Dear Andy, Bill, Jim, and Colby,
Please let me say that while I'm not sure exactly how I would fit into
some of the more confidential aspects of this project, I'm delighted
to offer suggestions or elaborations on general points of theory which
don't require knowledge of proprietary internal details.
One such interesting general point is Valid Tuning Range (VTR) for
what is termed the "schismatic temperament" -- or, as I would say, the
schismatic mapping.
Here I'll suggest that this mapping, whlle applicable for its original
purpose (approximating 5-limit ratios) in the immediate vicinity of
Pythagorean intonation or a rounded 702 cents, has a likely VTR from
19-EDO to 17-EDO, or a rounded 695-706 cents.
--------------------------------------------
1. Evaluating VTR for the schismatic mapping
--------------------------------------------
As I understand it, VTR pertains not to the absolute sizes of
intervals generated through a given mapping, but to their _relative_
sizes, or more precisely the ordering of their sizes.
One approach would be to note the ordering of relevant intervals in
the schismatic mapping when it is used around 702 cents for its
original purpose of obtaining just or near-just 5-limit intervals, and
then to test how far in either direction along the continuum that
ordering is preserved.
Here, for convenience, I will take a rounded 702 cents as the
generator for a traditional schismatic mapping, making the arithmetic
a bit simpler while arriving at a result virtually identical to that
of Pythagorean tuning (701.955 cents).
--------------------------------------------------------
Interval Example Cents 5-limit Cents
--------------------------------------------------------
Apotome C-C# 114 16:15 112
Diminished 3rd C#-Eb 180 10:9 182
Augmented 2nd Eb-F# 318 6:5 316
Diminished 4th C#-F 384 5:4 386
Augmented 5th F-C# 816 8:5 814
Diminished 7th F#-Eb 882 5:3 884
Augmented 6th Eb-C# 1020 9:5 1018
Diminished 8ve C#-C 1086 15:8 1088
-------------------------------------------------------
As I here define it, the VTR is the range over which these orderings
are preserved, with limits at the points where any relevant pair of
these intervals have identical sizes, so that any further motion in
that direction along the regular diatonic continuum of generators
would cause a reversal of sizes, and move us into a different
mapping.
----------------------------------------
1.1. The schismatic mapping at 696 cents
----------------------------------------
Let us first consider two interesting points where the mapping remain
valid, but yields very useful interval sizes other than those of the
5-limit. We begin at 696 cents, quite close to Zarlino's 2/7-comma
meantone, the first regular temperament to be defined mathematically
in terms of fractions of a syntonic comma (1558):
--------------------------------------------------------
Interval Example Cents 2-3-7 JI Cents
--------------------------------------------------------
Apotome C-C# 72 28:27 63
Diminished 3rd C#-Eb 240 8:7 231
Augmented 2nd Eb-F# 264 7:6 267
Diminished 4th C#-F 432 9:7 435
Augmented 5th F-C# 766 14:9 765
Diminished 7th F#-Eb 936 12:7 933
Augmented 6th Eb-C# 960 7:4 969
Diminished 8ve C#-C 1128 27:14 1137
-------------------------------------------------------
Note that while the interval sizes are quite different than around 702
cents, here approximating 2-3-7 or Archytan JI (named after the
diatonic of Archytas, as we'll shortly see) rather than 5-limit JI,
the ordering of sizes is the same. Thus we remain within the VTR.
Let's quickly consider the classic JI tunings that these two
realizations of the schismatic mapping approximation. Around 702 cents
our mapping approximates the Syntonic Diatonic of Ptolemy with its
prime factors of 2-3-5, later adopted in the 16th century by Fogliano
(1529) and Zarlino (1558) as a basic for contemporary vocal polyphony.
Here I'll give a version in the Ionian Mode, which Zarlino took as the
first of the 12 modes:
C D E F G A B C
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
0 204 386 498 702 884 1088 1200
9:8 10:9 16:15 9:8 10:9 9:8 16:15
204 182 112 204 182 204 112
Here I follow Zarlino in placing the smaller 10:9 tones at D-E and
G-A. In practice, of course, singers will dynamically adjust certain
notes to preserve concord, for example a 3:2 fifth at D-A. On
fixed-pitch instruments, as Zarlino notes, it is much more practical
to use temperament: 2/7-comma or 1/4-comma meantone for keyboards, and
12-TET for fretted instruments such as the lute.
