--------------------------------
Erv Wilson's Rast/Bayyati Matrix
and Variations in Zest-24
--------------------------------
by Margo Schulter
This article explores a fascinating tuning system developed and mapped
to a Bosanquet generalized keyboard by Erv Wilson, and here adapted to
the tempered scheme known as Zest-24 (Zarlino Extraordinaire Spectrum
Temperament). While this temperament is described in other papers, I
should briefly explain that it is based on a modified version of
Zarlino's 2/7-comma meantone tuning (1558), which seems to be the
first known regular temperament described in precise mathematical
terms in the Western European literature.
Zest-24 consists of two 12-note circles of fifths, each mapped in my
implementation to a conventional 12-note keyboard -- although Wilson's
work reminds us that a mapping to the Bosanquet keyboard might be much
more elegant! Each circle has eight fifths (F-C#) in Zarlino's regular
2/7-comma (695.810 cents, or 6.145 cents narrow of 3:2), and the other
four fifths equally wide (708.379 cents, or 6.424 cents wide of 3:2).
The two circles are placed at an interval of 50.276 cents, the
enharmonic diesis of Zarlino's regular 2/7-comma tuning.
The result is a 24-note system in which each step has a fifth and
fourth reasonably near 4:3 and 3:2 (albeit impure in one direction or
the other by some six cents, or sometimes indeed seven cents in an
implementation on a synthesizer tuning in 1/1024 octave steps), and at
the same time both the element of irregular temperament and the
possibility of combining notes from the two circles yield a rich
spectrum of intervals.
Erv's Wilson's Rast/Bayyati matrix, a tuning system based on chains of
neutral thirds, provides an opportunity to explore some aspects of
this spectrum. His presentation of this system is most generously made
available through the Anaphorian Embassy site:
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1. Some background: Rast and Bayyati
-------------------------------------
In Erv Wilson's _Rast/Bayyati Matrix_ (1992), he develops a system of
scale generation based on a chain of neutral thirds at 27:22 and 11:9,
or about 355 and 347 cents, the sizes featured in Mansur Zalzal's
famous lute tuning as described by al-Farabi. This scale demonstrates
a feature also central to Wilson's tuning: these two slightly unequal
sizes of neutral thirds together form a pure 3:2 fifth. In Near
Eastern theory, Zalzal's scale can be analyzed as having two conjunct
tetrachords with steps of a whole tone followed by two neutral
seconds, a tetrachord pattern known as Rast, as in Wilson's title.
Rast Rast
|------------------------|------------------------| tone |
1/1 9/8 27/22 4/3 3/2 11/8 16/9 2/1
0 204 355 498 702 853 996 1200
9:8 12:11 88:81 9:8 12:11 88:81 9:8
204 151 143 204 151 143 204
In modern Arabic music, the term Rast also refers to the principal
notes of a maqam or mode with two _disjunct_ Rast tetrachords with the
octave completed by a central whole tone between them, rather than by
an upper tone as in the conjunct tetrachord interpretation of Zalzal's
scale just given. Here we assume that steps in each tetrachord are
tuned as in the Zalzal/al-Farabi version:
Rast Rast
|------------------------| tone |-------------------------|
1/1 9/8 27/22 4/3 3/2 27/16 81/44 2/1
0 204 355 498 702 906 1057 1200
9:8 12:11 88:81 9:8 9:8 12:11 88:81
204 151 143 204 204 151 143
A Rast tetrachord thus has a whole tone in the lowest position
followed by two neutral seconds. The _Bayyati_ of Wilson's title
refers to a tetrachord with a different arrangement of these same
types of steps: two neutral seconds followed by a whole tone.
This tetrachord occurs both above and below the final or resting note
of the Arabic mode known as Maqam Bayyati, with the lower neutral
second of the Bayyati tetrachord above the final typically somewhat
smaller than the following one. Scott Marcus suggests a range for this
smaller step of about 135-145 cents; thus a tuning at 13:12 (139 cents)
would be near the lower end of the usual range, while the 88:81 shown
here at 143 cents is close to the upper end. The version below keeps
the 12:11 and 88:81 steps featured in Zalzal's tuning and also, as we
shall see, Wilson's matrix -- but reverses their order for this
tetrachord. In contrast, the Bayyati tetrachord below the final is
often understood to have the larger neutral second below the smaller,
so that the step a neutral third below the final is at a distance of
slightly less than 350 cents. Here this distinction is realized very
subtly by placing the larger 12:11 below the smaller 88:81 in the usual
Zalzalian order, with this neutral third at 11:9 or 347 cents. The
upper tetrachord is of a different type, Nahawand (tone-semitone-tone),
here realized with usual Pythagorean steps:
Bayyati Bayyati Nahawand
|-------------------|---------------------|----------------------| tone |
3/4 9/11 8/9 1/1 88/81 32/27 4/3 3/2 128/81 16/9 2/1
-498 -347 -204 0 143 294 498 702 792 996 1200
12:11 88:81 9:8 88:81 12:11 9:8 9:8 256:243 9:8 9:8
151 143 204 143 151 204 204 90 204 204
The use of inverse ratios and negative values in cents below the final
is meant to underscore a feature of maqam music: the frequent use of
patterns other than simple repetition at the octave. Thus in a
"textbook" form, as here, Bayyati has a minor sixth _above_ the final
(128/81), but a neutral third (9/11) below it. In the course of a
performance in Bayyati, the upper minor sixth sometimes shifts to a
neutral sixth, which Marcus explains can be viewed as a transposition
of the tetrachord below the final to the octave above.
When this inflection or transposition does occur, the neutral sixth
above the final is according to Marcus typically quite wide in
comparison to the 850-cent size of 24-tET or 24-EDO (a 24-tone "equal
temperament" or "equal division of the octave," thus tET or EDO), a
system often used in modern Arabic theory although performers and
theorists attuned to traditional techniques understand that steps and
intervals are in practice more flexible and may vary depending on
factors such as the maqam and the musical context. One possible ratio,
for example, might be 64/39, about 858 cents -- my guess; Marcus does
not attempt, at least in the account I have read, to quantify this
nuance regarding the inflected sixth step above the final.
My purpose of writing at this length is not only to explain the terms
Rast and Bayyati in the context of Wilson's system, but also to
emphasize the fluid nature of modality in maqam music. Thus not only
are some of the degrees of a maqam such as Bayyati subject to
expressive inflections, but it is routine for a piece in a given maqam
to mutate or "modulate" from time to time into other maqamat (the
plural for _maqam_). The modal mutations and inflections often
practiced in European medieval and Renaissance music, and also the
later use of modulation between major and minor keys, are analogous
aspects of another world musical tradition; and such changes of mode
are also common in the music of Southeast Asia, for example.
-------------------------------------------------
2. Wilson's 24-note tuning and a Zest-24 version
-------------------------------------------------
Returning to Wilson's paper, we find that his Rast/Bayyati matrix can
be realized in tunings of 7, 10, 17, or 24 notes, all featuring a
chain of 27:22 and 11:9 neutral thirds. Let us consider the 24-note
system, and its closest analogue in the Zest-24 temperament, found
starting on the step Bb* (the Bb key on the upper 12-note manual,
about 1046 cents with reference to C on the lower keyboard). Here
notes are assigned numbers from 0 for the "1/1" to 24 for the 2/1
octave, an approach like that of Manuel Op de Coul's outstanding Scala
program available for free on the Internet, and some of Wilson's
diagrams also. One might prudently add that Zest-24 accidental
spellings such as B-Gb or B-F# are equivalent and interchangeable.
See also Appendix 1 for Scala scale files for both systems.
Wilson's Rast/Bayyati Matrix Zest-24 (Bb*-Bb*)
1/1 0.000 # 0 Bb* 0.000 # 0
8192/8019 36.952 # 1 B 32.966 # 1
256/243 90.224 # 2 B* 83.241 # 2
12/11 150.637 # 3 C 153.914 # 3
9/8 203.910 # 4 C* 204.190 # 4
1024/891 240.862 # 5 Db 224.586 # 5
32/27 294.135 # 6 Db* 274.862 # 6
27/22 354.547 # 7 D 345.535 # 7
8192/6561 384.360 # 8 D* 395.810 # 8
128/99 444.772 # 9 Eb 441.345 # 9
4/3 498.045 #10 Eb* 491.621 #10
243/176 558.457 #11 E 537.155 #11
1024/729 588.270 #12 E* 587.431 #12
16/11 648.682 #13 F 658.103 #13
3/2 701.955 #14 F* 708.379 #14
4096/2673 738.907 #15 Gb 728.776 #15
128/81 792.180 #16 Gb* 779.052 #16
18/11 852.592 #17 G 849.724 #17
27/16 905.865 #18 G* 900.000 #18
512/297 942.817 #19 Ab 932.966 #19
16/9 996.090 #20 Ab* 983.241 #20
81/44 1056.502 #21 A 1041.345 #21
4096/2187 1086.315 #22 A* 1091.621 #22
64/33 1146.727 #23 Bb 1149.724 #23
2/1 1200.000 #24 Bb* 1200.000 #24
Note that some steps like 9/8 (#4) match almost exactly in the
tempered version. Others such as 128/99 (#9), 18/11 (#17), and 27/16
(#18) are reasonably close, with some fine difference of shading or
color. Not infrequently, however, we find differences on the order of
10-20 cents that can considerably alter the character of an interval.
Thus Wilson's 1024/891 (#5) at a rounded 241 cents beautifully
exemplifies the lower range of what I term the "interseptimal" region
between the large 8:7 tone and the small 7:6 minor third (respectively
231 and 267 cents). This intriguing ratio might serve as a very large
major second -- or a very small minor third, for example in a
cadential progression contracting to a unison. In this 24-note just
tuning, for example, we might use this three-voice progression where
the lower two voices resolve in this fashion, with some voices moving
by a 9:8 tone and others ascending or descending by the striking
diesis of 8192/8019, about 37 cents:
3/2 -- +37 -- 4096/2073 3/2 -- +204 27/16
1024/891 -- -204 -- 8192/8019 1024/819 -- -37 -- 9/8
1/1 -- +37 -- 8192/8019 or 1/1 -- +204 -- 9/8
702 739 702 906
241 37 241 204
0 37 0 204
In Zest-24, however, we have at step #5 instead a 225-cent interval, a
very reasonable approximation of 8:7 (231 cents), but quite distinct
from the special interseptimal qualities of 1024/891. As it happens,
we can find a near-just approximation of the 1024:891 -- but at other
locations, for example C-D* (241.897 cents). These permit a somewhat
distinct nuance of the just progressions above:
G 850 -- +50 -- G* 900 G 850 -- +192 -- A 1041
D* 396 -- -192 -- C* 204 D* 396 -- -50 -- D 346
C 154 -- +50 -- C* 204 or C 154 -- +192 -- D 346
Apart from such matters as the temperament of the fifth and also the
major seconds -- sometimes virtually a just 9:8, but often the
narrower 192-cent step of Zarlino's regular 2/7-comma meantone (a
facet of meantone not necessarily to be deemed advantageous, as Kraig
Grady has noted) -- a distinction between the just and tempered
versions of this progression is the size of the diesis or small
semitone step. In the just tuning it is 8192:8019, or 37 cents -- with
special melodic qualities comparable to those of the 49:48 diesis in a
tuning based on primes 2-3-7 (about 35.697 cents), e.g. 8/7-7/6 or
12/7-7/4.
