Hello, all.
One small but significant step in promoting a better
understanding and more frequent use of JI tunings is to celebrate
a most impressive JI system indeed: a circulating 53-note cycle
of pure 3:2 fifths or 4:3 fourths. This "Pythagorean 53" system
raises a few JI mysteries I would like here to resolve.
Cris Forster's _Musical Mathematics_ raises some of these points,
evidently leaving them open for fuller development -- unless that
development may occur in portions I have not yet read, which
remain voluminous! Please let me caution that Cris's treatment of
what follows would be much more rigorous, elegant, and concise
than what I'm about to present.
As Cris notes at p. 502, the Chinese mathematician Ching Fang
(fl. c. 45 B.C.E.) may have explored the cycle of 53 notes in 3:2
fifths, finding it so close to mathematical closure at 31 2:1
octaves that "we may think of this spiral as a circle."
However, as Cris also shows at pp. 736-737, we cannot literally
take the convenient approximation of "nine commas in a 9/8 tone"
as a mathematically accurate statement -- at least assuming that
these commas are all usual Pythagorean commas at 531441:524288 at
around 23.460 cents. Such a literal interpretation would give us
a tone at around 211.140 cents rather than 9/8 or 203.910 cents.
As Cris then explains, we can form a tone almost identical to the
just 9/8 tone from nine equal commas in the related system of
53-tone equal temperament, sometimes also known as a 53-tone
"equal division of the octave," thus 53-tET/EDO. Here each fifth
is tempered narrow by the minute amount of about 0.068 cents, and
identical commas of about 22.642 cents can be neatly multiplied to
generate all of the intervals in the system.
What happens, however, if we _don't_ temper, but use a just
Pythagorean cycle of 53 notes per octave? Understanding the
answer is one step to more persuasive and effective JI advocacy.
One barrier to such advocacy is the idea that there are "laws of
nature" or "practical realities" which preclude the real-world
use of JI to make music. The implication may sometimes seem,
especially to those exposed to the often triumphalistic
literature of "temperament as the only liveable solution," that
the more rich and beautiful a JI system seems, the more likely
that some fatal flaw will rule it out if only we look at the
mathematics critically rather than let ourselves be seduced by
some beguiling "numerology."
In the case of Pythagorean 53, or "Pyth-53" for short, there are
in fact no such barriers beyond those for any tuning system of
comparable size, just or tempered. However, there is a mental
barrier which must be surmounted, which I will call "the tyranny
of aliquot parts," the assumption that a viable tuning system
must be built from even or exact multiples of a single "atom" or
"elementary particle," like the comma of 53-tET/EDO at 1/53 of an
octave at 2:1.
Let us overcome this mental tyranny, not an inherent vice of
temperaments, but only of a mindframe viewing them as the only
viable alternatives.
------------------------------------------------------
1. The Pyth-53 9/8 tone as nine commas -- unequal ones
------------------------------------------------------
Cris's demonstrations at pp. 736-737 nicely show us that we
cannot accurately think of a Pyth-53 system as made up of 53
equal commas all at the familiar ratio of 531441:524288, the
familiar "12-comma" as we may call it, defined by the difference
between 12 fifths up each at 3:2 or 701.955 cents, or 8423.460
cents, and 7 octaves at 8400 cents, or 23.460 cents.
In fact, a Pyth-53 circle actually has two sizes of adjacent
steps or commas: this familiar 12-comma at 23.460 cents, and a
smaller comma at 19.845 cents, a "41-comma" as we shall see.
The regular Pythagorean intervals such as the diatonic semitone
or limma at 256/243 or 90.225 cents (4 commas), the 9/8 tone
(9 commas), and the 32/27 minor third (13 commas) involve a
mixture of these larger and smaller commas.
We can define the smaller comma at 19.845 cents as the difference
between 41 pure 3:2 fifths down or 4:3 fourths up and 17 octaves
at 2:1. In fact, starting at any point along a Pyth-53 circle,
we'll find that a complete circumnavigation of this system,
returning to that point, takes us through 41 regular commas
(12-commas) and 12 smaller commas (41-commas).
