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A Friendly Introduction to "Rank-3" Temperaments:
Designing a System with Three Degrees of Freedom
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With warmest thanks to Jake Freivald for his essay on "Three Similar
Temperaments," to Chris Vaisvil for his idea of focusing on "degrees
of freedom," and to Keenan Pepper for the quip that launched this
article, I will here attempt a friendly introduction to what may be
termed "Rank-3" tuning systems.
This is meant to be a newbie-friendly as well as oldster-friendly
presentation. The main theme is how a tuning system can be designed
with what Chris Vaisvil's post suggested might be called three
independent "degrees of freedom" -- that is, three different
dimensions or aspects of the system we can fine-tune in creative ways.
What follows is meant to address the basic concepts rather than the
tweaking, micro-optimizing, and haggling over thousandths of a cent.
Please let me add with great appreciation and admiration that George
Secor was already using all three degrees described here, and more,
when he designed his High Tolerance Temperament in 29 notes (HTT-29)
back in 1978. And Kraig Grady has remarked that Erv Wilson's writings
include many similar ideas.
People may joke that no one really knows what is going on with these
"Rank-3" temperaments, but actually the basics for at least some of
these systems are clear and easy.
Before getting into our degrees of freedom, I'd like to offer a link
to a wonderful article by George Secor himself on some uses for
another temperament he designed in 1978, his 17-tone well-temperament
(17-WT), that includes a number of scales available in the system
we're about to explore, as well as a few that depend on the specifics
of 17-WT, a great tuning in itself for newbies or others who can
comfortably deal with 17 notes per octave.
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1. Liberation Day: Understanding the three degrees of freedom
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To understand what the three degrees of freedom are, we'll build a
simple temperament that actually is musically quite effective, and
might be tuned on some synthesizer that rounds everything to the
nearest cent. This keeps the math easier!
Later on, we'll look at the "fine-tuning," but the basic ideas are
sometimes best understood when the numbers and calculations are kept
as simple as possible.
So let's assume that we have a synthesizer that can only tune to the
nearest rounded cent, or 1200 steps per octave -- actually, a bit
higher resolution than the one I use, although quite crude compared to
some state-of-the-art synths!
Recently there's been some discussion about a family of temperaments
that has come up here at various points through the years, for example
in 2000-2002: fifths are tuned slightly wider than their pure size of
3/2 or 702 cents, often around 704 cents; and major and minor thirds
are close to 14/11 (418 cents) and 13/11 (289 cents).
As Dave Keenan pointed out to me in the year 2000, a good way to look
at these temperaments is as variations on Pythagorean intonation where
fifths and fourths are pure, with major thirds at 81/64 (408 cents)
and minor thirds at 32/27 (294 cents). A big attraction of Pythagorean
in medieval European styles where it fits beautifully, and also for
many modern string players, is the narrow and expressive melodic
semitone at 256/243 or 90 cents, smaller than 12-equal at 100 cents.
Tuning fifths at a slightly stretched 704 cents rather than a pure 702
cents accentuates these aspects of Pythagorean intonation, but also
brings into play new ratios and possibilities. Let's first consider
the accentuated aspects.
If we tune the fifths at 704 cents, then each will be two cents wider
than pure, and a major third is formed from four of these fifths: thus
C-G-D-A-E, making it eight cents wider than the Pythagorean 81/64 at
408 cents, or 416 cents -- only about two cents from 14/11 at 418
cents. Major thirds in this region are generally active and intense:
fine for melody, or for styles of counterpoint or harmony like those
of 13th-14th century Europe where thirds are "partly concordant" and
often expand to smooth fifths, for example. They're points of action
and motion, not full and restful concords like the 5/4 in later music.
Likewise, with fifths at 704 cents and fourths at 496 cents --
assuming the common choice of a pure 2/1 octave at 1200 cents -- each
minor third is formed from three fifths down or fourths up, e.g. C-Eb
from Eb-Bb-F-C. With pure fifths and fourths, we'd get a Pythagorean
32/27 at 294 cents. Here each of the three fourth is about two cents
narrower, so our minor third is about six cents narrower: 288 cents or
so, very close to a just 13/11 at 289 cents.