In Zest-24, this same step is realized as a 50-cent semitone or diesis
(C-C* or D*-D), a less "special" interval which routinely serves as a
usual semitone in emulations of 2-3-7 JI where we might expect the
larger 28:27 semitone or thirdtone of Archytas (62.961 cents). Thus
each system has its own shadings, but those of Wilson's matrix are
especially notable.
It follows that our Zest-24 tuning on Bb* should be considered not so
much a tempered "implementation" of Wilson's system but rather one
possible variation on it. One of the important differences concerns
the prevailing sizes of the neutral thirds.
-----------------------------------------------
3. Zest-24 Neutral Thirds: A Shur/Nava matrix?
-----------------------------------------------
In Wilson's matrix, the 27:22 and 11:9 neutral thirds (355/347 cents)
define the contours of the tuning and provide fine ratios for much
maqam music lending itself to an intonational color like that of
Zalzal's scale. While Zest-24 can sometimes provide this kind of
division of the fifth into two slightly unequal neutral thirds, as at
Eb*-G-B* (358/350) or Ab-B*-Eb (350/358), typically the division is
more decidedly unequal.
The following table gives an overview of the situation, with each row
showing a chain of four intervals of seven tuning steps each; these
7-step intervals usually, although not always, are within the range of
what may broadly be termed neutral thirds (say 330-372 cents). We
might take the central part of this range, featured in Wilson's
tuning, as that roughly between 39:32 and 16:13, or 342-360 cents; the
outlying regions often featured in Zest-24 (330-342 and 360-372 cents)
could be called "semi-neutral" or "supraminor/submajor." Ratios near
the center of these supraminor/submajor zones are 17:14 and 21:17 at
336/366 cents.
As the table shows, the four 7-step intervals on each row add up to 28
steps, or a usual 4-step major second of some kind plus an octave; so
the next row starts at a major second or 4 steps higher. A number in
parentheses is added at the end of each row showing the size of the
next neutral third in the chain or circle. At the end of the first
row, for example, we see that the last third A-C* is 363 cents, and
that the next, starting the following row, will be 333 cents (C*-E).
This permits us more quickly to survey the division of fifths such as
A-E whose outer notes appear on different rows -- thus A-C*-E, 363/333
cents.
Zest 24: Cycle of Neutral Thirds and Major Seconds
0 346 708 1041 204
Bb* 346 D 363 F* 333 A 363 C* (333)
#0 #7 #14 #21 #4
204 537 900 33 396
C* 333 E 363 G* 333 B 363 D* (333)
#4 #11 #18 #1 #8
396 729 1092 225 587
D* 333 Gb 363 A* 333 Db 363 E* (346)
#8 #15 #22 #5 #12
587 933 83 441 779
E* 346 Ab 350 B* 358 Eb 338 Gb* (371)
#12 #19 #2 #9 #16
779 1150 275 658 983
Gb* 371 Bb 325 Db* 383 F 325 Ab* (371)
#16 #23 #6 #13 #20
983 154 492 850 1200
Ab* 371 C 338 Eb* 358 G 350 Bb* (346)
#20 #3 #10 #17 #24
As already noted, a few divisions are similar to Wilson's Rast/Bayyati
matrix (346/350, 350/346, 358/350, 350/358), but many favor the
outlying supraminor/submajor region (333/363, 363/333, 371/338), with
some having a quality somewhat intermediate (346/363, 363/346,
338/358, 358/338), with one of the thirds within the central neutral
region but the other in the supraminor or submajor zone. In the first
"near-equal" type of division, the two thirds are within 10-15 cents
of each other in size, while in the typical supraminor/submajor
divisions they differ by about 25-40 cents, giving them a degree of
"polarity." The intermediate shadings involve differences of about
17-20 cents, comparable to those of 39:32 and 16:13 (342/360) as rough
delimiters of the central neutral zone.
There are also, as appears in the fifth row of the table from the top,
a few locations where sonorities formed from two 7-step thirds overlap
with the 5-limit realm of meantone oriented to ratios of 5:4 and 6:5
(386/316 cents), and exemplified in Zest-24 as in Zarlino's regular
2/7-comma temperament by thirds of 383/313 cents. Thus Bb-Db*-F and
Db*-F-Ab* (325/383 or 383/325 cents) are compable to the 5-limit
approximations of 22-EDO (327/382 or 382/327 cents). The adjacent
sonorities to either side on this row, Gb*-Bb-Db* and F-Ab*-C at
371/325 and 325/371 cents, might be considered as transitional between
a classic meantone color like Zarlino's and the supraminor/submajor
or neutral realms favored by most 7-step thirds in Zest-24, and on
which we now focus.
While central neutral thirds like those of Wilson's Rast/Bayyati
matrix are certainly in evidence, the pervasive role of supraminor and
submajor flavors suggests an affinity to Persian music, which often
favors small neutral thirds in the general range of 335 cents, and
thus often large neutral thirds around 365 cents if we assume a usual
fifth at or close to 3:2 (702 cents). As recorded in the Scala scale
archive, Hormoz Farhat (persian-far.scl) suggests a typical division
of 335/365 cents, and Dariush Anooshfar (persian.scl) a very similar
division based on the just ratios of 243:200 and 100:81, or 337/365
cents (more precisely 337.148/364.807 cents) with a just 3:2 fifth.
Given this affinity, I am tempted to call the Zest-24 adaptation of
Wilson's tuning a "Shur/Nava matrix," Shur and Nava being two of the
modal families in the modern Persian dastgah system, where each family
or dastgah has a set of characteristic melodic themes or patterns
serving as a basic for artful elaboration and improvisation. Each
theme or _gusheh_ may have its own modal nuances, with a set of
distinct but related gusheh-ha (the plural form) making up a dastgah
family. In turn, the full range of dastgah-ha or families, each with
its own set of gusheh-ha, makes up the _radif_ or repertory of a
performer or ensemble.
Thus just as a "textbook" version of the principal notes in an Arabic
or Turkish maqam implies to a traditional performer a subtle set of
inflections and mutations to other maqamat which assume the
availability of other melodic steps in apt locations, so the principal
or introductory theme of a Persian dastgah like Shur or Nava, which we
are about to meet, is in classic practice a gateway to many gusheh-ha
calling for a variety of inflections.
These considerations lead to an important caution if one is attempting
to realize or approach a classic maqam or dastgah style: one must
carefully consider not only the usually quoted notes of a maqam or
introductory gusheh, but also likely inflections and modal mutations.
This suggests that if one is going to make such an attempt in Zest-24,
it is important to choose one's transpositions carefully. I have
identified a possible 17-note set starting on Eb which is rather like
Anooshfar's tuning for the tar in persian.scl, but of course differs
not only in its nuances but in the quite heavy tempering of the fifths.
For a brief description of this tuning set, see Appendix 2.
To illustrate the "Shur/Nava matrix" concept, I will give an example
of each dastgah in Zest-24, or, more precisely, a version of the main
notes for the introductory _gusheh_, known as the _daramad_. We begin
with Shur, the most popular of the modern Persian dastgah-ha, here
with the final or resting note placed on Bb, taking note of the seven
"usual" notes in an often quoted form of the mode and also two common
inflections. Step sizes in cents are shown among the seven usual
notes, and also between an inflected degree and its neighbors
(Eb-E*-Gb, 146/141; F-Gb*-Ab, 121/154).
829
Gb*
121 154
Bb B* Db Eb F Gb Ab Bb
0 134 275 492 708 779 983 1200
134 141 217 217 71 204 217
146 141
E*
637
The "usual" notes divide the lower tetrachord into two neutral seconds
making up a minor third (Bb-B*-Db) plus a whole tone (Db-Eb), a
pattern similar to that of the Arabic Bayyati, followed by the whole
tone Eb-F and then an upper tetrachord with a minor second followed by
two tones (F-Gb-Ab-Bb).
Sometimes, however, as an alternative to the minor sixth degree Gb at
779 cents above the final, a neutral sixth may be used (as in
Bayyati), here Gb* forming a fine supraminor or small neutral interval
of about 829 cents, close to 21:13 (830 cents). This alternative sixth
gives also gives rise to a small neutral third above the fourth degree
(Eb-Gb*) at 338 cents, almost identical to Anooshfar's 243:200.
The fifth degree is likewise fluid, with a lowered form here realized
by E* often used as the top note in a descending passage moving toward
the final. It may likewise occur in a descending passage as a note
following the regular sixth degree Gb (e.g. Gb-E*-Eb-Db...). This
alteration produces a large neutral or submajor third above the third
degree, Db-E*, at about 363 cents, again in the area of 365 cents or
so which Persian theorists find typical.
Other neutral thirds formed by usual or inflected degrees illustrate
the variability of the temperament. The standard second and fourth
degrees B*-Eb form a 358-cent third, near the upper end of the central
neutral range, while at Gb*-Bb (using the inflected sixth degree) we
find a 371-cent third in the wider portion of the submajor range,
close respectively to ratios of 16:13 and 26:21 (359 and 369 cents).
The lowered fifth degree and the seventh degree (E*-Ab) would form a
346-cent third very close to 11:9, evidently less idiomatic for
Persian music, although the use of E* mainly as the top tone of a
descending passage might make that interval less common. In a
Zalzalian setting like that addressed by Wilson's matrix, of course,
such intervals would be ideal.
We now turn to Nava, said to be less popular in modern practice than
Shur because it seems to have such a similar structure, with the lower
tetrachord of Shur (two neutral seconds followed by a tone) becoming
the upper tetrachord of Nava; however I find that this pattern has its
own charm, here with the final or resting note as A*.
A* B* C D* E* Gb G A*
0 192 262 504 696 837 958 1200
192 71 242 192 141 121 242
Here, as often happens in Persian music, the neutral sixth A*-Gb is
small, about 837 cents, and very close to the typical size suggested
by Persian theorists of around 835 cents, or the almost identical
ratio 81:50 (835.193 cents); this is a bit smaller than 13:8 (840.528
cents). Between the fourth degree D* and this neutral sixth step, we
have a small neutral or supraminor third of 333 cents, also as we have
seen quite idiomatic, as is the complementary 363-cent submajor third
Gb-A* above, making up the fifth D*-Gb-A*.
