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Epimores, Commas, and Tempering:
Arithmetic series in MET-24
-----------------------------------------
One important feature of just intonation is the subtly unequal melodic
steps resulting from the use of superparticular ratios and divisions
such as 14:13:12:11 or 12:11:10:9. These subtly graduated steps play
an especially important role in the arithmetic divisions often favored
on the monochord or flutes, based on equal differences of length.
In a JI rendition of these arithmetic series or divisions, one measure
of this subtlety is the gradation not only of melodic steps, but of
the unequal commas which mark their differences.
These dimensions of subtlety in just melodic intonation also provide a
measure for the compromises inevitably involved in temperament. Such a
measure may be especially appropriate in evaluating a tempered system
like MET-24 designed for Near Eastern or other musics using a variety
of neutral second steps, where melodic subtlety and variety is
especially prized.
Comparing not only the sizes of just and tempered steps and intervals,
whether melodic or harmonic, but also the resulting sizes of commas
permits a fine calibration of the compromises involved in the decision
to temper, however "gently" or carefully.
Here I will focus especially on arithmetic series, with the creative
explorations of Kathleen Schlesinger and Elsie Hamilton lending much
inspiration.
It may be helpful briefly to explain that the MET-24 system has two
12-note chains of fifths at 57.422 cents apart, with the fifths in
each chain alternating between sizes of 703.125 and 704.297 cents.
The resulting scheme is this:
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1. Equable divisions of the minor third
---------------------------------------
One of Ptolemy's diatonic tetrachords, the Equable Diatonic at
12:11:10:9, with its first three notes divides a 6:5 minor third into
epimoric or superparticular neutral second steps of 12:11:10. George
Secor has very aptly proposed that, drawing on the name of Ptolemy's
Equable Diatonic, we might more generally use the term "equable"
division to describe the division of a minor third into two neutral
second steps. While this paper focuses on arithmetic divisions,
Secor's concept embraces either arithmetic and harmonic divisions
alike (e.g. 12:13:14 as well as 14:13:12), and also tempered
approximations.
For the three simplest ratios for minor thirds, these equable
divisions include Ptolemy's division of 6:5 into 12:11:10; Ibn Sina's
division of 7:6 into 14:13:12; and Kathleen Schlesinger's division of
13:11, the mediant of 6:5 and 7:6, into 13:12:11.
Beginning with Ptolemy's 12:11:10 division of the 6:5 minor third, we
might analyze this division as follows, noting not only the melodic
steps and intervals, but the comma or difference between the unequal
steps of 12:11 and 11:10, at 121:120 or 14.367 cents:
0 150.637 315.641
1/1 12/11 6/5
|-----------------|----------------|
12:11 11:10
150.637 165.004
|-----------------|
121:120
14.367
Although MET-24 is mainly designed to represent primes 2-3-7-11-13,
the 11:10 neutral second is also a part of the design, so that
Ptolemy's Equable Diatonic may be found in two locations with adjacent
12:11 and 11:10 steps:
1/1 12/11 6/5
0 150.000 312.891
J -0.637 -2.751
|-----------------|----------------|
12:11 11:10
150.000 162.891
-0.637 -2.114
|-----------------|
121:120
12.891
-1.477
In MET-24, as it happens, all the steps and intervals of the JI
version are slightly narrowed or diminished, and likewise the 121:120
comma between the two melodic steps, here reduced from a just 121:120
or 14.367 cents to 12.891 cents (11/1024 octave). The rendition is
reasonably accurate, but with slightly less contrast between these
steps.
Let us next consider the 13:12:11 division:
0 138.573 289.210
1/1 13/12 13/11
|-----------------|----------------|
13:12 12:11
139.573 150.637
|-----------------|
144:143
12.064
In MET-24, the 13:12:11 division has two very slightly different
temperings, one more accurate for 13/11 and the other for the 13/12
step. Here is an example of the former, found at keyboard locations
E-F*-G, that is with F* showing F on the upper keyboard or chain of
fifths:
E F* G
1/1 13/12 13/11
0 139.453 289.453
J +0.880 +0.243
|-----------------|----------------|
13:12 12:11
139.453 150.000
+0.880 -0.637
|-----------------|
144:143
10.547
-1.518
Although all steps and intervals are within a cent of just, yet the
comma of 144:143 or 12.064 cents between 13:12 and 12:11 in the JI
version is reduced by more than 1.5 cents, to 10.547 cents (9/1024
octave), somewhat lowering the contrast between these melodic steps.
