-----------------------------------------------------------
The e-based tuning (704.607 cents) as a "bifurcated" 17-EDO
Amine Beyhom and ideas for a 17-step notation: Part 1
-----------------------------------------------------------
In June 2000, I came upon the idea of a regular tuning where the ratio
of logarithmic sizes between the whole tone and diatonic semitone
would be equal to Euler's _e_, approximately 2.71828. This tuning has a
fifth at about 704.607 cents, a diatonic semitone (e.g. E-F) of 76.965
cents, and a chromatic semitone (e.g. F-F#) of 132.248 cents.
At the time, my motivation for choosing the arbitrary mathematical
perimeter was simply to explore an interesting region of what I then
termed the "neo-Gothic spectrum" between Pythagorean and 22-EDO, and
since have come to call "neomedieval" so as to include medieval (and
later) Near Eastern styles as well as those of Gothic Europe. My first
excited report on the new temperament from July 2000 reflects at once
my long experience with medieval European styles and my lack of focus
at that time on Near Eastern modal systems and styles:
While my first explorations early that summer involved a 12-note
tuning, in October 2000 I realized that the region around 704.6 cents
did have a special property: a chain of 15 fifths up formed a
virtually just 7:4. This serendipitous touch that I had _not_
considered originally in plucking Euler's _e_ out of the air as a
mathematical parameter to define a tuning, was nevertheless most
felicitous given Leonhard Euler's championship of 7:4 as a concordant
ratio which might be used in practiced as a tuning for minor
sevenths.
Near the end of summer in 2001, a critical turning point came when I
found myself in contact with George Secor, whose inestimable guidance
and mentorship led my to delve into the realm of Near Eastern music
with its rich repertory of Zalzalian or neutral intervals also
abounding in the tuning system we were mainly exploring, his 17-tone
well-temperament (17-WT) of 1978. Our collaboration also lead to my
devising a 24-note system called Peppermint based on two 12-note
chains of a regular temperament (704.096 cents) proposed by Keenan
Pepper. In terms of the sheer variety of Zalzalian steps and intervals
available in "only" 24 notes, and the overall accuracy of prime
approximations, I would still recommend Peppermint as an ideal
neomedieval temperament.
The creative influence of Shaahin Mohajeri leads me to draw a parallel
between Peppermint and the varied schemes for tuning the frets of the
Persian tar. His skill as a composer and knowledge of the Persian
dastgah system plays an important part in my appreciation for this
music and the related Near Eastern maqam systems which often show
similar patterns of intonation.
The e-based system, in contrast, offers the advantages of a completely
regular (although noncirculating) temperament: there are fewer types
of intervals, but available at more locations with more transpositions
easily possible. How did I come, around 2008, to explore this system
anew with avidly renewed interest?
Here Ozan Yarman lent pivotal impetus by making me more aware of the
depth and intricacy of maqam music as actually realized more or less
fluently in a given tuning system, theoretical or practical. His
compendious 79-MOS for the qanun based on a slightly modified version
of 159-EDO at once inspired me to consider the art more deeply, and to
renew my interest in the medieval Near Eastern theorists, especially
the "Systematists" of the later 13th century such as Safi al-Din
al-Urmawi and Qutb al-Din al-Shirazi. Their tuning schemes, sometimes
predicated in part on a regular 17-note Pythagorean system, seemed to
have a curious modern counterpart in the regular 24-note e-based
temperament with its wealth of Zalzalian and septimal intervals.
My exploration of the e-based system was further advanced last summer
when I studied in more depth some writings of the Lebanese musician
and scholar Amine Beyhom, and became aware of a fascinating
development in some of his recent articles. In his PhD thesis of 2003
on _Modal Systematics_, he had followed the familiar modern Arab
theoretical model of 24 steps per octave, while stressing that this
scheme serves only to identify the _type_ of steps in a given maqam or
jins (i.e. "genus," for example a trichord, tetrachord, or
pentachord), not to specify a precise intonation. Going beyond a
caution that these 24 steps are indeed unequal in practice (at least
to informed traditional performers!), he suggests the use of plus or
minus signs to convey fine points of intonation.
In the more recent articles, he intriguingly follows this general
approach, giving some actual measurements of interval sizes used by
flexible-pitch performers of maqam music, but with a proposal that a
conceptual model of 17 steps per octave might better fit many Near
Eastern styles of intonation. His ideas seemed beautifully to tie in
with some main intonational themes I had been pursuing in the e-based
system, a happy cross-pollination, as George Secor might call it,
leading to this article.