This realization of Nava also features a small minor third and seventh
at 262 and 958 cents, which serve as narrow temperings of 7:6 and 7:4
(267 and 969 cents); the latter interval, some 10 cents smaller than
7:4, has both septimal and interseptimal qualities, like the 960-cent
interval of 5n-EDO. There are two steps of 242 cents (C-D*, G-A*), the
octave complement of the small minor seventh, and an interval which we
met in Section 2 as a close approximation of Wilson's 1024/891 at 241
cents. While in that context the interval served harmonically as a
very small minor third, here it takes the melodic role of a very large
whole tone, a kind of wide variation on an 8:7 step (231 cents). This
fluidity of roles and perceptions is characteristic of interseptimal
regions like those around 240-260 and 940-960 cents.
Summing up so far, we have found that while the 27:22 and 11:9 neutral
thirds of Zalzal's tuning and Wilson's matrix neatly fit many medieval
and modern Arabic styles, the smaller and larger neutral thirds of
Zest-24, also known as supraminor/submajor thirds, are very idiomatic
for Persian music, where contemporary theorists find sizes of around
335 and 365 cents typical. Thus as a kind of tempered variation on
Wilson's Rast/Bayyati matrix, the Zest-24 version might be termed a
Shur/Nava matrix after two of the modal families or dastgah-ha of
Persian music.
Having considered the neutral thirds of these systems, we now turn to
another vital aspect of Near Eastern music: the division of a minor
third into two neutral second steps.
----------------------------------------------
4. Cycles of Neutral Seconds and Minor Thirds
----------------------------------------------
Just as Near Eastern music prominently uses neutral thirds as routine
intervals, so a minor third is often divided into two neutral second
steps typically falling somewhere between roughly 14:13 (128 cents)
and 11:10 (165 cents). Just as tuning systems or musical styles may
lean toward different sizes of neutral thirds, so they may favor the
division of a minor third into more nearly equal or more contrasting
sizes of neutral seconds.
Thus Wilson's Rast/Bayyati matrix strongly favors the division of
Zalzal's scale, with subtly unequal steps of 12:11 and 88:81, or
151/143 cents, making up a 32:27 Pythagorean third at 294 cents. While
in the Rast tetrachords of Zalzal's scale (Section 1) the larger 12:11
step is placed below the smaller 88:81, the converse ordering of
143/151 cents could provide one idiomatic possibility for a modern
Egyptian performance of Maqam Bayyati as described by Scott Marcus.
Safi al-Din al-Shirazi, around 1300, describes a slightly more unequal
intonation through an arithmetic division of the 32:27, 64:59:54 or
141/153 cents (Scala archive, safi_diat2.scl).
However, modern Arabic theorists also sometimes mention more
contrasting divisions such as about 135/160 cents. If we assume
a Pythagorean norm for regular intervals such as minor thirds, which
Marcus finds an attractive thesis for the styles he is discussing,
then his suggestion of about 135-145 cents as an idiomatic range for
the neutral second above the final of Bayyati would result in
divisions ranging from around 135/159 to 145/149 cents.
Modern Persian music seems to favor a 135/160 style of division, with
these rounded values preferred in Farhat's scheme, and Anooshfar
specifying ratios of 27:25 and 800:729, or 133/161 cents. The
idiomatic Persian range for small neutral seconds extends to a bit
below 14:13 at 128 cents, which for a Pythagorean minor third would
imply a division like 128/166 cents, with a large neutral second of
around 11:10.
A scale of the great Persian philosopher and music theorist Ibn Sina,
known in medieval Latin Europe as Avicenna, shows a penchant both for
these small neutral second steps and for the 7:6 septimal minor third
which is divided arithmetically 14:13:12 or 128/139 cents (Scala
archive, avicenna_diat.scl).
1/1 14/13 7/6 4/3 3/2 21/13 7/4 2/1
0 128 267 498 702 830 969 1200
14:13 13:12 8:7 9:8 14:13 13:12 8:7
128 139 231 204 128 139 231
Here we have two disjunct tetrachords similar to the lower tetrachord
of Shur or the upper one of Nava, with two neutral seconds making up a
minor third, and followed by a tone to complete the fourth -- here the
large septimal tone at 8:7. In addition to suggesting a possible
precedent for modern Near Eastern styles like the Persian which prefer
smaller sizes of neutral seconds, this scale also includes a small
neutral sixth step at 21/13 (830 cents) of the kind typical of modern
Nava. The fifth between 4/3 and 2/1 is divided into neutral thirds of
63:52 (4/3-21/13) and 26:21 (21/13-2/1), or 332/370 cents, a division
close to the 335/365 or 337/365 cents reported by modern Persian
theorists.
Charting these divisions in Zest-24 is complicated by the fact that
minor thirds may be generated either within the notes of a single
12-note circle and manual, or by combining notes from the two
circles. Minor thirds of the first kind are 6-step intervals, while of
the second are 5-step intervals.
Six-step intervals have a rounded range of 275-313 cents, almost
spanning the region between the simplest ratios for minor thirds of
7:6 and 6:5 (267-316 cents). Five-step intervals at 225-267 cents
provide the closest approximations of the septimal 8:7 tone and 7:6
minor third, while also richly populating the intervening region of
interseptimal space.
Here our narrower focus is on the division of a minor third into two
steps both falling within the "neutral second" category, say roughly
125-165 cents. More generally, however, we will also consider related
types of divisions where one step has a size within this range, but
the other is large or small enough to come within a neighboring region
along the continuum.
We turn first to the cycle of 6-step minor thirds (275-313 cents)
found within each keyboard manual, with each third divided by a note
on the other manual into two 3-step intervals. From this perspective,
Zest-24 consists of three interlaced cycles of minor thirds, shown in
the three rows of the following table. Each row shows a chain of four
6-step minor thirds from each keyboard and the eight 3-step intervals
into which they divide each other. For convenience, the first 3-step
interval of a row is repeated at the end of the row with its size in
cents placed in parentheses; this may help in reading at a glance the
division of the last minor third on the lower chain of a row, for
example the A-Bb-C* division (159-154 cents) concluding the first row.
Cycle of Neutral Seconds and Six-Step Minor Thirds
|----- 275 -----|------- 313 -----|------ 313 -------|----- 300 ------|
0 154 275 441 587 729 900 1041 1200 154
Bb* 154 C 121 Db* 166 Eb 146 E* 141 Gb 171 G* 141 A 159 Bb* (154) C
#0 #3 #6 #9 #12 #15 #18 #21 #24 #27
|----- 287 ------|------ 287 ------|------ 313 -----|------- 313 ------|
|------ 313 -----|------ 313 ------|----- 275 -------|----- 300 ------|
33 204 346 492 658 779 933 1092 1233
B 171 C* 141 D 146 Eb* 166 F 121 Gb* 154 Ab 159 A* 141 B (171) C*
#1 #4 #7 #10 #13 #16 #19 #22 #25 #28
|------ 287 ------|----- 287 ------|-------313 -----|------- 313 ------|
|------ 313 -----|------ 313 ------|------ 275 ------|----- 300 ------|
83 224 396 537 708 850 983 1150 1233
B* 141 Db 171 D* 141 E 171 F* 141 G 134 Ab* 166 Bb 134 B* (141) Db
#2 #5 #8 #11 #14 #17 #20 #23 #26 #29
|------ 313 ------|------ 313 ------|------ 300 -----|------ 275 ------|
The 3-step category has various shadings within our neutral second
category at 134-166 cents, or from around 27:25 to 11:10, plus a few
outlying intervals at 121 and 171 cents, bridges to the adjoining and
sometimes overlapping realms of meantone and the equable heptatonic.
The 121-cent step is the regular diatonic semitone of Zarlino's
2/7-comma meantone, and typically occurs as a variety of 2-step
semitone on a single keyboard (e.g. E-F, B*-C*), but sometimes also
arises as a 3-step interval in divisions of smaller minor thirds, and
plays the role of a very small neutral second.
The 171-cent step, virtually identical to that of 7-EDO, sometimes
arises in the division of the large 313-cent minor third. This step,
found in certain transpositions of Near Eastern modes, can serve as a
bridge to the realm of what I term the "equable heptatonic," a style
of intonation with step sizes centering around 7-EDO or 171 cents
while varying up to something on the order of 25-30 cents in either
direction, say 140-195 cents. The equable heptatonic is briefly
discussed at the end of this section, and in Appendix 3.
We focus first on divisions with both steps within the neutral second
range of 14:13-11:10, represented here by sizes of 134-166 cents. The
275-cent third about eight cents wide of 7:6 admits of two such
divisions: F*-G-Ab* (141/134) and Bb-B*-Db (134/141). The first
permutation is rather like George Secor's 12:13:14 harmonic division
(139/128 cents), and the second like that of Ibn Sina's 14:13:12
arithmetic division shown above (128/139).
The 287-cent and 300-cent thirds provide the closest approximations of
the Pythagorean 32:27 (294 cents) which in practice as well as theory
seems to provide the basis for many Near Eastern divisions. The
287-cent third, slightly narrow of 13:11, admits near-equal divisions
of 146/141 (Eb-E*-Gb) or 141/146 (C*-D-Eb*), possibly fitting the
description of a very subtle Rast or Bayyati respectively. One might
compare the Zalzalian 12:11/88:81 division of Wilson's matrix at
151/143, or the simpler JI divisions of 11:12:13 and 13:12:11 at
151/139 and 139/141.
The 300-cent third, with an affinity for 32:27 and also 19:16 or 25:21
(298 and 302 cents), has divisions of 141/159 (G*-A-Bb*) and 159/141
(Ab-A*-B) that might fit either an Arabic style with more contrast
between step sizes or a Persian style: in the latter, around 140 cents
may mark the upper end of the preferred range for small neutral seconds.
The 134/166 (G-Ab*-Bb) and 166/134 (Ab*-Bb-B*) divisions could be seen
as variations on the 135/160 or 160/135 approach favored in Persian
and some Arabic styles, with the 32:27 third and the large neutral
second step tempered about six cents wide; the latter interval is very
close to a just 11:10 (165 cents).
The 313-cent third, very close to 6:5, is Zarlino's regular meantone
minor third, and lends itself to Near Eastern genres or styles
favoring some 5-limit colors. The divisions nearest equal, 159/154
(A-Bb*-C) and 154/159 (Gb*-Ab-A*), are very close to those of a
regular 31-note meantone at 1/4-comma (152 and 158 cents) or 31-EDO
with its equal neutral seconds of 4/31 octave or 155 cents. This is
the kind of intonation one would have if realizing a mode like Rast or
Bayyati on Nicola Vicentino's archicembalo of 1555 or Fabio Colonna's
Sambuca Lincea of 1618. The 166/146 (Db*-Eb-E*) and 146/166 (D-Eb*-F)
divisions have a smaller step at about the upper limit suggested by
Scott Marcus for the lower neutral third of Bayyati (135-145 cents),
and a larger step in the 11:10 region. These divisions might also be
regarded as tempered variations of Ptolemy's 12:11:10 (151/165) used
in his Equable Diatonic (12:11:10:9), or the converse 10:11:12
(165/151).
