Eb* -- Bb* -- F*/Gb - C*/Db --- Ab --- Eb
| | | | | | |
| | | | | | |
Gb --- Db --- Ab ----- Eb ---- Bb ---- F ---- C ---- G ---- D
696 708 709 709 | 708 | 696 | 696 | 695 |
| | | | |
G* ---- D* --- A* --- E* --- B*
695 696 696 696
This diagram shows the ten basic positions for 12:14:18:21 quads on
the lattice, so that the system might be styled a tempered "decaquad."
Here the eight fifths of the lower Zest-24 circle and keyboard are
represented by the middle row. In the left half of the diagram, the
five notes of the upper circle from Ab* to C* are brought into play in
forming approximate 7:6 minor thirds through a process of diesis
addition (the second strategy of Section 2.2.2); in the right half,
the five notes in this circle from G* to B* are used in diesis
subtraction (the third strategy). These strategies yield four quads
apiece. Additionally, two quads in the right center region of the
diagram (Bb-Db-F-Ab, F-Ab-C-Eb) result from septimal approximations
within a single chain of fifths, the lower one (the first strategy).
From another viewpoint, we could also speak of 11 quads, since the
lattice offers a choice between Gb-Ab*-Db-Eb*, shown at the far left
of the diagram, and the variant F*-Ab*-C*-Eb*, either of which could
represent 28/27-98/81-14/9-49/27. Both choices, the latter generated
within the upper circle of fifths alone, are shown in this version of
the diagram:
709
Ab* --- Eb*
| |
709 709 708 | 696 | 708 709
Ab* --- Eb* -- Bb* -- F*/Gb - C*/Db --- Ab --- Eb
| | | | | | |
| | | | | | |
Gb --- Db --- Ab ----- Eb ---- Bb ---- F ---- C ---- G ---- D
696 708 709 709 | 708 | 696 | 696 | 695 |
| | | | |
G* ---- D* --- A* --- E* --- B*
695 696 696 696
The eight fifths from the lower keyboard and nine from the upper one
give 17 fifths, each impure by about 6 or 7 cents, in all, providing
lots of stable trines to serve as modal centers or goals for directed
resolutions. Because of the dual roles played by certain Zest-24 notes
in the lattice, some fifths (e.g. Ab*-Eb*, Ab-Eb) appear more than
once in the diagram.
--------------------------------------------
3.2. Neutral intervals and tempered shadings
--------------------------------------------
In addition to presenting a "variegated" realization -- or artful
warping, one might say -- of the basic septimal intervals found in the
just Aaron/AKJ matrix, this 19-note superset in Zest-24 also makes
available shadings of more complex Aaron/AKJ ratios, including neutral
or semi-neutral ones. A few comparisons may suggest both the
similarities and differences between JI and tempered versions.
In Section 1, we considered a progression available within the 12-note
AKJ tuning -- and thus also, of course, within the just form of the
combined 16-note Aaron/AKJ matrix:
4/3 7/6
16/9 14/9
81/56 14/9
If we analyze this progression in terms of the melodic steps and
vertical intervals above the lowest voice in just ratios and cents,
with absolute lattice positions also indicated, we have:
4/3 7/6
(448:243) -- 7:8 (-231) -- (3:2)
1059 702
16/9 14/9
(896:729) -- 7:8 (-231) -- (1:1)
357 0
81/56 14/9
(1:1) -- 784:729 (+126) -- (1:1)
0 0
Here the overall color of the progression is influenced both by the
largish sizes of the neutral third and seventh at 357 and 1059 cents,
and by the 126-cent step in the lowest voice, in this context a very
small neutral second (or a "semi-neutral" or "supraminor" second, as
one might say) slightly narrow of 14:13 (128 cents).
Suppose we try the same progression in our 19-note Zest-24 superset of
the Aaron/AKJ matrix, using the corresponding lattice positions, and
choosing the Db rather than C* version of the 14/9 position, since it
is the former which supplies a near-3:2 fifth with the 7/6 position as
represented by Ab, the stable fifth required as the resolution of the
neutral sonority.
Bb -- -218 -- Ab
(1066) (708)
Eb -- -217 -- Db
(357) (0)
B* -- +141 -- Db
In this version the 357-cent neutral third is virtually identical to
the just 896:729, but the neutral seventh is somewhat wider at 1066
cents rather than the 1059 cents of 448:243. Also, the upper voices
descend by tempered steps of 217 or 218 cents, which as in 22-EDO can
represent either 8:7 (as here) or 9:8. The greatest difference may be
in the ascending melodic step of the lower voice: which has grown from
126 cents (near 14:13) to 141 cents, a bit larger than 13:12 (139
cents) and, curiously, almost identical to a just 243:224 (141 cents).
While a 126-cent step might be taken as either a very large semitone
or very small neutral second, 141 cents moves well into the lower
portion of the central neutral second region (say 135-160 cents).
Another type of neutral or semi-neutral progression also illustrates
some nuances of difference between the JI and tempered systems:
7/4 14/9
(49:27) -- 7:8 (-231) -- (3:2)
1032 702
7/6 28/27
(98:81) -- 7:8 (-231) -- (1:1)
330 0
27/14 28/27
(1:1) -- 784:729 (+126) -- (1:1)
0 0
The melodic steps in each voice are unchanged from those of the last
JI example, but with a different vertical color: here the 98:81 third
at 330 cents and 49:27 seventh at 1032 cents are just in the
transitional zone between a large minor interval (suggesting a wide
6:5 or 9:5 at 316 and 1018 cents respectively) and a small neutral,
semi-neutral, or supraminor one (e.g. 40:33 at 333 cents, or 20:11 at
1035 cents). Melodically, as we have noted, the 126-cent step in the
lowest voice in likewise in such a "fuzzy" or transitional zone.
Here there are actually two possible Zest-24 representations, since
the resolving fifth 28/27-14/9 has two "versions": Gb-Db or F*-C*. Let
us consider both versions.
Eb -- -217 -- Db
(1054) (708)
Ab -- -204 -- Gb
(345) (0)
E* -- +141 -- Gb
Here, as with our last example, a just 784:729 step of 126 cents is
transformed to a larger 141-cent step in Zest-24. Additionally, the
vertical color of the third and seventh are changed: from 330/1032
cents to 345/1054 cents in central neutral regions of the
spectrum. The 345-cent third is about midway between 39:32 (342 cents)
and 11:9 (347 cents), while the 1054-cent seventh is a bit wide of
11:6 (1049 cents). A possibly more subtle change is that while the
highest voice descends by a 217-cent step almost identical to the
218-cent motion in the previous tempered example, the motion of the
middle voice is reduced from 217 to 204 cents, a virtually just 9:8
step.
Our alternative version of this progression has the same vertical
intervals of the neutral third and seventh, but different melodic
steps:
Eb -- -237 -- C*
(1054) (696)
Ab -- -224 -- F*
(345) (0)
E* -- +121 -- F*
The upper voices now descend by steps of 224 and 237 cents, the two
closest Zest-24 approximations of a just 8:7, while the lowest voice
ascends by 121 cents, a step smaller than the just 784:729 at 126
cents and generally regarded as a large semitone rather than a small
neutral or supraminor second. In fact, this 121-cent step serves as
the usual diatonic semitone of 2/7-comma meantone (as the spelling
E*-F* indicates). The interesting question remains at to whether, in a
neomedieval or more specifically septimal as opposed to a Renaissance
meantone setting, the 121-cent step might take on some "semi-neutral"
qualities.
A final example may also suggest how harmonic and melodic colors can
vary between the just and tempered systems.
49/27 27/16
(49:27) -- 729:784 (-126) -- (3:2)
1032 702
3/2 27/16
(3:2) -- 9:8 (+204) -- (3:2)
702 702
98/81 9/8
(98:81) -- 729:784 (-126) -- (1:1)
330 0
1/1 9/8
(1:1) -- 9:8 (+204) -- (1:1)
0 0
We might describe this just sonority as a supraminor seventh quad,
with the 330-cent third and 1032-cent seventh just at the point of
transition from the large minor to the supraminor region. The larger
third between the two middle voices at 243:196 (372 cents) is likewise
at around the transition zone from a large semi-neutral or submajor
third to a small major third within an affinity for 5:4 (386 cents).
Here the resolving steps are an ascending 9:8 or a descending 784:729.
Our tempered equivalent in the Zest-24 lattice is:
Eb* -- -146 -- D
(1034) (695)
C -- +191 -- D
(696) (695)
Ab* -- -132 -- G
(325) (0)
F -- +192 -- G
The vertical intervals are not too far from their just values, and
take on some fine nuances. Thus F-Ab* at 325 cents, slightly smaller
than 98:81 at 330 cents or the corresponding 22-EDO third at 327
cents, seems more within the large minor domain of 6:5, from which it
is less than 10 cents wide, than the small supraminor realm of 63:52
(332 cents), for example. In contrast, the seventh F-Eb* at 1034 cents
does move more into the small supraminor realm, very closely
approaching a just 20:11, for example (as does the 1036-cent seventh
of 22-EDO). The 371-cent third Ab*-C between the middle voices might
also suggest a semi-neutral or submajor color (e.g. 26:21 at 370
cents), although, as with the minutely larger 243:196 of the just
version, a listener so inclined might also hear it as an inexact 5:4.
The melodic steps show a greater variation from their counterparts in
the JI version. Thus the two major second steps F-G and C-D have
meantone sizes of 191 or 192 cents, in contrast to a just 9:8 at 204
cents, while the descending steps Ab*-G and Eb*-D are increased from
126 cents in the JI version to respectively 132 and 146 cents, the
former between 14:13 (128 cents) and 13:12 (139 cents), and the latter
between 13:12 and 12:11 (151 cents). These steps might lead to more of
a semi-neutral or supraminor interpretation of the vertical color.
In the just Aaron and AKJ tunings or their merged 16-note union, as
well as larger supersets, and also in the tempered Zest-24 supersets
we are exploring, simple septimal ratios or their approximations are
the shaping factors -- but neutral or semi-neutral intervals an
important musical consequence. One result of the unequal temperament
of Zest-24 is that representations of both simple septimal and complex
neutral or semi-neutral intervals become less predictably accurate or
more "variegated."
-----------------------------------------------
3.3. Septimal and interseptimal colors: Gamelan
-----------------------------------------------
A notable contrast between the just Aaron/AKJ matrix in the 19-note
version we have arrived at and the Zest-24 approximation of that
matrix is the frequent representation in the latter of simple septimal
ratios by interseptimal shadings: for example, "7:6" minor thirds at
250 or 254 cents; "9:7" major thirds at 441/442 or 445 cents; and
"7:4" minor sevenths at 957/958 cents.
An interesting style to showcase these sometimes decidedly distinct JI
and tempered colors is the realm of gamelan. While traditional
Javanese and Balinese ensembles are tuned more by ear and cultured
taste than by any mathematical scheme, with the goal of giving each
ensemble its own _embat_ or intonational quality, musicians from other
parts of the world such as Lou Harrison have used JI ratios, simple
and complex, to devise tunings for which traditional gamelan musicians
have expressed approbation.
Here we focus specifically on slendro tunings, which divide an octave
(often a bit smaller or larger than 2:1) into five more or less subtly
unequal steps. Indeed we find in the Javanese literature as well as
other sources the view that 5-EDO, with its equal 240-cent steps,
might serve as a point of departure for the artfully nuanced
inequality of traditional ensemble tunings. Thus step sizes in the
range of around 240-260 cents often abound, as well as steps close to
the septimal ratios of 8:7 (231 cents) or 7:6 (267 cents), and others
somewhat smaller than the former or larger than the latter.
The JI variations on these diverse traditional tunings devised by Lou
Harrison, Jacques Dudon, Lydia Ayers, Bill Alves, and others often
feature steps of 8:7 and 7:6, as well as smaller steps of 9:8 in a
range evidently less often used but not unknown in traditional slendro
practice, where steps of around 190-205 cents have been reported. When
these JI tunings are realized in the Zest-24 version of the Aaron/AKJ
matrix, septimal ratios are often shifted toward an interseptimal
color, a kind of "imprecision" which may be not uncongenial in the
realm of gamelan.