Now let us consider the 14:13:12 division:
0 128.298 266.871
1/1 14/13 7/6
|-----------------|----------------|
14:13 13:12
128.298 138.573
|-----------------|
169:168
10.274
In MET-24, as with the 13:12:11 division, there are two slightly
different tempered forms of 14:13:12, with the following as an example
of the most common form:
F F# G*
1/1 14/13 7/6
0 125.391 264.844
J -2.908 -2.027
|-----------------|----------------|
14:13 13:12
125.391 139.453
-2.908 +0.880
|-----------------|
169:168
14.062
+3.788
A different situation obtains with the comma in this division: while
all steps and intervals are within three cents of just, the 169:168
comma between 14:13 and 13:12 at 10.274 cents is here _expanded_ to
14.062 cents (12/1024 octave), or 3.788 cents larger. The quite subtle
contrast between Ibn Sina's unequal steps, either of which might be
called a "2/3-tone," is thus somewhat exaggerated.
These equable divisions of the minor third may serve as an
introduction to the more extensive arithmetic divisions or series that
may arise in tetrachords, pentachords, and modes such as the
_harmoniai_ of Kathleen Schlesinger and the diaphonic, triaphonic, and
more complex cycles of John Chalmers. The rest of this paper considers
a few forms approximated in MET-24.
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2. Neutral second epimores, just and tempered: The Mytilene decatonic
---------------------------------------------------------------------
Named after the island of the poet Sappho, the Mytilene decatonic in
its just form consists of these steps:
0 128.3 266.9 417.5 498.0 626.3 702.0 830.3 968.8 1119.5 1200
1/1 14/13 7/6 14/11 4/3 56/39 3/2 21/13 7/4 21/11 2/1
|---------------------------|.............|------------------------|
14:13 13:12 12:11 22:21 14:13 117:112 14:13 13:12 12:11 22:21
128.3 138.6 150.6 80.5 128.3 75.6 128.3 138.6 150.6 80.5
In MET-24, a tempered form is the following:
C C# D* E F F# G G# A* B C
1/1 14/13 7/6 14/11 4/3 56/39 3/2 21/13 7/4 21/11 2/1
0 126.6 264.8 414.8 496.9 622.3 704.3 829.7 969.1 1119.1 1200
J -1.7 -2.0 -2.7 -1.2 -4.1 +2.3 -0.6 +0.3 -0.3 J
|---------------------------|..............|------------------------|
14:13 13:12 12:11 22:21 14:13 117:112 14:13 13:12 12:11 22:21
126.6 138.3 150.0 82.0 125.4 82.0 125.4 139.5 150.6 80.9
-1.7 -0.3 -0.6 +1.5 -2.9 +6.4 -2.9 +0.9 -0.6 +0.3
Doing some degree of justice to this scale would involve exploring its
various rotations and modes. Here, however, our very limited purpose
is to compare some details of the just and tempered intonations.
In the overview provided by the two diagrams above, a melodic quirk of
the tempered version stands out. In the JI version, there are five
distinct step sizes: the neutral second steps of 14:13, 13:12, and
12:11; the likewise epimoric semitone step at 22:21 (80.537 cents);
and finally the slightly smaller semitone of 117:112 (75.612 cents).
The difference between these two semitones is 352:351 or 4.925 cents.
In the tempered version, however, both the 22:21 steps, and much less
accurately the 117:112 step, are represented by an interval of 80.859
or 82.031 cents. The larger of these sizes, used here to represent
117:112, is a considerable 6.419 cents large. Thus a touch of melodic
variety is lost.
Looking in closer detail at a single tetrachord, we now focus on the
fine ordering of steps and commas:
0 128.298 266.871 417.508 498.045
1/1 14/13 7/6 14/11 4/3
|-----------------|----------------|---------------|--------|
14:13 13:12 12:11 22:21
128.298 138.573 150.637 80.537
|-----------------|---------------|...........|
169:168 144:143 126:121
10.274 12.064 70.100
The first four notes of this 4:3 division, 14:13:12:11, feature a
subtly graduated series of steps and commas: first the smaller 169:168
comma between 14:13 and 13:12; and then the slightly larger 144:143
between 13:12 and 12:11.
Quite distinct from these subtle distinctions of steps and commas in
the series 14:13:12:11 is the dramatic difference between 12:11 and
the 22:21 semitone which follows to complete the fourth -- a
difference large enough itself to constitute an excellent small
semitone, or better yet a thirdtone, at 126:121 or 70.100 cents.
This difference is shown with a cautionary dotted line, to distinguish
it from the preceding commas between the neutral second steps.