Let us consider an equivalent of this JI tuning in our 702-cent
schismatic mapping:
B C# Eb E F# Ab Bb B
0 204 384 498 702 882 1086 1200
204 180 114 204 180 204 114
In either the 5-limit JI or schismatic version, there are three types
of melodic steps: a larger tone at 9:8 or 204 cents; a smaller tone at
10:9 or 182 cents (180 cents in the schismatic version); and a rather
large semitone at 16:15 (112 cents), or 114 cents in the schismatic
version.
At 696 cents, the schismatic mapping approximates, as we have said, a
different kind of JI diatonic: the diatonic of Archytas, adopted by
Ptolemy as the "Tonic Diatonic." Here I give this tuning in a
permutation which might tend to maximize the degree of sensory
consonance in relation to the 1/1; this arrangement might be described
as a septimal (i.e. 2-3-7) JI version of the medieval/Renaissance
Dorian mode:
D E F G A B C D
1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1
0 204 267 498 702 906 969 1200
9:8 28:27 8:7 9:8 9:8 28:27 8:7
204 63 231 204 284 63 231
We have again three melodic step sizes: a 9:8 tone at 204 cents; a
larger 8:7 tone at 231 cents; and a very compact semitone at 28:27 or
63 cents, highly effective melodically, as George Secor has noted.
Now let us consider our equivalent for this Archytas or Tonic Diatonic
using the schismatic mapping at 696 cents:
Ab Bb B Db Eb F F# Ab
0 192 264 504 696 888 960 1200
192 72 240 192 192 72 240
Here the large tone at 240 cents is somewhat wide but not too far from
a just 8:7 at 231 cents, with the 72-cent semitone also a bit wider
than but fairly close to 28:27 at 63 cents. The biggest difference is
the substitution of the regular meantone major second or tone at 192
cents for a just 9:8 at 204 cents in our 2-3-7 JI version. In some
musical contexts, this might significantly change the artistic effect.
In the JI version, for example, we might have this three-voice
cadence with parallel thirds and sixths (and fourths between the upper
voices) leading to a stable 2:3:4 sonority:
E D E
B A B
G F E
Both G-B-E at a Pythagorean 0-408-906 cents with an 81:64 major third
and 27:16 major sixth (64:81:108), and F-A-D at a septimal 0-435-933
cents, with a 9:7 major third and 12:7 major sixth (7:9:12), are
rather active and outgoing sonorities, leading nicely to the
resolution in a usual 14th-century European fashion.
At 696 cents, however, the same progression becomes:
Bb Ab Bb
F Eb F
Db B Bb
Here the opening sonority Db-F-Bb is tuned in regular meantone at
0-384-888 cents, very close to a just 12:15:20 with a 5:4 major third
and 5:3 major sixth. This contrasts dramatically with the following
B-Eb-Ab at 0-432-936 cents, quite close to a just 7:9:12 -- where the
third and sixth are 48 cents larger! Such a dramatic difference might
be accepted or even relished as "going with the territory," but
produces a color distinct from that of the JI Tonic Diatonic version.
Interestingly, one element of this 696-cent or more generally meantone
schismatic mapping is already used in the Matrix/Thummer system: the
meantone augmented sixth (e.g. Ab-F#) as an equivalent of 7:4 in order
to approximate a seventh chord at 4:5:6:7 (0-386-702-969 cents) as
Ab-C-Eb-F# (here 0-384-696-960 cents, and 0-386-697-966 cents in
1/4=comma).
In the Tonic Diatonic or its 696-cent schismatic equivalent, however,
a different variety of sonority with a minor seventh is available:
2-3-7 JI version 696-cent schismatic version
C F#
A Eb
F B
D Ab
0-267-702-969 cents 0-264-696-960 cents
In JI, this chord with a 7:6 minor third, 3:2 fifth, and 7:4 minor
seventh has a ratio of 12:14:18:21 (0-267-702-969 cents). Musically,
like 4:5:6:7, it is a rather consonant sonority from a sensory
standpoint, but may be treated as stable or unstable depending on the
style. Just as 4:5:6:7 might serve either as a tuning of the dominant
seventh in a major/minor style (as proposed by Euler in 1764), or as
the norm of stable concord in Paul Erlich's decatonic system of
tonality based on a tuning around 22-EDO, so 12:14:18:21 may either
resolve to a yet more concordant sonority such as a 3:2 fifth, or
stand as a stable conclusion, as noted by Secor. Typical directed
resolutions would be:
2-3-7 JI version 696-cent schismatic version
C B F# F
A B Eb F
F E B Bb
D E Ab Bb
In these progressions, the 7:4 minor seventh contracts to a fifth and
the 7:6 minor thirds to unisons, with steps of 9:8 and 28:27 (204 and
63 cents) in JI, and 192 and 72 cents in our 696-cent tuning. These
cadences are very effective, and could be considered septimal
(i.e. 2-3-7) analogues of the tonal V7-I cadence.
