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Regions of the Interval Spectrum
Some Concepts and Names
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In naming categories of intervals, or regions of the spectrum in which
they are found, there may be many valid and desirable schemes
reflecting the diversity of viewpoints and styles to be found in world
musics. What I describe here is merely one possible solution, and one
influenced by my own musicmaking experience and philosophy which seeks
an equitable and inclusive balance between intervals at or near simple
integer ratios, and those having a more complex or active nature.
An invaluable resource in considering my own scheme is David Keenan's
"A Note on the Naming of Musical Intervals" (1999, 2001) available at:
. I am
much indebted to this thoughtful paper, and likewise to the system of
interval names used in Manuel Op de Coul's free software program Scala
with its many features for scale creation, analysis, and MIDI file
generation or retuning .
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1. Some ratios and points of reference
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In the approach followed here, the spectrum of harmonic intervals is a
continuum with each location or subtle shading having its own charm.
At the same time, certain ratios both simple and more complex can
serve as helpful reference points in viewing this spectrum in terms of
certain regions, e.g. "a large major third" or "a small minor seventh."
A main caution is that the borders are inevitably "fuzzy," so that one
region shades into another and suggested values in cents are more
illustrative than definitive. Also, interval perceptions can be
influenced by musical style and context. Often the specific interval
"maps" and concepts which have developed for a given world musical
tradition can be more informative than any generalized scheme such as
this.
For our generalized scheme, six types of ratios may assist in mapping
regions of interval space.
A _Pythagorean_ ratio is generated from a chain, often comparatively
short, of pure 3:2 fifths or 4:3 fourths, or prime factors of (2,3).
The Pythagorean major third at 81:64 (408 cents) and minor third at
32:27 (294 cents) are familiar examples. Broadly speaking, of course,
a "Pythagorean third" might be _any_ size of third generated from a
chain of pure fifths, no matter how long. Thus in a circulating
Pythagorean system with 53 notes per octave (a circle of 52 pure
fifths plus one narrow by about 3.62 cents) we would find many sizes
of thirds. Normally, however, a "Pythagorean" ratio implies the
shortest available chain of fifths for generating an interval in a
given category of interest.
A _pental_ (or "5-limit") ratio is generated from prime factors of
(2,3,5). The 5:4 major third at 386 cents, and 6:5 minor third at 316
cents, are familiar examples.
A _septimal_ ratio is generated from prime factors of (2,3,7). The 9:7
major third at 435 cents, and 7:6 minor third at 267 cents, are again
familiar examples.
A _pental-septimal_ ratio is generated from prime factors of (2,3,5,7).
The simplest ratios for a diminished fifth at 7:5 or 583 cents, and an
augmented fourth at 10:7 or 617 cents, are fine examples.
In addition to these four basic types of more or less simple ratios,
we have two types of ratios known as mediants which can also provide
some helpful points of orientation in mapping interval space.
The _classic mediant_ offers one helpful reference point within the
middle territory between a pental and a septimal ratio -- for example,
the small 5:4 and large 9:7 major thirds, at 386 and 435 cents. To
find this mediant, we add the numerators and denominators of the two
simpler ratios:
5 + 9 14
----- = --
4 + 7 11
Thus the ratio for a major third of 14:11 (418 cents) provides one
cardinal point of reference within the "middle realm" between the
small 5:4 and large 9:7. Another such point of reference in this
middle region of major thirds is the Pythagorean 81:64 at 408 cents.
Another and more complex type of mediant is the _noble mediant_
formulated by David Keenan and described in a paper which I was
honored to coauthor with him:
The noble mediant for two ratios such as 5:4 and 9:7 may approximate
the region of "metastability" or maximum complexity where an interval
is about equally far from the "gravitational influence" of either
ratio. It is calculated using the Golden Ratio of Phi, about 1.618034,
which is used to "weight" the numerator and denominator of the more
complex of the two simple ratios, here 9:7.
5 + 9 Phi
--------- = ~422.48 cents
4 + 7 Phi
Intervals tuned near a noble mediant, as Keenan suggests, may be
called "Nobly Intoned" (NI). This mediant, as a possible point of
rough equipose between a pental and a septimal ratio, for example, may
be of interest here as an indication of the region where a major
third, for example, may be approaching the "sphere of influence" of a
septimal ratio such as 9:7.