These descriptions are theoretical characterizations or free
associations, and they do not necessarily determine what I do at the
keyboard. Thus often I use the 146/141 division of Eb*-E-Gb for a Shur
tetrachord, although the converse ordering of step sizes would be more
idiomatic -- as well as the 134/141 division of Bb-B*-Db, the closest
approximation of Ibn Sina's 14:13:12 at 128/139. Here is a realization
of his scale opening with this division:
Bb B* Db Eb F Gb* Ab Bb
0 134 275 492 708 829 983 1200
134 141 217 217 121 154 217
Here the approximations of 7:6 and 7:4 are the best available within a
single circle of fifths (approaching the accuracy of 22-EDO), while the
other circle supplies neutral intervals of 134 and 829 cents for Ibn
Sina's 14/13 and 21/13 at 128 and 830 cents, the latter almost just.
Melodically, however, the 134/141 division of the lower tetrachord
shifts to 121/154 at F-Gb*-Ab in the upper tetrachord. One might wish
that the 121-cent step were a few cents larger, at least when writing
about the situation; but at the keyboard, it can seem merely another
of the myriad compromises of temperament, and one which simply results
in another intriguing shading. Consider, for example, this three-voice
progression based on an adaption of 13th-14th century European
polyphony to a Near Eastern modal context:
F 708 -- +121 -- Gb* 829
Db 275 -- -141 -- B* 133
Bb 0 -- +134 -- B* 133
The lower 275-cent minor third contracts to a unison with 134-cent and
141-cent steps, while the upper 433-cent major third -- the best
approximation of 9:7 in this tuning sytem -- expands to a 696-cent
fifth with steps of 141 and 121 cents. Overall, the effect seems to me
that of a progression with a desired neutral second kind of color,
possibly helped by the fact that the 121-cent step is only 7 cents or
so narrow of the ideal 14:13.
Thus when one is seeking a neutral second division for a small 6-step
minor third, a 121-cent step may sometimes take the role of small
neutral second. For 5-step minor thirds, as we shall see, this melodic
step takes center stage in analogous situations offering fascinating
shades of intonation.
At the beginning of this section, I commented that the 171-cent step
found in some divisions of the large 313-cent minor third may occur in
certain approximations or transpositions of Near Eastern modal
patterns, and can serve as a bridge to the realm of an equable
heptatonic realm of intonation. This realm centers around the region
of 7-EDO or 171 cents, mingling steps of this size with others we
might call middle to large neutral seconds or small whole tones. A
variety of Southeast Asian and African musics, for example, seem to
follow this type of approach while presenting different degrees of
"equability." Tunings may closely approximate 7-EDO, or moderately
vary from it in either direction by 25-30 cents, say.
Since the main focus of this paper is on neutral intervals, I will be
content here briefly to show how seeking out a Near Eastern tuning in
Zest-24 can sometimes lead one into equable heptatonic territory. Let
us consider this version of the "Iranian mode Segah from C" found in
the Scala scale archive (segah2.scl):
0 200 340 500 700 840 1000 1200
200 140 160 200 140 160 200
Like Zalzal's scale, this tuning might be viewed as having two
conjunct tetrachords with lower tone and two neutral seconds forming
an upper minor third (200-140-160); a 200-cent semitone completes the
octave:
|--------------------------|------------------------| tone |
0 200 340 500 700 840 1000 1200
200 140 160 200 140 160 200
Two differences are that the smaller neutral second precedes the
larger, and that the contrast of step sizes is greater (140/160) than
in Zalzal's tuning (12:11-88:81 or 151/143). We have, typically for
the Persian style, a small neutral or supraminor third at 340 cents,
slightly narrow of 39:32 (342 cents) and almost identical to a 28:23;
and an 840-cent neutral sixth at a virtually just 13:8.
Mapping this fine tuning to Zest-24 involves compromises whose
specifics vary depending on the chosen transposition. Here is one
version that I selected in Scala, focusing especially on the 140-cent
steps of the tetrachords and the small neutral third and sixth:
|--------------------------|-----------------------| tone |
G* A* B C* D* E F* G*
0 192 333 504 696 837 1008 1200
192 141 171 192 141 171 192
Here we have symmetrical tetrachords with middle steps of 141 cents, a
333-cent small neutral third of a somewhat different hue that the 340
cents of the original tuning, but likewise within the preferred
neighborhood of around 335 cents (our shading is a virtually just
40:33), and also an 837-cent neutral sixth close to the 840 cents of
our model.
While these intervals shape this choice of transposition, the
structure of the temperament introduces some fascinating variations.
The 300-cent minor thirds of the original version, near 32:27, grow to
313-cent meantone thirds -- and the large 160-cent neutral seconds to
171 cents. The resulting 141/171 divisions of these large minor thirds
(A*-B-C*, D*-E-F*) subtly alter the melodic qualities of the mode, and
also in a polyphonic style find expression in a beautiful progression
such as this:
A* 1392 -- +141 -- B 1533
F* 1008 -- -171 -- E 837
D* 696 -- +141 -- E 837
This three-voice cadential progression has a pleasant affinity to its
counterpart in Zalzal's scale or the Iranian Segah which suggested
this Zest-24 offshoot, but with the 171-cent step bridging us into a
another territory.
The nature of this territory may become clearer if we consider a mode
beginning on the third step of our Segah version:
B C* D* E F* G* A* B
0 171 363 504 675 867 1059 1200
171 192 141 171 192 192 141
This mode is rather similar to an African tuning as reported by Kevin
Volans (Scala scale archive, volans.scl):
0 171 360 514 685 860 1060 1200
171 189 154 171 175 200 140
Both tunings feature some 171-cent steps like those of 7-EDO, with
others varying in either direction by 25-30 cents. The African scale
has a small fifth of about 685 cents, almost exactly 4/7 octave, with
a narrower 675-cent fifth in the Zest-24 version. The 514-cent fourth
of the African tuning, almost precisely 3/7 octave, is likewise ten
cents narrower in the Zest-24 variation -- becoming a usual 504-cent
meantone fourth. The second and seventh steps are almost identical in
the two versions, with the third and sixth steps, 360 and 860 cents in
the African scale, a bit higher (363 and 867 cents) in Zest-24.
Thus the region of around 140-200 cents is a kind of common ground
both for certain Near Eastern modal patterns based on neutral seconds
and tones, and for African and Southeast Asian modal patterns, for
example, with an equable heptatonic outlook. However one seeks to
distinguish these two types of patterns, with a prominent use of some
step sizes at or very close to 7-EDO as one possible mark of the
equable heptatonic, Zest-24 sometimes bridges these worlds.
For a rather similar type of equable heptatonic scale found within
Wilson's Rast/Bayyati matrix, see Appendix 3.
--------------------------------------------
5. Five-step intervals and equable divisions
--------------------------------------------
In surveying the divisions of 6-step minor thirds into two 3-step
intervals, mostly neutral thirds plus the adjoining sizes of 121 and
171 cents, we have been focusing on tempered shadings of what George
Secor has aptly termed _equable_ divisions.
Secor borrows the term from Ptolemy's Equable Diatonic as it has been
called in English by Harry Partch and others, with its 12:11:10:9
tetrachord where a 6:5 minor third at a rounded 316 cents is
arithmetically divided into two superparticular ratios: 12:11:10, or
151/165 cents. The harmonic division of 10:11:12 may likewise be
styled as equable (165/151 cents).
Secor and I were looking for a term to describe similar divisions of
smaller minor thirds into two neutral second steps, such as the
arithmetic division of 7:6 at 267 cents by Ibn Sina, 14:13:12 or
128/139 cents, and its harmonic counterpoint nicely approximated in
Secor's 17-note well-temperament of 1978, 12:13:14 or 139/128 cents.
The term "equable" seemed to fit this division nicely, and also what
soon proved one of my favorites in JI and approximate tempered forms,
the division of 13:11 at 289 cents into an arithmetic proportion of
13:12:11 at 139/151 cents, a possible interpretation of the lower
three notes of a Bayyati tetrachord (Section 1), or the converse
harmonic proportion of 11:12:13 at 151/139 cents.
While these examples with 6:5, 7:6, and 13:11 have all involved
superparticular divisions, the arithmetic division of the more complex
Pythagorean minor third at 32:27 or 294 cents into 64:59:54 (141/153
cents) by Safi al-Din, or the converse harmonic division of 54:59:64
(153/141 cents), also nicely illustrates an equable division.
In our survey, we have encountered more or less close tempered
counterparts of all of these divisions, with some illustrative
examples offered in the following table:
---------------------------------------------------------------------
Just Minor 3rd Equable Divisions Illustrative
Ratio Cents Arithmetic Harmonic Zest-24 approximations
---------------------------------------------------------------------
6:5 316 12:11:10 10:11:12 313 D-Eb*-F Db-Eb*-E
151/165 165/151 146/166 166/146
.....................................................................
32:27 294 64:59:54 54:59:64 300 G*-A-Bb* Ab-A*-B
141/153 153/141 141/159 159/141
.....................................................................
13:11 289 13:12:11 11:12:13 287 C*-D-Eb* Eb-E*-Gb
139/151 151/139 141/146 146/141
.....................................................................
7:6 267 14:13:12 12:13:14 275 Bb-B*-Db F*-G-Ab
128/139 139/128 134/141 141/134
---------------------------------------------------------------------
In addition to these typical forms with two neutral second steps in
the range of about 14:13-11:10, we have already encountered among the
6-step minor thirds some allied forms involving a neutral second plus
a step of 121 cents or 171 cents. With smaller minor thirds, or
interseptimal intervals taking the role of minor thirds, in the 5-step
category, analogous equable divisions in Zest-24 typically involve
some type of small neutral second plus a 121-cent step. We now turn to
these divisions.
Five-step intervals, as briefly discussed above (Section 4), have a
range of 225-267 cents, thus including the closest approximations of
the large 8:7 tone at 231 cents (225 and 237 cents); a set of
intervals populating the interseptimal region between and more or less
distinct from 8:7 and 7:6 (242, 250, 254 cents); and the closest
approximations of the 7:6 minor third (262 and 267 cents).
In the following table, each row except the fifth and last shows a
chain of five 5-step intervals each divided in two potentially
"equable" ways: into a 2-step plus a 3-step, or a 3-step plus a
2-step, divisions shown respectively below and above the dashed
lines. Each chain on one of these rows thus moves us 25 tuning steps,
or one step plus a 24-step octave, which is where the next row
starts. However, the last row has only four 5-step intervals, which
complete a full 24-note circle. Note that two 5-steps make a usual
10-step fourth, with some divisions close to a septimal 8:7:6 or 6:7:8
(231/267 or 267/231 cents) and others having a near-equal quality
favored in many Javanese and Balinese slendro tunings and certain soft
diatonics.