Let us now consider the schismatic mapping near the other end of its
VTR, where different interval sizes and colors result: 705 cents.
----------------------------------------
1,2. The schismatic mapping at 705 cents
----------------------------------------
Here the lineup of relevant interval categories for the mapping is as
follows:
---------------------------------------------------------
Interval Example Cents 2-3-11-13 JI Cents
---------------------------------------------------------
Apotome C-C# 135 13:12 139
Diminished 3rd C#-Eb 150 12:11 151
Augmented 2nd Eb-F# 345 11:9 347
Diminished 4th C#-F 360 16:13 359
Augmented 5th F-C# 840 13:8 841
Diminished 7th F#-Eb 855 18:11 853
Augmented 6th Eb-C# 1050 11:6 1049
Diminished 8ve C#-C 1065 24:13 1061
---------------------------------------------------------
Again, while the interval sizes have changed, the ordering remains the
same, thus keeping us within the VTR for the schismatic mapping.
Using this mapping, we can approximate another traditional JI tuning,
this one however coming from the early medieval rather than ancient
Greek era: the `ud or lute tuning of Mansur Zalzal used in 8th-century
Baghdad, as reported by al-Farabi (?-950):
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
0 204 355 498 702 853 996 1200
9:8 12:11 88:81 9:8 12:11 88:81 9:8
204 151 143 204 151 143 204
As with the Syntonic Diatonic and the Archytan or Tonic Diatonic,
there are three step sizes: a 9:8 tone (common to all three tunings),
a larger neutral second at 12:11 or 151 cents, and a smaller neutral
second at 88:81 or 143 cents.
In a 705-cent schismatic mapping, there is a subtle change of interval
sizes and colors which may most notably affect the smaller neutral
second step, here reduced from 88:81 at 143 cents to 135 cents, or
slightly smaller than a just 13:12 at 139 cents. Thus some ratios of
13, while not an element of Zalzal's scale, are closely approximated
at some locations in this tempered version:
F# G# Bb B C# Eb E F#
0 210 360 495 705 855 990 1200
210 150 135 210 150 135 210
Small variations of this kind may be typical of medieval and modern
Near Eastern tuning practices: thus Scott Marcus reports that modern
performers of traditional Arab music in Egypt may favor in certain
contexts a smaller neutral second around 135-145 cents, with our
705-cent schismatic version of Zalzal's tuning illustrating the lower
end of this range, and al-Farabi's original version approaching the
higher end.
George Secor describes a different variety of 7-note tuning which
features a 13/8 "harmonic" neutral sixth above the 1/1:
1/1 13/12 11/9 4/3 3/2 13/8 11/6 2/1
0 139 347 498 702 841 1049 1200
9:8 44:39 12:11 9:8 13:12 44:39 12:11
204 209 151 204 139 209 151
Here, as in the older JI tunings we have considered so far, there are
three step sizes: a tone at 44:39 or 209 cents, slightly larger than
the classic 9:8 at 204 cents; and neutral second steps at 12:11 (151
cents) and 13:12 (139 cents). Our schismatic mapping at 705 cents can
quite closely approximate these sizes (here 210, 150, and 135 cents):
Bb B C# Eb F F# G# Bb
0 135 345 495 705 840 1050 1200
135 210 150 210 135 210 150
The versatile schismatic mapping can approximate 5-limit intervals at
702 cents; septimal (2-3-7 JI) intervals at 696 cents; or neutral
intervals (2-3-11-13 JI) at 705 cents. Let us now map the outer
boundaries of the VTR for this mapping.