Comparing the lower and upper 4:3 divisions in the MET-24 version,
which present some slight differences of temperament, will give some
idea of the nuances and compromises involved:
lower 4:3 division in MET-24
C C# D* E F
1/1 14/13 7/6 14/11 4/3
0 126.563 264.844 414.844 496.875
J -1.736 -2.027 -2.664 -1.170
|-----------------|----------------|---------------|--------|
14:13 13:12 12:11 22:21
126.563 138.281 150.000 82.031
-1.736 -0.291 -0.637 +1.494
|-----------------|---------------|...........|
169:168 144:143 126:121
11.719 11.719 67.969
+1.444 -0.346 -2.131
A quirk of this tempering is that the comma between the 14:13 and
13:12 steps is identical to that between the 13:12 and 12:11 steps,
11.719 cents or 10/1024 octave. In JI, of course, the first comma at
169:168 or 10.274 cents is slightly smaller than the second at 144:143
or 12.064 cents. Thus tempering again loses a subtle nuance of the
arithmetic series 14:13:12:11. The more dramatic difference between
12:11 and 22:21, at 126:121 or 70.100 cents, is slightly reduced to
67.969 cents, a step which, like the just 126:121, makes an excellent
thirdtone.
The upper 4:3 division (3/2-2/1) presents a generally similar picture,
but with small variations. To provide more complete information, this
diagram shows the location of each note both in reference to the 1/1
of the overall decatonic scale, and to the 3/2 step as the "local 1/1"
of this 4:3 division.
3/2 21/13 7/4 21/11 2/1
704.297 829.688 969.141 1119.141 1200
+2.342 -0.566 +0.315 -0.322 J
G G# A* B C
1/1 14/13 7/6 14/11 4/3
0 125.391 264.844 414.844 495.703
J -2.908 -2.027 -2.664 -2.342
|-----------------|----------------|---------------|--------|
14:13 13:12 12:11 22:21
125.391 139.453 150.000 80.859
-2,908 -0.880 -0.637 +0.322
|-----------------|---------------|...........|
169:168 144:143 126:121
14.062 10.547 69.141
+1.444 -1.518 -0.959
A curious quirk of the temperament is that while in the lower 4:3
division the difference between the 169:168 and 144:143 commas is
levelled, both being represented as 11.719 cents, here the 169:168 has
a tempered representation actually larger than that of 144:143,
respectively 14.062 cents and 10.547 cents, as compared with the just
sizes of 10.274 and 12.064 cents! Thus while the just and tempered
versions may be musically comparable, they are not identical.
Finally, we should also focus more closely on the middle 9:8 tone
which in the Mytilene diatonic connects the two 4:3 divisions, and
where the greatest disparities between the just and tempered versions
may be found:
JI version of middle 9:8 tone
498.045 626.343 701.955
4/3 56/39 3/2
|---------------|---------|
1/1 14/13 9/8
0 128.298 203.910
14:13 117:112
128.298 75.612
Middle 9:8 tone in MET-24
-1.170 -4.078 +2.342
496.875 622.266 704.297
4/3 56/39 3/2
F F# G
|---------------|---------|
1/1 14/13 9/8
0 125.391 207.422
J -2.908 +3.512
14:13 117:112
125.391 82.031
-2.908 +6.419
Most significantly, as already noted, MET-24 has reasonably accurate
representations of the 14:13 and 22:21 steps found in this diatonic,
but not of 117:112, with the usual 22:21 approximations of 80.859 or
here 82.031 cents serving as the closest available temperings. Also,
the 56/39 step at a just 626.343 cents is represented by a rather
narrower 622.266 cents, with the compressed sizes of both 4:3 and
14:13 leaving it at 4.078 cents smaller than just.
These nuances make the tempered version both distinct from and less
melodically varied than the JI version with its 22/21 and 117/112
steps. Also, as we have seen, some of the subtle graduations in the
series of superparticular neutral second commas are levelled or
reversed, making the MET-24 version a variation on the theme of the JI
version.
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3. Ptolemy's Equable Diatonic: An equable heptatonic mode
---------------------------------------------------------
Various musics, for example in Southeast Asia and Africa, use
heptatonic divisions of the octave where the steps are not too far
from equal. In a more strictly "near-equal" heptatonic tuning, all
intervals of four scale steps should be rather close to 4/7 octave,
permitting free transpositions; but tunings with somewhat larger
variations in step sizes are also favored in some of these traditions.
Ptolemy's Equable Diatonic with its 12:11:10:9 tetrachords is
sometimes mentioned as one example of how such equable but distinctly
unequal heptatonic divisions of the octave may be made.
If we choose a disjunct form with two 12:11:10:9 tetrachords plus a
middle 9:8 tone, then one way of viewing the resulting mode is as a
lower pentachord of 12:11:10:9:8 plus an upper 12:11:10:9 tetrachord.