These types of simple and complex ratios, while helpful in mapping the
spectrum, are by no means exhaustive. Thus 14:11, the classic mediant
between 5:4 and 9:7, can be described generally as one type of "middle
major third," but also more specifically as an "undecimal major third"
since it brings into play the prime factor of 11. This is the usage of
Scala, and nicely supplements the basic concepts of interval region
mapping we are about to explore.
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2. Major and minor thirds and sixths (36:35-plus regions)
---------------------------------------------------------
As here conceived, the term "major third" or "minor third" -- and
likewise "major sixth" or "minor sixth" -- describes a region of the
interval spectrum roughly delineated by the pental and septimal ratios
for that category of third or sixth, with at least a few extra cents
of leeway on either side.
Thus for a major third, the small or pental 5:4 at 386 cents and the
large or septimal 9:7 at 435 cents roughly delimit the region. For
major intervals, more generally, the pental ratio represents the lower
end of the range and the septimal ratio the upper range, with the
Pythagorean 81:64 (408 cents) and classic mediant 14:11 (418 cents)
ratios typifying the middle range.
In practice, as already noted, we will want at least a few extra cents
on either side of the region to recognize major thirds slightly
smaller than the pental 5:4, or larger than the septimal 9:7. One
possible range might be about 372-440 cents, with quite "fuzzy"
borders.
Similarly, for the minor third, the small or septimal 7:6 at 267 cents
and the large or pental 6:5 at 316 cents roughly delimit the region,
with the Pythagorean 32:27 at 294 cents and classic mediant 13:11 at
289 cents nicely representing the realm of middle minor thirds. Here,
as more generally with minor intervals, the septimal ratio marks the
lower end of the range and the pental ratio the upper range.
Again, some leeway is needed at either extreme of the range, with
something like 260-330 cents as a possible guide for the minor third
region, again with "fuzzy" borders.
The pental 5:4 and septimal 9:7 ratios for a major third differ by a
factor of 36:35, or 48.77 cents -- and likewise 7:6 and 6:5 for a
minor third. Thus we may refer to the region of major or minor thirds
as a "36:35-plus region" -- the "plus" referring to the extra leeway
on either extreme, making these regions actually about 20 cents wider,
if the approximate guidelines suggested here are followed.
Thus for major or minor thirds we have a rather encompassing region
with many shades of small, middle, and large varieties:
Major third region (~5:4-9:7)
pental Pyth CMed NI septimal
5:4 81:64 14:11 9:7
.......____|---------------|---------|----|----------|___...,
386 408 418 422 435
|--------------------|----------------------|---------------|
372? small 400? middle 423? large 440?
Minor third region (~7:6-6:5)
septimal NI CMed Pyth pental
7:6 13:11 32:27 6:5
....____|--------------|----|----|---------------|_____......
260? 267 284 289 294 316 330?
|--------------------|-----------------|--------------------|
small 280? middle 300? large
While it is easy to identify some characteristic examples of small,
middle, and large intervals in these regions, the question of just
where a small major or minor third may be approaching a "middle" size,
or a middle type a "large" size, is much more fuzzy. Here I have
suggested sizes of about 280-300 cents for middle minor thirds, and
400-423 cents for middle major thirds, but this is an approximate and
subjective judgment of finely shaded transitional zones.
The regions of major and minor sixths may be similarly sketched out,
with a mention of two criteria sometimes favored in coordinating these
categories with those of the thirds. People may desire either that
octave inversion should hold, so that an octave less a major third
will produce some kind of minor sixth; or that fourth or fifth
complementation should hold, so that a major or minor third plus a
fourth yields a major or minor sixth. Since fifths and fourths,
although frequently assumed to be at pure ratios of 3:2 and 4:3, are
often in practice tempered in either direction, these criteria can
have intricate ramifications in fine-tuning a mapping scheme.
Here, since the borders are taken to be inherently fuzzy, people
should feel free to adjust them according to either of these criteria.
Major sixth region (~5:3-12:7)
pental Pyth CMed NI septimal
5:3 27:16 17:10 12:7
.......____|---------------|---------|-----|--------|____....
884 906 918 923 933
|--------------------|-------------------|------------------|
870? small 900? middle 920? large 940?