Let's consider the familiar limma at 256/243. In current Turkish
and Syrian theory, and also in some late medieval European
Pythagorean theory, this interval is identified or approximated
as "4 commas."
In a Pyth-53 cycle, while the sizes of these 4 commas will always
be the same for any regular 256:243 limma, the order or sequence
in which we encounter these sizes may vary depending on our
location within the circle.
Let us assume, for example, that from the note we take as the 1/1
there are 40 regular fifths up and 12 regular fifths down. Note
that in a Pyth-53 circle, we have 52 such regular fifths, plus an
"odd" fifth narrower than the others by the comma of Mercator,
about 3.615 cents, the amount by which 53 pure fifths would
exceed 31 octaves at 2:1.
The "odd" fifth, at about 698.340 cents, is comparable to a fifth
tempered narrow by 1/6 of a usual Pythagorean comma (12-comma),
and well within the range of impurity accepted in European
temperaments, or occurring in the fifths or fourths formed
between some fret locations in `oud tunings of the Islamic
Renaissance (9th-15th centuries).[1]
To get our limma from our chosen 1/1, the simplest method doesn't
involve thinking in terms of commas, large or small, but simply
tuning 5 fifths down or fourths up, e.g. C-F-Bb-Eb-Ab-Db, and
thus the limma C-Db at 256/243 or 90.225 cents.
However, suppose we do wish to built our C-Db limma in commas.
One procedure would be to begin by moving 12 fifths up, which
will always give us the usual 12-comma at 23.460 cents --
assuming we do not encounter that one "odd" or "virtually
tempered" fifth in the 53-note circle. Here, since from our 1/1
there are 40 regular fifths up and 12 fifths down, we can move up
by 12 regular fifths.
Moving up 12 fifths, we arrive at a pitch a 12-comma higher,
which we might write as C+1r, our 1/1 at C plus "one
regular" comma or 12-comma at 23.460 cents.
Now let us move up another 12 fifths, so that we arrive at the
pitch C+2r or C plus two 12-commas, 46.920 cents, an interval we
can also call a "12-bicomma.".
Repeating this move yet again, we arrive at C+3r, our original
1/1 at C plus three 12-commas, a "12-tricomma" at 70.380 cents, or
36 fifths up in all.
As we know, a regular limma like C-Db is always five fifths down
or fourths up at 256/243 or 90.225 cents, and we are now 36
fifths up at 70.380 cents.
To complete the limma, we must now move from 36 fifths up to
5 fifths down, or by 41 fifths down or fourths up. We find that
this interval is our 41-comma or smaller comma at 19.845 cents.
And adding 70.380 cents (three 12-commas or 36 fifths up) to this
41-comma (19.845 cents) gives our desired regular limma C-Db at
256/243 or 90.225 cents.
We could also write this limma as C+3r+1s, where "3r" shows three
regular 12-commas, and "1s" a single smaller comma or 41-comma.
It thus follows that any regular 256/243 limma will consist of
three large commas and one small comma.
How about a 9/8 tone, traditionally described as having or
approximating a size of "9 commas"? Indeed, in a Pyth-53 circle,
it does have 9 commas, with a bit of traditional Pythagorean lore
helping us to determine the requisite numbers of each type.
We are told that a 9/8 consists precisely of two 256:243 limmas
plus a comma -- this comma being what we now may more
specifically identify as the usual 12-comma or larger comma at
531441:524288. This is the comma defining the difference between
the regular limma at 256:243 or 90.225 cents and the apotome or
chromatic semitone at 2187:2048 or 113.685 cents. formed
respectively from 5 fifths down (e.g. C-Db) or 7 fifths up
(e.g. C-C#).
Since we know that each of the two limmas in a tone consists of 3
large and 1 small commas, with the remaining comma large, we can
conclude that a regular 9/8 tone has in all 7 large and 2 small
commas. These yield the desired size of 203.910 cents.
We could thus write: 9/8 = 7r + 2s.