For example, consider this JI slendro tuning available within the
Aaron/AKJ matrix, with absolute lattice positions indicated, and then
just ratios and cents in terms of the lowest note of the scale (8/7):
8/7 4/3 14/9 7/4 1/1 8/7
1:1 7:6 49:36 49:32 7:4 2:1
0 267 534 738 969 1200
7:6 7:6 9:8 8:7 8:7
267 267 204 231 231
Here all melodic steps are small superparticular ratios, but the
rather complex ratios of 49:36 and 49:32, a 49:48 diesis (36 cents)
wider than 4:3 and 3:2 respectively, result. In Zest-24, obtaining a
reasonably close approximation of the 49:36 ratio (8/7-14/9) requires
that we map 14/9 to its Db rather than C* version:
G* Bb Db Eb F G*
0 250 524 741 957 1200
250 274 217 217 243
Among the intervals above the lowest note G*, Db at 524 cents is not
too far from 49:36 at about 10 cents wider, but much more closely
approximates the septimal ratio of 256:189 (525 cents) at a 64:63
comma rather than a 49:48 diesis larger than 4:3. Either size is
characteristic of gamelan, with wide fourths in this range evidently
being most common in pelog but sometimes also used in slendro. The
wide fifth at 741 cents, only about 3 cents larger than 49:32, is in
the upper portion of a range typical of slendro (with fifths from
around 3:2 or 702 cents to about 740 cents typical).
In Zest-24, the just ratios above G* of 7:6 (G*-Bb) and 7:4 (G*-F) are
realized at 250 and 957 cents, or about 17 cents and 11 cents narrow
of these septimal ratios, thus taking on a decidedly interseptimal
cast of a kind very typical of slendro.
Melodically, this narrow representation of 7:6 at 250 cents (G*-Bb)
and also the wide representation of 8:7 at 243 cents found at F-G*
(some 11 cents larger than just), show how these superparticular
septimal steps are often shifted into the range of 240-260 cents.
In contrast, the step Bb-Db at 274 cents more closely approximates 7:6
at about 7 cents wide, a realization close to that of 22-EDO at 273
cents or not quite 6 cents wide. As in 22-EDO, this interval is
generated from three narrowly tempered fourths within a single chain
or circle, there about 491 cents, and here 491 or 492 cents.
With Db-Eb-F, the original steps of 9:8 and 8:7 (204-231 cents) are
both represented by the intermediate 217-cent major second found
within a single Zest-24 chain of fifths, a pattern almost identical to
that of 22-EDO where 9:8 and 8:7 map alike to a 218-cent step. This
fits with a general situation where septimal approximations deriving
exclusively from wide fifths (708 or 709 cents) within a single circle
closely parallel 22-EDO with its 709-cent generators, while
approximations combining notes from the two circles may often lean in
an interseptimal direction (e.g. G*-Bb, F-G*).
Another JI slendro tuning, a mode found within Jacques Dudon's
"Slendro M" scale (slendro_m.scl in the Scala archives), also
illustrates these leanings while showing how Zest-24 colors may vary
depending on the transposition:
28/27 98/81 49/36 14/9 49/27 28/27
1:1 7:6 21:16 3:2 7:4 2:1
0 267 471 702 969 1200
7:6 9:8 8:7 7:6 8:7
267 204 231 267 231
In our Zest-24 version of the Aaron/AKJ matrix, this tuning maps as
follows:
Gb Ab* Bb* Db Eb* Gb
0 254 472 696 963 1200
254 218 224 267 237
Here, at Gb-Ab*, a 7:6 step is tempered at 254 cents, about 13 cents
narrow, and well within the interseptimal zone. Other intervals are
closer to septimal JI values, with Db-Eb* at 267 cents a virtually
pure 7:6; Gb-Bb* about 1.5 cents wide of 21:16 (471 cents); and Gb-Eb*
and Eb*-Gb only about 6 cents respectively narrow of 7:4 and wide of
8:7. The 224-step Bb*-Db at 224 cents, or 7 cents narrow of 8:7, gives
the next most accurate realization of this interval in Zest-24. The
representation of a just 9:8 step at Ab*-Bb*, 218 cents, again shows
how this step size (217 or 218 cents in Zest-24) formed within a
single 12-note circle by two wide fifths can often, as in 22-EDO,
represent either 9:8 or 8:7.
Suppose we try the identical JI version of this tuning transposed up a
fifth to start at 14/9 in our expanded 19-note version of the just
Aaron/AKJ lattice:
14/9 49/27 49/48 7/6 49/36 14/9
1:1 7:6 21:16 3:2 7:4 2:1
0 267 471 702 969 1200
7:6 9:8 8:7 7:6 8:7
267 204 231 267 231
In Zest-24, preferring the Db version of 14/9 in order to obtain a
reasonable representation of 21:16, we have:
Db Eb* F* Ab Bb* Db
0 267 484 708 976 1200
267 217 224 268 224
Here the two 7:6 steps Db-Eb* and Ab-Bb* at 267 and 268 cents are
virtually just, while 8:7 is consistently represented at 224 cents.
The most notable compromise involves the tuning of 21:16 (Db-F*) at
484 cents, or about 13 cents wide -- although one might argue that
this tempered size, like the just 21:16, fits very nicely within the
continuum of narrow fourths so characteristic of slendro.
In the previous Zest-24 transposition, we had the opening JI steps of
7:6 and 9:8 (267-204 cents) forming the 21:16 above the starting note
realized as Gb-Ab*-Bb* (254-218 cents), with the 254-cent step about
13 cents narrow of 7:6 and the 218-cent step an almost equal 14 cents
wide of 9:8 -- thus yielding a nearly just 21:16. From a septimal
perspective, this is a substantial compromise in the tuning of 7:6 --
albeit one fitting the frequent predilection of traditional slendro
for step sizes in the region of 240-260 cents.
In this transposition, we have Db-Eb*-F* at 267-217 cents, with the
opening 7:6 step almost just and the following 9:8 step again
represented by an interval of 217 or 218 cents, about 13 cents wide,
making 21:16 wide by a nearly identical amount at 484 cents. Either
type of "compromise" -- a 7:6 narrow by 13 cents in the last example,
or a 21:16 wide by 13 cents in this -- seems happily to fit the fluid
and diverse intonational realm of slendro, if not necessarily
exemplifying an accurate "near-JI" tuning.
To this point, we have explored how JI slendro tunings found on the
Aaron/AKJ lattice in its just form, and more specifically those with
simple superparticular steps, map to our Zest-24 lattice. Another
approach is to take the tempered lattice itself as our starting point,
seeking out tunings whose intervals may or may not be conceived in
terms of such intervals. For example:
F* Ab Bb* Db Eb* Gb
0 224 492 716 983 1220
224 268 224 267 237
All melodic steps could be interpreted in a superparticular septimal
fashion, with 267 or 268 cents a virtually just 7:6; and 224 or 237
cents a reasonably close 8:7. Although not so accurate, the interval
above the lowest step of F*-Eb* at 983 cents, much like 982 cents in
22-EDO, is a routine tempered equivalent of 7:4.
However, the wide fifth F*-Db at 716 cents (about 14 cents, or roughly
half of a 64:63 comma, wide of 3:2), and the wide octave F*-Gb at 1220
cents, represent a calculated tempering of these intervals "at about
10-20 cents wider than a pure 3:2 or 2:1" rather than any specific
septimal ratios. In appropriate timbres, this style of tuning brings
about the beautiful "shimmering" effect so characteristic of gamelan.
More specifically, the availability in our tempered Aaron/AKJ matrix of
F* or Gb to represent 28/27, and either C* or Db for 14/9, here serves
mainly not to promote the more accurate tuning of septimal intervals
or a choice of two comparable approximations, but rather to achieve a
most deliberate _inaccuracy_, from a JI viewpoint, in tuning the
simple ratios of 3:2 and 2:1. The point is made most dramatically by
the choice of one "version" of the 14/9 step (F*) for the lowest note
and the other (Gb) for the octave, thus stretching the latter interval
by about 20 cents.
Substituting Gb for F* as the lowest note, and otherwise leaving the
tuning unaltered, yields a near-septimal tuning with its own
attractions, including an almost just 21:16 as well as the closest
Zest-24 approximation of 7:4 at 963 cents:
Gb Ab Bb* Db Eb* Gb
0 284 472 696 963 1200
204 268 224 267 237
Indeed we have a rather close rendering of a septimal slendro tuning
by Lou Harrison and Jacques Dudon (slendro_7_4.scl in the Scala
archive), here given with 19-note Aaron/AKJ lattice positions:
28/27 7/6 49/36 14/9 49/27 28/27
1:1 9:8 21:16 3:2 7:4 2:1
0 267 471 702 969 1200
9:8 7:6 8:7 7:6 8:7
204 267 231 267 231
This tuning differs from the permutation of Dudon's "Slendro M" we
considered above only in the order of the first two steps, with 9:8
preceding rather than following 7:6 (204-267 rather than 267-204
cents). As it happens, this ordering meshes with the structure of our
Zest-24 lattice (using Gb for 28/27), so that these steps map to
Gb-Ab-Bb* at 204-268 cents, with 9:8, 7:6, and the resulting 21:16 all
within about 1.5 cents of just.
Our discussion of slendro may suggest the interplay of two creative
elements in musical intonation: purposeful design and artful
indirection. In the Zest-24 version of the Aaron/AKJ matrix, these
elements can conspire to weave a kaleidoscopic reality evoking gamelan
tunings both traditional and new.
------------------------------------------------------
4. Tempering the Aaron/AKJ lattice: A comparative view
------------------------------------------------------
Having considered some aspects of the 19-note Aaron/AKJ matrix as
realized in Zest-24, we may gain some perspective by exploring how
this septimal system might be realized in some other tempered
tunings. Such a survey provides an opportunity to showcase some of the
different strategies for generating septimal intervals (Section 2.2.2)
and compare fine points of structure.
Here is the just version of the lattice:
98/81 ---- 49/27 ---- 49/36 ---- 49/48
330 1032 534 50
| | | |
| | | |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
63 765 267 969
| | | |
| | | |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
996 498 0 702 204 906
| | | | |
| | | | |
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
231 933 435 1137 639
As we saw in Section 3.1, this lattice as realized in the unequally
tempered Zest-24 maps as follows, with the interval sizes used to
represent the best approximation of a 12:14:18:21 septimal quad
varying as we move around the system, and therefore helpful to specify
on a diagram:
98/81 ---- 49/27 ---- 49/36 ---- 49/48
Ab* Eb* Bb* (F*)
325 1034 543 50
| 963 | 976 | 976 |
| 254 267 | 267 268 | 268 267 |
| 442 | 441 | 441 |
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb/F* Db/C* Ab Eb
70/50 766/746 274 983
| 963 | 982 | 983 |
| 267 254 | 274 274 | 274 287 |
| 442 | 434 | 422 |
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
983 492 0 696 192 887
| 957 | 959 | 959 | 957 |
| 250 262 | 262 262 | 262 262 | 262 261 |
| 445 | 434 | 434 | 434 |
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
G* D* A* E* B*
243 938 434 1130 626
For equal temperaments, a simpler notation may suffice, since the
sizes of intervals used for the closest representation of 12:14:18:21
generally remain constant. At the same time, a reminder as to the just
size of lattice ratios in cents may be helpful, with numbers in
parentheses serving this purpose. Here is the same Zest-24 system
notated in this simpler style:
(330) (1032) (534) (36)
98/81 ---- 49/27 ---- 49/36 ---- 49/48
Ab* Eb* Bb* (F*)
325 1034 543 50
| | | |
| | | |
(63) (765) (267) (969)
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb/F* Db/C* Ab Eb
70/50 766/746 274 983
| | | |
| | | |
(996) (498) (0) (702) (204) (906)
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
983 492 0 696 192 887
| | | | |
| | | | |
(231) (933) (435) (1137) (639)
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
G* D* A* E* B*
243 938 434 1130 626
Especially in an equally-tempered system where all interval sizes are
available at all locations, specifying note spellings may not be
strictly necessary; however, it can be helpful in suggesting the
"style" of a given tuning. Given that a number of meritorious
notations are available, I will explain my choices for each system.
One way of surveying a few equal temperaments is to consider the three
strategies of septimal generation discussed in Section 2.2.2., all of
which come into play in our Zest-24 lattice.
---------------------------------------------------
4.1. A single-chain solution: Strategy 1 and 22-EDO
---------------------------------------------------
A very economical and efficient strategy for generating approximate
septimal ratios is to use a single chain of fifths somewhere in the
neighborhood of 6-7 cents wide. This is what happens in the more
remote portion of each 12-note Zest-24 circle (Db-Ab-Eb-Bb-F), with
its fifths at around 708 or 709 cents. In 22-EDO with its similar
fifth size of 13/22 octave or 709.091 cents, the following version of
our 19-note lattice results, realized using only tempered 15 steps as
listed in the accompanying Scala file.
! aaron-akj-22edo.scl
!
22-tET/EDO version of aaron-akj superset
15
!