Cycle of Five-Step Intervals and Potentially Equable Divisions
#0 #3 #5 #8 #10 #13 #15 #18 #20 #23 #25
Bb* 154 C 71 Db 171 D* 96 Eb* 166 F 71 Gb 171 G* 83 Ab* 166 Bb 83 B
0 154 225 396 492 658 729 900 983 1150 1233
|----- 225 -----|---- 267 ----|---- 237 ----|---- 254 -----|---- 250 ----|
0 83 225 346 492 587 729 850 983 1092 1233
Bb* 83 B 141 Db 121 D 146 Eb* 96 E* 141 Gb 121 G 134 Ab* 108 A* 141 B
#0 #2 #5 #7 #10 #12 #15 #17 #20 #22 #25
#1 #4 #6 #9 #11 #14 #16 #19 #21 #24 #26
B 171 C* 71 Db* 166 Eb 96 E 166 F* 71 Gb* 154 Ab 108 A 158 Bb* 83 B*
33 204 275 396 537 658 779 933 1041 1150 1283
|----- 242 -----|---- 262 ----|---- 242 ----|---- 262 -----|---- 242 ----|
33 154 275 346 537 587 779 900 1041 1150 1283
B 121 C 121 Db* 121 D* 146 E 121 F 121 Gb* 121 G* 141 A 108 Bb 141 B*
#1 #3 #6 #8 #11 #13 #16 #18 #21 #23 #26
#2 #5 #7 #10 #12 #15 #17 #20 #22 #25 #27
B* 141 Db 121 D 146 Eb* 96 E* 141 Gb 121 G 134 Ab* 108 A* 141 B 121 C
83 225 346 396 587 729 850 983 1091 1233 1354
|----- 262 -----|---- 242 ----|----- 262 ----|---- 242 -----|---- 262 -----|
83 204 346 441 587 587 850 933 1091 1200 1354
B* 121 C* 141 D 96 Eb 146 E* 121 F* 141 G 83 Ab 158 A* 108 Bb* 154 C
#2 #4 #7 #9 #12 #13 #17 #19 #22 #24 #27
#3 #6 #8 #11 #13 #16 #18 #21 #23 #26 #28
C 121 Db* 121 D* 141 E 121 F 121 Gb* 121 G* 141 A 108 Bb 134 B* 121 C*
154 275 396 537 658 779 900 1041 1150 1283 1404
|----- 242 -----|---- 262 ----|----- 242 ----|----- 250 ----|----- 254 -----|
154 224 396 492 658 729 900 983 1150 1233 1404
C 71 Db 171 D* 96 Eb* 166 F 71 Gb 171 G* 83 Ab* 166 Bb 83 B 171 C*
#3 #5 #8 #10 #13 #15 #18 #20 #23 #25 #28
#4 #7 #9 #12 #14 #17 #19 #22 #24
C* 141 D 96 Eb 146 E* 121 F* 141 G 83 Ab 159 A* 108 Bb*
204 346 441 537 708 850 933 10 1200
|----- 237 ----|----- 267 -----|----- 225 ----|----- 267 ----|
204 275 441 537 708 779 933 1041 1200
C* 71 Db* 166 Eb 96 E 171 F* 71 Gb* 154 Ab 108 A 159 Bb*
#4 #6 #9 #11 #14 #16 #19 #21 #24
Our reference to "potentially equable" divisions of 5-step intervals
is shown to be the better part of valor, since these 2-3 or 3-2
divisions vary dramatically in their proportions. For the 254-cent
interseptimal minor third Bb-C* at the end of the fourth row, for
example, almost identical to a just 22:19, the Bb-B*-C* division at
134/121 cents nicely fits our expectations of a small neutral second
plus an almost-neutral step of 121 cents. However, the Bb-B-C*
division at 83/171 cents, offers a fascinating but very different
pattern approaching that of a semitone plus a small tone; one might
compare the 83/165 cent division of the 248-cent minor third (or very
large major second) available in 29-EDO (i.e. 2-5 tuning steps). While
we here focus on divisions of the first type, the others should not be
neglected in practice or theory, and indeed would be a fine topic for
another paper!
Starting with our 267-cent third, virtually identical to 7:6, we find
that the most equable divisions available are Db-D-Eb* at 121/146
cents, and Eb-E*-F* at 146/121 cents. The second division figures
prominently in an improvisation I did to honor the then newly born
Baran (a name meaning "rain"), daughter of Shaahin Mohaajeri, an
Iranian musician and theorist.
The improvisation is based in part on a septimal kind of Shur favoring
the neutral sixth degree, here very close to a just 13:8 (841 cents):
Eb E* F* Ab Bb B* C* Eb
0 146 267 492 708 842 963 1200
146 121 225 217 134 121 237
In a polyphonic context, this modal pattern lends itself to
progressions like this:
C* 963 -- -121 -- B* 842
Bb 708 -- +134 -- B* 842 Bb 708 -- +134 -- B* 842
F* 267 -- -121 -- E* 146 F* 267 -- -121 -- E* 146
Eb 0 -- +146 -- E* 146 Eb 0 -- +146 -- E* 146
The Eb-F*-Bb* sonority of the first or three-voice version is close to
6:7:9 (0-267-702 cents), with the lower minor third contracting to a
unison and the upper major third F*-Bb at 441 cents, about 6 cents
wide of 9:7, expanding to a fifth. These progressions involve a
121-cent step (F*-E*) plus a small neutral second of 134 or 146 cents.
The second version adds a fourth voice at 963 cents, about 6 cents
narrow of 7:4, for a sonority close to 12:14:18:21 (0-267-702-969
cents). The outer minor seventh Eb-C* contracts to a fifth E*-B* by
way of a 146-cent ascend of the lowest voice and a 121-cent ascent of
the new fourth voice. The upper pair of voices introduces a new type
of interval, a 254-cent minor third Bb-C* which contracts to unison
via a 134-cent ascend ot the next to highest voice and the 121-cent
descent of the highest voice. This 254-cent third, shortly considered
further, is itself in the interseptimal range, but here may lend more
of a subtle nuance to an overall impression of septimal color conveyed
by the near-just approximations of 7:6 and 7:4 above the lowest note.
Like the 267-cent third, the next largest 5-step third at 262 cents
may also be considered a tempering of 7:6, here a bit less than five
cents narrow. This interval is identical to the augmented second which
would occur in a single chain of fifths in Zarlino's regular 2/7-comma
meantone (e.g. Bb-C#), but in the irregular Zest-24 it occurs only as
an interval mixing notes from the two 12-note chains or circles
(e.g. E*-G). In either system, it is equal to a 313-cent meantone
third (e.g. E*-G*) less the 50.28-cent diesis. In Zest-24, this diesis
is the distance between the two 12-note circles; in a regular 24-note
version of Zarlino's tuning, it would be the distance between two
regular 12-note chains of meantone fifths (actually making up a single
24-note chain).
This beautiful 262-cent third lends itself to an equable division of
the 141/121 variety in this septimal variation on Shur, with some
alternative inflections also shown:
837
Gb
141 121
A* B C D* E* F G A*
0 141 262 504 696 766 958 1200
141 121 242 192 71 192 242
141 121
E
646
In the "standard" form with neutral second and minor sixth, we have a
lower tetrachord A*-B-C-D* (0-141-262-504 cents) featuring a 141/121
division plus a 242-cent step here used as a large whole-tone (C-D*).
The upper tetrachord of E*-F-G-A* has a near-14:9 minor sixth, and the
958-cent minor seventh which we have noted may be taken either as a
7:4 approximation or as a distinctly interseptimal interval. Both the
lower 141-121-242 and upper 71-192-242 tetrachords of this form
feature a 262-cent third plus an upper step of 242 cents (A*-C-D* or
E*-G-A*). In descending passages, the fifth degree E* may be lowered
to E at 646 cents, typically as a top tone or tone following the
regular sixth degree F in a descending phrase (Section 1).
When we use the alternative version of the sixth degree at Gb, a small
neutral sixth of 837 cents above the final, then we have available a
version of the upper tetrachord, E*-Gb-G-A*, identical to the lower at
141-121-242 cents. This inflection of the sixth degree facilitates an
equable variety of cadence with 141/121 divisions such as this:
E* 696 -- +141 -- Gb 837
C 262 -- -121 -- B 141
A* 0 -- +141 -- B 141
Here the lower minor third A*-C contracts to a unison via 141/121
motions, while the upper major third at 434 cents likewise expands to
a fifth.
From the standpoint of the preferred tunings of the lower tetrachord
of Shur described by theorists such as Farhat and Anooshfar at around
135-160-204 cents, this interpretation diverges not only in its
septimal-flavor intervals but in the 141/121 order of the lower two
steps with the larger preceding the smaller. Curiously, however,
another Persian theorist, Dariush Talai, describes a rather different
type of tetrachord of which our 141-121-242 might be considered a kind
of permutation.
Talai, in his _A New Approach to the Theory of Persian Art Music_
(Mahur, Tehran, 1993, p. 16), suggests for the Chahargah tetrachord a
tuning with steps of about 140-240-120 cents (0-140-380-500 cents).
One traditional interpretation of this tetrachord, known as Hijaz in
the Arabic or Turkish maqam tradition, has a lower neutral second, a
middle step around 7:6, and a smallish upper semitone. Around 1300,
Qutb-al-Din specifies 33:36:42:44 or 1/1-12/11-14/11-4/3, a division
of 0-151-418-498 cents with steps of 151-267-81 cents. Farhat also
suggests a middle step around 7:6, describing this interval as a "plus
second" rather smaller than a usual minor third. A version suggested
by his account of average tar and sehtar tunings would be about
0-135-410-500 cents, or steps of 135-275-90 cents. Anooshfar's
rational 17-note tar tuning offers 1/1-27/25-81/64-4/3 at a very
similar 0-133-408-498 cents or steps of 133-275-90 cents.
Talai's 140-240-120 cent division has a lower neutral third step
comparable to these versions, but with a lower major third above the
lowest note of 380 cents, close to 5:4 (386 cents); a larger upper
semitone step of 120 cents; and a smaller middle step of 240 cents,
which we might describe as a narrower form of "plus second" than
Farhat's 7:6, but also nicely answering to this term.
A Chahargah tetrachord very close to Talai's is available in Zest-24
using the usual meantone major third and diatonic semitone at 383 and
121 cents, so that a meantone style of spelling is apt:
A* B C#* D
0 141 383 504
141 242 121
In this kind of setting with a near-5:4 major third, the 121-cent step
(or Talai's 120 cents) is very likely to be heard as a wider version
of a 16:15 semitone (112 cents) -- as also, of course, in a typical
16th-century European context of the kind which prompted Zarlino's
2/7-comma tuning. However, in the 141-121-242 permutation of our
septimal Shur with its 262-cent minor third, the different musical
context may facilitate an appreciation of this step as something like
a very narrow 14:13.