Note that this understanding nicely accounts for the anomaly
revealed by Cris (pp. 736-737) which results if we try to derive
a 9/8 tone using the familiar 12-comma at 23.460 cents as an
aliquot part, multiplying it by nine. As mentioned above, this
procedure would give us a size of 9r, or 211.140 cents, exceeding
9/8 by 7.230 cents. This discrepancy turns out to be precisely
twice the comma of Mercator, which among other things defines the
difference between a regular or 12-comma and a small or 41-comma.
Changing two of the regular commas to small ones, 7r + 2s, will
resolve this discrepancy and yield our desired 9/8 tone.
----------------------------------------------------
2. Creative variations in Pyth-53: Some minor thirds
----------------------------------------------------
From what we now know about the 9/8 tone and 256/243 limma or
diatonic semitone, we can also analyze the "elementary particles"
or adjacent tuning steps or commas for another familiar interval,
the regular minor third at 32/27 or 294.135 cents. This interval
results from a chain of 3 fifths down or fourths up, for example
C-F-Bb-Eb.
Such a minor third is equal to a 9/8 tone plus a 256/243 limma,
and is familiarly estimated in the popular Turkish and Syrian
53-comma system as having a size of "13 commas."
More specifically, we have:
9/8 tone = 7r + 2s
256/243 limma = 3r + 1s
----------------------------
32/27 min3 = 10r + 3s
Thus a usual 32/27 minor third is equal to 10 regular or larger
12-commas plus 3 smaller 41-commas.
Another type of minor third, also described at times in Turkish
theory as an "augmented" interval, is a smaller type neatly
referred to in _MM_ as a "triple limma," and slightly larger than
a pure 7:6 at 266.871 cents. It is in fact equal in theory to
precisely thrice a regular limma at 256/243 or 90.225 cents, and
thus to 15 fifths down or fourths up, 270.675 cents.
Interestingly, Karl L. Signell found by stroboscope measurements
of flexible-pitch performances that Turkish musicians indeed tend
to intone this near-septimal variety of minor third at a bit
higher than 7:6, favoring a range of around 270-273 cents (see
his _Makam: Modal Practice in Turkish Art Music_, p. 157).
If one limma equals 3r + 1s, then a literal triple limma is equal
to 9r + 3s, which indeed gives us our desired 270.675 cents.
Recognizing that this beautiful interval includes nine 12-commas
plus three 41-commas resolves the anomalous result to which Cris
alerts us if we try to derive this triple limma from 12 usual
12-commas, or 12r. Thus we would arrive at 281.520 cents, a size
exceeding the actual triple limma by three commas of Mercator,
the difference between 12r and 9r + 3s.
The triple limma is, from another point of view, equal to a
regular 9/8 tone (2 fifths up) plus a very useful small semitone,
or thirdtone, we may call the "17-diesis," the amount by which 17
fourths down exceed 10 octaves at 2:1, about 66.765 cents,
slightly larger than the septimal semitone or thirdtone of
Archytas at 28/27 or 62.961 cents. Since the 17-diesis is
slightly largely than 28/27, by about 3.804 cents, the triple
limma is larger than 7/6 by this same amount, one variety of
"septimal schisma."
What are the "elementary particles" or adjacent Pyth-53 steps
making up this 17-diesis? One way of answering this question is
to think of this interval as 12 fifths down -- lowering the pitch
by a usual 12-comma -- plus another 5 fifths down, raising it by
a usual 256/243 limma. Thus we have:
12 fifths down (12-comma) -1r
5 fifths down (256/243 limma) 3r + 1s
---------------------------------------------------
17 fifths down (17-diesis) 2r + 1s
The 17-diesis at 2r + 1s is thus exactly one 12-comma smaller
than a usual limma, or 66.765 cents.
Note that there is another way in Pyth-53 of obtaining a small
minor third at "12 commas." which brings into play another subtly
different flavor of thirdtone interval we already have briefly
encountered in forming the regular limma from four unequal
commas.
To find this alternative "12 comma" minor third, we may begin
with a usual 9/8 tone at two fifths up, or 7r + 2s.