This brings us to the narrow and expressive melodic semitone for which
Pythagorean intonation is so prized, at 256/243 or 90 cents with the
fifths pure. We get a diatonic semitone, e.g. B-C, from a chain of
five fourths up or fifths down, here B-E-A-D-G-C. Since each fourth is
two cents narrow, our semitone is a full 10 cents narrower, at around
80 cents or so. This is an extra-expressive melodic step, and has a
virtually just size of 22/21 (81 cents). The 22/21 step appears in a
tuning of Ptolemy, and in various medieval Near Eastern tunings. It's
around the size of some of the narrow semitones favored by string
players, when tend to be even more compact and incisive than the
Pythagorean step of 90 cents.
While we're talking basic melody, let's not forget the whole-tone
step, e.g. C-D. In Pythagorean, this is a pure 9/8 (204 cents), formed
from two pure 3:2 fifths, here C-G-D. With the fifths at 704 cents,
each two cents wide, our whole tone or major second is four cents
larger, around 208 cents. The nice contrast between the generous wide
tone at 208 cents and the compact semitone at 80 cents is a winning
feature of this temperament, accentuating a bit further the contrast
in Pythagorean tuning (9:8 and 256:243, or 204 cents and 90 cents).
At this point, we've already flexed two of our three degrees of
freedom, which it's a good time to explain:
(1) PERIOD: This is the 2/1 octave at 1200 cents; and
(2) LINEAR GENERATOR: Here this is the 704-cent fifth, or the
496-cent fourth, which repeat to build up a "line" or
chain of notes defining either the whole tuning or a
portion of it; and
(3) SPACING: This applies if we decide to expand our tuning
by having more than one line or chain of fifths, each of
which uses the same period (2/1 octave) and generator
(704 or 496 cents), and defines the distance or spacing
between these identical chains.
We've set a period of 2/1 or 1200 cents, and chosen a linear generator
of 704 cents or 496 cents (some people like to consider the fifth as
the generator, others the fourth, and either is a logical choice).
Note that choosing a period and linear generator leaving open the size
of length of our "line" or chain of fifths or fourths: how many notes
do we want, or can our synthesizer or keyboard comfortably handle?
Common choices with our period and linear generator would be 12 or 17
notes. Here we'll take 12, because it's familiar, friendly to
conventional keyboards, and large enough to give an idea of some of
the possibilities and new features of our tuning.
So at this point we'll put our first two degrees into action by tuning
a 12-note scale, starting the chain at Eb and ending at G#.
128 288 626 832 992
C# Eb F# G# Bb
0 208 416 496 704 912 1120 1200
C D E F G A B C
Before we get into our third degree of freedom, let's take another
look at what we already have with a single chain. There's a saying
that a good mix should include "something old, something new,
something borrowed, something blue."
We could say that the similarity to traditional Pythagorean tuning
systems or intonational tendencies of performers such as string
players, for example, is something old; tempering the fifths slightly
in the wide direction is something new; and the near-just 22/21 step
is something borrowed from Ptolemy, or from medieval Near Eastern
theorists such as Qutb al-DIn al-Shirazi around 1300.
However, what about something blue? If we're looking for intervals
which some listeners might associate with the Blues, there is a family
we haven't yet considered, which whether or not it is truly "Bluesy"
does relate to Persian and some other Near Eastern styles.
One good introduction on the Persian and possibly Bluesy side is this
mode found starting on G:
0 128 288 496 704 784 992 1200
G G# Bb C D Eb F G
128 160 208 208 80 208 208
This is a type of modality called Shur, which is said to express the
experience of intense spiritual love. The first step C-C#, technically
a chromatic semitone, has a size of 128 cents, virtually identical to
a just 14/13. The second, G#-Bb, is 160 cents, and the two together
bring us to Bb at a 288-cent minor third near 13/11; then a usual
208-cent whole tone completes the fourth, G-G#-Bb-C.
The open steps of 128-160 cents are small and large forms of what are
called middle or neutral seconds, larger than usual semitones but
smaller than usual whole tones. This lower fourth or tetrachord of our
mode has the common Persian pattern of a minor third, often sung or
played as here a bit smaller than the Pythagorean 32/27 (294 cents),
divided into a small and large neutral step, the smaller one first as
we ascend.