12:11:10:9:8 pentachord 12:11:10:9 tetrachord
|-------------------------------------|--------------------------|
0 150.637 315.641 498.045 701.955 852.592 1017.596 1200
1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1
|--------|--------|---------|---------|-------|--------|---------|
12:11 11:10 10:9 9:8 12:11 11:10 10:9
150.637 165.004 182.404 203.910 150.637 165.004 182.404
|--------|---------|---------| |-------|---------|
121:120 100:99 81:80 121:120 100:99
14.367 17.399 21.506 14.367 17.399
The subtly graduated epimore or superparticular steps of this tuning
are accompanied by a similarly graduated set of superparticular
commas: 121:120, 100:99, and 81:80.
In MET-24, we find this tempered version at a single location:
12:11:10:9:8 pentachord 12:11:10:9 tetrachord
|-------------------------------------|--------------------------|
B* C# Eb E* F#* G# Bb B*
1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1
0 150.000 312.891 495.703 703.125 853.125 1016.016 1200
J -0.637 -2.751 -2.342 -1.170 -0.533 -1.581 J
|--------|--------|---------|---------|-------|--------|---------|
12:11 11:10 10:9 9:8 12:11 11:10 10:9
150.000 162.891 182.891 207.422 150.000 162.891 183.984
-0.637 -2.114 +0.409 +3.512 -0.637 -2.114 +1.581
|--------|---------|---------| |-------|---------|
121:120 100:99 81:80 121:120 100:99
12.891 19.922 24.609 12.891 21.094
-1.477 +2.522 +3.103 -1.477 +3.694
While the intervals and melodic steps are generally rather accurate,
the subtle sequence of commas is somewhat altered, with 121:120 a bit
smaller than just but 100:99 and 81:80 both enlarged.
Since the representation of 12:11:10 in MET-24, and more generally of
the 12:11 and 11:10 steps, does not change, the main variable
intervals here are the 10:9 step (at 182.891 or 183.984 cents) and the
4:3 fourths and 3:2 fifths at 495.703 or 496.875 cents, and 703.125 or
704.297 cents. Because of the variability of 10:9, the size of the
100:99 comma likewise changes in the lower tetrachord or pentachord
and upper tetrachord.
The tempered 121:120 and 100:99 commas do not occur as direct steps on
the keyboard, but the comma of 23.438 or 24.609 cents (20 or 21 steps
of 1024-EDO), with the latter form here representing 81:80, does, and
also represents the Archytan or septimal comma of 64:63 (27.264
cents). In typical Near Eastern contexts with neutral steps of special
interest, this comma also represents the difference between 11:10 and
13:12 (66:65 or 26.432 cents), or 12:11 and 14:13 (78:77 or 22.339
cents).
A due caution is also in order here: one should not draw a conclusion
from the above example that MET-24 is generally a good source of close
approximations for 5-limit intervals such as 6:5. It isn't, with its
main focus on primes 2-3-7-11-13, quite unlike George Secor's High
Tolerance Temperament (HTT) family of tunings, for example, with
comprehensive support for primes 2-3-5-7-11-13 as typified by a
near-just ogdad at 4:5:6:7:9:11:13:15.
However, the design of MET-24 does include support for certain neutral
intervals combining factors of 5 and 11, notably the 11:10 epimore
step and such related ratios as 40:33, 33:20, and 20:11. In this
framework a few approximations of simple 5-limit ratios also occur,
such as 6/5 in the above tempering of Ptolemy's Equable Diatonic, and
also a 5:4 approximation at 6/5-3/2 with a size of 390.234 cents
(3.921 cents wide of a just 386.314 cents), with a slightly larger
form at 391.406 cents or 5.093 cents wide found elsewhere at MET-24
(at Bb-C#*).
From a musical and historical perspective, it seems fitting that a
temperament focusing on the equable 14:13:12 and 13:12:11 divisions of
the minor third should also render homage to Ptolemy and his germinal
12:11:10:9 division lending its Equable Diatonic name to the others.
The presence of a tempered version of this classic tuning is a partly
unanticipated but totally welcome consequence of the tuning.
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4. Conclusion
-------------
The subtle graduation of melodic steps is a special attraction of JI;
comparing not only step and interval sizes, but also the sizes of
resulting commas, is one fine gauge for the altered melodic
proportions brought about by tempering.
Arithmetic or harmonic series of epimore steps such as 12:11:10,
13:12:11, or 14:13:12 are settings where this sense of melodic
proportion may be especially important. The MET-24 system, as a
temperament which seeks close approximations of the neutral epimore
steps 14:13, 13:12, 12:11, and 11:10, may serve as an illustration of
the compromises which predictably occur when the just fabric is
altered, however carefully.
Such an approach should invite a renewed appreciation of the unique
attractions of just tuning, as well as the potentials and inevitable
compromises of temperament.
In peace and love,
Margo Schulter
mschulter@calweb.com
First version: 14 October 2012
Revised: 5 December 2012