Now we move up another 36 fifths, adding an interval we have
called a "12-tricomma" or 3r at 70.380 cents. Thus we have:
2 fifths up (9/8 tone) 7r + 2s
36 fifths up (12-tricomma) 3r
-------------------------------------
38 fifths up 10r + 2s
Our alternative 12-comma third is thus equal to a 9/8 tone at
203.910 cents, plus a 12-tricomma at 70.380 cents, or 274.390
cents, 10r + 2s. This compares with the more familiar triple
limma at 9r + 3s or 270.675 cents, a difference of a comma of
Mercator, or 3.615 cents.
Both subtly different sizes are musically useful; but how do we
know, when reading about a "12-comma" interval in Turkish or
Syrian theory, that the more familiar size of 270.7 rather than
274.4 cents is likely meant?
The answer is that, all things being equal, we tend to assume
that intervals like a "13-comma" minor third have their usual
sizes from the shortest available chain of fifths, here 32/27
from three fifths down or fourths up; and that a difference of a
comma between two more or less usual intervals will tend to
involve the regular 12-comma at 23.460 cents rather than the less
familiar 41-comma at 19.845 cents.
Thus a usual triple limma at "12 commas," formed from 15 fifths
down or fourths up, is smaller than a regular "13-comma" minor
third at 32/27 from 3 fifths down or fourths up by a comma of
12 fifths down or fourths up -- the usual 531441:524288 or 23.460
cents.
By making these assumptions that regular diatonic intervals like
the 13-comma minor third have their usual sizes, and that many
other intervals in the Turkish or Syrian system like the 12-comma
triple limma differ from these regular diatonic intervals by a
familiar comma of 531441:524288, musicians can accurately
calculate or estimate these sizes without necessarily delving
into the fine points of a Pyth-53 cycle.
The fact that, from a Pyth-53 viewpoint, almost all of the
regular diatonic as well as variant intervals consist of a
mixture of 12-commas and 41-commas does not interfere with such
routine calculations or estimates. However, the recognition of
this more complex reality does serve to clarify why the usual
calculations succeed uneventfully, but an attempt to measure
Pyth-53 intervals (including the familiar ones) in terms of the
usual 12-comma as an aliquot part would result in anomalies of
the kind that Cris has documented.
---------------------------------------------------------
3. A subtle Pyth-53 advantage: the matter of Turkish Rast
---------------------------------------------------------
While either 53-tET/EDO or Pyth-53 can nicely realize the regular
diatonic intervals, schismatic 5-based and 7-based
approximations, and neutral or middle intervals used in Turkish
music, there is a possible advantage for Pyth-53 which perhaps
has not been so often noted.
Here Ozan Yarman would have me add that neither of these systems,
although adequate, can fully satisfy the desire of Turkish and
other Near Eastern musicians for a great variety of neutral or
Zalzalian steps. Thus I focus only on a subtle and entertaining
nuance, not a "total solution" for Near Eastern intonation of the
kind which no fixed-pitch instrument with a practical number of
steps can provide, unless we are speaking of something like a
MIDI tuning standard which approaches the smooth continuum of
flexible=pitch intonation.
In Turkish practice and theory, there are actually two different
varieties of Maqam Rast, each with many possible shadings.
The first variety favors a third very slightly smaller than a
simple 5/4 at 386.314 cents, with the Pyth-53 ratio of 8192/6561
or 384.360 cents as one possible tuning going back to the Islamic
Renaissance. In modern Turkish practice, there is a tendency to
prefer a yet slightly lower tuning, somewhere around 380-382
cents. Here the traditional 8192/6561 is the best choice in
Pyth-53.
Incidentally, we can express this interval in terms of large and
small commas by noting that, at 8 fifths down or fourths up, it
is a 12-comma smaller than the usual major third at 81/64 or 4
fifths up, 407.820 cents. Since a 9/8 tone is 7r + 2s, and an
81/64 third or ditone equal to precisely two 9/8 tones, this
regular major third is equal to 14r + 4s.
It follows that the "17-comma" major third at 8192/6561 is equal
to one 12-comma less than this, or 13r + 4s.
However, our focus of attention here is on the second variety of
Turkish Rast, favoring a large neutral or "submajor" third at
somewhere around 21/17 (365.825 cents) or 26/21 (369.750 cents),
for example. This type of large neutral third is theoretically
described as having a size of around "16 commas."