Minor sixth region (~14:9-8:5)
NI or
septimal CMed Pyth pental
14:9 11:7 128:81 8:5
....____|-------------|-------|----------------|_____.......
765 782 792 814
|------------------|-----------------|---------------------|
760? small 777? middle 800? large 828?
While much effort could be devoted -- and rightly so -- to
appreciating some of the fine nuances of these fuzzy borders and
transitions, the basic regions and subregions of major and minor
thirds and sixths (small, middle, and large) can be helpful in
describing the resources of some common tuning systems.
Thus a 12-note regular meantone tuning around 1/4-comma or 2/7-comma
will have both small and large major thirds -- 386 and 427 cents in
1/4-comma, or 383 and 434 cents in Zarlino's 2/7-comma -- but no
middle sizes. Likewise minor thirds are either large or small: in
1/4-comma, 269 or 310 cents; and in 2/7-comma, 262 or 313 cents.
In contrast, a 12-note set of the regular Wilson/Pepper temperament
with fifths at about 704.096 cents (2.14 cents wide) will have a
wealth of middle major thirds around 416 cents, quite close to the
classic mediant of 14:11 -- but no small or large sizes. Similarly,
middle minor thirds at around 288 cents are very close to the classic
mediant between 7:6 and 6:5 at 13:11 or 289 cents, the tridecimal
(13-based) minor third -- but we have no large or small minor thirds.
George Secor's 17-note well-temperament has middle and large major
thirds ranging from 418 to 429 cents, and likewise small and middle
minor thirds from 278 to 287 cents -- but no large minor or small
major thirds near pental ratios.
Many intonational systems, especially larger ones, survey small,
middle, and large varieties of major and minor thirds and sixths.
This may also happen in a smaller system such as Zest-12, where an
irregular 12-note circle based on 2/7-comma meantone with eight narrow
and four equally wide fifths produces major thirds at 383, 396, 408,
421, and 435 cents; and minor ones at 274, 287, 300, and 313 cents.
Such a system can be attractive if one is drawn to the middle sizes of
major and minor thirds, but less so if one strongly prefers that the
simplest ratios of 5:4 and 9:7, or 6:5 and 7:6, be approximated at as
many locations within the tuning as possible.
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3. Neutral or "interpental" thirds and sixths (25:24-minus regions)
-------------------------------------------------------------------
Having considered major and minor thirds and sixths, we may proceed in
a natural and intuitive way to neutral thirds and sixths situated
somewhere between our minor and major categories.
Thus a neutral third should have a size somewhere between that of a
large minor third at 6:5 or 316 cents, and a small major third at 5:4
or 386 cents. Since these large minor and small major thirds both have
pental ratios, we may also describe neutral thirds as "interpental,"
or occupying a region between these pental points of demarcation.
The ratios of 6:5 and 5:4 differ by 25:24, or 70.67 cents; but, in
practice, the neutral region is rather smaller, since these rather
simple pental ratios have considerable "spheres of influence"
including intervals somewhat larger than 316 cents or smaller than 386
cents. We may thus speak of a "25:24-minus" region of neutral thirds,
say around 330-372 cents, with musical context as a factor in shaping
perceptions around the fuzzy transitional regions.
Within the neutral third region, we may speak of small, middle or
central, and large subregions. One possible guideline is that middle
or central neutral thirds range from around 39:32 (342 cents) to 16:13
(359 cents) -- or, if we like, a round 360 cents for the upper border,
again a fuzzy zone. The simplest neutral third ratio of 11:9 (347
cents), and also Zalzal's famous lute fret as estimated in the
tradition of al-Farabi at 27:22 (355 cents), come within this central
region.
Small neutral thirds of around 330-342 cents are sometimes called
supraminor, while large major thirds at around 360-372 cents are also
known as submajor. The simplest ratios of 17:14 (336 cents) and 21:17
(366 cents) nicely exemplify these small and large subregions.
Neutral third region
63:52 17:14 NI 39:32 11:9 27:22 16:13 21:17 26:21
..--|------|---|----|-----|-------|-----|-------|-----|--..
332 336 339 342 347 355 359 366 370
|-------------------|--------------------|---------------|
330? small 342 middle 360 large 372?