54.54545
218.18182
272.72727
327.27273
436.36364
490.90909
545.45455
654.54545
709.09091
763.63636
927.27273
981.81818
1036.36364
1145.45455
2/1
(330) (1032) (534) (36)
98/81 ---- 49/27 ---- 49/36 ---- 49/48
Bbb Fb Cb Gb
327 1036 545 55
| | | |
| | | |
(63) (765) (267) (969)
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gb Db Ab Eb
55 764 273 982
| | | |
| | | |
(996) (498) (0) (702) (204) (906)
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
982 491 0 709 218 927
| | | | |
| | | | |
(231) (933) (435) (1137) (639)
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
G D A E B
218 927 436 1145 655
To emphasize the simplicity of the system, I have used straightforward
diatonic spellings throughout. Septimal approximations are
consistently realized as usual diatonic intervals: thus F-Ab at 273
cents for 7/6; F-Eb at 982 cents for 7/4; F-A at 436 cents for 9/7;
and so on. A 12:14:18:21 quad is consistently represented by a tuning
of 0-273-709-982 cents, as the diagram shows, apart from some quirks
of the rounding process (e.g. Cb-Gb looks like a 710-cent fifth, but
is actually 709.09 cents, with these two notes at 545.4545... and
54.5454.. cents above F, rounding to 545 and 55 cents).
To realize the just 19-note lattice, only 15 notes of 22-EDO are
required, since several notes play dual roles. We have G at 218 cents
as 9/8 and 8/7; D at 927 cents as 27/16 and 12/7; Eb at 982 cents as
7/4 and 16/9; and also Gb at 55 cents as both a usual 28/27 semitone
and a 49/48 diesis.
The versatile 55-cent semitone can sometimes be viewed as serving
simultaneously as a diatonic 28:27 step (the much-favored semitone or
thirdtone of Archytas) and a tempered 49:48 -- with respective just
sizes of 63 and 36 cents. For example, the step D-Eb is a routine
diatonic semitone; at the same time, with reference to F, it moves
from D at 927 cents, a rather close approximation of 12:7 (6 cents
narrow), to Eb at 982 cents (13 cents wide), a less precise but still
persuasive equivalent of 7:4. In just intonation, a 12:7-7:4 step
(933-969 cents) would have a size of 49:48 or 36 cents. Here it is
about 19 cents larger, an expansion equal to the extra melodic space
created by the narrow tempering of 12:7 and the wide tempering of 7:4.
While used in medieval Near Eastern tunings (e.g. al-farabi_diat.scl
in the Scala archive) as well as modern ones (e.g. kring2.scl,
identical to a tuning of Robert Walker), a just 49:48 step with its
small size may seem rather "extraordinary" for a usual semitone.
Seeing how Walker's tuning, included within the above lattice, can be
implemented in 22-EDO may suggest the utility of this "melodic
tempering" as well as the special qualities of the JI version:
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
0 231 267 498 702 933 969 1200
8:7 49:48 8:7 9:8 8:7 49:48 8:7
231 36 231 204 231 36 231
F G Ab Bb C D Eb F
0 218 273 491 709 927 982 1200
218 55 218 218 218 55 218
In Zest-24 we can generate an approximation of the same JI scale with
the minor third and seventh degrees above the lowest note, as well as
the fourth and fifth, generated within a single circle; but the major
second and sixth degrees arising from the other 12-note circle.
Ab Bb Bb* Db Eb F F* Ab
0 218 268 492 709 926 976 1200
218 50 224 217 217 50 224
This last version is in some ways intermediate between the JI and
22-EDO versions. In JI, we have distinct 8:7 and 9:8 whole-tone steps
(a 64:63 or 27-cent difference) both contrasting most dramatically
with the 49:48 dieses or semitones at 36 cents. In 22-EDO, we have two
standardized step sizes of 218 and 55 cents, with the whole tone
(which might stand for either 9:8 or 8:7) at precisely four times the
size of the semitone (which might represent either 49:48 or 28:27,
again a difference of 64:63).
In Zest-24, as in the JI version, we have essentially three sizes of
steps, whole tones at 217/218 cents and 224 cents (the former almost
identical to 22-EDO), and semitones at 50 cents, about midway between
49:48 and 28:27. However, the distinction between the 224-cent step
often used to represent 8:7, and the 217/218-cent step which, as in
22-EDO, may represent either 9:8 or 8:7, is considerably muted
compared to the 27-cent distinction in JI (204 or 231 cents). We have
a mixture in which some septimal representations such as 8:7 at Ab-Bb
(218 cents) and 12:7 at Ab-F (926 cents) are tempered about as in
22-EDO; others such as 7:6 at Ab-Bb* (268 cents) or Eb-F* (267 cents)
are virtually just; and still others such as 7:4 at Ab-F* (976 cents)
are roughly midway between JI and 22-EDO (here 969 and 982 cents). Yet
other intervals are notably more _inaccurate_ in Zest-24 from a
septimal JI perspective: thus 9:7 at Bb*-Eb (441 cents), as compared
to a just 435 cents, or the minutely larger 436 cents in 22-EDO.
Another point of difference between these systems in that while the
19-note JI matrix includes 7 instances of the narrow 21:16 fourth, for
example (moving two 3:2 fifths to the right, or a 9:8 major second,
plus one 7:6 minor third up), 22-EDO does not have any close
representation of this more complex septimal interval. Rather the
single tempered size of 491 cents -- 7 cents narrow of 4:3 or 20 cents
wide of 21:16 -- takes the place of either ratio on the lattice.
In Zest-24 we have a curiously mixed situation where the same lattice
relationship of two fifths to the right and one 7:6 third up may
produce either a virtually just 21:16 at 472 cents (e.g. Gb-Bb*);
a wider approximation at 484 cents (Db-F*) still distinct from that of
a 4:3 fourth; an interval identical to a usual 4:3 representation, at
491/492 cents, much like the 491-cent fouirth of 22-EDO (e.g. Bb-Eb);
or an interseptimal interval of around 454/455 cents (e.g. G*-C),
almost identical to a just 13:10 in a territory about midway between
the 9:7 major third and the 21:16 fourth.
Thus 22-EDO, which has no intermediate sizes between 436 and 491
cents, is a more parsimonious system that either septimal JI or
Zest-24; while supporting fewer complex types of septimal intervals
(or others, such as 13:10), it is amazingly efficient at filling in
all the simple ratios and quads of the 19-note Aaron/AKJ matrix using
only a single chain of fifths with 15 notes per octave.
Further, if the goal is to have an elegantly ordered and predictable
realization of these basic ratios which holds across all portions of
the matrix, as it does on the original JI lattice, then 22-EDO is a
more logical solution than the "semi-ordered chaos" of Zest-24.
As our lattice shows, 22-EDO can nicely represent some complex
septimal ratios as well, with 98/81, for example, realized at 327
cents (less than 3 cents narrow of just); 49/27, 49/36, and 49/48 all
remain within our 20 cent tolerance, with the last ratio near this
limit (a 55-cent step about 19 cents wide of a just 36 cents).
Thus while the 22-EDO web or network of near-3:2 fifths and near-7:6
minor thirds is infinitely extensible -- ultimately by assigning more
and more multiple roles to existing notes on the lattice, of course,
since there only 22 steps in this closed system -- there are limits on
the ability accurately, or distinctly, to represent complex septimal
ratios. Again, this is hardly surprisingly giving the modest size of
this very compact and efficient system.
Using some terms proposed above (Section 2.2.3), we might say that
22-EDO as a realization of septimal JI has relative congruity over an
infinite lattice: that is, we can indefinitely expand the lattice
horizontally (to use longer chains of fifths) and vertically (to use
longer chains of 7:6 thirds), while remaining within any specified
tolerance that can be met by intervals or sonorities in the immediate
area of the chosen 1/1. Fifths will always be about 7 cents wide;
minor thirds about 6 cents wide of 7:6; major thirds about 1 cent wide
of 9:7; and minor sevenths about 13 cents wide of 7:4.
The absolute congruity of the lattice at the "within about 20 cents of
just" standard we have been using, however, holds over a limited area
which happens closely to fit that of our 19-note Aaron/AKJ matrix. If
we attempted to extend the lattice further in the rightward direction
from Gb at 49/48 (about 19 cents wide), we would have Db as 49/32, a
just interval of about 738 cents represented by a 764-cent interval at
26 cents wider -- and one cent narrow of a just 14/9, which Db
represents most excellently.
Philosophically, this raises the question of whether and to what
degree one seeks in a tempered version of a JI system more or less
faithfully to represent not only the simple but the more complex
integers: in mapping a lattice degree of 49/32, for example, does one
wish to savor the unique flavor of a 49:32 interval itself, or merely
to have a web of more basic ratios (e.g. 3:2, 7:6, 9:7, 7:4) reaching
out to and including the position where a 49:32 above the 1/1 would
occur in septimal JI?
Just intonation ideally satisfies both goals: the simple ratios are
available throughout the lattice, and in pure form, while the complex
ratios generated within whatever size of system is chosen are also
present and precise. A relatively compact equal temperament such as
22-EDO excels in providing reasonable and predictable representations
of as many basic septimal ratios and sonorities as possible with as
few notes as possible. A relatively compact but irregular system like
Zest-24 provides highly variable and not-so-predictable realizations
of _some_ simple and complex septimal ratios.
Three examples will show how cadences in 22-EDO and those in Zest-24
deriving septimal intervals from a single chain of fifths can be
strikingly similar, but with the irregular nature of the latter system
also in evidence.
22-EDO Zest-24
Ab -- -218 -- Gb Ab -- -224 -- F*
(982) (709) (982) (708)
F -- +55 -- Gb F -- +50 -- F*
(709) (709) (708) (708)
Db -- -218 -- Cb Db -- -224 -- Bb*
(273) (0) (274) (0)
Bb -- +55 -- Cb Bb -- +50 -- Bb*
Here the vertical intervals of this intensive resolution from a
12:14:18:21 quad to a fifth all correspond within a couple of cents.
While the intervals of the quad itself in Zest-24 are all generated
from the single lower circle of fifths, an intensive resolution within
our Aaron/AKJ lattice requires that we cadence on Bb*-F* in the upper
circle. We reach this goal via large whole-tone steps of 224 cents and
semitone steps of 50 cents (Bb-Bb* and F-F*, using the diesis between
the circles), similar to the 22-EDO steps of 218 cents and 55 cents
and indeed yet a bit more polarized.
In a remissive resolution of this quad, Zest-24 can more consistently
follow the 22-EDO strategy of using regularly spelled intervals built
from relatively short chains of fifths in a single circle -- not only
for the quad itself, but for the resolving steps.
22-EDO Zest-24
Ab -- -55 -- G Ab -- -82 -- G
(982) (709) (982) (696)
F -- +218 -- G F -- +192 -- G
(709) (709) (708) (696)
Db -- -55 -- C Db -- -70 -- C
(273) (0) (274) (0)
Bb -- +218 -- C Bb -- +204 -- C
In 22-EDO all intervals have identical size, this time with ascending
whole tones and descending semitones (218/55 cents); but in Zest-24
the whole tones are now smaller at 192 or 204 cents, and the semitones
larger at 70 or 82 cents, the smallest available within a single chain
of fifths. This is true because in this irregular system, each circle
has only four wide fifths, while a chain of five are required to
generate a diatonic semitone such as Db-C or Ab-G. With four wide
fifths and one narrow one, we get a 70-cent step, in George Secor's
view close to the optimal size for neomedieval music; with three wide
and two small fifths, we get an 82-cent step. These semitones are also
close to superparticular ratios of 25:24 (71 cents) and 22:21 (81
cents) respectively. The 204-cent step is a virtually just 9:8, and
the 192-cent step a smaller meantone variety of major second.
In addition to these melodic changes, the unequal temperament of
Zest-24 can also cause variations in vertical interval sizes:
22-EDO Zest-24
Eb -- -218 -- Db Eb -- -217 -- Db
(982) (709) (983) (696)
C -- +55 -- Db C -- +70 -- Db
(709) (709) (696) (696)
Ab -- -218 -- Gb Ab -- -204 -- Gb
(273) (0) (274) (0)
F -- +55 -- Gb F -- +70 -- Gb
The Zest-24 quad F-Ab-C-Eb at 0-274-696-983 again has a lower minor
third and seventh within a cent or two of 22-EDO values; the upper
minor third C-Eb at 287 cents, and the middle major third Ab-C at 422
cents, however, are decidedly further from septimal JI or 22-EDO,
although very characteristic of neomedieval style. Melodically, the
two 70-cent semitones might be regarded as just about ideal, and not
too far from 28:27 at 63 cents. The 217-cent tone is almost identical
to 22-EDO, and the 204-cent tone also fits in nicely. However, it is
22-EDO which more closely resembles the JI version of the lattice in
consistently mapping the same ratios to the same interval sizes.
Having explored the admirably efficient harmonic geometry of 22-EDO,
which can represent our 19-note lattice using only 15 contiguous notes
in a single chain of fifths, we now consider a second strategy.