The lower fourth or tetrachord G-G#-Bb-C with its neutral second steps
is the heart of Shur, and the upper steps often vary, but let's take
the "standard" tetrachord C-D-Eb-F, or 208-80-208 cents, with its wide
whole-tone steps and the narrow semitone D-Eb at 80 cents, the only
usual semitone in a mode otherwise built from whole-tones and neutral
steps.
Another Near Eastern mode, this one with a brighter mood, is Arab
Rast, here in a tuning which might be close to that in parts of Syria,
or among some Turkish musicians. This time we start from B:
0 208 368 496 704 912 1072 1200
B C# Eb E F# G# Bb B
208 160 128 208 208 160 128
In a Rast tetrachord such as B-C#-Eb-E we have a tone, a large neutral
step, and then a small neutral step rising to the fourth, or in this
tuning 208-160-128 cents. Then we have a tone E-F# from the fourth to
the fifth step, and another Rast tetrachord F#-G#-Bb-B.
The large neutral or "submajor" third at 368 cents, typical of a
Syrian Rast in places like Aleppo, for example, has a somewhat major
quality -- but quite different from usual major thirds at ratios such
as 5/4, 14/11 (as here), or 9/7. We could likewise describe the 14/13
step at 128 cents as a "supraminor" second: it sounds rather like a
semitone, and yet different -- a contrast enhanced when it is compared
to our regular semitone at 80 cents. While the standard modern form of
Rast doesn't have any semitone steps -- only tones and large and small
neutral seconds, a delightful variation sometimes called by the Arabs
Suz-i Dilara (actually just one facet of a more complicated modal
development) does:
0 208 368 496 704 912 992 1200
B C# Eb E F# G# A B
208 160 128 208 208 80 128
We again have our lower Rast tetrachord B-C#-Eb-E with its bright Rast
third at 368 cents, then the whole-tone step between the fourth and
the fifth, and then a tetrachord of F#-G#-A-B or 208-80-208 cents,
with the expressive 80-cent semitone G#-A from the major sixth to the
minor seventh of the mode. The Arabs call this kind of tetrachord
Nahawand or Busalik.
This mode also has another tetrachord on the fourth step that Arab
musicians may bring out and explore: E-F#-G#-A, or 208-208-80 cents,
rising by two 208-cent tones to a bright major third at 416 cents,
close to 14/11, and then by an 80-cent semitone to the fourth.
While either the first version of Rast with a high neutral seventh, or
the second with a minor seventh, can be treated as a mode in its own
right, performers often use the first form with the neutral seventh
when ascending, but the minor seventh in descending. This is just a
beginning to the melodic nuances and refinements of Near Eastern
tunings and modal systems.
At this point, we've brought two degrees of freedom into play:
choosing the 2/1 octave as our period, and 704 or 496 cents as our
linear generator. But what about _spacing_, that third degree yet
unexplored?
The idea is actually simple. If a single 12-note tuning set, or
"line" or "chain" of generators, is fun, wouldn't another chain but
yet merrier, giving us yet more intervals? It would: essentially,
we'll "clone" our 12-note chain and place the second copy at some
pleasant spacing from the first.
Let's look again at our 12-note layout, and consider what new kinds of
intervals we might want and how much spacing between our first 12-note
chain and the new one we would need to get them, either just or close
to just.
128 288 626 832 992
C# Eb F# G# Bb
0 208 416 496 704 912 1120 1200
C D E F G A B C
One interval we might like is the 7/4 minor seventh, the "harmonic
seventh" at 969 cents with a history going back at least to the Greek
theorist Archytas; and with it the small 7/6 third. There are various
approaches to this, but say we note the major sixth C-A at 912 cents.
If we place our second chain at about 57 cents higher than this one,
we'll get an interval of 912 + 57 = 969 cents.
Since we have nine major sixths in all in our 12-note chain, we'll get
nine just or near-just 7/4 sevenths! And there's the 7/6 minor third,
also, at 267 cents. Since our major seconds are 208 cents, we'd need a
distance of about 59 cents to get this 7/6 third just, 208 + 59 = 267
cents.
Let's look at another kind of harmonic interval often mentioned: the
"harmonic sixth" at 13/8 or 841 cents. We don't happen to have a
regular minor sixth from our 1/1 of C on the keyboard diagram above,
but we can find one at E-C, at (1200 - 416) cents or 784 cents, close
to the just ratio of 11/7. To get a just 13/8 at each location where
we now have a minor sixth -- eight of them -- we would need a spacing
of about 57 cents, since 784 + 57 = 841 cents.