-------------------------------------------
2. The limits of the VTR: 19-TET and 17-TET
-------------------------------------------
We found that the schismatic mapping is valid at 696, 702, and 705
cents -- and now wish to locate the limits of the VTR. Let us first
consider the set of relevant intervals in 19-TET, with a generator of
694.74 cents:
--------------------------------------------------------
Interval Example Cents JI Cents
--------------------------------------------------------
Apotome C-C# 63 28:27 63
Diminished 3rd C#-Eb 252 15:13 248
Augmented 2nd Eb-F# 252 15:13 248
Diminished 4th C#-F 442 9:7 435
Augmented 5th F-C# 758 14:9 765
Diminished 7th F#-Eb 948 26:15 952
Augmented 6th Eb-C# 948 26:15 952
Diminished 8ve C#-C 1137 27:14 1137
-------------------------------------------------------
Here we have reached a boundary where the diminished third at C#-Eb
has a size identical to that of the augmented second at Eb-F# at 252
cents. If we seek a JI equivalent involving prime factors no higher
than 13, then a good choice would be 15:13 at 248 cents; 22:19 at 254
cents would by somewhat closer.
Similarly, the diminished seventh F#-Eb at 948 cents is identical to
the augmented sixth Eb-C#. Here 26:15 (952 cents) is a reasonable JI
equivalent, although the 19-prime 19:11 (946 cents) would be closer.
These intervals are what I have termed "interseptimal," located
between the two simple septimal ratios of 8:7 (231 cents) and 7:6 (267
cents) at 252 cents; or between 12:7 (933 cents) and 7:4 (969 cents)
at 948 cents. See
An interesting musical property of this 19-TET mapping is that the
252-cent interval might represent either a very wide major second or a
very small minor third; and likewise 948 cents might be used as a very
large major sixth or a very narrow minor seventh. These two
alternative resolutions illustrate this creative ambiguity:
C# C C# E
Bb C Bb B
Eb F Eb E
In the first alternative, the outer interval Eb-C# at 948 cents acts
like a narrow minor seventh, contracting to a fifth, with steps of a
regular meantone major second at 189 cents (Eb-F) and a small semitone
C#-C (63 cents). At the same time, the upper interval Bb-C# at 252
cents acts like a small minor third contracting to a unison via the
same melodic step sizes (Bb-C, C#-C).
In the second resolution, Eb-C# instead acts as a wide major sixth,
expanding to the octave of the 2:3:4 sonority E-B-E, while the upper
Bb-C# acts like a wide major second expanding to a fourth. These
progressions, as in the first resolution, involve steps of 189 and 63
cents.
It is perhaps psychologically convenient that 19-TET should be the
lower limit of the VTR for the schismatic mapping, since it might also
be considered the lower limit of the meantone range proper, if we
regard meantone temperament as primarily a strategy for optimizing JI
factors 2-3-5 among regular diatonic intervals. Around 19-TET, or the
virtually identical 1/3-comma meantone (694.79 cents), we get a pure
6:5 minor third at 316 cents. Tempering the fifth more than in
1/3-comma or 19-TET (where the minutely greater temperament produces
precise mathematical symmetry, although 1/3-comma will circulate
without any problem, as will 1/4-comma in 31 notes vis-a-vis 31-TET)
moves the fifth, major third, and minor third alike away from their
values in the 5-limit, and thus moves into another realm where
different motivations prevail.
Let us now consider the upper limit of the VTR for the schismatic
mapping: 17-TET, with a generator at 705.88 cents.
--------------------------------------------------------
Interval Example Cents JI Cents
--------------------------------------------------------
Apotome C-C# 141 13:12 139
Diminished 3rd C#-Eb 141 13:12 139
Augmented 2nd Eb-F# 353 27:22 355
Diminished 4th C#-F 353 27:22 355
Augmented 5th F-C# 847 44:27 845
Diminished 7th F#-Eb 847 44:27 845
Augmented 6th Eb-C# 1059 24:13 1059
Diminished 8ve C#-C 1059 24:13 1059
-------------------------------------------------------
Here we may in brief describe what has happened by saying the neutral
second, third, sixth, and seventh have single values at 141, 353, 847,
and 1059 cent. The apotome is identical to the diminished third; the
augmented second to the diminished fourth; the augmented fifth to the
diminished seventh; and the augmented sixth to the diminished octave.
In the "neomedieval" portion of the continuum which runs from around
702 to 709 cents, or Pythgorean to 22-TET, 17-TET marks not only the
upper limit of the VTR for the schismatic mapping, but also a divide
between two approaches to approximating septimal (2-3-7) intervals
where they are available in reasonably representative versions within
this part of the spectrum.