------------------------------------------------------------
4.2. Neomedieval intervals and dieses: Strategy 2 and 63-EDO
------------------------------------------------------------
A second strategy for generating septimal approximations in Zest-24 is
to take an interval somewhere between Pythagorean and septimal found
in a single circle of fifths and to add or subtract a 50-cent diesis,
the distance between the two 12-note circles and keyboards. For
example, starting from Db-Eb at 217 cents, about midway between the
Pythagorean 9:8 (204 cents) and the septimal 8:7 (231 cents), we add
the diesis to form Db-Eb* at 267 cents, a virtually just 7:6.
Likewise, the most accurate Zest-24 approximations of 8:7 result from
taking a 274-cent third (e.g. F*-Ab*) about 7 cents wide of 7:6, or a
287-cent third (e.g. C*-Eb*) about 7 cents narrow of the Pythagorean
32:27 (294 cents), and subtracting a diesis to form F*-Ab at 224 cents
or C*-Eb at 237 cents, respectively about seven cents narrow and six
cents wide of 8:7 at 231 cents.
An equally tempered system where this method can be used consistently
to generate the 19-note Aaron/AKJ matrix is 63-EDO, with a fifth of
704.762 cents, or about 2.81 cents wide of 3:2.
! aaron-akj-63-edo.scl
!
19-note Aaron/AKJ superset in 63-EDO
19
!
38.09524
57.14286
209.52381
228.57143
266.66667
323.80952
438.09524
495.23810
533.33333
647.61905
704.76190
761.90476
914.28571
933.33333
971.42857
990.47619
1028.57143
1142.85714
2/1
(330) (1032) (534) (36)
98/81 ---- 49/27 ---- 49/36 ---- 49/48
G**/G#v D**/D#v A**/A#v E**/E#v
324 1029 533 38
| | | |
| | | |
(63) (765) (267) (969)
28/27 ---- 14/9 ------ 7/6 ------ 7/4
F* C* G* D*
57 762 267 971
| | | |
| | | |
(996) (498) (0) (702) (204) (906)
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
990 495 0 705 210 914
| | | | |
| | | | |
(231) (933) (435) (1137) (639)
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
Abd Ebd Bbd Fd Cd
228 933 438 1143 648
Before considering some matters of spelling, we may note that the ten
12:14:18:21 quads of the lattice at a tempered 0-267-705-971 cents are
on the whole much more accurate than in either Zest-24 or 22-EDO. The
fifths are not quite three cents wide, as also the 7:4 minor sevenths;
7:6 is virtually just at a rounded 267 cents; and 9:7, the least
accurate interval, just over three cents wide. The near-just 7:6 at
266.667 cents, about 0.20 cents narrow, is shared with the outstanding
36-EDO system we will consider later, and more generally with any
9n-EDO (an equal temperament of the 2:1 octave with a size divisible
by 9).
In the above notation, a usual flat or sharp lowers or raises a note
by 7 tuning steps, the chromatic semitone (e.g. Bb-B) formed by seven
fifths up, at 133 cents, a small neutral second very typical of
neomedieval temperaments with regular fifths around 704-705 cents, and
here about midway between 14:13 and 13:12 (128 and 139 cents). In
63-EDO, this step is equal to almost precisely half of a just 7:6
third. The usual diatonic semitone (e.g. D-Eb) at 76 cents has a size
of 4 tuning steps, with the 210-cent major second (e.g. F-G) equal to
the sum of these two semitones (e.g. F-Gb-G), or 11 tuning steps.
The difference between the 4-step diatonic and 7-step chromatic
semitones is the enharmonic diesis of the temperament at 3 steps, or
57 cents. This interval, shown here by an asterisk (*) in the
ascending direction or a "reversed flat" (d) sign in the descending
direction, is used to represent the somewhat larger 28:27 semitone or
thirdtone at 63 cents.
Thus a 7:6 third on the lattice is spelled as a usual 210-cent major
second plus a 57-cent diesis, e.g. F-G*, at 267 cents. A 7:4 minor
seventh at 971 cents is likewise formed by a regular 914-cent major
sixth (around 56:33 at 916 cents, for example) plus this diesis,
e.g. F-D*. A 14:9 minor sixth is equal to a 705-cent fifth plus
diesis, or 762 cents, e.g. F-C*. The diesis itself, as we have seen,
approximates the 28:27 minor second, e.g. F-F*.
Major septimal intervals are formed by diesis subtraction. A 9:7 third
is represented by a 495-cent fourth less the 57-cent diesis, or 438
cents (e.g. G*-C); and a 12:7 sixth by a 990-cent minor seventh less
diesis (e.g. G*-F) at 933 cents. To form an 8:7 tone, we subtract the
diesis from a regular minor third at 286 cents (close to 33:28 at 284
cents), arriving at a size of 229 cents (e.g. D*-F). The 27:14 major
seventh at 1137 cents is represented by an octave less diesis or 1142
cents, e.g. F*-F.
This style of notation might especially suit an arrangement where a
subset of 63-EDO is mapped to two 12-note keyboards with the same
steps and intervals (e.g. Eb-G#) at a 57-cent diesis apart. For a
deeper lattice as as Aaron/AKJ with several layers of 7:6 thirds and
other septimal intervals, however, the piling up of multiple diesis
signs (e.g. F*-G** for the 7:6 third at 28/27-98/81) may be less than
elegant.
One possible solution, seen on the uppermost chain of fifths in the
lattice, is to use a "v" sign to show a note lowered by a single
tuning step of 19 cents, the difference between a "double diesis" at
6 steps (G**) or 114 cents and a usual chromatic semitone, indicated
by a sharp, at 7 tuning steps or 133 cents (G#). We could thus write
the 7:6 third above F* as either G** or G#v.
Whatever notation one might choose -- and the sagittal notation
developed by George Secor, David Keenan, and others is one extremely
refined and versatile system for a host of just and tempered tunings
-- 63-EDO offers an outstanding representation of many septimal
intervals, simple and complex alike.
For example, the narrow fourth at 21:16, e.g. 4/3-7/4 at Bb-D*, is
consistently represented at 476 cents, about 5 cents wide of just;
49:32, e.g. 4/3-49/48 at Bb-E**/E#v, at 743 cents, wide by almost the
same amount (since the difference between 21:16 and 49:32 is a 7:6,
here virtually just).
If we imagine some subset of 63-EDO mapped to a series of 12-note
keyboards at a diesis apart, then the style of some common septimal
progressions would resemble that used in some regions of Zest-24, as a
comparison suggests.
63-EDO Zest-24
D* -- -210 -- C* F* -- -217 -- Eb*
(971) (705) (976) (709)
C -- +57 -- C* Eb -- +50 -- Eb*
(705) (705) (709) (709)
G* -- -210 -- F* Bb* -- -218 -- Ab*
(267) (0) (268) (0)
F -- +57 -- F* Ab -- +50 -- Ab*
Here we have in each system a tempered 12:14:18:21 quad with virtually
just 7:6 thirds resolving intensively to a fifth via descending
whole-tone and ascending diesis motions -- the latter at 57 cents in
63-EDO and 50 cents in Zest-24. While these minor thirds are
comparably pure (with Zest-24 Ab-Bb* slightly less accurate at 268
cents in contrast to the others in both systems at 267 cents), the 9:7
and 7:4 are notably more accurate in 63-EDO, at each about 3 cents
wide, than in Zest-24 where they are respectively 6 and 7 cents wide.
Here are remissive resolutions of the same quads, this time with
ascending whole tones and descending dieses:
63-EDO Zest-24
D* -- -57 -- D F* -- -50 -- F
(971) (705) (976) (708)
C -- +210 -- D Eb -- +217 -- F
(705) (705) (709) (708)
G* -- -57 -- G Bb* -- -50 -- Bb
(267) (0) (268) (0)
F -- +210 -- G Ab -- +218 -- Bb
Our choice of the quad F-G*-C-D* (1/1-7/6-3/2-7/4 on the lattice) for
63-EDO, but the more remote Ab-Bb*-Eb-F* (7/6-49/36-7/4-49/48) for
Zest-24 may reflect the fact that while diesis addition or subtraction
is an impressively accurate method for generating septimal intervals
anywhere in the former system, it is _one_ of the methods used in
Zest-24, available only for some positions or regions within the
lattice.
Thus in 63-EDO, a 7:6 third calls for a 210-cent major second (two
fifths up) plus a 57-cent diesis (12 fifths up), both available
anywhere within this equal temperament. In Zest-24, 7:6 is
approximated most accuracy through a similar addition of a major
second at 217 or 218 cents formed by two wide fifths up, plus a
50-cent diesis. However, given the unequally tempered 12-note circles,
there are only three 217/218-cent major seconds in each circle; and to
add a 50-cent diesis to such a major second, we must be at a location
on the _lower_ chain or keyboard.
We accordingly have these virtually pure 7:6 thirds at 13 locations on
the Aaron/AKJ lattice as realized in 63-EDO, but only 3 locations as
it is realized in Zest-24.
From a structural perspective, there is another important difference
between these similar strategies of diesis addition or subtraction as
applied in the two systems. In 63-EDO, the 57-cent enharmonic diesis
is the difference between 12 fifths and 7 pure octaves. Since these
fifths at 704.762 cents are wider than 700 cents, the diesis is in the
_positive_ direction: that is, 12 fifths exceed 7 pure octaves, so
that, for example, Gb is lower than F# by a 57-cent diesis.
In contrast, Zest-24 has two 12-note circles, each built from two
basic varieties of families of fifths. Eight narrow or meantone fifths
(two at 695 cents and six at 696 cents) are balanced by four wide or
neomedieval fifths (two at 708 cents and two at 709 cents). Strictly
speaking, there is no enharmonic diesis in such a circle: the same
note serves as both Gb and F#, for example, and the 12 fifths (narrow
and wide) are together equal precisely to 7 pure octaves.
The 50-cent diesis between the circles is rather defined as the
difference which would result if we took the average size of the eight
narrow fifths in a circle (about 695.801 cents, almost identical to
Zarlino's 2/7-comma meantone at 695.810 cents), and compared 12 of
these fifths to 7 pure octaves -- with the fifths falling short of the
octaves by about 50.391 cents (or 50.276 cents in Zarlino's tuning).
It is thus, at least conceptually, a _negative_ diesis, as results in
a regular temperament with fifths smaller than 700 cents.
As used musically, however, the character or role of this 50-cent
diesis depends on where we are in the system and what type of interval
we seek to approximate. In a meantone region where narrow fifths
predominate, it often acts as a negative diesis, as we shall see; in a
neomedieval region with mostly wide fifths, it acts and feels much
like the positive diesis in a regular system like 63-EDO.
Here there may be some humor: we are "taming" or tempering out the
diesis in each 12-note circle by balancing out narrow and wide fifths;
and then "reintroducing it into the wild" by choosing it as the
interval between the two circles! Further, while the 50-cent diesis is
conceptually based on the average size of the _narrow_ fifths, it is
often used in connection with intervals derived from a chain of wide
fifths in order to realize approximate septimal ratios.
Having considered the "positive" side of the diesis strategy, we now
turn to its "negative" but not inauspicious aspects.
------------------------------------------------------------
4.3. Meantone intervals and dieses: Strategy 3 and 31-EDO
------------------------------------------------------------
A third strategy for septimal approximations in Zest-24 is to take a
meantone interval formed from a chain of narrow fifths within a circle
and to add or subtract the 50-cent diesis between the circles. For
example, the minor third E-G at 313 cents less a diesis yields E*-G at
262 cents, about 5 cents narrow of 7:6. Similarly, the major third F-A
at 383 cents plus a diesis yields F-A* at 434 cents, only 1.5 cents
narrow of 9:7.
A familiar and historically fascinating equal temperament exemplifying
this strategy is 31-EDO, described in mathematical terms by Lemme
Rossi in 1666 and enthusiastically advocated by Christiaan Huygens in
1691. Here is the 19-note Aaron/AKJ lattice in this tuning, with two
styles of notation offered for some notes:
! aaron-akj-31-edo.scl
!
19-note Aaron/AKJ matrix in 31-EDO
19
!
38.70968
77.41935
193.54839
232.25806
270.96774
348.38710
425.80645
503.22581
541.93548
619.35484
696.77419
774.19355
890.32258
929.03226
967.74194
1006.45161
1045.16129
1122.58065
2/1
(330) (1032) (534) (36)
98/81 ---- 49/27 ---- 49/36 ---- 49/48
Gx/Ad Dx/Ed Ax/Bd Ex/F*
348 1045 542 39
| | | |
| | | |
(63) (765) (267) (969)
28/27 ---- 14/9 ------ 7/6 ------ 7/4
F# C# G# D#
77 774 271 968
| | | |
| | | |
(996) (498) (0) (702) (204) (906)
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
1006 503 0 697 194 890
| | | | |
| | | | |
(231) (933) (435) (1137) (639)
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
Abb/G* Ebb/D* Bbb/A* Fb/E* Cb/B*
232 929 426 1123 619
As this lattice shows, 12:14:18:21 quads are realized at 0-271-697-968
cents. The fifth at 696.774 cents is narrow by about 5.181 cents. The
7:6 minor third is realized at a rounded 271 cents, or 4 cents wide;
the 7:4 minor seventh at 968 cents, only about 1 cent narrow; and the
9:7 major third at 426 cents or about 9 cents narrow, the least
accurate of these intervals, but nevertheless more accurate than the
422-cent or 445-cent thirds sometimes representing 9:7 in the Zest-24
realization.