At this point we could have two years of discussion on the perfect
spacing to the nearest hundredth of a cent or better -- but let's just
do it, and settle on 58 cents as a safe, middle-of-the-road choice
that won't get any of these intervals we're seeking (7/4, 7/6, or
13/8) very far from just.
So we've flexed all three degrees of freedom!
(1) PERIOD: the 2/1 octave (1200 cents).
(2) LINEAR GENERATOR: The 704-cent fifth or 496-cent fourth.
(3) SPACING: 58 cents.
One question in drawing a diagram of our system is whether and how to
show which of the two chains a note belongs to: here I'll use an
asterisk (*) symbol for notes on our new and upper chain. This symbol
raises a note by our spacing, 58 cents. Voila!
186 346 674 890 1050
C#* Eb* F#* G#* Bb*
58 266 474 554 762 970 1120 1200
C* D* E* F* G* A* B* C*
128 288 626 832 992
C# Eb F# G# Bb
0 208 416 496 704 912 1120 1200
C D E F G A B C
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2. Ratios of 7: The 7/6, 7/4, and 21/16
---------------------------------------
How about that 7/6 minor third and 7/4 minor seventh? Let's try them,
and another interval in the same family that's quite unique, by
locating Erv Wilson's 1-3-7-9 hexany, a JI tuning here slightly
tempered:
C D D* E* G A* C
0 204 266 474 704 970 1200
208 58 208 230 266 230
If the usual 80-cent step is quite compact and efficient, our 58-cent
step from D to the near-7/6 third D* is strikingly narrow; a just
equivalent would be the famous 63-cent thirdtone step of Archytas,
tempered here five cents narrower yet. When played over a drone, this
six-note mode has a near pure 7/6 at D* and 7/6 at A* that may have a
special blend and resonance.
Now for the interval that's "quite unique": the narrow fourth C-E* at
474 cents, very close to the just but complex ratio of 21/16. The
strong beating of this interval may not be especially attractive in
itself, but let's try this chord progression, played very slowly, with
the first chord used by LaMonte Young, Keenan Pepper, and others:
A* A
G A
E* D
C D
The first chord, meant to be held and dwelt on -- or better yet dwelt
_in_ for a while as an awesome sonic space -- has the narrow 21/16
fourth; a regular fifth at 704 cents close to 3/2; and a harmonic
seventh just a cent or so from 7/4. In this rich setting with the many
simpler and blending ratios, the 21/16 seems (at least to me) to
illuminate and add energy to this concord. At the same time, energy
can suggest eventual motion, here to a simple fifth on D (a near-3/2).
I've written out the voice-writing with unisons, but on two 12-note
keyboards, or a generalized keyboard in whatever mapping fits best,
it's simply a question of playing and holding the first four-note
chord, and then moving to the simple two-note sonority D-A.
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3. Shifting into neutral gear: 13/8 and more
--------------------------------------------
Each chain gives some neutral intervals, for example the large and
small neutral steps at 160 and 128 cents, and the 368-cent bright
neutral or submajor third we used for a flavor of Syrian Rast. But now
we have yet more. To show what's out there, without worrying about all
the ratios and approximations, here some different shadings of neutral
seconds, thirds, sixths, and sevenths, including the ones we've
already encountered:
Neutral or middle seconds: Neutral or middle sixths
small C-C# 128 cents small C-G# 832 cents
medium small E-F* 138 cents medium small E-C* 842 cents
medium large D*-E 150 cents medium large D*-B 854 cents
large C#-Eb 160 cents large C#-Bb 864 cents
Neutral or middle thirds: Neutral or middle sevenths:
small Eb-F# 336 cents small Eb-C# 1040 cents
medium small E-G* 346 cents medium small C-Bb* 1050 cents
medium large G*-B 358 cents medium large C*-B 1062 cents
large B-Eb 368 cents large B-Bb 1072 cents
How about the 13/8, here tempered at 842 cents? Let's here it in a
couple of tunings based on some tetrachords of Ibn Sina, the first a
tuning of Jacques Dudon, here slightly tempered, called Ibina. For
those curious, I will give the just ratios and cents above the notes.