Part I focuses on the structure of the e-based temperament and the
three main interval families in a 24-note system, while Part II will
explore a slightly modified version of Beyhom's 17-step notation as
applied to some medieval and modern Near Eastern tunings as realized
in the e-based system.
--------------------------------------------------
1. Overview of the e-based tuning: a 17-step ethos
--------------------------------------------------
In its 24-note version, the e-based tuning has a character or ethos
largely shaped by three distinctive "colors" or families of intervals
met within a 17-note chain of fifths. The choice of a 24-note size
serves to make these intervals available at more locations, and also
more incidentally to introduce some rarer intervals which, along with
some others arising only in a larger tuning set (e.g. 29, 34, 46, 63,
or a circulating 109), will be briefly mentioned later in this
article.
A Scala file follows:
! eb24.scl
!
Regular "e-based" tuning, Blackwood's R or tone/limma at e, ~2.71828
24
!
55.28289
132.24835
187.53125
209.21382
264.49671
286.17928
341.46217
418.42763
473.71052
495.39309
550.67598
627.64145
682.92434
704.60691
759.88980
836.85526
892.13815
913.82072
969.10362
990.78618
1046.06908
1123.03454
1178.31743
2/1
My keyboard layout places this regular chain of 23 fifths on two
12-note manuals each with an accidental range of Eb-G#, indeed the
range on the Halberstadt organ which since 1361 has become a byword
for this arrangement. Corresponding notes on the two keyboards differ
by the diesis (12 fifths up) of 55.283 cents, an interval often
serving as an expressive narrow semitone. Since the tuning is entirely
regular, it is quite possible to use conventional spellings for all
notes, an approach which, as Aaron Johnson has pointed out in the
context of 31-EDO, can be useful with some musical software permitting
multiple sharps and flats but lacking support for "microtonal"
symbols. Here such a notation is shown together with a pragmatic
keyboard notation using an asterisk (*) to show a note on the upper
keyboard raised by the 55-cent diesis:
187.531 341.462 682.924 892.138 1046.069
C#*/Bx Eb*/D#3 F#*/Ex G#*/F#x Bb*/A#
_132.2|77.0_77.0|132.2_ _132.2|77.0_132.2|77.0_77.0|132.2_
C*/B# D*/Cx E*/Dx F*/E# G*/Fx A*/Gx B*/Ax C*/B#
55.283 264.497 473.145 550.676 759.890 969.104 1178.317 1255.283
209.214 209.214 76.965 209.214 209.214 209.214 76.965
--------------------------------------------------------------------------
132.248 286.179 627.641 836.855 990.786
C# Eb F# G# Bb
_132.2|77.0_77.0|132.2_ _132.2|77.0_132.2|77.0_77.0|132.2_
C D E F G A B C
0 209.214 418.428 495.393 704.607 913.821 1123.035 1200
209.214 209.214 76.965 209.214 209.214 209.214 76.965
The 24-note e-based tuning, like 17-note circles including 17-EDO and
more accurate unequal temperaments such as Secor's 17-WT, features
approximations of primes 2-3-7-11-13; additionally, there some
intervals close to ratios of 23. These prime factors, occurring early
and often, help shape the three main families of intervals: regular or
"accentuated Pythagorean" (1-6 fifths up or down); Zalzalian or
neutral (6-11 fifths); and septimal (12-16 fifths).
Intervals formed from chains of 1-6 fifths up or down, together with
the unison and 2:1 octave, make up the regular diatonic family. The
"accentuated Pythagorean" qualities of this family nicely fit much of
the polyphonic music of 13th-14th century Europe. Major and minor
thirds at 418 and 286 cents (near 14:11 and 13:11 or 33:28), and
sixths at 914 and 782 cents (near 22:13 or 56:33 and 11:7) very
effectively resolve to stable concords. The limma or diatonic semitone
at 76.965 cents, yet more compact and incisive than the Pythagorean
step of 256:243 (90.224 cents), facilitates many of these efficient
resolutions: it is almost exactly a just 23:22 (76.956 cents), and
often represents nearby ratios such as 22:21 (80.537 cents) or 117:112
(75.612 cents). The augmented fourth at 628 cents and diminished fifth
at 572 cents (near 23:16 or 56:39, and 32:23 or 39:28) are also
members of this family, and may occur as vertical intervals or melodic
outlines in European polyphony around 1200, for example. In many Near
Eastern contexts, however, they take on an affinity with the intervals
of our next family, and so may conveniently be considered as members
of it also.
