----------------------------------------------------
The Turquoise-17 System and 33:36:39:42:44
Homage to Scott Dakota and the 13/11 minor third
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Scott Dakota's presentation of his 24HzMasterMind JI system caught my
attention, including his remark about the one 1/1-13/11-3/2 sonority
in the system.
This make me think about Turquoise-17, a near-just tuning I originally
developed back in 2011, which I recalled had 1/1-13/11-3/2 or 22:26:33
in lots of locations -- 9 out of 17, as Scala helpfully informed me.
A bit of musical context: in my everyday composition or improvisation,
22:26:33 (0-289.2-702.0 cents), or its tempering in this version of
Turquoise-17 at 0-288.3-703.1 cents or 0-289.5-704.3 cents (such are
the minute variations of 1024-ed2, my synthesizer's resolution), is a
standard cadential sonority where the lower minor third resolves to a
unison and the upper 33:26 major third (at 412.7 cents, or here a
tempered 414.8 cents) expands to a fifth. For example:
G# A
E D
C# D
Really, it's a slight modernization or tweaking of your classic 14th
century European Pythagorean tuning, where C#-E-G# would be
1/1-32/27-3/2, or 54:64:81, at 0-294.1-702.0 cents, with the upper
major third at 81:64 (407.8 cents). We might call this a touch of
neoclassicism, or more specifically neomedieval neoclassicism.
However, as I learned during the years 2000-2002 with help in the
latter part of that period from my mentor George Secor, getting primes
11 and 13 into act opens the way for other relationships not so
familiar in the European tradition, but quite familiar to the Near
Eastern tradition at least since Ibn Sina (c. 980-1037). While there
are scattered references to prime 13 in Greek theory and the earlier
Near Eastern theory of al-Farabi (c. 870-950), it was Ibn Sina who
made prime 13 central to his scheme of tetrachords and his approach to
tuning the `oud (of which the European lute is one offshoot).
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1. A Catalytic Reaction: Bleu + Jacques Dudon ---> Turquoise-17
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Along with some discussions where Jacques Dudon shared his wisdom and
ideas, a catalyst to Turquoise-17 was a system called Bleu presented
on the Tuning List around this time, a prime 2-3-7-11-13 temperament
where the idea seemed to be to disregard lots of commas that I enjoy
observing.
For Turquoise, my nucleus, as with Scott Dakota's 24HzMasterMind
tuning, was a relatively simple sequence of ratios: the division of
the fourth into 33:36:39:42:44 or 12:11-13:12-14:13-22:21, with steps
in a JI version of 150.6-138.6-128.3-80.5 cents, and notes at
1/1-12/11-13/11-14/11-4/3 or 0-150.6-289.2-417.5-498.0 cents. This
could be summed up as a series 11:12:13:14 plus a 22:21 step
completing the 4:3 fourth.
In the version of Turquoise-17 I arrived at by the end of July 2011,
as a subset of my new MET-24 temperament actually very close to the
relevant portion of George Secor's High Tolerance Temperament (HTT)
family of systems such as 29-HTT, the 12:11 and 13:12 steps are always
within a cent of just, as is 13/11. The 14:13 steps are not quite so
accurate at 125.4 or 126.6 cents, being narrow by 1.7 or 2.9 cents of
a just 128.3 cents. As for 33/26 (412.745 cents) and 14/11 (417.508
cents), two reasonable just interpretations for the regular major
third at 414.8 cents, they are almost equally impure, but with a
slight leaning toward 33/26 (at 2.098 cents wide), as compared to
14/11 (at 2.664 cents narrow).
The 17-note structure of Turquoise grows out of three tempered series
of 33:36:39:42:44 on the 1/1, 4/3, and 3/2 steps. Here are two
slightly different versions of this tetrachord, illustrating the
variations caused by the granularity or limited resolution of
1024-ed2. I show note locations in MET-24, approximate just ratios and
just values in cents, and tempered values and deviations from just:
In this MET-24 notation, I take the upper chain of fifths as the
reference for the 1/1 step at G, and use the symbol "d" (like an Arab
half-flat) to show a note on the lower chain, lowered by the spacing
of 57.422 cents.
150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
0 150.6 289.2 417.5 498.0
1/1 12/11 13/11 14/11 4/3
33 36 39 42 44
G Ad Bb B C
0 150.0 288.3 414.8 495.7
0 -0.6 -0.9 -2.7 -2.3
150.0 138.3 126.6 80.9
-0.6 -0.3 -1.7 +0.3
150.6 138.6 128.3 80.5
12:11 13:12 14:13 22:21
0 150.6 289.2 417.5 498.0
1/1 12/11 13/11 14/11 4/3
33 36 39 42 44
C Dd Eb E F
0 150.0 289.5 414.8 496.9
0 -0.6 +0.2 -2.7 -1.1
150.0 139.5 125.4 82.0
-0.6 +0.9 -2.9 +1.5
Starting with this foundation, I added other notes to build a 17-note
system where each step is a small, middle, or large thirdtone, with
the vagaries of 1024-ed2 inducing more slight variations in sizes:
57.4, 68.0 or 69.1, and 80.9 or 82.0 cents.