In the range from 702 to 706 cents (or more precisely 17-TET),
septimal approximations, where available, involve rather long chains
of fifths. Optimal spots for such purposes are right around or very
slightly above Pythagorean, where chains of 14, 15, or 16 generators
approximate 7:4, 7:6, and 9:7. Exactly at Pythagorean with a 3:2
fifth, these intervals will vary from just by the "septimal schisma"
of about 3.80 cents. By making the fifth minutely wider, we can temper
out this small schisma: at 702.21 cents, with the fifth larger than
pure by 1/15 of this schisma, 7:6 is pure, and 7:4 and 9:7 impure by
the same tiny amount as the fifth, about 0.25 cent.
A second "sweet spot" is found at around 704.6 to 705 cents, where
chains of 13, 14, or 15 generators produce 9:7, 7:6, and 7:4. Here the
overall accuracy cannot be quite so great as in the niche just above
Pythagorean, where the fifth is virtually just, since the tempering of
the fifth at around 3 cents wide of pure must make some of the
septimal ratios impure by at least this amount. An optimal strategy
might be setting 7:6 pure, using a generator of 704.78 cents, with 9:7
and 7:4 impure by the same amount as the fifth, about 2.82 cents.
Starting at around 17-TET, however, another strategy prevails: simply
tempering out the septimal comma so that regular intervals approximate
septimal (2-3-7) ratios, much as the syntonic comma is tempered out in
meantone to approximate 5-limit intervals. At 17-TET, we are on the
threshold of this strategy: the minor and major thirds and minor
seventh at 282, 423, and 988 cents have a rather different musical
quality than 7:6, 9:7, and 7:4 at 267, 435, and 969 cents, one which
may be more evocative of the "middle" territory between Pythagorean
and septimal flavors, a territory also characteristic of neomedieval
style.
At around 707 cents, these regular intervals become more convincingly
"septimal," as in the nearer transpositions of George Secor's 17-note
well-temperament (where the larger fifths are 707.22 cents). In the
territory around 708-709 cents, these approximations become more
accurate. At 22-TET (709.09 cents), major thirds at 436 cents are very
slightly wide of a just 9:7; minor thirds at 273 cents about 6 cents
wide of 7:6; and minor sevenths at 982 cents about 13 cents wide of
7:4.
Above 17-TET, the schismatic mapping is no longer valid, which is to
say that it is replaced by a related by different mapping we will
briefly consider below.
----------------------------------------------------------------
3. An overview, the schismatic and "reverse schismatic" mappings
----------------------------------------------------------------
Having taken "snapshots" of the schismatic mapping as it operates at
696, 702, and 705 cents, and ascertained its limits of validity as
19-TET and 17-TET, we may get an overview of this mapping by sampling
the values of our relevant intervals at generator sizes located along
the VTR for this mapping. Here we start at 19-TET and conclude at
17-TET, sampling intermediate generators at reference points about one
cent apart:
------------------------------------------------------------------------------
Intvl Example 19-TET 696 697 698 699 700 701 702 703 704 705 17-TET
------------------------------------------------------------------------------
Apot C-C# 63 72 79 86 93 100 107 114 121 128 135 141
dim3 C#-Eb 252 240 230 220 210 200 190 180 170 160 150 141
Aug2 Eb-F# 252 264 273 282 291 300 309 318 327 336 345 353
dim4 C#-F 442 432 424 416 408 400 392 384 376 368 360 353
Aug5 F-C# 758 766 776 784 792 800 808 816 824 832 840 847
dim7 F#-Eb 948 936 927 918 909 900 891 882 873 864 855 847
Aug6 Eb-C# 948 960 970 980 990 1000 1010 1020 1030 1040 1050 1059
dim8 C#-C 1137 1128 1121 1114 1107 1100 1093 1086 1079 1072 1065 1059
------------------------------------------------------------------------------
This overview of the schismatic mapping suggests some general regions
along the VTR for approximating different flavors of intervals.
At 19-TET or around 695 cents, we generally get interseptimal flavors,
although the apotome or chromatic semitone at 63 cents provides a
virtually just 28:27, and likewise the diminished octave a virtually
just 27:14.
Around 696-697 cents, or in the general zone of 2/7-comma, 1/4-comma,
and 31-TET, we get the best septimal (2-3-7) approximations for this
mapping, with 9:7 virtually just around 2/7-comma, and 7:4 around
31-TET.
At 697-699 cents, we get intervals in the intermediate range between
Pythagorean and septimal -- also a staple of regular neomedieval
tunings around 702-706 cents, where intervals of this type are the
usual diatonic sizes.