Intervals with chains of 7-11 fifths make up the Zalzalian or neutral
family, a basic element of Near Eastern modal and intonational
systems. The name Zalzalian, of course, honors Mansur Zalzal, the
'oudist of 8th-century Baghdad famed for adding his _wusta_ or middle
finger fret at a neutral third measured by al-Farabi at 27:22 (355
cents) and by Ibn Sina at 39:32 (342 cents). This family includes
Zalzalian seconds at 132 and 154 cents (near 14:13 and 12:11 or
128:117); thirds at 342 and 363 cents (near 28:23 or 39:32, and 16:13
or 121:98); sixths at 837 and 859 cents (near 13:8 and 23:14);
sevenths at 1046 and 1068 cents (near 11:6 and 13:7); and Zalzalian
"tritonic" intervals of 551 and 649 cents, within a cent of the 11:8
"superfourth" and 16:11 "subfifth" as David Keenan has styled them.
In many Near Eastern contexts, the 6-fifth intervals of the augmented
fourth and diminished fifth at 628 and 572 cents may also be
considered as Zalzalian, having a musical kinship with their 11-fifth
relatives at 649 and 551 cents respectively, as will be discussed
shortly.
Intervals from chains of 12-16 fifths make up the septimal family of
intervals approximating ratios from the factors of 2-3-7-9. The diesis
of 55.283 cents (12 fifths up) serves as a very narrowly tempered
variant of the septimal thirdtone at 28:27 (62.961 cents), playing a
dramatic and effective role in the resolution of many other intervals
in this family. The septimal major third at 440.11 cents is some five
cents wide of 9:7, but the minor third at 264.50 cents is only about
2.37 cents narrow of 7:6, and the 969.10-cent minor seventh is a
virtually just 7:4 (968.825 cents). Rounding out this family is the
colorful narrow fourth at 473.71 cents, not quite three cents wide of
a just 21:16. These intervals and their octave complements or
inversions occur in various medieval Near Eastern tuning systems or
modes, and additionally provide one interpretation for the enhanced
cadential intonations of 14th-century European polyphony as described
by Marchettus of Padua and practiced by a modern performer such as
Christopher Page.
In this scheme of three main families there are two areas of overlap.
The first, as noted, involves intervals of 6 fifths up or down, which
enjoy "dual citizenship" in both the regular diatonic and Zalzalian
families, a point influencing the 17-step notation a la Beyhom we
shall soon meet. The second, more of a theoretical nicety, involves
intervals of 12 fifths up or down at 55 and 1145 cents, which
typically represent the septimal ratios of 28:27 or 27:14 (heavily
tempered at 7.68 cents narrow and wide respectively), but can also
represent the Zalzalian ratios of 33:32 and 64:33 (53 and 1147 cents).
A quick way of illustrating both points is to consider two versions of
a pentachord called Buzurg (likely Persian) or Buzruk (Arabic). The
first, from the "systematists" Safi al-Din and Qutb al-Din, nicely
illustrates the use of the 628-cent tritone (6 fifths up) as a
Zalzalian interval, here representing 56:39 (626 cents):
1/1 14/13 16/13 4/3 56/39 3/2
0 128 359 498 626 702
14:13 8:7 13:12 14:13 117:112
128 231 139 128 76
F# G* Bb B C* C#
0 132 363 495 628 705
132 231 132 132 77
Here I show both the original just intonation (JI) version and the
e-based realization to acknowledge the compromises of a regular
temperament: not only do we lose the pure fifths and fourths of the
original, but also the subtle distinction between 14:13 (128 cents)
and 13:12 (139 cents), both rendered as an identical 132 cents. In
both versions, a fourth plus a small Zalzalian second results in an
interval of 56:39, or a tempered 628 cents, which itself shares in the
pervasive Zalzalian flavor.
A variation on this medieval tuning of Buzurg or Buzruk proposed by
the Lebanese scholar Nidaa Abou Mrad illustrates how the 55-cent
diesis, although naturally placed in our septimal family as a highly
tempered equivalent of 28:27, may also represent the Zalzalian
"quartertone" at 33:32 (53.273 cents).