After the 262-cent third, the next largest 5-step interval is the
254-cent third almost identical to a just 22:19, and which we met in
the version of Shur for _Baran: Gift of Rain_ discussed near the
beginning of this section, where it occurs in the upper tetrachord.
There the overall color of the scale was septimal, with a 267-cent
third in the lower tetrachord; but we can highlight the 254-cent third
and its interseptimal color by making the upper tetrachord of that
scale the lower tetrachord of this:
Bb B* C* Eb F Gb* G* Bb
0 134 254 492 708 829 950 1200
134 121 237 217 121 121 250
The 254-cent third and 950-cent seventh move us clearly into
interseptimal territory, by comparison to the 7/6 and 7/4 of Ibn
Sina's scale at 267 and 969 cents (Section 4), let alone the usual
modern Persian tuning of these steps as described by Farhat and
Anooshfar at around the Pythagorean ratios of 32/27 and 16/9 at 294
and 996 cents. It is thus well to emphasize that this is a kind of
innovative or "xenharmonic" variation on Shur rather than an
interpretation of medieval or modern Persian practice.
At the same time, the neutral second and sixth degrees at 134 and 829
cents maintain a note of familiarity, the first close to Farhat's
average size of 135 cents or Anooshfar's 27/25 at 133 cents, and the
second to Ibn Sina's 21/13 at 830 cents.
The equable division of the 254-cent third, Bb-B*-C* in the lower
tetrachord or a rounded 134/121 cents, has as a possible rational
model the harmonic division of the rather smaller 15:13 minor third
(or very large major second) at about 248 cents, 13:14:15 or a rounded
128/119 cents. A more complex harmonic division which this tempered
version fits quite closely is that of the 22:19 minor third at 254
cents into 38:41:44, or 132/122 cents.
While I find this division charming in a purely melodic context, it
can also be "verticalized" in a polyphonic setting where the steps of
134/121 cents are used in directed resolutions of interseptimal
sonorities:
G* 950 -- -121 -- Gb* 842
F 708 -- +121 -- Gb* 829 F 708 -- +121 -- Gb* 842
C* 254 -- -121 -- B* 134 C* 254 -- -121 -- B* 146
Bb 0 -- +134 -- B* 134 Bb 0 -- +134 -- B* 146
In the three-voice progression, the lower 254-cent third contracts to
a unison while the upper 454-cent major third almost identical to
13:10 (identical to the augmented third of Zarlino's regular 2/7-comma
meantone, e.g. Eb-G# in a 12-note chain of meantone fifths), another
interseptimal interval par excellence, expands to a fifth by equal
ascending and descending steps of 121 cents (C*-F to B*-Gb*).
The four-voice version adds a 950-cent minor seventh contracting to a
fifth by steps of 134 cents up and 121 cents down (Bb-G* to B*-Gb*),
and also a 242-cent interval between the highest pair of voices here
taking the role of a very small minor third and contracting to a
unison (F-G* to Gb*-Gb*) through equable and indeed equal motions of
121 cents.
This version of Shur includes both the 242-cent interval F-G* used
vertically as a small minor third in the last progression, and also a
250-cent step G*-Bb between the seventh degree and octave, which in
this melodic capacity takes the role of a very large major second.
These are the two remaining interval sizes for our survey of equable
divisions among 5-step intervals which may regularly take the role of
minor thirds -- the 225-cent and 237-cent steps being more allied with
the 8:7 tone, although the latter can also have some interseptimal
qualities.
The 250-cent interval differs from all others in the 242-267 cent
range in not having a division involving the 121-cent step; rather,
its equable division is 141/108 (G*-A-Bb) or 108/141 (G#*-A*-B). Here
are two striking progressions involving the former division, with the
141-cent motions of the two outer voices in the three-voice
progression lending a special "neutral" quality:
F 958 -- -121 -- E 837
D* 696 -- +141 -- E 837 D* 696 -- +141 -- E 837
Bb 250 -- -108 -- A 141 Bb 250 -- -108 -- A 141
G* 0 -- +141 -- A 141 G* 0 -- +141 -- A 141
The four-voice version has a somewhat different shade of color because
of the introduction of the 958-cent minor seventh G*-F between the two
outer voices, and also the 262-cent third between the upper voices,
the former with some affinity for 7:4 and the latter in the septimal
orbit of 7:6 -- in contrast to the 250-cent third at G*-Bb, close to
15:13 (248 cents), and near the middle of the interseptimal region
between 8:7 (231 cents) and 7:6 (267 cents).
In either version, the small minor third G*-Bb contracts to a unison
while the large major third Bb-D* at 446 cents expands to a fifth by
way of ascending 141-cent and descending 108-cent steps. Although, as
mentioned, the 121-cent step does not occur in these two-voice
resolutions, the fourth voice introduces it in the second version as a
descending motion, where it joins with a 141-cent motion in another
voice in resolving the outer 958-cent minor seventh to a fifth and the
upper 262-cent third to a unison.
Incidentally, the 250-cent step may also very pleasingly occur in a
variation on the Persian Chahargah tetrachord of the general kind
described by Talai and discussed earlier in this section (140-240-120)
with a major third of around 380 cents, or near 5:4, and an upper
semitone thus not too far from 16:15.
G Ab* B C
0 134 383 504
134 250 121
Again, a middle step of around 240-250 cents seems musically
interchangeable with the step at or close to 7:6 described elsewhere
in the medieval and modern Persian literature; Zest-24 includes both
types of interpretations, the latter generally favoring a larger major
third above the lowest note of the tetrachord (e.g. 408 or 421 cents,
corresponding to the 81:64 Pythagorean third specified or approximated
in the modern tunings of Farhat and Anooshfar, and the 14:11 of
Qutb-al-Din around 1300 at 418 cents).
The 242-cent interval, as we have already seen, has an equable and
indeed equal division into two 121-cent steps. We have already
encountered this division in a version of Shur discussed earlier in
this section featuring a 254-cent third in the lower tetrachord. Here
is the upper tetrachord of that scale, which raises an interesting
melodic point:
F Gb* G* Bb
0 121 242 492
121 121 250
My impression upon playing this tetrachord is that of a soft diatonic,
with the 242-cent step as a very small minor third and the upper
250-cent step as a very large tone completing the fourth, here the
small size of usual fourth at 492 cents. Since two 121-cent steps seem
to make a small minor third, this suggests that they may be serving in
effect as something like very small neutral seconds. Here I use
"diatonic" in its Classic Greek sense to mean a division of the
tetrachord where no adjacent interval is larger than a "tone" of some
kind, or about half the size of the fourth. Mathematically, the
250-cent step is actually a bit larger than half of the 492-cent
fourth; but musically, it seems to me open to a "tonelike" perception.
In contrast, a tetrachord with the same lower two steps but a large
or meantone fourth at 504 cents seems to have an effect more like a
chromatic genus, not surprisingly since the upper step of the
tetrachord has grown from 250 to 262 cents, close to the 7:6 of
Ptolemy's Intense Chromatic (1/1-22/21-8/7-4/3, 0-81-231-498 cents,
with steps of 22:21-12:11-7:6 or 81-151-267 cents), for example:
C Db* D* F
0 121 242 504
121 121 262
In Zest-24, the 242-cent interval occurs as a 5-step interval mixing
notes from the two keyboards. In Zarlino's regular 2/7-comma meantone,
it is also the diminished third arising within a single 12-note chain
of meantone fifths, and equal precisely to two diatonic semitones at
121 cents each (e.g. C#-Eb from C#-D-Eb). In an extended version of
Zarlino's meantone, say 24 or 31 notes (the latter a Moment of
Symmetry, with the tuning circulating in 119 notes almost identical to
119-EDO), the 242-cent interval may also be built as in Zest-24 from a
192-cent tone (Zarlino's regular mean-tone) plus a 50-cent diesis
(e.g. F-G* from F-G-G*).
While the equable 121/121 division of Zarlino's regular tuning is not
available within a single 12-note circle of Zest-24, it is happily
available if we use notes from both circles or keyboard manuals. As in
an extended version of that regular tuning, there is often an
intriguing choice of divisions as illustrated by some alternative
resolutions for the three voice sonority F-G*-C with a lower 242-cent
interval F-G* acting as a small minor third and an upper 454-cent
interval G*-C acting as a large major third:
C 696 -- +50 -- C* 746 C 696 -- +71 -- Db 766
G* 242 -- -192 -- F* 50 G* 242 -- -171 -- Gb 71
F 0 -- +50 -- F* 50 F 0 -- +71 -- Gb 71
C 696 -- +121 -- Db* 817 C 696 -- +192 -- D 887
G* 242 -- -121 -- Gb* 121 G* 242 -- -50 -- G 192
F 0 -- +121 -- Gb* 121 F 0 -- +192 -- G 192
Here our equable division appears as the third of four possibilities.
The first is a 50/192 division with the small 50-cent semitone or
diesis ascending and the 192-cent major second descending. Thus the
lower minor third contracts to a unison while the upper major third
expands to a fifth. The small melodic semitone or diesis at 50 cents
is very effective in these resolutions, and may somewhat resemble the
narrow cadential diesis described by Marchettus of Padua in his
_Lucidarium_ of 1317-1318.
The second division is 71/171, again with the smaller interval
ascending -- here the 71-cent semitone at a just 25:24 which is the
chromatic semitone of Zarlino's meantone (e.g. F-F# or C-C# in a usual
meantone spelling), although here often given a diatonic spelling such
as F-Gb or C-Db because in a style oriented mainly to Pythagorean,
septimal, or interseptimal regions it often acts as a regular diatonic
step. The accompanying 171-cent step in the ascending direction is the
intriguing interval very close to 1/7 octave. This resolution seems
rather like the first, with a different shading, with the 50-cent step
more concise and economic, and the 71-cent step more "roomy."
The third division is our equable 121/121, with all voices moving by
121-cent steps. Do these steps, here unmixed with others in the
neutral second range -- unlike in many other divisions we have
considered such as 121/146, 121/141, and 121/134 -- take on a certain
"quasi-neutral" character of their own? Do they give the impression of
large semitones, as in meantone? Or is the musical situation beyond
such familiar categories?
The fourth division of 50/192 uses the same steps as the first, but in
with the small step descending and the large ascending. Like the first
progression, it is very concise and efficient, and one wonders whether
Marchettus or some of his followers in the 14th century may have sung
cadences involving descending semitones and minor thirds contracting
to unisons in a manner something like this -- although he writes of
the narrow cadential diesis only in connection with progressions
involving actual or anticipated resolutions by expansion (e.g. major
third to fifth or major sixth to octave) with ascending semitones
involving sharps.