(supraminor) (central) (submajor)
Here it may be noted that the 11:9 third, typical of the central
region, is the classic mediant of 5:4 and 6:5. The Noble Intonation
(NI) mediant between these ratios is around 339 cents, which might
give the upper range of small neutral or supraminor thirds a rather
ambiguous or complex quality.
For neutral sixths between the pental ratios of 8:5 or 814 cents for a
large minor sixth, and 5:3 or 884 cents for a small major sixth, a
generally similar situation holds:
Neutral sixth region
21:13 NI 34:21 13:8 44:27 18:11 64:39 21:17 104:63
.__|-----|-|---------|-----|-------|-----|------|-----|--...
830 833 834 841 845 853 858 864 868 870?
|-------------------|--------------------|-----------------|
828? small 840 middle 858 large
(supraminor) (central) (submajor)
Interestingly, the classic mediant between the pental major and minor
sixths at 5:3 and 8:5 is 13:8 (841 cents), located near the lower end
of the central neutral sixth range. This "harmonic neutral sixth" or
"tridecimal neutral sixth" (since its ratio is 13-based) may have a
special attractive force because the upper tone is a harmonic of the
fundamental. Not too far from this ratio is the Nobly Intoned or NI
small neutral or supraminor third at around 833 cents, defined by the
Golden Ratio of Phi itself! More generally, the region offers a wealth
of subtle shadings.
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4. Major seconds and minor sevenths (36:35-plus regions)
--------------------------------------------------------
Major seconds and minor sevenths may be convenient to consider next,
since they follow much the same pattern as major and minor thirds and
sixths.
For major seconds, we have three handy ratios of orientation, all of
them superparticular in form (n+1:n). These are the small or pental
tone at 10:9 or 182 cents; the middle or Pythagorean tone at 9:8 or
204 cents; and the large or septimal tone at 8:7 or 231 cents. As it
happens, the classic mediant of the pental 10:9 and septimal 8:7 is
also equal to the Pythagorean 9:8, our exemplary "middle tone."
As with our regions for major or minor thirds and sixths, the pental
and septimal values of 10:9 and 8:7 differ by 36:35 or about 49 cents,
but the major third region is actually somewhat larger -- say 180-240
cents. Indeed, intervals somewhat smaller than 180 cents or larger
than 240 cents might also sometimes function as "major seconds" -- but
these adjoining regions have their own qualities which might best be
considered as distinctive, as discussed below.
Within the major second region, in addition to the three simplest
ratios of the small 10:9, middle 9:8, and large 8:7, there are some
other main points of interest. A small 10:9 plus a middle 9:8 tone
will form a pure pental 5:4 major third, as will two "mean-tones" each
equal to the geometric mean or average of these unequal ratios,
approximately 193.16 cents. Likewise, a middle 9:8 plus a large 8:7
tone will form a pure septimal 9:7 major third, as will two
"eventones" each equal to the average of 9:8 and 8:7, around 217.54
cents. We may refer to these 193-cent and 218-cent sizes as the
"pental meantone" and "septimal eventone" respectively. The classic
mediants of 10:9 and 9:8 at 19:17 (193 cents), and 9:8 and 8:7 at
17:15 (217 cents), closely approximate these sizes.
Thus we have something like this:
Major second region (~10:9-8:7)
pental 5:4 Pyth or 9:7 septimal
10:9 meantone CMed 9:8 eventone 8:7
.__|--------|---------|-----------|-------------|____.....
182 193 204 218 231
|-----------------|-------------------|------------------|
180? small 200? middle 220? large 240?
One might mention incidentally that the Phi-weighted mediant between
10:9 and 8:7 is around 198.39 cents; but it is not clear whether or
how significant this might be for any special quality of "complexity"
near this tuning, which is also not too far from the ratio of 9:8.
David Keenan (personal correspondence, 19 June 2008) helpfully
explains that in fact 10:9 and 8:7 do _not_ form a noble mediant,
which would require that a pair of ratios meet an important
mathematical test. With two ratios a:b and c:d -- here 10:9 and 8:7 --
the difference between the cross-products of the outer and inner
terms, a*d and b*c, or here 10*7 and 9*8, must differ by 1; the
difference here is 2 (72 - 70). Thus the "noble mediant" concept is
inapplicable to this pair of ratios.