0 139 347 498 702 841 996 1200
1/1 13/12 11/9 4/3 3/2 13/8 16/9 2/1
E F* G* A B C* D E
0 138 346 496 704 842 992 1200
Here we have three intervals I call "medium small" above the 1/1: the
13/12 neutral second; the 11/9 neutral third; and the 13/8 harmonic
neutral sixth, which can interesting to listen to as one explores this
mode above a drone.
Jacques Dudon's just ratios illustrate an interesting point about this
temperament: he has a 13/12 neutral second followed by an 11/9 neutral
third, connected by a melodic step a bit larger than a just 9:8 tone
at 204 cents: actually a 44:39 step at 209 cents. It happens that our
slightly large tone at 208 cents matches almost perfectly!
Another fine point is that we have an 11/9 third in the lower
tetrachord, with an interesting pattern of 1/1-13/12-11/9-4/3, or
medium small neutral step, tone, medium large neutral step. In JI,
the small and large neutral steps have sizes of 13/12 and 12/11, or
138.6 and 150.6 cents -- I'm only using tenths of a cent to avoid a
rounding error and show that they add up to around 289 rather than 290
cents -- while we have 138 and 150 cents, adding up to 288 cents. So
the just and tempered melodic steps are very close!
But to finish our fine point, we have an 11/9 (347 cents) in the lower
tetrachord of the just version, and a 13/8 (841 cents) in the upper
tetrachord, that intriguing harmonic sixth. In JI, the fourth between
these two steps is a bit narrower than a pure 4/3 at 498 cents -- more
specifically, about 493 cents, at the more complex ratio of 117/88.
This fifth is thus, in JI, narrow by about 5 cents, although the
others are pure.
In our tempered version, every fourth including this one (G*-C*) is
equally impure at 496 cents or two cents narrow. While both versions
are fine, this a typical contrast: JI systems have more variety of
pure and (here mildly) impure intervals, while temperaments tend to
"even things out."
Let's conclude with a quick look at a Near Eastern tetrachord known in
the Arab world and Turkey as Hijaz, and in Iran as Chahargah. This is
often played in 12-equal to suggest an "exotic" setting, with steps
like these: D-Eb-F#-G, or 0-100-400-500 cents, with a lower semitone,
a middle augmented second at 300 cents (identical to the minor third
in this system), and upper semitone.
Some Arab performers and scholars call this a "piano Hijaz": the
traditional shades of intonation are much more subtle. Let's try a few
classic forms and variations, with the caution that in the Near East
as well as elsewhere, tastes may vary!
The first form comes from around 1300, as recorded by the Persian
philosopher and musician Qutb al-Din al-Shirazi.
0 151 418 498
1/1 12/11 14/11 4/3
C* D E* F*
0 150 416 496
We have a 150-cent neutral second step, a 7/6 small minor third at 266
cents notably smaller than our regular 288-cent minor third, and a
narrow, 80-cent semitone almost exactly matching Qutb al-Din's just
step of 22:21. Note that that lower neutral step is almost twice as
large as the upper semitone, in his original tuning or this tempered
version. Tuning the middle step around 7/6, or at any rate smaller
than a usual minor third, is considered a mark of stylishness by some
Arab and Iranian writers, although forms also occur where this step is
at or around the size of a regular minor third.
This kind of tuning is still favored in Iran, where we also find a
slightly different shading with a smaller neutral step around the
ratio of 13/12. Note that, in practice, there are a wide variety of
tunings and shadings, with just ratios more of a guide or method of
description than an exact measurement:
E F* G# A
0 138 416 496
This tuning is actually the same as the last, _except_ that the second
step is 12 cents lower, and thus the middle larger: about 278 cents.
This might be considered rather large by Persian standards, but still
within a recorded range of variation for performances by some leading
musicians measured by Jean During.
A third form, described by some Turkish theorists who like a high
major third, has a small neutral step, a regular minor third, and
again an incisive semitone somewhat smaller than the Pythagorean 90
cents. All the notes and intervals for this form occur within a single
chain of fifths:
C C# E F
0 128 416 496
Note that even in this form, the lower neutral step at 128 cents is
notably larger than the upper 80-cent semitone -- a difference of 48
cents! Incidentally, a JI explanation of this might be 28:26:22:21.