The three-voice progression with an equable 121/121 division has
four-voice extensions which can take on different colors as the extra
voice introduces new intervals and resolutions. One striking form has
two 242-cent thirds both resolving equably with 121-cent motions in
all voices:
D* 938 -- -121 -- Db* 817
C 696 -- +121 -- Db* 817
G* 242 -- -121 -- Gb* 121
F 0 -- +121 -- Gb* 121
The sonority F-G*-C-D* at 0-242-696-938 cents seems, at least in the
string kind of timbre I am using and find congenial, to have a
shimmering sound at once dense and spacious and beckoningly
modernistic. The outer 938-cent interval F-D*, although within five
cents of a just 12:7 at 933 cents (this interval is the octave
complement or inversion of our 262-cent third), seems to act something
like a very small minor seventh, contracting to the fifth Gb*-Db* by
the same equable 121-cent motions which help the two minor thirds to
contract to unisons and the middle major third to expand to a fifth.
Perhaps one might recall the medieval European concept of a minor
seventh as a minor-third-plus-fifth or _semiditonus cum diapente_:
here F-D* is indeed equal to the 242-cent minor third F-G* plus the
696-cent fifth G*-D*. This sound structure in the opening sonority,
plus the contraction of the outer voices to a fifth by contrary
motion, can together convey the impression of a "minor seventh" type
of category.
From this four-voice progression we might derive an octave scale with
two tetrachords we considered at the beginning of this discussion on
the 242-cent third, the first more of a soft diatonic and the second
more of a chromatic:
|------- 492 cents -----| tone |------ 504 cents ------|
F Gb* G* Bb C Db* D* F
0 121 242 492 696 817 938 1200
121 121 250 204 121 121 262
As there explained, the two tetrachords at 121-121-250 and 121-121-262
cents differ only in the size of the fourth -- and therefore also of
the upper step at 250 or 262 cents.
While in the special context of the above four-voice progression the
interval F-D* at 938 cents can take on some of the qualities of a
minor seventh, I should add that more typically it serves as a fine
septimal-flavor major sixth close to 12:7, as in this equable
progression also featuring some 121-cent steps:
D* 938 -- +141 -- E 1079
C 696 -- -121 -- B 575
A* 434 -- +141 -- B 575
F 0 -- -121 -- E -121
One note has been changed in the first sonority: instead of G* at a
242-cent minor third above F, we have A* at 434 cents, a major third
very close to a just 9:7, with D* a fourth higher, thus yielding a
tempered approximation of 14:18:21:24 (1:1-9:7-3:2-12:7, 0-435-702-933
cents).
The near-9:7 major third F-A* expands to the fifth E-B, and the
near-12:7 major sixth F-D* to the octave E-E, while the middle
262-cent minor third A*-C, close to 7:6, contracts to a unison on B.
The upper 242-cent interval C-D*, our special focus here, resolves
like a large 8:7 tone (it is not quite 11 cents wider), expanding to
the upper fourth B-E of the stable near-2:3:4 sonority E-B-E at which
we arrive. All of these resolutions involve equable motions of a
121-cent descent in one voice and a 141-cent ascent in another.
Thus F-D* at 938 cents, which took rather the role of a very small
minor seventh in the previous progression, here acts more routinely as
a large major sixth; and C-D* at 242 cents likewise shifts in role
from a very small minor third to a large major second. This kind of
role shift is typical for an interval like our 242-cent minor third or
major second with its distinctly interseptimal qualities, but seems
more remarkable for an interval like our 938-cent major sixth, as we
should usually describe it, within 5 cents of 12:7. One might guess
that the subtle zone of transition from a basically "septimal" to a
more distinctly "interseptimal" color could fall about 5-10 cents wide
of a ratio like the 8:7 major second or 12:7 major sixth, or narrow of
a ratio like the 7:6 minor third or 7:4 minor seventh.
Returning to our main focus on the 242-cent interval in its capacity
as a small minor third, we may recall an earlier remark that in
multi-voice polyphony this third can take on various colors depending
on the other intervals with which it is combined in a vertical
sonority. Thus its effect when combined with a 938-cent major sixth
acting as a kind of "quasi-seventh" contracting to a fifth was
especially striking or avant garde:
D* 938 -- -121 -- Db* 817
C 696 -- +121 -- Db* 817
G* 242 -- -121 -- Gb* 121
F 0 -- +121 -- Gb* 121
The following transposition, also featuring a lower 242-cent third
resolving by equable 121-cent steps, has an allied but somewhat
different shading, likewise alluring:
Ab* 950 -- -134 -- G 817
Gb 696 -- +121 -- G 817
Db* 242 -- -121 -- C 121
B 0 -- +121 -- C 121
Here the outer minor seventh B-Ab* contracting to the fifth C-G is at
950 cents, almost equidistant between 12:7 and 7:4, so that it can
readily serve either in this capacity or as a very large major sixth.
Both this interval and the upper 254-cent minor third Gb-Ab*
contracting to a unison on G resolve equably by steps of 121 cents
ascending and 134 cents descending. The middle 454-cent major third
Db*-Gb expands to the fifth C-G by 121-cent steps, as in the previous
progression.
This four-voice progression suggests an octave scale with the lower
121-121-262 tetrachord having a chromatic quality and the upper
121-134-250 tetrachord seeming to me to have a quality somewhere
between a soft diatonic and a chromatic.
|------- 504 cents -----| tone |------ 504 cents ------|
B C Db* E Gb G Ab* B
0 121 242 504 696 817 950 1200
121 121 262 192 121 134 250
The lower tetrachord B-C-Db*-E has a 121-121-262 pattern identical to
that of the upper tetrachord C-Db*-D*-F in a scale derived from the
previous progression, with the upper 262-cent step large enough to
give a clear chromatic feeling.
The upper tetrachord of this scale, Gb-G-Ab*-B, has a 121-134-250
pattern very similar to the lower tetrachord of the previous scale
F-Gb*-G-Bb at 121-121-250 cents, which has a smaller middle step and
fourth (121 and 492 cents as compared with 134 and 504 cents), the two
forms otherwise being identical. Curiously, while I find 121-121-250
somewhat on the soft diatonic side, 121-134-250 seems to have a
notably mysterious air somewhere between a diatonic and a chromatic, a
quality which might have special musical possibilities. Each version
has its own attractions.
To conclude this discussion of our 242-cent minor third or major
second, I should note that like the 1024/819 of Wilson's Rast/Bayyati
matrix at 241 cents (Section 2), it is about a third of the way from
8:7 at 231 cent to 7:6 at 267 cents. Zephyr 24, a tuning system
including the 1-3-7-9-11-13 eikosany plus the simple factors 1, 3, 9,
and 27, has a ratio sharing the same neighborhood, 1152:1001 at about
243 cents. Slightly smaller than these is an interval occuring in
septimal JI, 147:128 at 239.607 cents, wider than 8:7 by 1029:1024
(about 8.433 cents), and almost identical to a 5-tET step at 240
cents.
This survey of 5-step intervals in Zest-24 has been a very partial
one, with a specific focus on tetrachords, scales, and polyphonic
progressions involving equable divisions of these intervals bringing
into play neutral seconds or the sometimes allied 121-cent steps.
A study of these and other possibilities may lead to a further
appreciation, through composition and improvisation as well as
analysis, of the fascinating ambiguities of categorical perception
already recognized by scholars such as Erich M. von Hornbostel and
Mieczyslaw Kolinski; see the latter's "Gestalt Hearing of Intervals"
in Gustave Reese and Rose Brandel, eds., _The Commonwealth of Music:
In Honor of Curt Sachs_ (The Free Press, New York, 1965), pp. 368-374,
kindly brought to my attention by Douglas Leedy.
From this perspective, this paper may have combined two themes which
often intersect in my musical life, although not necessarily in
everyone else's. The first theme is the world of Near Eastern musics
and offshoots, as reflected in Wilson's Rast/Bayyati matrix and its
roots in the tuning of Zalzal rich with neutral thirds and seconds.
The second theme is the world of what I term interseptimal intervals,
denizens of realms such as those around 250, 450, 750, and 950 cents;
these intervals are relished in Javanese and Balinese gamelan and some
African musics, for example, and figure prominently in Kolinski's
paper.
To sum up, I hope that paper will suggest the richness of the
Rast/Bayyati matrix and of Erv Wilson's many other tuning systems and
perspectives on the sonorous universe of musical ratios both rational
and irrational.
--------------------------------------------------------------------
Appendix 1: Scala files for Wilson's Rast/Bayyati matrix and Zest-24
--------------------------------------------------------------------
Please let me emphasize that while Erv Wilson has most discerningly
mapped and described the Rast/Bayyati matrix, this Scala version of
his just tuning is strictly my responsibility, including any errors I
may have introduced. My warmest thanks to Erv, Kraig Grady, and others
who have made possible and generously shared the treasures of the
Anaphorian Embassy site, .
! wilson-RastBayyati24.scl
!
Erv Wilson scale from Rast/Bayyati matrix (27/22, 11/9)
24
!
8192/8019
256/243
12/11
9/8
1024/891
32/27
27/22
8192/6561
128/99
4/3
243/176
1024/729
16/11
3/2
4096/2673
128/81
18/11
27/16
512/297
16/9
81/44
4096/2187
64/33
2/1
! zest24.scl
!
Zarlino Extraordinaire Spectrum Temperament (two circles at ~50.28c apart)
24
!
50.27584
25/24
120.94826
191.62069
241.89653
287.43104
337.70688
383.24139
433.51722
504.18965
554.46549
574.86208
625.13792
695.81035
746.08619
779.05173
829.32757
887.43104
937.70688
995.81035
1046.08619
1079.05173
48/25
2/1
----------------------------------------------------------
Appendix 2: A 17-note subset in Zest-24 for Persian styles
----------------------------------------------------------
As mentioned in Section 3, Zest-24 includes a 17-note set somewhat
resembling the beautiful just tar tuning by Dariush Anooshfar in the
Scala scale archive (persian.scl). The following table presents the
two tunings side by side, and may facilitate an appreciation of both
the similarities and the compromises entailed in the tempered
version. For the Zest-24 version, a more or less conventional
"keyboard map" spelling is given at the left, and a possible spelling
adapting Persian notation at the right. In the Persian notation, a
koron 'p' typically lowers a note by about 50-70 cents, or somewhere
between a quartertone and a thirdtone, while a sori '>' raises it by a
similar amount, which may vary according to the taste of the performer
or the exigencies of tuning a fixed-pitch instrument.