! met24-turquoise_Gup.scl
!
Turquoise 17 temperament in MET-24
17
!
67.96875
150.00000
207.42188
288.28125
357.42188
414.84375
495.70313
564.84375
645.70312
703.12500
785.15625
853.12500
910.54688
992.57813
1060.54688
1117.96875
2/1
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2. Overall Philosophy: Lots of usual fifths, and two others also
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One medieval European, or more generally Pythagorean aspect of my
usual intonational philosophy is a quest for lots of just, or often
"near-just," fifths and fourths. As George Secor taught me, a really
"near-just" fifth should be more accurate than 12n-ed2's 700 cents,
that is, closer than 1.955 cents to a pure 3:2. In 1024-ed2, my
solution in MET-24 and its implementation of Turquoise-17 is to use
chains of fifths alternating sizes of 600 and 601 tuning steps, or
703.125 cents and 704.297 cents. The first is really "near-just" at
around 1.17 cents wide, while the second at around 2.34 cents wide is
what I'd call "close enough, and about the best I can do at this
resolution." One test is that regular tones or major seconds are at
207.4 cents, or 3.512 cents wide; not so close as George Secor's HTT
family with fifths of 703.579 cents (3.247 cents wide), but still
closer than 200 cents in 12n-ed2 (3.910 cents narrow).
In Turquoise-17, there are fifths at these usual sizes at 15 of the 17
locations. The exceptions are G#d-Eb (~27/26-11/7) at 717.2 cents,
curiously a near-just 286/189; and F#-Dd at 727.7 cents, close to
32:21 (729.2 cents).
Apart from helpfully balancing out the rest of the 17-note cycle,
these two fifths and their complementary fourths play musically
valuable roles in three ways. First, it's their presence that makes
possible septimal approximations: for example, three locations with
minor thirds at 264.8 cents (narrow of 7:6 by 2.0 cents); and three
at 275.4 or 276.6 cents (wide of 7:6 by 8.5 or 9.7 cents, and right
around the 276 cents measured in one Persian tuning of Dariuche
Safvate in 1966).
Secondly, the 21:16 has wonderful uses in contexts like this
Turquoise-17 sonority, for which I'll show approximate ratios in terms
of the system's usual 1/1, G, as well as for this pentatonic scale
based on the series favored by Jacques Dudon and others for a JI
slendro at 12:14:16:18:21:24.
G: 12/11 14/11 16/11 18/11 21/11 24/11
1/1 7/6 4/3 3/2 7/4 2/1
12 14 16 18 21 24
0 266.9 498.0 702.0 968.8 1200
Ad B Dd Ed F# Ad
0 264.8 495.7 703.1 968.0 1200
0 -2.0 -2.3 -1.1 -0.9 0
When 12:14:16:18:21 is played as a simultaneous five-note sonority, a
wonderful concord results, with the 21:16 blending in and at the same
time adding a bit of energy and interest to the overall effect.
The other "wolf" fourth at 482.8 cents has an idiomatic melodic use in
a variation on what Nelly Caron and Dariuche Safvate describe as a
traditional version of the Avaz or satellite modal family in Persian
music of Bayat-e Tork (the "Song or Verses of the Turks," meaning the
ethnic Turks of Iran), and call "the Old Tork."
Eb F F# G#d Bb C C#d Eb
0 207,4 332.8 482.8 703.1 910.5 979.7 1200
207.4 125.4 150.0 220.3 207.4 69.1 220.3
Caron and Safvate suggest that a fourth of around 484 cents was
typical for the Old Tork, but went out of fashion in a quest for more
regular fourths and fifths. While this overall tuning is not
necessarily typical of Iranian practice, the large whole-tone steps at
around 220 cents, and also the narrow minor thirds at 275.4 or 276.6
cents (F-G#d and Bb-C#d), are characteristic of some Persian practice
leaning toward what might be called "minor thirds and sevenths a bit
larger than septimal," say by around 10 cents.
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3. More Philosophy: Some maqamistic touches
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While a keyboard tuning as small as 17 notes can hardly capture the
various fine nuances of flexible-pitch Maqam performance, it can at
least represent some of the possible distinctions and shadings, giving
a kind of sampler of Zalzalian or middle intervals, sometimes also
known as "neutral" intervals.
The nuclear series of 33:36:39:42:44 itself includes some elements and
distinctions of this type, with Turquoise-17 making available various
orderings or permutations of these steps and intervals.
For example 33:36:39:44 or 1/1-12/11-13/11-4/3, e.g. G-Ad-Bb-C (at a
tempered 0-150.0-288.3-495.7 cents or 150.0-138.3-207.4 cents), with
its 11:12:13 division of the 13/11 minor third, might be a tuning of
Arab Huseyni. In contrast, the closely related 52:48:44:39 or
1/1-13/12-13/11-4/3, e.g. Bd-C-Dd-Ed, with its 13:12:11 division (here
138,3-150.0 cents) nicely fits Arab Bayyati or Persian Shur.