Two styles of spelling are used, the first based on sharps and flats
only (sometimes multiple), and the second largely on the notation of
Nicola Vicentino (1555) for a system based on his archicembalo or
"superharpsichord" tuned in a 31-note meantone circle, likely
somewhere around 31-EDO or the almost identical 1/4-comma meantone
(fifths at 696.578 cents, or 5.377 cents narrow).
In both systems, a flat or sharp lowers or raises a note by two tuning
steps or 77 cents, the chromatic semitone (e.g. F-F#), equal precisely
to 2/5 of a regular whole tone or major second at five steps or 194
cents (e.g. F-G). The difference between these intervals is the usual
diatonic semitone at 3/5 tone or 116 cents (e.g. F#-G). In contrast to
the situation in 63-EDO with its smaller diatonic semitone and larger
chromatic one (76/133 cents), as happens more generally in regular
tunings with fifths larger than 700 cents, here the diatonic semitone
is the larger (116/77 cents).
The difference between these two semitones defines the enharmonic
diesis, one tuning step or 38.710 cents, precisely 1/5-tone. Vicentino
placed a dot over a note to show its raising by a diesis, with the
asterisk (*) serving a similar function in ASCII. This style of
notation can be especially intuitive if one maps 31-EDO (or 1/4-comma)
to an instrument with manuals having the same patterns of steps and
intervals a diesis or "fifthtone" apart -- as happened on some
portions of Vicentino's two-manual archicembalo.
In short, a diatonic semitone (F-Gb) is three tuning steps; a
chromatic semitone (F-F#) two steps; and a diesis (F-F* or F#-Gb) one
step -- respectively a rounded 116, 77, and 39 cents.
To clarify the logic of this system and also some of the alternative
spellings possible, let us consider the fine near-7:6 third F-G# at
1/1-7/6 on the lattice. Here the regular whole tone F-G at 194 cents
contains five tuning steps, while the sharp adds a 77-cent chromatic
semitone of two tuning steps -- 271 cents, or seven such steps, in
all. This "minimal third," as Vicentino calls it, is thus identical to
an augmented second: a major second plus the small or chromatic
semitone at 2/5-tone. It is perhaps not inappropriate that one of the
prime intervals of septimal JI should be approximated by seven degrees
of 31-EDO.
To appreciate the strategy of diesis subtraction, let us start with
the regular meantone minor third at D-F (27/16-1/1) on the lattice,
generated by a chain of three narrow fifths (or wide fourths). This
third at 310 cents, about 6 cents narrow of 6:5 (316 cents), is equal
to a 5-step major second or whole tone at 194 cents plus a 3-step
diatonic semitone at 116 cents (D-E-F or D-Eb-F). This make eight
tuning steps in all -- one 39-cent step or diesis more than our 7-step
septimal minor third at 271 cents.
Thus by subtracting a diesis, D*-F, we obtain a 7:6 third (12/7-1/1).
Using only sharps and flats, we could also write this third as Ebb-F.
Here Eb-F is a 5-step tone, with the extra flat adding a 2-step or
chromatic semitone -- arriving at a 7-step or septimal minor third.
The first spelling emphasizes the diesis subtraction, while the second
conveys that the interval is an augmented second (regular major second
plus chromatic semitone) in the same style as our earlier F-G#.
Note the 31-EDO equivalence of D* and Ebb. If we start with D-E as a
5-step tone, then to raise D by a 1-step diesis is the same as to
lower E by two 2-step chromatic semitones, or 4 steps in all. Ones
preferences might depend on various factors, including the layout of
any keyboard mappings one might be using for 31-EDO or the similar
1/4-comma meantone.
A 7:4 minor sixth at 968 cents, virtually just, can similarly be
produced by subtracting a diesis from a usual meantone minor seventh
at 1006 cents such as D-C on the lattice -- thus D*-C (12/7-3/2). This
interval can also be analysed as an augmented sixth, the spelling of
choice at a location like Bb-G# (4/3-7/6). With the latter interval,
we have a meantone major sixth Bb-G at 890 cents, plus a 77-cent
chromatic semitone G-G# -- adding up, when rounding quirks are allowed
for, to 968 cents.
Our first 7:4 approximation at D*-C might also be spelled Ebb-C, a
regular major sixth Eb-C plus an extra flat or 2-step semitone. Again,
raising a note by a diesis is equivalent to lowering the note a usual
tone higher by a double flat or 4 steps (i.e. D*=Ebb).
The 14:9 minor sixth is approximated by a usual fifth plus a chromatic
semitone, or 697 plus 77 cents, yielding a 774-cent interval, for
example F-C# (1/1-14/9). This septimal minor sixth is identical to an
augmented fifth, as this spelling shows, and also to a regular
meantone minor sixth at 813 cents less a diesis (e.g. A*-F, 9/7-1/1).
This interval is about 9 cents wide of a just 14:9, and 8 cents narrow
of 11:7, having an intermediate location between these ratios. It is
very close to 25:16 (773 cents), which would be pure in 1/4-comma. The
778-cent minor sixth of Zest-24 is also in this general region (at a
point rather closer to 11:7 at 782 cents), and both are fine intervals
for neomedieval music, if not the most distinctively "septimal."
Diesis addition likewise allows us to take a usual meantone major
third at 387 cents -- not quite a cent wide of 5:4 -- such as F-A
(1/1-9/7), and obtain F-A* at a diesis larger or 426 cents, rather
narrow of a just 9:7 at 435 cents, but in the general vicinity.
This same interval is found with another style of spelling at C#-F
(14/9-1/1), for example, marking it as a diminished fourth: a usual
fourth C-F less a diesis.
A septimal major sixth (e.g. F-D*) at 929 cents is formed from a
meantone major sixth (e.g. F-D) at 890 cents plus a 39-cent diesis.
This interval may also be spelled as a diminished seventh (e.g. F#-Eb)
or regular 1006-cent minor seventh (F#-E or F-Eb) less a 77-cent
chromatic semitone. This near-12:7 sixth is only about 4 cents narrow
of just.
A septimal major seventh (e.g. F#-F) at 1123 cents, about 14 cents
narrow of a just 27:14, is synonymous with an octave diminished by a
chromatic semitone -- or a meantone major seventh at 1084 cents (about
4 cents narrow of a just 15:8 at 1088 cents) plus a diesis, e.g. F-E*.
In this system, a 28:27 septimal semitone or thirdtone is realized as
a 2-step or chromatic semitone at 77 cents (e.g. F-F#), about 14 cents
wider than the just value. This situation contrasts with that in
22-EDO where the regular 55-cent diatonic semitone (e.g. E-F) is used;
or in 63-EDO, where the 57-cent enharmonic diesis serves this purpose
(e.g. F-F*); or in regions of Zest-24 where the 50-cent diesis often
fills this role (e.g. Ab-Ab*).
While the 77-cent step is actually further mathematically from a just
28:27 at 63 cents than these narrow equivalents, there is a very
cogent opinion that if one tempers this septimal step by more than two
or three cents, it is preferable to do so in the wide direction.
George Secor has suggested an optimal melodic range for semitones in a
neomedieval context of 60-80 cents, with our 77-cent step happily
fitting in this range. In contrast, the 55-cent step of 22-EDO, and
yet more so the 50-cent step of Zest-24, may more ambiguously suggest
either a semitone or some kind of smaller interval.
In a septimal context, then, it is the smaller chromatic semitone of
31-EDO which effectively becomes the usual step -- much like the
larger diatonic step in a meantone style. Likewise, in the regions of
Zest-24 where meantone-style diesis addition or subtraction applies,
the 70-cent chromatic semitone of Zarlino's 2/7-comma (or its 1024-EDO
version) often serves as a usual minor second step. Let us compare,
for example, these remissive resolutions of the quad D*-F-A*-C in the
two systems:
31-EDO Zest-24
C -- -77 -- B* C -- -70 -- B*
(968) (697) (959) (696)
A* -- +194 -- B* A* -- +192 -- B*
(697) (697) (696) (696)
F -- -77 -- E* F -- -70 -- E*
(271) (0) (262) (0)
D* -- +194 -- E* D* -- +192 -- E*
Here the same spellings nicely serve both progressions, which vary
only in the nuances of temperament. The 70-cent semitone of Zest-24 is
somewhat smaller than the 77 cents of 31-EDO, with either representing
a 28:27 step. The near-just 31-EDO representation of 7:4 compares with
the somewhat interseptimal color of Zest-24 at almost 10 cents
narrower; more subtly, 7:6 is about 4 cents wide in 31-EDO but 5 cents
narrow in Zest-24, although still, at 262 cents, in the close septimal
neighborhood of 7:6. The middle 9:7 major third at 426 cents in 31-EDO
and 434 cents in Zest-24 is the one interval of the 12:14:18:21 quad
more accurate in the latter system.
Diesis addition and subtraction operate similarly in both systems --
as long as we stay within regions of Zest-24 where intervals are
generated using narrow fifths only, and apply diesis alterations to
these intervals -- vertical or melodic. When large fifths enter the
picture, intonational variations occur -- subtle or otherwise.
For example, consider this intensive resolution with the septimal
quads themselves formed as in the last example:
31-EDO Zest-24
G -- -194 -- F G -- -192 -- F
(968) (697) (959) (708)
E* -- +77 -- F E* -- +70 -- F
(697) (697) (696) (708)
C -- -194 -- Bb C -- -204 -- Bb
(271) (0) (262) (0)
A* -- +77 -- Bb A* -- +59 -- Bb
In 31-EDO, all intervals are identical to those of the previous
example, except that now we have ascending 77-cent semitones and
descending 194-cent whole tones. In Zest-24, the 12:14:18:21 quad is
the same in its vertical intervals; but this time the resolution
brings us into the realm of wide fifths, landing on Bb-F at 708 cents
or 6 cents larger than 3:2. While it is an interesting question
whether listeners might hear a subtle contrast between fifths
comparably tempered in the narrow and wide directions, our main focus
here is on the melodic consequences.
The upper pair of voices in Zest-24 again use the steps of the
192-cent meantone major second and the 70-cent semitone, this time
with the former descending (G-F) and the latter ascending (E*-F).
As the spelling indicates, the 70-cent step may be formed by the
121-cent meantone diatonic semitone E-F less a 50-cent diesis.
The melodic intervals of the two lower voices, however, are formed by
chains of fifths including the wide Bb-F. Thus the descending whole
tone C-Bb at 204 cents (a virtually just 9:8) is about 12 cents larger
than G-F, while the semitone A*-Bb at a rounded 59 cents is almost 12
cents smaller than E*-F at 70 cents. The latter interval results from
applying diesis subtraction to the step A-Bb, a semitone derived from
a chain of four narrow fifths plus the wide Bb-F (Bb-F-C-G-D-A), and
thus tempered at 109 cents rather than the 121 cents of E-F.
These melodic shadings in the representation of the 9:8 and 28:27
steps seem more subtle than seismic. In a neomedieval setting, one
might welcome the larger 204-cent tone, which fits both with the
septimal JI model and the general tendency in this style to use more
ample sizes of major seconds at around 9:8 to 8:7 (204-231 cents).
Either 59 or 70 cents (more precisely 58.6 and 70.3 cents), the two
most accurate approximations of 28:27 in Zest-24, may be regarded as
an apt tunings, with the former only slightly smaller than Secor's
optimal range of 60-80 cents.
It is easy, however, while engaged in routine progressions involving
meantone-style diesis alterations, to move into territory where more
dramatic changes occur in the size of melodic steps. It is easy, that
is, in Zest-24 -- since, of course, in a regular system like 31-EDO,
as in the JI version of the septimal Aaron/AKJ matrix, 9:8 and 28:27
will always be represented by intervals of the same size. Let us
consider an intensive resolution for our earlier quad D*-F-A*-C where
the remissive resolutions in 31-EDO and Zest-24 were so similar:
31-EDO Zest-24
C -- -194 -- Bb C -- -204 -- Bb
(968) (697) (959) (709)
A* -- +77 -- Bb A* -- +59 -- Bb
(697) (697) (696) (709)
F -- -194 -- Eb F -- -217 -- Eb
(271) (0) (262) (0)
D* -- +77 -- Eb D* -- +46 -- Eb
Here the suggested note spellings in the two systems are the same, but
the melodic results are distinct. In 31-EDO, the vertical and melodic
interval sizes are again identical to those of previous examples, this
time with ascending 2-step or chromatic semitones (D*-Eb, A*-Bb) and
descending 5-step whole tones (F-Eb, C-Bb). All these intervals are
built exclusively from the narrow meantone fifths of the tuning (the
only kind of near-3:2 fifths), sometimes altered by the 1-step diesis.