1/1 12/11 27/22 4/3 16/11 3/2
0 151 355 498 649 702
12:11 9:8 88:81 12:11 33:32
151 204 143 151 53
C* D E F* G G*
0 154 363 495 649 705
154 209 132 154 55
Here Mrad suggests how this pentachord might be realized by a musician
using the steps of Zalzal's tuning according to al-Farabi (9:8, 12:11,
and 88:81) -- plus a "minimal diesis" of 33:32! The tempered 55-cent
diesis nicely approximates this interval. In the absence of intervals
supporting a septimal interpretation (e.g. the 8:7 step in the
previous example), the 55-cent diesis may take on this special role as
a member of the Zalzalian family.
A 17-note chain of fifths suffices to produce all the intervals of our
three main families. From this perspective, while adding a few rarer
intervals to the mix, a 24-note system may serve mainly to make the
intervals of these families -- regular, Zalzalian, and septimal --
available at more locations. For example, a 17-note tuning would have
our virtually pure 7:4 available at only two locations, while a
24-note tuning expands this to nine locations.
Additionally, going to 24 notes brings explicitly into play an
interval exerting its influence "behind the scene," as it were, in
shaping the structure of a 17-note set. Let us now meet that
interval.
----------------------------------------
1.1. The 17-comma and a 17-step notation
----------------------------------------
In circulating 17-tone systems such as 17-EDO or George Secor's 17-WT,
17 fifths are equivalent to 10 octaves. In the e-based tuning,
however, 17 fifths at 704.607 cents each fall short of 10 octaves by
about 21.683 cents.
This small interval, called the 17-comma, thus makes its appearance
when the e-based system is expanded beyond 17 notes, where it results
from a chain of 17 fifths down (e.g. E*/Dx-F). While this interval may
serve as a direct melodic step or microtonal inflection, or as a
vertical sonority (e.g. a "complex unison" in gamelan), its role in
the 24-note e-based system is far more pervasive, in practice and
theory.
The 17-comma spells variety by defining the subtle difference between
two intervals of the same general type, such as smaller and larger
Zalzalian seconds at 132 and 154 cents (e.g. G#-A*/Gx and G#-Bb), or
septimal and regular minor thirds at 264 and 286 cents (e.g. C-D*/Cx,
C-Eb). We may note that these "kindred but distinct" pairs of
intervals have "17-complementary" chains of fifths: thus 7 fifths up
or 10 down for our Zalzalian seconds at 132/154 cents; and 14 fifths
up or 3 down for septimal and regular minor thirds at 264/286 cents.
This variety often means a special element of artistic choice when
two forms of a given step or interval differing by a 17-comma are
available from the same location, as often happens in our 24-note
system. Consider, for example, how the two interpretations of a
Buzurg/Buzruk pentachord given above could both be realized above a
final or resting note of G# -- in comparison to the single version
with its uniform interval sizes available at this or any other
location in 17-EDO. The comparison provides an apt occasion for
introducing some symbols for a 17-step notation.
Safi al-Din/Qutb al-Din version
G# A*/Gx C C# D*/Cx D#/Eb*
0 132 363 495 628 705
0 2s 5L 7 9s 10
132 231 132 132 77
2s 3+ 2s 2s 1
Mrad version
G# Bb C C# Eb D#/Eb*
0 154 363 495 649 705
0 2L 5L 7 9L 10
154 209 132 154 55
2L 3 2s 2L 1-
17-EDO (either version)
G# Bb/Gx C C# Cx/Eb D#
0 141 353 494 635 706
0 2 5 7 9 10
141 212 141 141 71
2 3 2 2 1
Using both the conventional and pragmatic keyboard spellings for the
notes on the upper manual in the e-based versions may make the
notation a bit more cluttered, but helps to show how two spellings
located a 17-comma apart in this tuning are equivalent in 17-EDO.
Thus in our two e-based versions, G#-Gx/G* (7 fifths up) yields a step
of 132 cents, and G#-Bb (10 fifths down) a step of 154 cents; in
17-EDO, either spelling yields an identical size of 141 cents.
The 17-EDO version is a good place to start in becoming acquainted
with the 17-step notation which is a main focus of this article. In
this equal temperament, there are only 16 interval sizes other than
the unison and 2:1 octave, consisting of from 1 to 16 steps of 1/17
octave. Thus 1 step defines the 71-cent limma or diatonic semitone
(also the diesis); 2 steps, the 141-cent Zalzalian second; and 3
steps, the major second or whole tone. We find 5 steps in a 353-cent
Zalzalian third, 7 in a 494-cent perfect fourth, 9 in a 636-cent
augmented fourth (or fifth-less-diesis), and 10 in a 706-cent perfect
fifth.