Those are string lengths we might use to obtain a tuning like this,
with steps of 28:26 or 14:13, 26:22 or 13:11, and 22:21, or 128-289-81
cents, giving us 0-128-418-498 cents. Here we have 128-288-80 cents,
melodically very close.
Let's conclude with two final forms of Hijaz, the first of which I
often find the most striking of all, and the second close to a variety
called Zirkula described by the Lebanese composer and theorist Amine
Beyhom. I'm not really sure if anyone else in the Near East or
elsewhere tunes the first version quite like this, but here it is:
0 81 347 498
1/1 22/21 11/9 4/3
E F G* A
0 80 346 496
This has the same steps as Qutb al-Din's Hijaz, but in the reverse
order, with the narrow semitone first, then the 7:6 third, and finally
the neutral second at 12:11. One result is that instead of having a
major third step at a bright 14/11, we have a medium small neutral
third at 346 cents, a near-just 11/9, with a medium large neutral step
of 12:11 up to the fourth instead of that 80-cent semitone.
Here's a version of Beyhom's Zirkula, which is a variation on Rast.
Let's first find a Rast tetrachord that goes with Beyhom's shade of
tuning that he found is preferred in Lebanon, and the try the Zirkula
variation or modulation.
According to Beyhom, the Lebanese taste is for something like
200-155-145 cents or 200-355-500 cents. Our 358-cent or medium large
neutral third should be a reasonable fit, as here shown in a standard
form of Rast with the two disjunct tetrachords, including a major
sixth and neutral seventh step:
D* E* F# G* A* B* C# D*
0 208 358 496 704 912 1062 1200
Here we have neutral steps of 150 and 138 cents, with the larger step
first -- and order preferred in an Arab or Turkish Rast. This subtle
contrast, here 12 cents, is much to taste for many Arab musicians and
listeners, although Beyhom notes that folk or popular styles often
prefer more polarity between large and small neutral steps.
For our Zirkula, as Beyhom explains, we change only one step in a Rast
tetrachord: the second step, which changes from a whole-tone step
(D*-E*) to a usual semitone (D*-Eb*), maybe somewhere around the
Pythagorean 90 cents in many Arab performances, and here a yet
narrower 80 cents:
D* Eb F# G* A* Bb* C# D
0 80 358 496 704 784 1062 1200
Here our Rast tetrachord modulates to Zirkula, with a semitone (here
at 80 cents), a middle step at 278 cents, and the same medium small
neutral step (here around 13:12) as in Rast. The Rast third, here at
358 cents, also stays the same.
There's also a possible distinction between the lower tetrachord in
Zirkula as shown, and the upper one, where the seventh step might be
made higher than the usual Rast neutral step (here 1062 cents) in
order to emphasize the pull up to the octave of the final or resting
note. In this tuning, the contrast might be a bit more than Beyhom
bargained for, since we get a Qutb al-Din Hijaz at its most ebullient.
If we want to try it, here's how:
D* Eb F# G* A* B C#* D
0 80 358 496 704 854 1120 1200
We can note the contrast between the 138-cent step of F#-G* in
approaching the fourth degree, and the 80=cent at C#*-D. How Amine
Beyhom might view this juxtaposition, I'm not sure: he cites Qutb
al-DIn's tuning with much approval and notes its use in Iran, and
emphasizes the pleasing nature of the Zirkula pattern: but can the two
tastefully fit together like this? With a tuning this adventurous, one
must use discretion, but have lots of fun exploring.
Note that this is not a circulating system: the available intervals
change as we move around the system, with medium small neutral steps
available from notes on the lower chain for example, and medium large
intervals from notes on the upper chain. Deciding which interval to
use and where to find it is a matter of experience and practice, but
the possibilities remain vast.
-------------
4. Conclusion
-------------
The purpose of this article is to show what a "Rank-3" system is, and
how the three degrees of freedom are the period (often 2/1), the
linear generator (here 704 or 496 cents), and the spacing between
tuning chains (here 58 cents).
Contrary to some recent humorous remarks here which had the virtue (I
hope) of getting me to write this, this "Rank-3" or "three degrees of
freedom" approach is not obscure or incomprehensible.