In Zest-24, however, the same intensive cadence moves us into the
realm of large fifths. As with the last example, the fact that we
happen to be landing on a fifth (Eb-Bb) tempered wide rather than
narrow by 6 or 7 cents (here the greater amount at 709 cents) seems in
itself inconsequential: what we focus on are the melodic consequences.
This time both whole-tone motions involve larger steps than the
192-cent meantone sizes we saw in the remissive resolution of this
quad, very similar to the 194-cent steps of 31-EDO. The highest voice
(C-Bb) has a 204-cent step, a virtually just representation of 9:8;
while the second to lowest (F-Eb) has a yet more ample 217-cent step,
which routinely represents either 9:8 or 8:7 (much as in 22-EDO).
What might be taken as problematic are not these larger tones, quite
the norm in a neomedieval setting, but the resulting semitone sizes.
At D*-Eb in the lowest voice, we have a step of about 46 cents --
notably smaller than the 57-cent diesis of 63-EDO, the 55-cent
semitone of 22-EDO, or indeed the 50-cent diesis between the two
keyboards of Zest-24 often used as a small semitone. In medieval
European terms, it is almost identical to an interval known as a
_diesis_ or _diaschisma_ (terms with various historical uses) and
equal to half of a usual Pythagorean diatonic semitone at 256:243 or
90.224 cents -- and thus about 45.112 cents. If one follows Jay Rahn's
suggested reading of a treatise by Marchettus of Padua (1318), then
this theorist may have advocated the use of a small cadential step
around 37:36 (47.434 cents), comparable to our 46-cent step.
The other semitone motion, A*-Bb at 59 cents, is, as has been noted,
close both to a just 28:27 (63 cents) and to Secor's optimal range of
60-80 cents. The use of a 46-cent step to represent this septimal
ratio is, however, rather remarkable!
The cause of this quirk or adornment -- and I would call it both -- is
the unequal temperament of Zest-24 and the resulting "variegation of
melodic steps." Here the unaltered semitone D*-Eb* in the upper
circle, formed from two large and three small fifths (Eb-Bb-F-C-G-D),
has a size of 96 cents -- not too far from the Pythagorean 256:243 at
90 cents, and considerably smaller than the 121-cent step built from
five narrow fifths (e.g. B-C, E-F). When diesis subtraction is applied
to this 96-cent step, we get a 46-cent interval (D*-Eb) actually
smaller than the 50-cent diesis itself.
More moderately, as has been noted, the 59-cent semitone A*-Bb
similarly results from applying diesis subtraction to A*-Bb* at 109
cents, rather than to a meantone step like E-F at 121 cents, which
would produce a 70-cent semitone parallel to the 77 cents of 31-EDO.
Between the 31-EDO and Zest-24 versions of this progression it is
possible to see a contrast which might be described, drawing on the
16th-century meantone roots of both tuning systems, as one between
"classic" and "manneristic" tendencies. While the style of polyphony
we are addressing seems more closely related to the 14th or 21st
century than to this colorful era, the 31-EDO realization of our
septimal lattice has a poised and lucid quality, like that of classic
16th-century counterpoint, which contrasts with the "artful
distortion" and indeed caprice of Zest-24 with its variable interval
sizes, analogous to the arbitrary dissonances and chromatic idioms of
16th-century mannerism.
While the 46-cent step at D*-Eb is striking, we might also try what
would seem a more "classic" form of this progression using the
unaltered version of this semitone, D*-Eb* at 96 cents. While notably
larger than 28:27 at 63 cents, this step is quite comparable to the
usual Pythagorean semitone at 90 cents, and is often used in
neomedieval styles in Zest-24. Here it may be instructive also to see
what happens in 31-EDO when the unaltered D-Eb, a 3-step or diatonic
semitone at 116 cents, is used in a similar variation:
31-EDO Zest-24
C -- -155 -- Bb* C -- -154 -- Bb*
(968) (697) (959) (709)
A* -- +116 -- Bb* A* -- +109 -- Bb*
(697) (697) (696) (709)
F -- -155 -- Eb* F -- -166 -- Eb*
(271) (0) (262) (0)
D* -- +116 -- Eb* D* -- +96 -- Eb*
In 31-EDO, we see a more simple and regular version of the musical
mathematics that govern this progression in both systems. A 7-step or
septimal minor third at 271 cents (D*-F or A*-C) is contracting to a
unison, with the lower voice of the interval ascending by a 3-step
semitone at 116 cents (D*-Eb* or A*-Bb*). Thus to reach a unison, the
other voice must descend by a 4-step interval (F-Eb* or C-Bb*) equal
to the usual 5-step or 194-cent major second less a diesis, or 155
cents. This step is a neutral second about 4 cents larger than 12:11
at 151 cents, one of the many simple just ratios for which 31-EDO
offers fine approximations.
In Zest-24, a similar if not so regular situation emerges. Here the
septimal minor thirds D*-F and A*-C are 262 cents. In the resolution
of D*-F, the lowest voice ascends by our 96-cent semitone D*-Eb*,
leaving for the upper voice of this third a descending step F-Eb* at
166 cents, or the 217-cent semitone F-Eb less a 50-cent diesis. Thus
we arrive at a unison. The 166-cent step is a large neutral second
very close to Ptolemy's 11:10 (165 cents). In our septimal lattice, it
often plays the role of 54:49 at 168 cents, for which it also offers a
very close representation, a ratio defining the differene, for
example, between a 7:6 minor third at 267 cents and a 9:7 major third
at 435 cents.
The upper minor third A*-C resolves to a unison via an unaltered
semitone A*-Bb* at 109 cents in the lower voice of this third, like
the 116 cents of 31-EDO rather close to 16:15 at 112 cents and more
typical of a Renaissance meantone than a neomedieval style; and a
descending neutral second step C-Bb* at 154 cents, interestingly
almost identical to the 155 cents of 31-EDO, and equal to a 204-cent
major second less the 50-cent diesis.
In both systems, these progressions with ascending semitone and
descending neutral second steps combine elements of a usual intensive
cadence with semitone and whole-tone steps, and an equable cadence
where a minor third, for example, resolves to a unison via two neutral
second steps (see Section 1). When combined with the ascending
semitones, the rather large sizes of neutral seconds used (155 cents
in 31-EDO, 154 or 166 cents in Zest-24) favor a generally intensive
impression; George Secor has suggested that when a step reaches a size
of about 12:11 or wider, it is more likely to be heard as a kind of
small whole-tone, as it were.
From one perspective, these examples in 31-EDO and Zest-24 have showed
some ways of moving by stepwise contrary motion from a septimal minor
third to a unison, in effect dividing the interval -- consistently 271
cents in the former system, and 262 cents in the latter at the
specified locations -- into two melodic steps. The felicitous 194/77
division in 31-EDO and kindred 192/70 division in Zest-24 seem ideal,
with 204/59 (more precisely 203.9/58.6 for a 262.5-cent third, to
clarify a rounding anomaly) in the latter system also close to Secor's
optimal melodic semitone size of 60-80 cents.
In this not-so-predictable system we also encounter 217/46, a caprice
of tuning structure which might be elevated to a manneristic virtue.
Both systems additionally offer an intriguing type of hybrid division,
155/116 in 31-EDO and 166/96 or 154/108 in Zest-24, coupling a
semitone with a neutral second step.
Today, as in 1555 when Nicola Vicentino published his treatise
exploring some of the intriguing interval categories and shadings
available in a 31-note meantone circle (possibly 1/4-comma or 31-EDO),
these gradations and choices are intriguing. Let us conclude this
discussion of 31-EDO by considering the realizations of some more
complex septimal ratios.
A charm of the tuning is that its 39-cent diesis offers an excellent
rendition of the 49:48 step, facilitating a melodically near-just
version of a special diatonic scale used by Robert Walker and others
for which we have given versions in 22-EDO and Zest-24 (Section 4.1).
Just values in cents are shown in parentheses.
(0) (231) (267) (498) (702) (933) (969) (1200)
F G* G# Bb C D* D# F
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
0 232 271 503 697 929 968 1200
8:7 49:48 8:7 9:8 8:7 49:48 8:7
232 39 232 194 232 39 232
(231) (36) (231) (204) (231) (36) (231)
Like a just 49:48 at 36 cents, the diesis has a "glissando" quality,
as Jacky Ligon has described it, with an effect quite distinct from
that of a usual semitone. In 1618, Fabio Colonna describes a "sliding
of the voice" in diesis or fifthtone steps as one of the techniques
available in a 31-note meantone cycle (again possibly either 1/4-comma
or 31-EDO). While variations on this septimal scale with semitones at
around 50-55 cents have their own attractions, the special "sliding"
quality of the JI version and this close 31-EDO realization should not
be missed.
We find that this tuning maps the 98/81 and 49/27 steps (330 and 1028
cents in JI) well into the regions of neutral thirds and sevenths
respectively at 348 and 1045 cents, the first a virtually just 11:9
and the second close to 11:6 (347 and 1049 cents). From one point of
view, this is a subtle distortion of the JI matrix, changing the color
of these intervals; from another, it might be seen as a welcome
addition of some non-septimal but very popular ratios to the mix.
Another complex septimal interval, 21:16 at 471 cents, is tuned in
31-EDO at 465 cents or 6 cents narrow -- producing, as it happens, a
virtually just 17:13 (less a tenth of a cent narrower at a rounded 464
cents). Both the representation of 21:16 by an interval in its fairly
close neighborhood, and the particularities of its shading, open the
way for much exploration.
Christiaan Huygens, in his systematic and enthusiastic presentation of
31-EDO in 1691, also offered what may be one of the first statements
in the Western European musical tradition advocating the practical
value of septimal intervals such as 7:6, 7:4, and 12:7, and their use
in this "new harmonic cycle" as one of its ornaments.
The "meantone-style" strategy for obtaining septimal approximations in
Zest-24 can be seen as a curious tributary of this current of thought.
------------------------------------------------
4.4. Another road: 36-EDO and the 33-cent diesis
------------------------------------------------
Having surveyed the three main strategies for obtaining tempered
septimal intervals in Zest-24 and their operation also in other tuning
systems, we conclude by considering a variation on the theme of diesis
alterations: the amazingly accurate realization of our Aaron/AKJ
matrix available in 36-EDO.
To this point, we have considered equal temperaments which can be
generated from a single circle of fifths, whether wide as in 22-EDO
(Section 4.1) and 63-EDO (Section 4.2), or narrow as in 31-EDO
(Section 4.3).
Unlike these equally tempered systems, but like Zest-24, 36-EDO is
built not from a single circle of fifths, but from a series of closed
12-note circles -- three of them placed at 33-1/3 cents apart, and
each tempered in 12-EDO. In Zest-24, we have two unequally tempered
circles at 50.391 cents apart.
A characteristic distinguishing 36-EDO from the other systems we have
considered so far, just or tempered, is that a chain of 12 fifths
generates only one size of semitone, the equal semitone at 100 cents,
which might play either a diatonic or chromatic role depending on the
circumstances. This is different than 22-EDO with its diatonic and
chromatic semitones at 55/163 cents, the latter actually a large
neutral second; or 63-EDO at 76/133 cents; or, conversely, 31-EDO
where the diatonic semitone is the larger at 116/77 cents.
Another way of putting this is that the equal diatonic/chromatic
semitone of 36-EDO is equal to precisely half of the 200-cent major
second. In this system, as we are analyzing it, the three 12-note
circles are placed at intervals of 33.333 cents equal to precisely a
third of this 100-cent semitone, or a sixth of a tone. We shall here
consider this "sixthtone" as a very small diesis, although it could be
considered a very large comma, and may indeed be said to play both
roles.
In 36-EDO, as in 12-EDO, the fifths at 700 cents or 1.955 cents narrow
of 3:2 generate intervals in a 12-note circle not too far from
Pythagorean, but with major intervals somewhat smaller and minor
intervals larger. Thus the 400-cent major third might represent the
Pythagorean 81:64 at 408 cents, or rather more inaccurately the pental
or 5-based 5:4 at 386 cents; likewise, the 300-cent minor third is
fairly close to the Pythagorean 32:27 at 294 cents, and rather more
distant from 6:5 at 316 cents.
As it happens, addition or subtraction of a 33-cent diesis provides an
exquisitely accurate strategy for transforming these versatile
intervals into near-just septimal ones. Thus a 400-cent major third
plus the diesis yields a 433-cent version of 9:7 (435 cents); and a
300-cent minor third less a diesis, a virtually just 7:6 at 267 cents.