In other words, this third will be 2 commas smaller than a usual
major third at 81/64 or 18 commas, and 1 comma smaller than the
5-based approximation at 8192/6561 or 17 commas.
How might we obtain this 16-comma third in a Pyth-53 circle? One
method is to go up 4 fifths, thus obtaining a regular 81/64 major
third, and then seek to lower the pitch by two commas.
The most obvious way to do this is to go down by 24 regular
fifths, or two 12-commas, thus lowering the pitch of 81/64 by
46.920 cents. We have:
4 fifths up (81/64 Maj3) 14r + 4s
24 fifths down (12-bicomma) -2r
---------------------------------------
20 fifths down 12r + 4s
The resulting 16-comma third at 12r + 4s has a size of 360.900
cents, slightly larger than 16:13 at 359.472 cents. This Pyth-53
size, like 16:13, might be perceived as more of a large neutral
third still in the "central" region between around 39/32 (342.483
cents) and 16/13, rather than a "submajor" third with a more
polarized color.
That is to say that neutral thirds at around 360/342 cents, as in
Ibn Sina's `oud tuning using 39/32 and generating 16/13 also, may
have the "musically near-equivalent" quality of less contrasting
neutral thirds in the central region, such as al-Farabi's ratios
of 27/22 (354.547 cents) and 11/9 (347.408 cents).
For much Arab music, 16/13 or the like is an ideal neutral third
step for Rast; while this shade of intonation is also heard in
Turkish music, a somewhat higher neutral third is often
preferred. Can Pyth-53 help us more closely to approach this
ideal?
If we are playing Rast on the 1/1 of our Pyth-53 system as
defined above with 40 regular fifths up and 12 regular fifths
down, an arbitrary choice of course, then in fact we will be
unable to follow the procedure just outlined of going up a
regular 81/64 major third and then 24 regular fifths down --
there aren't enough regular fifths in that direction!
We can however, go up by four 5ths or a regular 81/64 third from
our 1/1 -- 18 commas (14r + 4s) -- and then go down by two commas
by following a different procedure, with a subtly different
result.
That procedure, having gone up four fifths and arrived at 81/64,
is to go up 24 fifths -- adding two regular 12-commas -- and then
up another 5 fifths, thus _subtracting_ a regular 256/243 limma
at 90.225 cents. Let's summarize these moves:
4 fifths up (81/64 major third) 14r + 4s +407.820c
24 fifths up (12-bicomma) 2r + 46.920c
5 fifths up (243/128 Maj7) -3r -1s - 90.225c
-----------------------------------------------------------------
33 fifths up 13r + 3s +364.515c
Here I have treated the last move of 5 fifths up, producing a
regular 243/128 major seventh at 1109.775 cents, as equivalent to
falling short of the octave by a regular 256/243 limma -- in
other words, as a ratio of 243/256.
From another perspective, we can combine the moves of 24 fifths
up and 5 fifths up into a single interval of 29 fifths up, the
"29-diesis," as we may call it. Here we have:
24 fifths up (12-bicomma) 2r + 46.920c
5 fifths up (243/128 Maj7) -3r -1s - 90.225c
-----------------------------------------------------------
29 fifths up (29-diesis) -1r -1s - 43.305c
The 29-diesis, the amount by which 29 pure 3:2 fifths fall short
of 17 octaves at 2:1, thus is 43.305 cents, equal to a regular
12-comma plus a smaller 41-comma. Subtracting this diesis from a
regular 81/64 third yields a size of 364.515 cents -- quite close
to the auto-peak envelope value of 365.2 cents measured by Ozan
Yarman, Can Akkoc, and a number of colleagues in a study of some
recorded flexible-pitch Turkish performances of this maqam.
Nearby JI ratios are 100/81 (364.807 cents) and 121/98 (364.984
cents), with 21/17 larger by about 1.310 cents.