Our two e-based versions, in contrast, show two types of subtle
variations each reflected in their 17-step notations. The first type
involves the distinction between smaller and larger forms of Zalzalian
intervals, here most notably the seconds at 132 or 154 cents. These
may be considered as "small" and "large" counterparts of the 2-step
interval in 17-EDO at 141 cents, or 2s and 2L for short. Our first
e-based version uses the smaller 132-cent step (2s) only, representing
either 14:13 or 13:12, while the second version uses the 154-cent step
(2L) as the first and fourth melodic intervals of the pentachord. The
penultimate note of the pentachord stands at an interval above the
final G# of 628 cents in the first version (9s), and 649 cents (9L) in
the second, representing 56:39 and 16:11 in the original JI versions;
in any 17-EDO version, this 9-step interval is a uniform 635 cents, a
fine representation of the intermediately sized 13:9.
The second refinement of the 17-step notation uses a plus/minus
modification (+/-) to distinguish a regular interval, shown simply by
a number of steps without any sign, from a kindred interval of the
septimal family at a 17-comma wider (+) or narrower (-). In the first
version, the second melodic step of the pentachord is a septimal 8:7
tone shown as "3+"; in the second version, it is a regular 9:8 tone at
a tempered 209 cents, shown simply as "3". Likewise, the last melodic
interval in the first version of the pentachord is a regular limma or
diatonic semitone at 77 cents (simply "1"); but in the second it is
the narrower diesis at 55 cents, which may represent a septimal 28:27
or, as here, a Zalzalian 33:32 (marked as "1-").
Generally, as illustrated in the 17-step notations of these two
e-based versions, regular intervals are shown as a simple number of
steps without any marking; Zalzalian intervals are shown as small or
large (s/L); and septimal intervals are shown as a comma narrower or
wider (-/+) than their regular counterparts. For the purposes of this
notation, intervals of 628 and 572 cents (6 fifths up or down) are
treated as Zalzalian and grouped with the related forms derived from
a chain of 11 fifths: thus 551/572 cents as 8s/8L; and 628/649 cents
as 9s/9L. Intervals from 12 fifths up or down at 55 and 1145 cents,
whether in a septimal context (28:27 and 27:14) or a Zalzalian one
(33:32 and 64:33), take the septimal symbols: 1-/16+.
Another helpful point is that regular major intervals (seconds,
thirds, sixths, and sevenths) have wide (+) septimal forms, as does
the perfect fifth; while regular minor intervals have narrow (-)
septimal forms, as does the perfect fourth. This and other points
may become clearer as we explore the 17-step notation in more detail.
----------------------------------------------------------
1.2. 17-EDO "bifurcation": Interval catalogue and overview
----------------------------------------------------------
In 17-EDO, there are 16 categories of intervals other than the unison
and octave, each with a single uniform size throughout the tuning: for
example, the 5-step Zalzalian or neutral third at 352.942 cents. In a
17-note e-based tuning, each of these categories is in effect
"bifurcated" into a pair of kindred but distinct intervals with sizes
differing by the 17-comma of 21.683 cents. For example, we have
smaller and larger Zalzalian thirds at 341.462 and 363.145 cents,
shown as "5s" and "5L" in our 17-step notation.
From this "bifurcation" of the 16 categories in 17-EDO, we arrive in a
17-note e-based system at 32 intervals other than the unison and
octave, grouped into 16 kindred pairs. The following table shows each
of the 16 category in 17-EDO in order of increasing size, and the
corresponding pair of e-based intervals.
In reading the table, two points may be helpful. First, all 17-EDO
intervals are generated from a 17-note circle of fifths at 10/17
octave or 705.882 cents, and have sizes which are multiples of the
1-step minor second (also the diesis) at 1/17 octave or 70.588 cents.
Secondly, given that 17-EDO consists of a 17-note circle of fifths, a
given interval can be generated by moving along the circle in either
an upward (sharpward) or downward (flatward) direction: our 5-step
Zalzalian third at 353 cents, for example, from either 8 fifths down
or 9 fifths up (-8/+9 fifths). More generally, moving by n fifths in
one direction or (17-n) in the other produces the same interval.