For example, we found that a generator of 704 cents gave us a major
third of 416 cents near 14/18 (418 cents), and our fourth of 496 cents
gave us a minor third of 288 cents, near 13/11 (289 cents). Once we
know the just sizes in cents, this is mostly simple arithmetic, and
will get a newbie or oldster close enough for joyful musicmaking.
Similarly, we looked at our major sixth at 912 cents, and found
that spacing a second tuning chain at 57 cents would give us a 7/4
minor seventh at 969 cents. We also looked at the ideal spacing for
other new intervals like the 13/8 neutral sixth, and decided on a
spacing of 58 cents to get both these intervals within a cent or two
just.
Basically, this is a form of a temperament from 2002 called
Peppermint. In 2000, Keenan Pepper proposed a temperament defining the
first two of our degrees of freedom: a 2/1 period, and a generator of
704.096 cents. In 2002, I added the third degree of freedom by adding
a second chain at 58.680 cents apart. That's the story.
Here the parameters are rounded to 2/1, 704 or 496 cents, and 58 cents
for the spacing.
A final or reported caution, and word of encouragement: just as
important as knowing how to tune or navigate a system like this is now
how and why to use it, or least which music most naturally fits and
which is generally best approached with other tunings, unless one is
deliberately being "experimental" or maybe teaching a lesson in what
does _not_ fit.
A good term for this kind of 14/11-and-13/11-temperament might be
"para-Pythagorean," a term John Chalmers mentioned to me around 1998.
Large major thirds, small minor thirds, incisive semitones, and also
augmented and diminished intervals like the 128-cent and 160-cent
neutral steps (e.g. in the tetrachord C-C#-Eb-F or B-C#-Eb-E) that
we found even within a single 12-note tuning chain. Expand a chain to
17 notes, or add a second chain as here, as get a cornucopia of these
neutral steps!
The term "para-Pythagorean" tells about some of these exciting things,
with the neutral steps an element _not_ found in a 12-note Pythagorean
tuning, and so part of the "para" that makes things a bit different.
It also cautions that chord progressions of major-minor tonality
assuming major thirds around 5/4, or at least not too much larger than
the 400 cents of 12-EDO, will _not_ be at home here, just as a Bach
prelude would not be at home on a church organ tuned with typical
gamelan fifths. This is no flaw of gamelan or Bach, just the reality
that not all musical elements smoothly or wisely mix.
This not to exclude the experimental, nor demonstrations of how Chopin
sounds curious but not optimal in 7-EDO, for example, while Chopin in
the right well-temperament or Thai music in 7-EDO or something close
to it is beautiful.
Here are some things we can do with this Peppermint-like tuning:
* Delve into some Arab or Persian music, maybe especially the
latter, where minor thirds around 13/11 and semitone steps
around 80 cents are usual practice, and the different
possible neutral shadings permit lots of choices as well
creative ear training.
* Delve into medieval European music meant for Pythagorean
tuning, and maybe into some of the mysteries of 14th-century
intonation where the small 58-cent thirdtones (or
quartertones?) of this tuning might be especially apt,
depending on whose theories or modern interpretations one
follows.
* Read George Secor's "17-puzzle" article linked to at the
beginning of this article, and explore some of his chord
progressions and seven=note scales which can be found in this
system also. Generally, things written for a 17-note
circulating system like Secor's may fit here also, although
things may get a bit complicated since his tuning is fully
transposable while this one isn't. But his article is a great
way to get a good general sense of direction. He writes about
melody and harmony in really insightful ways.
* Explore pure melody of the kind a string player might relish,
as long as major thirds near 5/4 aren't called for; with a
single unaccompanied melodic line, this issue may not arise.
* Explore something like Erv Wilson's 1-3-7-9 hexany, or
ancient Greek tunings based on ratios of 2-3-7-11 (e.g. the
Diatonic and Chromatic of Archytas, or Ptolemy's Intense
Chromatic). These are available in near-just form, and they
wonderfully expand one's musical experience. Note that while
the Greek tunings tend to focus on pure melody (although a
drone might bring out some concordant ratios), the 1-3-7-9
hexany has lots of harmonic material like triads (e.g. 4:6:7,
C-G-A*) and tetrads (e.g. 16:21:24:28, C-E*-G-A*) which in
this temperament will be very nicely in tune, if not quite
just.