Thus in a Pyth-53 circle the larger 16-comma third at 364.5 cents
might more closely approach the Turkish ideal of a "submajor"
flavor of Rast (13r + 3s), while the smaller 16-comma third at
360.9 cents would nicely fit a rather bright Arab Rast or low
Turkish Rast. While these two sizes may be regarded as musically
interchangeable for many purposes, someone playing a Pyth-53
instrument might choice to make a deliberate distinction at
times, a choice afforded by this most sophisticated JI system.
In considering this fine distinction on the order of 3.6 cents,
another element may enter in: the perception of the melodic
interval leading from the neutral third to the 4/3 fourth of
Rast. Since in a neutral third flavor of Rast this third will be
at some version or isotope of "16 commas," and a regular fourth
consists of 22 commas (more specifically 17r + 5s), this small
neutral second step must be equal to some version of "6 commas,"
as Turkish or Syrian theory tells us.
Let us consider first the lower version of our 16-comma third at
360.9 cents, and chart the steps and melodic intervals of a Rast
tetrachord:
1/1 9/8 ~16/13 4/3
commas: 0 9 16 22
(7r + 2s) (12r + 4s) (17r + 5s)
cents: 0 203.9 360.9 498.0
5ths up/down: 0 +2 -20 -1
9:8 ~128:117 ~13:12
commas: 9 7 6
(7r + 2s) (5r + 2s) (5r + 1s)
cents: 203.9 157.0 137.1
5ths up/down: +2 -22 +19
These steps are notably almost identical to those of Ibn Sina's
Mustaqim tetrachord at 1/1-9/8-39/32-4/3 or 9:8-13:12-128:117 at
203.9-138.6-155.6 cents -- except that the smaller neutral second
step precedes the larger in his Mustaqim, but follows it in a
modern Arab or Turkish Rast. If our two neutral steps were
arranged in the converse order, we would have a fine Mustaqim
indeed, one of the delightful features of Pyth-53!
Here the upper melodic step at 137.1 cents, very close to the
138.6 cents of a just 13:12 (138.573 cents), may have a tendency
to be perceived as rather different from either a usual semitone
or a usual whole tone, a distinctly "other" category.
Now let us consider the Rast tetrachord in Pyth-53 with the other
isotope of our "16-comma" third:
1/1 9/8 ~21/17 4/3
commas: 0 9 16 22
(7r + 2s) (13r + 3s) (17r + 5s)
cents: 0 203.9 364.5 498.0
5ths up/down: 0 +2 +33 -1
9:8 ~56:51 ~68:63
commas: 9 7 6
(7r + 2s) (6r + 1s) (4r + 2s)
cents: 203.9 160.6 133.5
5ths up/down: +2 +31 -34
In this version, only the third step of the tetrachord has
changed, the neutral third now at a higher 364.5 cents.
Consequently, the upper melodic neutral second step is smaller,
at about 133.530 cents, comparable to the step of 68:63 or
132.220 cents (the difference between 4/3 and 21/17), and almost
identical to the complex 5-based value of 27/25 at 133.238
cents.
At least for some listeners, these alternative "6-comma" steps at
133.5 or 137.1 cents may be near the line where a small neutral
or "supraminor second," say around 14:13 or 128.298 cents, shifts
toward the central neutral region represented, for example, by
Ibn Sina's 13:12 used, for example, in his Mustaqim. Tetrachords
featuring both 14:13 and 13:12, also much favored by Ibn Sina and
cited some two centuries later by Safi al-Din al-Urmawi, may make
the most of this subtle melodic distinction.
Thus it is possible that one quality of a typical Turkish
"submajor" flavor of Rast is that the step from the large neutral
third to the fourth, say around 14:13 (the difference of 4/3 and
26/21), is small enough to have some "semitonal" qualities. If,
as Easley Blackwood has suggested, a step size somewhere around
135 cents may mark the rough transition from the supraminor to
the more central neutral zone, then the subtle difference of
Pyth-53 may be a valuable resource in the interpretation of maqam
music.
Likewise, for Persian music, Hormoz Farhat's suggested size for a
small neutral second at around 135 cents is based in part on an
averaging on some tars and setars. While a value at around 13/12
seems very popular -- much as in Ibn Sina's time a millennium
ago, judging from his writings -- a smaller size closer to 14/13
has also been recorded in measurements of certain instruments.