The 1000-cent minor seventh (near the Pythagorean 16:9, 996 cents)
less a diesis produces a 967-cent approximation of 7:4 (969 cents).
These and related diesis alterations populate our Aaron/AKJ lattice
with admirable simplicity, efficiency, and accuracy -- with a special
accent on the last.
! aaron-akj-36-edo.scl
!
36-tET/EDO version of aaron-akj superset
19
!
33.33333
66.66667
200.00000
233.33333
266.66667
333.33333
433.33333
500.00000
533.33333
633.33333
700.00000
766.66667
900.00000
933.33333
966.66667
1000.00000
1033.33333
1133.33333
2/1
(330) (1032) (534) (36)
98/81 ---- 49/27 ---- 49/36 ---- 49/48
Ab* Eb* Bb* F*
333 1033 533 33
| | | |
| | | |
(63) (765) (267) (969)
28/27 ---- 14/9 ------ 7/6 ------ 7/4
Gbd Dbd Abd Ebd
67 767 267 967
| | | |
| | | |
(996) (498) (0) (702) (204) (906)
16/9 ------ 4/3 ------ 1/1 ------ 3/2 ------ 9/8 ----- 27/16
Eb Bb F C G D
1000 500 0 700 200 900
| | | | |
| | | | |
(231) (933) (435) (1137) (639)
8/7 ----- 12/7 ------ 9/7 ----- 27/14 ---- 81/56
G* D* A* E* B*
233 933 433 1133 633
Here the symbols (*) and (d) are used to show a note raised or lowered
by a 33-cent or sixthtone diesis. Note that two of these dieses, or a
regular 100-cent semitone less a diesis, form a 67-cent "thirdtone"
which serves both mathematically and musically as a superb realization
of the just 28:27 at 63 cents.
Like the 39-cent diesis of 31-EDO, the 33-cent step here marks the
difference between a regular major or minor interval and its septimal
counterpart. Additionally, as in 31-EDO, it serves in effect as a
tempered 49:48 (36 cents), marking the difference between two septimal
ratios such as 8:7 and 7:6 -- here realized at 233.33 and 266.67 cents.
Interestingly, we find one location in Zest-24 where an interval of
around the same as the 33-cent diesis of 36-EDO shares these roles:
the step G*-Ab (8/7-7/6) at about 31.641 cents. In both systems, these
small steps mark the difference between a regular 300-cent minor third
and a near-just 7:6, and also represent the 49:48 step.
The relevant relationships, in JI and these two tempered systems, can
be illustrated by considering the lattice steps 8/7, 7/6, and 49/36.
36 267
|-- 49:48 --|---------------- 7:6 --------------------|
JI 8/7 ------- 7/6 ------------------------------------ 49/36
|--------------------- 343:288 -----------------------|
303
G* 33 Abd 267 Bb*
|-- 49:48 --|---------------- 7:6 --------------------|
36-EDO 8/7 ------- 7/6 ------------------------------------ 49/36
|--------------------- 343:288 -----------------------|
300
G* 32 Ab 268 Bb*
|-- 49:48 --|---------------- 7:6 --------------------|
Zest-24 8/7 ------- 7/6 ------------------------------------ 49/36
|--------------------- 343:288 -----------------------|
300
From a septimal JI perspective, the 300-cent third in either tempered
system might represent the ratio of 343:288 at 303 cents, equal to a
7:6 third at 267 cents plus a 49:48 diesis at 36 cents. In 36-EDO,
this division, available throughout the tuning and lattice, is
realized at G*-Abd-Bb* with its steps of 33/267 cents; as expected,
the regular 300-cent G*-Bb* less the 33-cent diesis G*-Abd yields
the virtually just 7:6 third Abd-Bb* at 267 cents.
In Zest-24, we happen to find the same pattern at G*-Ab-Bb*, where the
300-cent third G*-Bb* less the 32-cent diesis (as we might call it)
G*-Ab yields Ab-Bb* at 268 cents, not quite as accurate as 36-EDO, but
still within 1.5 cents of a just 7:6.
This parallel between the two systems has a charming serendipity. In
36-EDO, the regular 300-cent third and the 33-cent diesis between the
circles of fifths are both basic and pervasive constants, shaping the
system and its realization of septimal relationships. In Zest-24, the
300-cent minor third G-Bb is simply one of several sizes found in each
circle, while the 32-cent step G*-Ab results from taking an 82-cent
semitone G-Ab (again one of several sizes) and subtracting the 50-cent
diesis between the circles.
A progression using these near-7:6 thirds at 7/6-49/36 on the lattice
as realized in the two temperaments may give a quick view of some
melodic and vertical features:
36-EDO Zest-24
F* -- -67 -- Fd F* -- -50 -- Bb
(967) (700) (976) (709)
Ebd -- +200 -- Fd Eb -- +217 -- F
(700) (700) (709) (709)
Bb* -- -67 -- Bbd Bb* - - -50 -- Bb
(267) (0) (268) (0)
Abd -- +200 -- Bbd Ab -- +218 -- Bb
Vertically, while the 7:6 minor thirds are almost as accurate in
Zest-24 as in 36-EDO, the latter system is notably more accurate for
the other intervals: both fifths and fourths, and septimal ratios. Its
regular fifths at 700 cents, only about 2 cents from just, facilitates
near-perfection not only for the 267-cent thirds but the 967-cent
minor seventh, narrow by about the same amount as the fifth. In
Zest-24, while the lower minor third Ab-Bb* is a near-just 268 cents,
this interval plus the 708-cent fifth Bb*-F at 6 cents wide produce a
minor seventh Ab-F* at 976 cents or some 7 cents larger than 7:4.
Melodically also, 36-EDO with its consistent steps of 200/67 cents in
septimal resolutions such as this (a remissive resolution of a
12:14:18:21 quad), closely approximates the JI steps of 9:8 and 28:27
at 204/63 cents. In Zest-24, while the 217/218-cent steps representing
either 9:8 or 8:7 (as in 22-EDO) might be considered a fine nuance,
the 50-cent semitones are almost 13 cents smaller than 28:27. This is
a routine feature of the temperament, but contrasts with the more
ample 67 cents of 36-EDO.
As mentioned above, either the 33-cent diesis of 33-EDO or the 32-cent
step which parallels it in Zest-24 may represent a 49:48 step, These
radical realizations of a scale used by Robert Walker and others with
steps of 9:8, 8:7, and 49:48 (see Sections 4.1 and 4.3 above) nicely
dramatize this role:
36-EDO
(0) (231) (267) (498) (702) (933) (969) (1200)
F G* Abd Bb C D* D# F
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
0 233 267 500 700 933 967 1200
8:7 49:48 8:7 9:8 8:7 49:48 8:7
233 33 233 200 233 33 233
(231) (36) (231) (204) (231) (36) (231)
Zest-24
(0) (231) (267) (498) (702) (933) (969) (1200)
F G* Ab Bb C D* Eb F
1/1 8/7 7/6 4/3 3/2 12/7 7/4 2/1
0 243 274 492 696 938 983 1200
8:7 49:48 8:7 9:8 8:7 49:48 8:7
243 32 218 204 241 46 217
(231) (36) (231) (204) (231) (36) (231)
While either realization is impressive in its small diesis steps
serving as "semitones" (8/7-7/6, 12/7-7/4), the 36-EDO example is
additionally exemplary in its regularity. Each just ratio has one
mapping: 8:7 at 233 cents; 9:8 at 200 cents; and 49:48 at 33 cents.
In Zest-24, while 9:8 happens to be represented at a virtually just
204 cents, 8:7 may map to either 217/218 or 241/243 cents -- either
much more inaccurate than the 233 cents of 36-EDO. The diesis steps
likewise are 32 cents at G*-Ab, but 46 cents at D*-Eb. These
variations might be deemed pleasant and engaging; but it is clearly
36-EDO which more closely approximates the structure of the original
JI matrix.
In this brief appraisal of 36-EDO, another point may be worth making:
accurate representations of ratios such as 12:7 and 7:4 are available
at every location in the complete temperament, and at every point
where these intervals occur in the Aaron/AKJ lattice. In Zest-24, one
or both of these ratios may be represented at some locations but not
others -- sometimes with results rather different than those of
36-EDO, but rather similar to those of another system based on 12-EDO
circles.
For example, with the lattice 1/1 (F in either temperament) as the
lowest note, we can build either a 14:18:21:24 major sixth quad
or 12:14:18:21 minor seventh quad in either system,
here shown resolving intensively:
36-EDO Zest-24
D* -- +67 -- Eb D* -- +46 -- Eb
(933) (1200) (938) (1200)
C -- -200 -- Bb C -- -204 -- Bb
(700) (700) (696) (709)
A* -- +67 -- Bb A* -- +59 -- Bb
(433) (700) (434) (709)
F -- -200 -- Eb F -- -217 -- Eb
Ebd -- -200 -- Dbd Eb -- -217 -- Db
(967) (700) (983) (696)
C -- +67 -- Dbd C -- +70 -- Db
(700) (700) (696) (696)
Abd -- -200 -- Gbd Ab -- -204 -- Gb
(267) (0) (274) (0)
F -- +67 -- Gbd F -- +70 -- Gb
In the first resolution, the lower 9:7 major third has an almost
identical size in the two systems: 433.333 cents in 36-EDO, and
433.594 cents in Zest-24; the spellings, under the notations followed
here, are also identical. Here the virtually just 12:7 at 933 cents in
36-EDO contrasts with the rather wider Zest-24 interval of 938 cents.
While the 67-cent semitones of the former system are used as expected,
the latter again shows its penchant for very small semitones at 59
cents and 46 cents.
In the second, the melodic semitone steps of 67 cents and 70 cents
respectively are close to ideal in both systems from the viewpoint of
the just 28:27. Vertically, while the septimal intervals generally are
much more accurate in 36-EDO, the characteristic Zest-24 realizations
of 7:6 at 287 cents (C-Eb), and 9:7 at 422 cents (Ab-C), underscore
the variability of the temperament. Here the spellings differ, with
all Zest-24 intervals generated in a single chain of fifths, rather
than by diesis alterations.
Shifting to the lattice location of 8/7, however, we encounter the
usual representation of a 12:14:18:21 quad in 36-EDO but a less
conventional approximation in Zest-24:
36-EDO Zest-24
F -- -67 -- E* F -- -70 -- E*
(967) (700) (957) (696)
D* -- +200 -- E* D* -- +192 -- E*
(700) (700) (695) (696)
Bb -- -67 -- A* Bb -- -59 -- A*
(267) (0) (250) (0)
G* -- +200 -- A* G* -- +191 -- A*
Here the note spellings are identical, but the musical effects
distinct. In 36-EDO, G*-Bb as usual is equal to a 300-cent third less
a 33-cent diesis or 267 cents. In Zest-24, G*-Bb is equal to a
300-cent third less a 50 cent diesis -- or 250 cents, much closer to
15:13 (248 cents) or 37:32 (251 cents) than to 7:6.
The greater accuracy of 36-EDO from a septimal point of view in only
part of the story. Moving outside the strict scope of our septimal
lattice, let us consider these three-voice progressions in the two
systems, the first involving a 12:7 major sixth expanding to an octave
intensively, and the second a 7:4 minor seventh contracting to a
fifth remissively.
36-EDO Zest-24
Bbd -- +67 -- Bb* Bb -- +50 -- Bb*
(933) (1200) (946) (1200)
G* -- -200 -- F* G* -- -192 -- F*
(700) (700) (696) (708)
C* -- -200 -- Bb* C* -- -204 -- Bb*
Bb -- -67 -- A* Bb -- -59 -- A*
(967) (700) (946) (696)
G* -- +200 -- D* G* -- +191 -- A*
(700) (700) (696) (696)
C* -- +200 -- D* C* -- +191 -- D*
In 36-EDO, the major sixth sonority C*-G*-Bbd at 0-700-933 cents and
the minor seventh sonority C*-G*-Bb at 0-700-967 cents have, of
course, distinct spellings and sounds, with the highest voice at a
33-cent diesis higher in the second sonority, thus representing the
distinction of 49:48 between 12:7 in the first sonority (14:21:24 at
0-702-933 cents in JI)) and 7:4 in the second (4:6:7 at 0-702-969
cents). In the first resolution we have an outer 933-cent sixth and
upper 233-cent major second expanding to octave and fourth; in the
second, we have a 967-cent seventh and 267-cent third contracting to
fifth and unison, as expected.
In Zest-24, however, the same spellings and notes are used -- C*-G*-Bb
(0-696-946 cents) for both progressions! With reference to C*, Bb at
946 cents is the best representation available for either 12:7 or 7:4;
and likewise, above G*, the best representation of either 8:7 or 7:6.