* With due awareness that this isn't an authentic gamelan
tuning, I've used this kind of tuning to explore some
features of slendro and pelog, often as interpreted in some
JI variations by Lou Harrison or Jacques Dudon: for example,
the large major thirds at 416 cents, and yet larger ones at
426 cents (e.g. F#-Bb*) and 438 cents (e.g. D*-G), can sound
beautiful in a pelog mode. Maybe it's best to look at this,
not as learning gamelan, but as paying a tribute to it by
admiring some of the features which this tuning can emulate
to a degree.
* If you love 20th century classical music or certain styles of
jazz that focus on "quartal/quintal" harmony -- lots of
fourths, fifths, and chords like C-F-G (the "sus4") or D-G-C
(a "fourth chord"), you may love this temperament! The fifths
are fourths are almost exactly as impure as in 12-equal, and
the 9:8 major seconds or 16:9 minor sevenths (208 and 992
cents) likewise almost exactly the same amount from just.
You get these familiar 12-equal features, plus some septimal
and neutral "Blues" intervals opening up totally new worlds!
If you love a D-G-C or 1-4-7 chord (9:12;16 in JI) as played
in 12-eq -- and I do! -- just try D-G-C-E* with a near-pure
7/3 minor tenth and that wonderful 21:16 at C-E* (in JI
terms, 9:12:16:21 -- but you don't have to know the ratios to
be awed by it!). The main caution for 12-equal people is that
G#-Eb and G#*-Eb* are _not_ regular fifths; people used to
historical European systems like Pythagorean or meantone may
not be surprised by this.
* Look into some JI or related avant-garde ideas that don't
depend on ratios of 5, like the 8:11:13 chord (e.g. C-F*-A*)
or a 7:9:11:13 (B*-E-G*-Bb*); Secor's article may get you
started. Also, as points out, 7-based minor intervals and
chords like 12:14:18:21 (C-D*-G-A*) can be very sweet and
appealing to people who don't have a special penchant for
xenharmonics or microtonality.
* While many features of the tuning might be conducive to
Debussy or Bartok, that doesn't mean that their music, meant
for a circulating 12-equal, will fit. It does mean that you
can look to their music for ideas and use them as a basic for
new music designed to fit the tuning and harness some of its
special resources. This is not a system to make 12-equal
obsolete!
Those are some possibilities: the other side of this is that
para-Pythagorean is simply not meant for use, at least in usual
harmonic timbres, with 16th-19th century European music intended to
have major thirds close to 5/4. There are a long list of things (some
of them above, some still to be discovered and fleshed out) that this
tuning can do well; and a vast list of tunings that are superior where
a 5/4 third is supposed.
For a bit of daring experimentation with later 16th-17th century
music, I'd recommend a highly colorful or "contrastive"
well-temperament or modified meantone system. The latter, in a
17th-century French manner, may indeed have a few remote major thirds
at around 14/11 or so, and people used them to create tension,
surprise, and drama as composers and players skirted the bounds of
tolerable intonational license. Those same instruments, however, had
an abundance of thirds near 5/4 in the nearer transpositions, so the
contrast was one between sweet concord and daring near-dissonance.
In contrast, a tuning system like this has 14/11 and 13/11 for its
norm, so that when applied to such music (or almost any 16th-19th
century European classical music!), it very efficiently mistunes
almost every harmonic interval. This does justice neither to the
tuning nor to the music. It also gives the impression that beautiful
intervals are themselves "out of tune" when they are simply being used
in an "out of place" context.
This isn't to exclude "the exception that proves the rule," only to
mark out clearly a zone of caution. Nor is to say what may happen in
inharmonic and possibly customized timbres, a la Bill Sethares, for
example.
Remember, one tuning can't and shouldn't attempt to do everything!
This one leaves lots of room for 12-equal, meantone (which I also
love), unequal well-temperaments of the 18th-century variety, tunings
of Erv Wilson, the xylophone tunings of Africa, and so much else!
My advice is to start by focusing on what a tuning does well, pick
some aspect of it you like, and put your main energy into that.
And have fun, because if you're doing it right, you will!
Peace and love,
Margo Schulter
19 October 2012