Thus Pyth-53 makes available a fine approximation of 13/12, plus
a smaller step almost identical to 27/25, affording creative
choice.
------------------------------------------------
3. Conclusion: 53-complements and subtle nuances
------------------------------------------------
One way of summing up the situation in Pyth-53 is to say that
each basic category of interval, aptly identified as in Turkish
or Syrian theory simply by specifying a generic number of commas
(e.g. "16 commas" for either large neutral third, or "6 commas"
for either small neutral second step), has two versions or
isotopes differing in size by the comma of Mercator, 3.615
cents.
Further, a pair of these isotopes are mutual "53-complements":
thus a 16-comma or large neutral third from 20 fifths down at
360.9 cents (12r + 4s) has as its complement the similar but
distinct third from 33 fifths up at 364.5 cents (13r + 3s).
More generally, an interval formed from n fifths up is
complemented by one from (53 - n) fifths down, and vice versa.
For example, let us consider the 53-complement of a 9/8 tone from
two fifths up, which should be formed by a chain of (53 - 2) or
51 fifths down.
Now 48 fifths down (or fourths up) lowers the pitch by four usual
12-commas (4r), while another 3 fifths down or fourths up raises
the pitch by a regular minor third at 32/27 or 294.1 cents. The
two moves together produce this result:
48 fifths down (4 12-commas) -4r - 93.840 cents
3 fifths down (32/27 min3) 10r + 3s +294.135 cents
-----------------------------------------------------------
51 fifths down 6r + 3s +200.295 cents
While a usual 9/8 tone has 7r + 2s, this 53-complement also has
"9 commas," but 6r + 3s, or 200.295 cents. It is equal, in other
words, to a usual 32/27 minor third less four 12-commas, an
interval of 93.840 cents, and larger than the usual 256/243 limma
at 90.225 cents (3r + 1s).
Note that while the usual limma is 5 fifths down, its complement
is 48 fifths up, again an illustration of the 53-complement rule.
Having pointed out these subtle distinctions and choices in
Pyth-53, I should emphasize that no dramatic and unpleasant
surprise awaits the musician who simply moves freely around the
circle, disregarding these small inequalities.
Indeed, the charm of the tuning is that one is free either to
disregard them or, in certain styles and contexts, to make the
most of them!
Also, nothing here precludes the choice of 53-tET/EDO. My purpose
is only to show that Pyth-53 has its own creative possibilities,
and follows mathematical patterns just as logical and practical
as those of an equal temperament.
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Note
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1. Consider, for example, the possibility of setting the mujannab
or neutral second fret in Safi al-Din al-Urmawi's second `oud
tuning at 4608/4235 (146.134 cents), an impressive ratio which
describes a fret placed at the midway point between the frets for
limma (256/243) and regular tone (9/8), with a neutral third fret
at 72/59 (344.738 cents). This neutral third fret on the Bamm or
lowest string would result in a neutral sixth on the Mathlath
(open 4/3) string at a 4:3 fourth higher, or 96/59, 842.783
cents. The fifth from 4608/4235 on the Bamm to 96/59 on the
Mathlath would be equal to 4235:2832 or 696.649 cents, narrow of
3:2 by 4238:4235 or 5.306 cents, a somewhat greater impurity than
that of the "odd" fifth in Pyth-53 at 3.615 cents narrow. A
mathematical curiosity of the `oud fretting scenario above is
that it produces a "justly tempered" fifth almost identical to
that of the 16th-century European temperament of 1/4-comma
meantone at 696.578 cents, or 5.377 cents narrow. This curious
free association prompts mention of the fact that just as Pyth-53
is a fine JI system featuring a very practical musical circle
despite the lack of precise mathematical closure, so a 31-note
cycle of 1/4-comma meantone, very possibly used in the era
1555-1620 on Italian keyboards such as Nicola Vicentino's
archicembalo and arciorgano (1555 and 1561), and Fabio Colonna's
Sambuca Lincea (1618), likewise provides a fine circulating
system without a need for precisely equal temperament, although
31-tET/EDO shares many of the same attractions.
Most appreciatively,
Margo
mschulter@calweb.com