For minor sevenths, rather similar pattern holds, except that the
simple harmonic ratio of 7:4 (969 cents), the "harmonic minor
seventh," may have a great attracting power, thus exerting its
influence over a wider portion of the spectrum than with the more
complex 9:7 septimal major third, for example. As with the 9:8 middle
tone, so with the 16:9 minor seventh (996 cents), the Pythagorean
ratio is also the classic mediant between septimal and pental values,
here 7:4 and 9:5 (1018 cents).
Minor seventh region (~7:4-9:5)
CMed or
septimal Pyth pental
7:4 16:9 NI 9:5
...._______|---------------------|-----|-----------|_____....
969 996 1002 1018
|-----------------------|-------------|---------------------|
960? small 987? middle 1000? large 1025?
The Noble Intonation value is around 1001.61 cents, as with the major
second not too far from the classic mediant and also Pythagorean
value, here 16:9.
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5. Minor seconds and major sevenths (405:392-plus regions)
----------------------------------------------------------
One way to seek an approximate range for a minor second region is to
take the septimal ratio of 28:27 (63 cents), the favored small
semitone or thirdtone of Archytas, as near the lower end; and the
pental-septimal ratio of 15:14 (119 cents) as near the upper end.
These ratios differ by a factor of 405:392 or 56.48 cents. Allowing,
as usual, for some leeway at each end, we thus have a "405:392-plus"
region of minor seconds.
One rough guideline might be a range of about 60-125 cents. While
smaller steps may sometimes function as semitones or minor seconds,
they have their own qualities which might fit the label of "diesis,"
as discussed below. Likewise, while a step as large as 27:25 (133
cents) serves as a minor second in pental just intonation, this size
is also a fine example of a small neutral (or supraminor) second, to
be considered shortly.
Within this range of about 60-125 cents, we have some cardinal ratios
of orientation: the small or septimal minor second at 28:27 or 63
cents; the middle Pythagorean semitone at 256:243 or 90 cents; and the
large or pental semitone at 16:15 or 112 cents, with 15:14 at 119
cents near the upper end of the region. In addition to the Pythagorean
value, the middle region is also represented by the classic mediant
between the septimal 28:27 and pental 16:15, 22:21 or 81 cents.
Minor second region (~28:27-15:14)
pental-
septimal CMed Pyth pental septimal
28:27 22:21 256:243 16:15 15:14
..__|-------------|-------|---------------|-------|___....
63 81 90 112 119
|----------------|----------------|----------------------|
60? small 80? middle 100? large 125?
We may also note that the difference between the middle 9:8 tone and a
large or pental 16:15 semitone yields an excellent small semitone at
25:24 or 71 cents; and the difference between 9:8 and the Pythagorean
middle semitone at 256:243 yields a large semitone at 2187:2048 or 114
cents, quite close to the pental 16:15.
The major seventh region may be regarded as rather similar to this,
but with small pental and large septimal ratios, as holds for major
categories generally.
Major seventh region (~28:15-27:14)
pental-
septimal pental Pyth CMed septimal
28:15 15:8 243:128 21:11 27:14
..____|-------|--------------|--------|--------------|___..
1081 1088 1110 1119 1137
|----------------------|---------------|------------------|
1075? small 1100? middle 1120? large 1140?
Small major sevenths are represented by the pental-septimal 28:15 at
1081 cents and pental 15:8 at 1088 cents; middle major sevenths by the
Pythagorean 243:128 at 1110 cents and the classic mediant between the
pental 15:8 and septimal 27:14 at 21:11 or 1119 cents; and large major
sevenths by the septimal 27:14 at 1137 cents.
-----------------------------------------------------
6. Neutral seconds and sevenths (28:27-minus regions)
-----------------------------------------------------
One way to approach a neutral second region is to regard it as having
a range between the large septimal-pental minor second at 15:14 or 119
cents, and the small or pental tone at 10:9 or 182 cents. These
intervals differ by a factor of 28:27 or 63 cents.
In practice, however, the range is somewhat smaller, say the 125-170
cents which Hormoz Farhat suggests as the domain of the neutral second
in Persian music based on the dastgah system of modality. The handy
superparticular ratio of 14:13 (128 cents) favored by Ibn Sina may
nicely represent the neighborhood near the lower end of the range, and
Ptolemy's 11:10 (165 cents) the neighborhood near the upper end.