In contrast, in the e-based system, such "17-complementary" chains as
8 fifths down or 9 fifths up will produce a kindred pair of intervals
differing by a 17-comma: here respectively our 5L/5s thirds at 363/341
cents. We might describe this 17-comma as the engine of bifurcation.
----------------------------------------------------------------------
Table of "Bifurcated" 17-EDO Counterparts in 704.607-cent tuning
----------------------------------------------------------------------
17-EDO steps 5ths type cents 704.607: symbol 5ths cents ~JI
----------------------------------------------------------------------
1- +12 55.28 32:31
1 -5/+12 min2 70.59
1 -5 76.97 23:22
----------------------------------------------------------------------
2s +7 132.25 14:13
2 +7/-10 Zal2 141.18
2L -10 153.93 12:11
----------------------------------------------------------------------
3 +2 209.21 44:39
3 +2/-15 Maj2 211.76
3+ -15 230.90 8:7
----------------------------------------------------------------------
4- +14 264.50 7:6
4 -3/+14 min3 282.35
4 -3 286.18 13:11
----------------------------------------------------------------------
5s +9 341.46 28:23
5 -8/+9 Zal3 352.94
5L -8 363.14 21:17
----------------------------------------------------------------------
6 +4 418.43 14:11
6 +4/-13 Maj3 423.53
6+ -13 440.11 9:7
----------------------------------------------------------------------
7- +16 473.71 21:16
7 -1/+16 4 494.12
7 -1 495.39 4:3
----------------------------------------------------------------------
8s +11 550.68 11:8
8 -6/+11 dim5 564.71
8L -6 572.36 32:23
----------------------------------------------------------------------
9s +6 627.64 23:16
9 +6/-11 Aug4 635.29
9L -11 649.32 16:11
----------------------------------------------------------------------
10 +1 704.61 3:2
10 +1/-16 5 705.88
10+ -16 726.29 32:21
----------------------------------------------------------------------
11- +13 759.89 14:9
11 -4/+13 min6 776.47
11 -4 781.57 11:7
----------------------------------------------------------------------
12s +8 836.86 13:8
12 +8/-9 Zal6 847.06
12L -9 858.54 23:14
----------------------------------------------------------------------
13 +3 913.82 22:13
13 +3/-14 Maj6 917.65
13+ -14 935.50 12:7
----------------------------------------------------------------------
14- +15 969.14 7:4
14 -2/+15 min7 988.24
14 -2 990.79 39:22
----------------------------------------------------------------------
15s +10 1046.07 11:6
15 -7/+10 Zal7 1058.82
15L -7 1067.75 13:7
----------------------------------------------------------------------
16 +5 1123.03 44:23
16 +5/-12 Maj7 1129.41
16+ -12 1144.72 31:16
----------------------------------------------------------------------
In many ways 17-EDO is an admirably compact system: all intervals are
available from any location in the system, something not true in a
17-note or even a 24-note e-based temperament. The circle neatly
provides a full complement of regular diatonic intervals, including
the augmented fourth and diminished fifth at 635 and 565 cents with
their frequently Zalzalian character, plus a Zalzalian second, third,
sixth, and seventh, making up an attractive set for neomedieval music
in an "accentuated Pythagorean" style a la 13th-14th century Europe or
a Near Eastern style.
From a Near Eastern perspective, a main weakness of this solution is
that we are restricted to one Zalzalian interval in each category, a
limitation shared in common with 24-EDO when literally applied.
However, as Beyhom and others have shown, we can take a conceptual map
of either 17 or 24 steps per octave as a convenient basis for a
flexible model of intonation where the steps and intervals are open to
nuances and shadings. The e-based tuning is one possibly way to
realize certain aspects of such an approach on a fixed-pitch
instrument, with a 24-note tuning having the advantage of making
available at more locations the intervals found within a 17-note chain
of fifths and shown in the above table.
Having catalogued these intervals, we can get an overview of the order
in which they are generated by a chain of fifths in the following
diagram showing the three main families we earlier surveyed:
small/large (s/L)
Zalzalian (Neutral)
{------------------------}...
+1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16
705 209 914 418 1123 628 132 837 341 1046 551 55 760 264 969 474
10 3 13 6 16 9s 2s 12s 5s 15s 8s 1- 11- 4- 14- 7-
-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16
495 991 286 782 77 572 1068 363 859 154 649 1145 440 936 231 726
7 14 4 11 1 8L 15L 5L 12L 2L 9L 16+ 6+ 13+ 3+ 10+
{------------------}... {-------------------}
Regular diatonic Septimal
No modifier plus/minor (+/-)
Intervals of +/- 6 fifths use Zalzalian s/L, and may also be regular
Intervals of +/- 12 fifths use septimal +/-, and may also be Zalzalian
Here the upper half of the table shows intervals generated from upward
chains of fifths, while the lower half focuses on downward chains.