One famous tetrachord that helped me find the 33:36:39:42:44 division
back in 2002 is Qutb al-Din al-Shirazi's Hijaz around 1300 at
33:36:42:44 or 1/1-12/11-14/11-4/3, e.g. G-Ad-B-C (tempered at
0-150.0-414.8-495.7 cents). A simple ascending version of Hijaz shows
this tetrachord and two possible other tetrachords also based on or
closely allied with the 33:36:39:42:44 series. Again, just values are
shown in the upper portion of the diagram, and tempered values below.
Hijaz 33:36:42:44 Huseyni 33:36:39:44
|------------------| |-----------------|
0 150.6 417.5 498.0 702.0 852.6 991.2 1200
1/1 12/11 14/11 4/3 3/2 18/11 39/22 2/1
|------------------|
Rast 39:44:48:52
G Ad B C D Ed F G
0 150.0 414.8 495.7 703.1 853.1 992.6 1200
While the superparticular ratios have a special intellectual elegance,
and receive a special honor in classic Near Eastern theory of the
10th-15th centuries (with Ptolemy as an important influence), more
important in practice is some variety of shadings in steps and
intervals. The goal of Turquoise-17 is to maximize this subtle variety
while also maximizing the number of near-pure fifths and fourths.
This is a situation where a 24-note regularized keyboard like the
MET-24 layout can help in distinguishing the 15 usual fifths from the
two special ones at 717 and 728 cents. In this layout, as in a
standard European medieval Pythagorean or Renaissance meantone tuning,
all fifths found within either chain are usual, except for a
diminished sixth at G#-Eb. Fourths or fifths formed by mixing notes
from different keyboards will be "special," as with G#-Ebd (717 cents)
and F#-Dd (728 cents).
Of course, a mapping to a generalized keyboard could provide these
advantages and more!
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4. Conclusion: Consequences of neomedievalism
---------------------------------------------
The basic philosophy of using 13/11 and 14/11 as regular thirds is
that medieval Pythagorean intonation, as an approach to 12th-14th
century polyphony, is fine; but one interesting 21st-century option is
to "modernize" it a bit. I had reached this point by 2000. It serves
the music well, and introduces what I thought of at time as
"intriguing neutral or semi-neutral thirds" at ratios like 17:14 or
21:17, for example (336.1 and 365.8 cents). Turquoise-17 has one small and
middle thirds at 332.8 cents (Eb-F#), close to 63/52 (332.2 cents),
and two large middle thirds at 370.3 cents, close to 26/21 (369.7
cents), both 13-based ratios that occur in Ibn Sina.
However, it was under the aegis of George Secor's mentorship in
2001-2002 that I came to appreciate the possibility of dividing 13/11
into 11:12:13, and likewise 14:11 into 11:12:13:14, etc. In such
divisions, as in Ptolemy's Equable Diatonic at 12:11:10:9 (which
George also called to my attention in suggesting that we describe
12:13:14 or 11:12:13 also as an "equable" division), the subtly
unequal middle or Zalzalian steps become a main melodic focus, which
is precisely the situation that obtains in the art of Maqam.
Another point illustrated by Turquoise-17 is the way that not
attempting to support prime 5 in a given system may open other
possibilities. The intricacies of primes 2-3-7-11-13 are the creative
terrain here. In contrast, many styles of Maqam music do give a vital
role to prime 5, whether 5/4, for example, is seen as a modification
of a usual major third generally understood to be somewhere around
Pythagorean; or, as in the Turkish tradition which can cite some
theory from Safi al-Din al-Urmawi and Qutb al-Din al-Shirazi in the
period around 1250-1300, taken as a large middle third in Maqam Rast,
for example.
Thus the goal is not to cover all the territory of Maqam, but to
present a useful subset where a variety of steps and intervals can
permit at least a degree of ear training and creative choice.
I should also caution that while this 17-note system is fine for most
13th-century European polyphony, where a range for the upper chain of
fifths at Eb-F# should be fine (and pieces using C# can often be
transposed down a fifth or up a fourth), the break in the chain at
F#-C# (with F#-Dd near 32/21 as the closest replacement) could cause
problems for 14th-century music. The full MET-24 would remedy the
problem by providing two near-Halberstadt keyboards (Eb-G#) -- if we
reasonably assume that the Halberstadt organ in 1361 was tuned in
regular Pythagorean with the Eb-G# range shown in a later diagram, and
which indeed fits much of the music of the era.
To sum up, Turquoise-17 is a kind of compromise between the JI ideal
and the desire for lots of near-just fifths; and between the desire to
observe lots of commas with reasonable accuracy, and to arrive at a
relatively predictable and symmetrical 17-note thirdtone system.
Margo Schulter
mschulter at calweb dot com
3 December 2014