At 700 cents, we have a special situation where schismatic intervals
are identical to regular diatonic ones (e.g. the augmented second to a
regular minor third at 300 cents, and the diminished fourth to a
regular major third at 400 cents). These regular and schismatic
intervals might be taken either as somewhat subdued approximations of
Pythagorean intonation, or rather inaccurate 5-limit approximations
(more persuasive in less assertively harmonic timbres, for example a
16th-century lute or 20th-century piano).
Around 701-703 cents we have the "schismatic" mapping in its familiar
5-limit sense, with an optimal niche in the zone at or just below
Pythagorean if it is these 5-limit intervals which are to be optimized
(e.g. 53-TET).
Starting around 29-TET (703.45 cents), we enter a zone where the
schismatic mapping shifts from the 5-limit to the neutral region -- a
zone extending to 17-TET, the upper limit of the VTR. Note in 29-TET,
the augmented second at 331 cents and diminished fourth at 372 cents
might be considered either rather inaccurate 5-limit approximations,
or as reasonable approximations of small and large neutral thirds, for
example 63:52 (332 cents) and 26:21 (370 cents). By 704 cents, the
neutral character of these "schismatic" thirds at 336 and 368 cents is
more pronounced, and by 705 cents were are into the "central" neutral
region of thirds from around 39:32 (342 cents) to 16:13 (359 cents).
At 17-TET, the 706-cent fifth is divided into two precisely equal
neutral thirds at 353 cents -- or, more precisely, 705.88 cents and
352.94 cents.
In fact, 17-TET marks the end not of the great region where augmented
and diminished intervals yield neutral sizes, but simply of the
portion where the augmented second is smaller than the diminished
third. Here it may be instructive to see what happens between 702 and
709 cents (or 22-TET at 709.09 cents), and a bit beyond that.
------------------------------------------------------------------------------
Intvl Example 702 703 704 705 17-TET 707 708 22-TET 710 711 712
------------------------------------------------------------------------------
Apot C-C# 114 121 128 135 141 149 156 164 170 177 184
dim3 C#-Eb 180 170 160 150 141 130 120 109 100 90 80
Aug2 Eb-F# 318 327 336 345 353 363 372 382 390 399 408
dim4 C#-F 384 376 368 360 353 344 336 327 320 312 304
Aug5 F-C# 816 824 832 840 847 856 864 873 880 888 896
dim7 F#-Eb 882 873 864 855 847 837 828 818 810 801 792
Aug6 Eb-C# 1020 1030 1040 1050 1059 1070 1080 1091 1100 1110 1120
dim8 C#-C 1086 1079 1072 1065 1059 1051 1044 1036 1030 1023 1016
------------------------------------------------------------------------------
At 17-TET, we have not only the upper limit of the schismatic mapping
but also the lower limit for the VTR of mapping we might describe as
"reverse schismatic," with the order of certain pairs of intervals
reversed. Thus the diminished third is now larger than the apotome;
the augmented second than the diminished fourth; the augmented fifth
than the diminished seventh; and the augmented sixth than the
diminished octave.
From 17-TET to around 708 cents, this mapping produces neutral
intervals of various shadings -- a kind of "mirror reversal" of the
schismatic mapping around 703.45-705.88 cents (or 29-TET to 17-TET).
As we approach 709 cents or 22-TET, we enter a zone where the reverse
schismatic mapping -- like the schismatic mapping around 702 cents --
produces 5-limit approximations. Indeed the 382-cent augmented second
of 22-TET is very close to the 384-cent diminished fourth at 702
cents, with the 22-TET augmented sixth at 1091 cents likewise
comparable to the 702-cent diminished octave at 1086 cents. The
diminished fourth of 22-TET at 327 cents, however, is considerably
larger and further from a just 6:5 (316 cents) than the Pythagorean
schismatic equivalent at 318 cents, approaching the lower end of the
neutral zone at around 330 cents. As the fifth is tempered more and
more heavily in the wide direction, here about 7.14 cents (comparable
to 19-TET at 7.22 cents in the opposite direction), such divergences
among at least some intervals of a given "family" of approximations
from JI values (here 5-limit) become inevitably greater and greater.
By conventional standards of historical European keyboard
temperaments, about 7 cents may be the maximum desirable tempering of
the fifth in a regular system, with 1/3-comma meantone or 19-TET as a
16th-century precedent. Moving further along the continuum is to enter
a "xentonal" region where other assumptions, aesthetics, and timbral
conditions may often prevail.