Each half has an upper line showing the number of fifths up (+) or
down (-); a middle line showing interval sizes in cents; and a lower
line showing the 17-step notation for each interval.
The scope of each of the three families is also shown, with dotted
lines showing that a given family extends to an interval also
belonging to the next family and using the 17-step modification
symbols of that family: thus the "regular but also Zalzalian"
intervals of +/- 6 fifths with their s/L modifications; and the
"septimal but also Zalzalian" intervals of +/- 12 fifths with their
+/- notations.
-------------------------------------------------------
2. More remote intervals: beyond 17 fifths, or 24 notes
-------------------------------------------------------
In addition to the three main families we have surveyed, all found
within a 17-note version of the e-based temperament, a 24-note tuning
gives rise to some more remote intervals which here, as promised, will
be briefly surveyed. A tuning set larger than 24 would generate yet
more sizes of intervals, some of which might be of special interest to
a musician such as Ozan Yarman.
We have already encountered the 17-comma of 21.68 cents generated by a
chain of 17 fifths down -- with 17 fifths up producing an octave less
this comma, or 1178.32 cents. In a septimal context, these intervals
might be interpreted as 64:63 and 63:32 at 27.26 and 1172.74 cents.
While a special comma sign such as "k" might be desirable, we could
show these intervals within our 17-step system of notation as
respectively a "0+" to show a regular unison or "0-step" interval plus
a comma, and "17-" to show a usual 17-step interval (the octave) less
a comma.
The remaining "remote" intervals generated in our 24-note system from
chains of 18-23 fifths have the common property of being regular
diatonic intervals (here including the augmented fourth or diminished
fifth) altered by a 17-comma in a direction _opposite_ to that of the
septimal family (12-16 fifths). In other words, the perfect fifth,
major intervals, and also the augmented fourth are in these remote
variations a comma narrower; while the perfect fourth, minor
intervals, and diminished fifth are a comma wider. These intervals can
thus nicely be expressed by the same +/- notation we use for the
septimal family, but with opposite signs for septimal and remote
variations on the same regular interval.
Starting from an interval in many ways more pervasive than "remote,"
the 17-comma itself, we thus have the following intervals:
----------------------------------------------------------------------
fifths cents 17-step symbol type
----------------------------------------------------------------------
+17 1178.32 0+ octave less 17-comma
-17 21.68 17- 17-comma
----------------------------------------------------------------------
+18 682.92 10- narrow fifth
-18 517.08 7+ wide fourth
----------------------------------------------------------------------
+19 187.53 3- narrow major second
-19 1012.47 14+ wide minor seventh
----------------------------------------------------------------------
+20 892.14 13- narrow major sixth
-20 307.86 4+ wide minor third
----------------------------------------------------------------------
+21 396.75 6- wide major third
-21 803.25 11+ wide minor sixth
-----------------------------------------------------------------------
+22 1101.35 16- narrow major seventh (~17:9)
-22 98.65 1+ wide minor second (~18:17)
-----------------------------------------------------------------------
+23 605.96 9- narrow augmented fourth
-23 594.04 8+ wide diminished fifth
-----------------------------------------------------------------------
The notation for the intervals at +/- 23 fifths at 606 and 594 cents
follows a convention that when a usually Zalzalian number of steps
normally taking the s/L modifications is instead follows by +/- sign,
this notation means the larger Zalzalian size plus a comma or the
smaller size minus a comma. Thus 9- means 9s (a 628-cent augmented
fourth) less a 22-cent comma; while 8+ means 8L (a 572-cent diminished
fifth) plus a comma, yielding 606 and 594 cents respectively.
Here I have taken note of possible just ratios only for the intervals
of +/- 22 fifths, which yield virtually just versions 17:9 and 18:17,
the latter a ratio famously present in some lute fretting schemes. The
wide minor and narrow major thirds from -20 and +21 fifths, at 308 and
397 cents, are not too far from the related ratios of 81:68 (the wusta
or middle finger placement known as _Fars_ or "Persian," at 302.86
cents); and 34:27 at 399.09 cents, or 4:3 less 18:17. The longest
chains in this tuning, at +/- 23 fifths, produce intervals rather
close to 17:12 and 24:17 at a virtually precise 603 and 597 cents.