Anooshfar's tar tuning Zest-24 (Eb-Eb*)
1/1 0.000 # 0 Eb 0.000 # 0 Eb
256/243 90.224 # 1 E/Fb 95.810 # 1 Fb
27/25 133.238 # 2 E* 146.086 # 2 Fp
9/8 203.910 # 3 F 216.759 # 3 F
32/27 294.135 # 4 Gb 287.431 # 4 Gb
243/200 337.148 # 5 Gb* 337.707 # 5 Gp
81/64 407.820 # 6 G 408.379 # 6 G
4/3 498.045 # 7 Ab 491.621 # 7 Ab
25/18 568.717 # 8 Ab* 541.897 # 8 Ab>
36/25 631.283 # 9 A* 650.276 # 9 Bbp
3/2 701.955 #10 Bb 708.379 #10 Bb
128/81 792.180 #11 B/Cb 791.621 #11 Cb
81/50 835.193 #12 B*/Cb* 841.897 #12 Cp
27/16 905.865 #13 C 912.569 #13 C
16/9 996.090 #14 Db 983.241 #14 Db
729/400 1039.103 #15 Db* 1033.517 #15 Dp
243/128 1109.775 #16 D 1104.190 #16 D
2/1 1200.000 #17 Eb 1200.000 #17 Eb
In this brief appendix, it may suffice to point out one important type
of compromise, and to comment quickly on Zest-24 note spellings. While
many step and interval sizes are similar in the two 17-note systems,
the _ordering_ of intervals can make a difference. Thus if we wish to
play a version of Shur on the 1/1 step, Anooshfar's just tuning offers
an ideal 133/161 division of the 32:27 or 294-cent minor third. In the
Zest-24 version, while the corresponding 287-cent minor third Eb-Gb
near 13:11 is not too far from 32:27, the 146/141 division Eb-E*-Gb
is rather less idiomatic. In relative terms, the lower step should be
smaller than the upper; and in absolute terms, it should probably be
around 125-140 cents, with 146 cents a bit large (see also Section 4).
Thus with Anooshfar's tar tuning, as with Wilson's Rast/Bayyati
matrix, the Zest-24 version is not so much an approximation as a
tempered variation.
On the matter of Zest-24 spellings, I should explain that one's choice
may be influenced by the musical context. In Anooshfar's tunings,
Pythagorean major and minor intervals are the norm -- as Marcus
suggests also for the Egyptian practice he studied. The table thus
prefers regular diatonic spellings for corresponding Zest-24 intervals
with respect to Eb as the reference note or "1/1," the intervals
formed with it by notes on the lower keyboard. In the Persian-style
notation, this preference applies consistently, and occasionally
results in remote accidental spellings: the minor second or limma
Eb-Fb and minor sixth Eb-Cb. The goal is to maximize consistency and
clarity -- within the limits set by an irregular temperament.
In the "keyboard map" spellings, the table offers a choice of E/Fb for
the minor second step and B/Cb for the minor sixth step. The first
form is more familiar in staying within the accidental range of Gb-A#
often used for notating 17-note systems, and the second more
diatonically correct or regular.
Here are Scala scale files for both Anooshfar's tar tuning and the
Zest-24 variation.
! persian.scl
!
Persian Tar Scale, from Dariush Anooshfar, Internet Tuning List 2/10/94
17
!
256/243
27/25
9/8
32/27
243/200
81/64
4/3
25/18
36/25
3/2
128/81
81/50
27/16
16/9
729/400
243/128
2/1
! zest24-persian_Eb.scl
!
Version somewhat like Darius Anooshfar's persian.scl, Eb-Eb
17
!
95.81035
146.08618
216.75861
287.43104
337.70688
408.37931
491.62069
541.89653
650.27584
708.37931
791.62069
841.89653
912.56896
983.24139
1033.51722
1104.18965
2/1
-----------------------------------------------------------------------
Appendix 3: An equable heptatonic scale in Wilson's Rast/Bayyati matrix
-----------------------------------------------------------------------
At the end of Section 4, I mentioned that Wilson's Rast/Bayyati matrix
includes a scale of what I term the "equable heptatonic" type, with
all seven steps having sizes within 30 cents or so of 1/7 octave at
about 171 cents, and some steps at or very near this 7-EDO size.
This scale takes for its reference note or "1/1" degree #8 of WIlson's
24-note matrix, a Pythagorean diminished fourth of 8192/6561 or about
384 cents above Wilson's 1/1, a ratio smaller than that of a 5:4 major
third by the 32805:32768 schisma, about 1.954 cents. Here are a Scala
scale file and a scale listing generated from this fine software
program by Manuel Op de Coul with the SHOW SCALE command:
! wilson-RastBayyati24-heptatonic.scl
!
Heptatonic from Wilson's Rast/Bayyati matrix like volans.scl
7
!
1594323/1441792
27/22
177147/131072
531441/360448
18/11
81/44
2/1
0: 1/1 0.000 unison, perfect prime
1: 1594323/1441792 174.097
2: 27/22 354.547 neutral third, Zalzal wosta of al-Farabi
3: 177147/131072 521.505 Pythagorean augmented third
4: 531441/360448 672.142
5: 18/11 852.592 undecimal neutral sixth
6: 81/44 1056.502 2nd undecimal neutral seventh
7: 2/1 1200.000 octave
A centerpiece of the scale is a step of 1594343/1441792 at 174.119
cents, or about 2.668 cents wide of a 7-EDO step. The most common step
size, found twice, is 65536:59049 at 180.450 cents, the Pythagorean
diminished third (e.g. C#-Eb in a regular chain of pure 3:2 fifths)
equal to two limmas or diatonic semitones at 256:243 or 90.224 cents
each. There is also a step at 72171:65536 or 166.958 cents.
These three step sizes of around 167, 174, and 180 cents, each within
10 cents of 7-EDO, provide a middle ground for the other interval
sizes of the Zalzalian 88:81 (143.498 cents) and 12:11 (150.637 cents)
on the narrow size, and 9:8 (203.910 cents) on the wide side.
A Scala table made with SHOW /LINE INTERVALS conveniently displays the
steps and intervals from the vantage point of each scale degree or
rotation of the notes:
0.0 : 174.1 354.5 521.5 672.1 852.6 1056.5 1200.0
174.1 : 180.4 347.4 498.0 678.5 882.4 1025.9 1200.0
354.5 : 167.0 317.6 498.0 702.0 845.5 1019.6 1200.0
521.5 : 150.6 331.1 535.0 678.5 852.6 1033.0 1200.0
672.1 : 180.4 384.4 527.9 702.0 882.4 1049.4 1200.0
852.6 : 203.9 347.4 521.5 702.0 868.9 1019.6 1200.0
1056.5: 143.5 317.6 498.0 665.0 815.6 996.1 1200.0
1200.0
The difference between the smallest step size at 88:81 or 143 cents
and the largest at 9:8 or 204 cents is, interestingly, identical to
that of Zalzal's scale; and the 12:11 step at 151 cents is also, of
course, common to both scales. This is a range of about 60 cents,
comparable to the 140-200 cents reported by Kevin Volans for the
African scale (Scala archive, volans.scl) discussed in Section 4.
Thus if we considered only the range between smallest and largest
steps, and suggested that they should remain within about 30 cents of
7-EDO (a total range of up to about 60 cents), then not only this
scale from Wilson's matrix, but also Zalzal's, could be called equable
heptatonic.
The former scale, however, meets an additional criterion I suggest:
that some of the intervals should be at or very close to 7-EDO size,
say within 5-10 cents. In Zalzal's scale, we have a jump from 12:11 to
9:8, with no intermediate steps in this 7-EDO neighborhood. The scale
from Wilson's matrix, in contrast, has four of its seven steps within
this general neighborhood: 167 cents, 174 cents, and two occurrences
of 180 cents.
These middle or near-7-EDO sizes provide a "bridge" of the kind I
discussed in Section 4 from Zalzalian and other typical Near Eastern
modal patterns contrasting neutral seconds and whole tones as more or
less discrete categories, to equable heptatonic patterns where the
middle range between and overlapping with these categories is also
populated or "filled in." The result is a more or less smooth
continuum of sizes centered at or near 7-EDO, so that step sizes to
either side of this central 7-EDO region, like those shared with
Zalzal's scale in this heptatonic from Wilson's matrix, are set -- and
heard -- in a different perspective.
We might thus have three criteria for an equable heptatonic scale, the
first two more definitional and the third more preferential, applying
specifically to tunings where some step sizes vary considerably from
7-EDO.
(1) All step sizes should be within about 30 cents of 7-EDO
(171.429 cents), thus keeping within an overall region of
about 60 cents or less.
(2) Some of these step sizes should be within about 5-10 cents
of 7-EDO, thus populating a "central" zone of the region.
(3) Step sizes might ideally form a rather smooth continuum with
gradual transitions, e.g. the 143-151-167-174-180-204 cent
sizes of the equable heptatonic from Wilson's matrix.
For 7-EDO itself and scales closely approximating it, only the first
two criteria are relevant; but for others with a wider spread of step
sizes, the third criterion suggests one guide for assessing the
degree of "equability."
Thus we might describe the scale from Wilson's matrix of having a wide
range (88:81 to 9:8, or about 60 cents, more precisely 729:704 or
60.412 cents) with a smooth continuum of step sizes.
In contrast, the Zest-24 equable heptatonic presented in Section 4 has
a somewhat narrower range (about 141.3-191.6 cents, or 50.3 cents,
more precisely the 50.276-cent diesis), but in a "jumpier"
distribution with only three step sizes of 141-171-192 cents.
Additionally, while the scale from Wilson's matrix has four of the
seven steps within 10 cents of a 7-EDO size, the Zest-24 scale has
only two such steps, both 171 cents.
Here are a Scala scale file and interval table:
! zarte24-volans_B.scl
!
Equable heptatonic like volans.scl (reported African scale)
7
!
171.22411
362.84480
504.18966
675.41376
867.03446
1058.65515
2/1
Equable heptatonic like volans.scl (reported African scale)
1/1 : 171.2 362.8 504.2 675.4 867.0 1058.7 2/1
171.2 : 191.6 333.0 504.2 695.8 887.4 1028.8 2/1
362.8 : 141.3 312.6 504.2 695.8 837.2 1008.4 2/1
504.2 : 171.2 362.8 554.5 695.8 867.0 1058.7 2/1
675.4 : 191.6 383.2 524.6 695.8 887.4 1028.8 2/1
867.0 : 191.6 333.0 504.2 695.8 837.2 1008.4 2/1
1058.7: 141.3 312.6 504.2 645.5 816.8 1008.4 2/1
2/1
More generally, in Zest-24 an equable heptatonic scale often, as here,
has these three step sizes, a situation I call a TOS or "Threefold
Order of Symmetry" by analogy with an MOS or Moment of Symmetry with
two such sizes. This 141-171-191-cent kind of structure may have a
somewhat different flavor than either a tuning with all steps at or
quite close to 7-EDO, or one with a comparable range but a smoother
continuum of interval sizes, as with the scale from the Wilson matrix.
Finally, I should emphasize that these remarks about an equable
heptatonic style of tuning -- or range of styles -- are meant to be
tentative and flexible. Thus a scale might have some steps smaller
than 140 cents and still present elements of an equable heptatonic
flavor -- as with some intonational styles of Southeast Asia said to
mix or hybridize elements of a more or less near-7-EDO outlook and of
a diatonic orientation contrasting tones and semitones. Any attempts
at categorization should serve to define not locked compartments but
open bridges along a continuum.
Margo Schulter
25 January 2007
Revised 11 October 2007