Between 22-TET (709.09 cents) and 712 cents, various intervals move
through a 5-limit region, and some are still in this range at 712
cents. The apotome has grown from 164 cents in 22-TET, very close to
Ptolemy's large neutral second at 11:10 (165 cents), to a near-just
10:9 at 184 cents, actually a bit larger than the just ratio (182
cents). The diminished fourth at 304 cents is still not too far from
6:5 (316 cents), actually somewhat closer than in 12-TET. However, the
augmented second at 408 cents is a virtually just Pythagorean major
third at 81:64, while the augmented sixth at 1120 cents, very close to
a just 21:11 (1119 cents), is about ten cents larger than the
Pythagorean major seventh at 243:128 or 1110 cents. The tempering of
the fifth at a full 10 cents wider than pure helps to account for some
of these colorful variations between related categories of intervals
in the reverse schismatic mapping.
What might be the upper limit of the VTR for the reverse schismatic
mapping? Starting from 17-TET, we can survey its values in order to
seek an answer to this question. Here interval categories are ordered
so that the smaller category of a relevant pair appears first in the
table:
------------------------------------------------------------------------------>
Intvl Example 17-TET 707 708 22-TET 710 711 712 713 714 715 716 717
------------------------------------------------------------------------------>
dim3 C#-Eb 141 130 120 109 100 90 80 70 60 50 40 30
Apot C-C# 141 149 156 164 170 177 184 191 198 205 212 219
dim4 C#-F 353 344 336 327 320 312 304 296 288 280 272 264
Aug2 Eb-F# 353 363 372 382 390 399 408 417 426 435 444 453
dim7 F#-Eb 847 837 828 818 810 801 792 783 774 765 756 747
Aug5 F-C# 847 856 864 873 880 888 896 904 912 920 928 936
dim8 C#-C 1059 1051 1044 1036 1030 1023 1016 1009 1002 995 988 981
Aug6 Eb-C# 1059 1070 1080 1091 1100 1110 1120 1130 1140 1150 1160 1170
------------------------------------------------------------------------------>
-------------------------------
Intvl Example 718 719 720
-------------------------------
dim3 C#-Eb 20 10 0
Apot C-C# 226 233 240
dim4 C#-F 256 248 240
Aug2 Eb-F# 462 471 480
dim7 F#-Eb 738 729 728
Aug5 F-C# 944 952 960
dim8 C#-C 974 967 960
Aug6 Eb-C# 1180 1190 1280
-------------------------------
Since the ordering of intervals remains valid to the upper regular
diatonic limit of 5-TET or 720 cents, we can conclude that the reverse
schismatic mapping has a VTR running from 17-TET to 5-TET.
Over this imposing range of the spectrum we find neutral intervals
from 17-TET or 706 cents to about 708 cents; 5-limit approximations
around and somewhat beyond 22-EDO or 709 cents; and approximations in
the range from Pythagorean to septimal, for the most part, between
about 712 and 715 cents. The latter value yields an apotome at 205
cents or a near-just 9:8 tone, and an augmented second at 435 cents or
a virtually just 9:7.
At 716-717 cents, the diminished fourth provides very nice
approximations of a 7:6 minor third, while some other intervals move
into the interseptimal region -- specifically the augmented second and
diminished seventh, respectively 444 and 756 cents with a 716-cent
generator; and 453 and 747 cents at 717 cents, ratios very close to
13:10 and 20:13 (454 and 746 cents). The diminished third, as we move
beyond 715 cents, shrinks to less than 50 cents -- about the minimum
value for a routine "semitone" in neomedieval styles where about 50-90
cents may be the norm.
At we more and more closely approach 720 cents, the impending convergence
on 5-TET values becomes more clear. Interestingly, 719 cents offers an
augmented second at a virtually just 21:16 at 471 cents, and a diminished
octave at 967 cents, very close to 7:4 at 969 cents. Curiously, the
literal tritone in this tuning at 709 cents is almost identical to the
22-TET fifth!
At the limiting value of 720 cents or 5-TET, the diminished third
becomes a unison, and the augmented sixth a 1200-cent octave.