The remote thirds and sixths in this family could also be described as
having a "meantone" quality, allowing one to produce, at a few
locations, plausible approximations of Near Eastern interpretations of
ratios of 5. This liberty is to me most pleasing, and most in keeping
with the spirit of a 24-note tuning, when other and often more
characteristic intervals are also present to give the intonation a
special quality.
For example, Amine Beyhom describes a striking Turkish intonation he
measured for a Hijaz tetrachord -- about 130-265-90 cents or
0-130-395-485 cents -- which I found myself emulating as follows:
Bb B C#* D*
0 132 397 474
132 264 77
The narrow fourth, close to 21:16 in this version as opposed to the
somewhat more nuanced 485 cents in the Turkish version, and the middle
step at 7:6, contribute to my fascination with this emulation. The
397-cent third might represent 5:4 (not unlikely in a Turkish
tradition), 34:27, or simply "the 395 cents that the performer used."
One application for the virtually just 18:17 step (-22 fifths) was
inspired by a suggestion of the Syrian musician and theorist Tawfiq
al-Sabbagh, as translated and explained by the Lebanese performer and
teacher Ali Jihad Racy, that a Maqam called Sikah Arabi should be
played in the manner of "Turkish Huzam," with steps measured in
Pythagorean of 53-EDO commas at 6 9 5 12 5 9 7. What I arrived at was
this, with steps shown in approximate 53-commas and 17-step notation:
Bb* B* C#* Eb F* F#* G#* Bb*
0 132 341 440 705 837 1046 1200
0 6 15 19 31 37 46 53
0 2s 5s 6L 10 12s 15s 17
132 209 99 264 132 209 154
6 9 4 12 6 9 7
2s 3 1+ 4- 2s 3 2L
Here I will offer no analysis except to say that it is interesting all
the melodic steps except the 99-cent or 18:17 step at C#*-Eb come from
the three main families; and that I only became aware of some of the
other remote intervals (here from 20-23 fifths) present when I tried
"SHOW INTERVALS" in Scala. Also, I have used some quite inaccurate
rounded numbers of 53-commas: 99 cents is close to 4.5 commas, and
likewise 440 cents to 19.5 commas. My purpose, perhaps a la Beyhom, is
to convey the type of interval rather than offer an exact measurement.
Finally, it may be noted that going beyond 24 notes would immediately
introduce a new interval: a double diesis (24 fourths up) equal to
110.566 cents, closely approximating a just 16:15 (111.731 cents). A
chain of 25 fifths down, subtracting this interval from a regular
fourth, would result in a major third at 384.827 cents, about 1.49
cents narrow of 5:4. A chain of 26 fifths up would yield a minor third
at 319.780 cents, about 4.14 cents wide.
For a musician who seeks a fixed-pitch system of intonation with a
versatile choice of intervals including accurate ratios of 5, how
would a 46-note or larger e-based system rate?
Judging by my dialogues with Ozan Yarman, and his eloquent writings,
two vital aspects of his 79-MOS based on a subtly modified 159-EDO for
the qanun and other instruments would not be present. First, the
system would not fully circulate until one reached 109 notes, at which
point it would almost identical to a complete set of 109-EDO.
Secondly, while a 24-note e-based system is intended to focus mainly
on the families of regular, Zalzalian, and septimal intervals
generated within the first 17 fifths, plus the subtle 17-comma as the
instrument of artful discretion, a performer seeking 5-based intervals
as primary ingredients might find the decidedly non-meantone structure
of the e-based tuning as a weighty impediment.
Both considerations are central to the design of Ozan Yarman's 79-MOS:
not only are many 5-based intervals present, but they are often
available from meantone-like chains where no fifth is impure by more
than 1/3 Holdrian comma (about 7.55 cents).
Thus a 17-step notation for the 24-note e-based system not only
follows up a brilliant suggestion by Beyhom, however modestly and
inexpertly, but fits the primary purpose of the tuning as it has
developed over the last 10 years: a "bifurcated" variation on 17-EDO
realizing the admirable ethos of that system while ornamenting it with
diversity and variety.
Most respectfully,
Margo Schulter
mschulter@calweb.com
Most appreciatively,
Margo Schulter
mschulter@calweb.com