Thus the outer C*-Bb at 946 cents acts like a very large major sixth
expanding to an octave in the first progression, and a very small
minor seventh contracting to a fifth in the second; and G*-Bb at 250
cents as either a major second expanding to a fourth or a minor third
contracting to a unison.
In its regions favoring this kind of interseptimal color with "dual
roles" for intervals around 250 or 950 cents, Zest-24 resembles
another system of the same size with two 12-note circles at a distance
of 50 cents apart -- 24-EDO.
24-EDO Zest-24
Bb -- +50 -- Bb* Bb -- +50 -- Bb*
(950) (1200) (946) (1200)
G* -- -200 -- Eb* G* -- -192 -- F*
(700) (700) (696) (708)
C* -- -200 -- Bb* C* -- -204 -- Bb*
Bb -- -50 -- A* Bb -- -59 -- A*
(950) (700) (946) (696)
G* -- +200 -- D* G* -- +191 -- A*
(700) (700) (696) (696)
C* -- +200 -- D* C* -- +191 -- D*
In 24-EDO, as in Zest-24, the 50-cent diesis often serves as a small
semitone in directed progressions -- a role also served in
interseptimal contexts by other steps in the latter systems such as
the 59-cent semitone in the second example. The spellings and their
logic in the two systems are similar, with some varying melodic
nuances and vertical shadings. Thus the 950-cent interval of 24-EDO
might represent 64:37 (949 cents) or 26:15 (952 cents); while its
946-cent neighbor in Zest-24 is almost identical to 19:11. In either
temperament, G*-Bb is precisely 250 cents.
Just as 36-EDO offers pervasively accurate septimal approximations, so
24-EDO offers these charming interseptimal sonorities predictably at
all locations. In Zest-24, one has a world whose inconstant atmosphere
offers a sampling of both these climates, each with its own virtues
and possibilities.
This observation applies not only to basic septimal intervals or their
interseptimal alternatives, but also to neutral or semi-neutral
categories. Thus in 33-EDO, 98/81 in our lattice at 330 cents (F-Ab*)
is represented as 333 cents -- the virtually just 7:6 at 267 cents
plus the slightly wide 28:27 at 67 cents. This is a fine semi-neutral
or supraminor third which might also represent 63:52 (332 cents),
40:33 (333 cents), or 17:14 (336 cents). This interval is formed by a
usual 300-cent minor third (e.g. F-Ab) plus the 33-cent diesis
In Zest-24 an almost identical size at a rounded 333 cents is
available, but at different locations (in the lattice, D*-Gb or A*-Db)
and through a different strategy of generation. Here this supraminor
third is equal to a meantone major third at 383 cents less the 50-cent
diesis, a relationship which will become clearer if we use meantone
spellings (D*-F#, A*-C#).
In Zest-24, as in 36-EDO, the sonority 9/7-14/9-7/6 (0-330-1032 cents
in JI) features this fine 333-cent small neutral third:
36-EDO Zest-24
Dbd/C#d -- -133 -- B* Db/C# -- -141 -- B*
(1033) (700) (1029) (695)
Gbd/F#d -- -133 -- E* Gb/F# -- -141 -- E*
(333) (0) (333) (0)
D* -- +200 -- E* D* -- +192 -- E*
In 36-EDO, the 333-cent third is some 3 cents wide of 98:81, while the
1033-cent seventh is only about 1.5 cents wide of 49:27, and also very
close to 20:11 (1035 cents). In Zest-24, the almost identical 333-cent
third plus a narrow 696-cent fifth produce a smaller 1029-cent seventh
about three cents smaller than 49:27, and curiously almost identical
to the interval of 6/7 octave (1028.571 cents) found in 7-EDO and its
equally tempered relatives with sizes of 7n-EDO.
In 36-EDO, the small neutral second steps at 133 cents, about midway
between 14:13 and 13:12, are equal to twice the tempered 28:27 at 67
cents; in JI, the corresponding smaller step of 784:729 or 126 cents
is likewise equal to twice a just 28:27 at 63 cents (Section 3.2).
Each upper voice descends by this 133-cent step while the lowest
ascends by a 200-cent tone. In Zest-24, the whole-tone step is a
somewhat smaller 192 cents (as in Zarlino's 2/7-comma meantone), and
the neutral seconds larger at 141 cents -- very close, as it happens,
to a different septimal ratio, 243:224 (141 cents).
For both systems, I have included some alternative spellings, which
draw on the same basic equivalences as 12-EDO (e.g. Db=C#, Gb=F#).
Note that the 133-cent step of 36-EDO is equal to a regular 100-cent
semitone plus a 33-cent diesis (e.g. G-Ab*), or a 200-cent tone less
two such dieses (e.g. C#d-B*, with the higher note of the tone C#-B
lowered and the lower note raised by a diesis). The Zest-24 steps of
141/142 cents are formed from a meantone major second of 191/192 cents
less a 50-cent diesis.
Suppose we transpose this same progression up a major second so that
the opening neutral sonority becomes 27/14-7/6-7/4, again 0-330-1032
cents in JI and 0-333-1033 in 36-EDO. To keep the resolution within
our Aaron/AKJ lattice, this time we use ascending neutral second and
descending whole-tone steps, cadencing on the fifth 28/27-14/9.
36-EDO Zest-24
Ebd/D#d -- -200 -- C#d/Dbd Eb/D# -- -217 -- C#/Db
(1033) (700) (1054) (696)
Abd/G#d -- -200 -- F#d/Gbd Ab/G# -- -204 -- F#/Gb
(333) (0) (345) (0)
E* -- +133 -- F#d/Gbd E* -- +141 -- F#/Gb
While the vertical intervals and melodic steps in 36-EDO are identical
to those of the last example, in Zest-24 the vertical color moves into
the central neutral range. Thus we have a neutral third at 345 cents,
not far from 11:9 (347 cents), and a neutral seventh at 1054 cents,
about 5 cents wide of 11:6 (1049 cents). The former interval is about
15 cents wide of a just 98:81, and the latter 22 cents wide of 49:27.
Melodically, the neutral second step is of the same 141/142-cent type
used in the last example, while the whole-tone motions are larger at
204 and 217 cents, the usual neomedieval sizes in a single chain of
fifths. Note that the neutral third and seventh are formed by
subtracting a 50-cent diesis from the major third E-G#/Ab at 395 cents
and the major seventh E-D#/Eb at 1104 cents -- respectively about 5
cents smaller and 4 cents larger than 12-EDO sizes.
It is interesting to compare this last neutral progression in Zest-24
to a version in 24-EDO:
24-EDO Zest-24
Eb/D# -- -200 -- C#/Db Eb/D# -- -217 -- C#/Db
(1050) (700) (1054) (696)
Ab/G# -- -200 -- F#/Gb Ab/G# -- -204 -- F#/Gb
(350) (0) (345) (0)
E* -- +150 -- F#/Gb E* -- +141 -- F#/Gb
In both systems, the melodic neutral second step and the vertical
neutral third and seventh of this progression are each equal to a
major interval of the same category less a 50-cent diesis. For 24-EDO,
where the relevant major categories always have sizes of 200, 400, and
1100 cents, this means uniform sizes of 150, 350, and 1050 cents
(nicely approximating 12:11, 11:9, and 11:6). Furthermore, since
subtracting a diesis from a major interval has an identical effect to
adding a diesis to a minor interval of the same category (a minor
second, third, or seventh at 100, 300, or 1000 cents), these neutral
sizes are uniform from any position on either 12-note circle.
In Zest-24, this example places us in a region where the vertical
neutral intervals have sizes rather close to 24-EDO at 345 and 1054
cents, while the melodic step at 141 cents is significantly smaller,
close to 13:12 (139 cents) rather than 12:11 (151 cents). As we move
around the system these sizes can change kaleidoscopically -- as do
the sizes of major or minor intervals within a single chain of fifths
which are altered by a diesis to produce these neutral or semi-neutral
categories.
Each system has its own character and style. Thus 36-EDO uniformly
yields near-just septimal intervals, and also small neutral intervals
(e.g. 133, 333, 1033 cents); and 24-EDO, interseptimal intervals about
midway between 8:7 and 7:6, or 12:7 and 7:4 (250 and 950 cents), as
well as central neutral intervals (e.g. 150, 350, 1050 cents). In
Zest-24, sizes and colors may resemble those of either equally
tempered system, as well as a variety of intermediate or outlying
shades as we move about the 12-note circles.
The question of 21:16 (471 cents) also illustrates these contrasts. In
36-EDO, this septimal ratio is represented by a 500-cent fourth less a
33-cent diesis, or 466.67 cents, about 4 cents narrow of just. This
tempered size of 467 cents is also equal to two 233-cent or large
major seconds each representing 8:7 -- and about 4 cents larger than a
just 64:49 formed from two pure 8:7 steps (462 cents). In this
situation, the 33-cent diesis can represent either 64:63 (27 cents) or
49:48 -- the difference between a 4:3 fourth (498 cents) and 21:16 or
64:49 respectively.
Note that the 467-cent approximation of 21:16 or 64:49 is equal to
either a 500-cent fourth less diesis or a 433-cent septimal major
third plus diesis (e.g. G*-C or Gbd-Bb* on the 36-EDO lattice). As the
latter spelling shows, 467 cents is also equal to a regular 400-cent
third (e.g. Gb-Bb) plus two dieses, thus Gbd-Bb*.
In 24-EDO, the closest size to 21:16 is the quite different interval
of 450 cents, equal either to a 400-cent major third plus a 50-cent
diesis or a 500-cent fourth less the diesis (F-A* or F*-Bb). This is
an interseptimal interval in the middle range between the septimal
ratios of 9:7 and 21:16, i.e. between a septimal major third and a
narrow fourth. It might represent 22:17 (446 cents) or 13:10 (454
cents), for example, with a rather different color than anything found
in our 19-note Aaron/AKJ lattice in JI (with 9:7 at 435 cents or 64:49
at 462 cents as the closest values) or 36-EDO (with 433 or 467 cents).
In Zest-24, we find that the representation of a 21:16 (one 7:6 third
up on the lattice and two 3:2 fifths to the right) varies with
position. Thus we find Gb-Bb* (28/27-49/36) at 472 cents or near-just;
and Db-F* (14/9-49/48) at a wider but still kindred 484 cents. Moving
down from the top to the middle portion of the lattice, we see that
Eb-Ab (16/9-7/6) and Bb-Eb (4/3-7/4) are identical to usual fourths at
491 cents -- a size and situation like that of 22-EDO (Section 4.1).
In the bottom portion of the lattice, G*-C (8/7-3/2) at 454 cents or
D*-G (12/7-9/8) at 455 cents is close to 24-EDO, and offers a
virtually just 13:10.
This survey of the Zest-24 lattice also shows how getting an accurate
approximation of 21:16 requires a rather specific strategy: adding a
50-cent diesis to a major third at 422 or 434 cents, yielding 472 or
484 cents. To add a 50-cent diesis, we must start with a major third
on the _lower_ circle or keyboard, where we have two 422-cent thirds
(Gb-Bb and Ab-C) and one 434-cent third (Db-F). Thus our strategy
applies to only 3 of the 24 positions of the tuning -- two of which
represent 21:16 relationships on our 19-note lattice.
In contrast, of course, 36-EDO yields its 467-cent version of 21:16
predictably and uniformly throughout the lattice; while 24-EDO offers
its quite different but also enchanting 450-cent interval and other
interseptimal colors. With Zest-24, we get something of both worlds.
-----------------------
5. Conclusion: An envoi
-----------------------
While this article has revolved around two just tunings of Gene Ward
Smith and their merger to create larger supersets, I would suggest
that it may also have touched on a theme very creatively developed by
the composer for whom these tunings are named: Aaron K. Johnson
himself.
A fascinating attraction of Aaron's software utilities for scoring
compositions in equal temperaments or JI systems is the use of
"Gaussian rhythm," in which a MIDI file generated from a score can be
"humanized" by introducing very small variations in the timing of
notes according to the probabilities of a normal curve. This algorithm
has an impressively subtle (or subtly impressive) effect in his
realization, for example, of a famous chromatic setting for the
harpsichord or virginals by the English composer John Bull around
1600, here tuned in 19-EDO:
Indeed, so successful was this tool that when Aaron introduced another
new piece, musicians asked if it had likewise been "humanized" through
Gaussian rhythm -- to which he replied that it likely sounded
humanized because it had been played at the keyboard by a human,
namely himself.
Might mapping a JI structure like the Aaron/AKJ lattice to an
irregular temperament like Zest-24 similarly involve a certain kind of
"randomization" -- using intonational colors now rather closely
related to those of their just equivalents, and now quite tangibly
different? I conclude with this open question.
Margo Schulter
mschulter@calweb.com
27 January 2008
Revised 18 June 2008