Thus the reverse schismatic mapping between 17-TET and 5-TET provides
a spectrum of intervals ranging from neutral to 5-limit to Pythagorean
and septimal to interseptimal. At 5-tET, the intervals of 240 and 960
cents embody the interseptimal region, with the first having qualities
of either a very wide major second or a very narrow minor third, and
the second likewise of either a very wide major sixth or a very narrow
minor seventh. In larger tunings such as 20-TET, these possible
identities may be defined in contrapuntal terms, as when a 240-cent
interval acts as a "quasi-third" by contracting to a unison, one voice
moving by 180 cents and the other by a 60-cent semitone; or when by a
similar motion a 960-cent "quasi-sixth" expands to an octave.
We have now accounted for the schismatic mapping with its VTR from
19-TET to 17-TET; and the "reverse schismatic" mapping with a VTR from
17-TET to 5-TET. This raises the question: what of the territory
between 7-TET and 19-TET?
-------------------------------------------------
4. Inframeantone: a mapping below the schismatic?
-------------------------------------------------
We have seen that 19-TET is the lower limit of the VTR for the
schismatic mapping, the point at which a diminished third and an
augmented second (e.g. C#-Eb and Eb-F#) have identical sizes of about
252 cents. In the schismatic mapping, a diminished third is smaller
than an augmented second -- for example, 240 and 264 cents at a
generator of 696 cents, and 141 and 353 cents at the mapping's upper
limit of 17-TET.
What happens below 19-TET, the lower limit for this mapping and also
for what might be called "meantone proper" where the fifth is narrowed
in order to optimize tempered ratios within the 5-limit such as 3:2,
5:4, and 6:5? Here, again, it may be instructive to sample some
interval sizes from 7-TET (685.71 cents) to 19-TET (694.74 cents):
----------------------------------------------------------------------
Intvl Example 7-TET 687 688 689 690 691 692 693 694 19-TET
----------------------------------------------------------------------
Apot C-C# 0 9 16 23 38 37 44 51 58 63
Aug2 Eb-F# 171 183 192 201 210 219 228 237 246 252
dim3 C#-Eb 343 330 320 310 300 290 280 270 260 252
dim4 C#-F 514 504 496 488 480 472 464 456 448 442
Aug5 F-C# 686 696 704 712 720 728 736 744 752 758
Aug6 Eb-C# 857 870 880 890 900 910 920 930 940 948
dim7 F#-Eb 1029 1017 1008 999 990 981 972 963 954 948
dim8 C#-C 1200 1191 1184 1177 1170 1163 1156 1149 1142 1137
----------------------------------------------------------------------
Here it appears that we have a mapping with an ordering valid from
7-TET to 19-TET -- might it be called "infraschismatic"? In an
overview like this, interval values sweep through wide swaths of the
spectrum.
The diminished third, for example, a neutral third at 7-TET with a
size of 343 cents (very close to 39:32 at 342 cents), shrinks at 687
cents to 330 cents, at the lower end of the neutral third region; it
then moves through the 5-limit region, by 690 cents reaching 300
cents, more on the outskirts of the Pythagorean zone near a just 32:27
(294 cents). At 690-693 cents we then move through this Pythagorean
zone and the territory between Pythagorean and septimal, by 693 cents
reaching a size of 270 cents, very close to 7:6 (267 cents) -- and by
19-TET, the fine interseptimal 252 cents, identical in size to the
augmented second, a convergence defining the upper limit of the VTR.
--------------------------------------------
5. Conclusions: mappings for 7-10 generators
--------------------------------------------
Possibly the best way to describe the infraschismatic mapping (7-TEt
to 19-TET), the schismatic mapping (19-TET to 17-TET), and the reverse
schismatic or ultraschismatic mapping (17-TET to 5-TET) as is mappings
of "7-10 generators," since the relevant intervals are generated from
chains of this length.
Given the diversity of interval regions generated within the territory
of each mapping, a term like "schismatic" may be more of an historical
curiosity than a comprehensive description of what these mappings are
about. We get neutral intervals, major and minor intervals of various
colors, and interseptimal intervals -- as well as, sometimes, what we
might call perfect fourths and fifths at or near a just 4:3 or 3:2, as
with the 496-cent diminished fourth and 704-cent augmented fifth at a
generator size of 688 cents.
While the range from around 695 to 709 cents, or from 19-TET to
22-TET, might be considered familiar, the regions below and above
deserve their own thorough exploration in their own terms. A
realization that each interval or generator region has its own charm
is a vital element in this process.
Most appreciatively,
Margo Schulter
10 September 2008