A middle or central subregion might run from about 135 to 160 cents,
with these approximate sizes often preferred according to Farhat in
Persian music, and also reported by some Arab musicians to be common
in the maqam traditions. Within this central portion of the range we
find the superparticular ratios of 13:12 (139 cents) and 12:11 (151
cents), the latter used in Zalzal's scale according to al-Farabi along
with the more complex ratio of 88:81 or 143 cents.
One general rule is that two neutral seconds should add up to some
kind of minor third (Section 2 above). Thus from 14:13 and 13:12 we
get the small or septimal 7:6 third; from 13:12 and 12:11, the middle
or tridecimal minor third at 13:11; and from 12:11 and 11:10, the
large or pental minor third at 6:5. In Near Eastern traditions where
many instruments are tuned in pure fifths and fourths, the Pythagorean
size of middle minor third at 32:27 or 294 cents may often serve as a
guide, with Farhat's rounded 135-cent and 160-cent steps adding up to
something around this value -- and Zalzal's 88:81 and 12:11 (143/151
cents) yielding it precisely.
Thus we have a situation something like this:
Neutral second region
14:13 13:12 88:81 12:11 11:10
..___|-------------|-----|-------|----------------|-----..
128 139 143 151 165
|--------------|--------------------------|--------------|
125? small 135? middle 160? large 170?
(supraminor) (central) (submajor)
From here, it is intuitive to posit a similar situation for neutral
sevenths, which may have a range roughly from 20:11 (1035 cents) to
13:7 (1072 cents), with a few cents leeway at either end, say
something like 1030-1075 cents:
Neutral seventh region
20:11 51:28 11:6 24:13 13:7
..____|-----|--------------|------------|------------|_____..
1035 1038 1049 1061 1072
|-----------------|-----------------=------|----------------|
1030? small 1043? middle 1065? large 1075?
(supraminor) (central) (submajor)
Representative of the small neutral seventh subregion are the 20:11,
and also 51:28 (the small 17:14 neutral second plus a 3:2 fifth) at
1038 cents. An interval of special prominence in the middle or central
range is 11:6 at 1049 cents, on account of its simple ratio as a zone
of harmonic "gravitation." Also within the central subregion is 24:13
(an octave less 13:12) at 1061 cents. In the large neutral seventh
subregion, 13:7 at 1072 cents plays a prominent role.
The reader may have noticed that there is a gap between the high end
of our neutral second region at around 170 cents, or a bit larger than
11:10, and the start of the major second region with the small or
pental tone at 182 cents, or a very slightly smaller value around 180
cents. This small intermediate region of around 170-180 cents marks
the heart of an "equitable heptatonic" zone where steps have a size at
or near 1/7 octave or 171 cents, to be discussed below, and which
deserves its own special recognition.
------------------------------------------------------
7. Perfect fifths and fourths (4096:3959-plus regions)
------------------------------------------------------
It may seem curious to have surveyed an assortment of major, minor,
and neutral intervals before coming to those intervals which adorn so
many world musics: fifths and fourths. In medieval European and also
Balinese or Javanese gamelan polyphony, for example, these intervals
serve as stable concords and resting points toward which often complex
contrapuntal textures are drawn. Comparing the pure 3:2 fifths and 4:3
fourths of a medieval European Pythagorean tuning with typical gamelan
values for the corresponding concordant interval category of _kempyung_
will reveal a variance of often around 10-30 cents between these
systems. We thus have a fine illustration of how intonation can vary
according to taste, and also timbral conditions.
Generally the zone of a perfect fourth or fifth might be taken as
extending for about a 64:63 zone, equal to the septimal comma or comma
of Archytas at 27.26 cents, on either side of 4:3 or 3:2. This zone
would thus have a size equal to twice that of the comma, or of the
square of its 64:63 ratio: 4096:3959, or 54.53 cents. In practice, we
may find it convenient to round this to an even 60 cents, 30 cents on
either side.
Thus the zone of perfect fourths would have a range of about 468-528
cents, 30 cents to either side of 4:3 at 498 cents; and likewise the
perfect fifth one of about 672-732 cents, centering around 3:2 at 702
cents. Thus gamelan tunings often seem to prefer values around 680
cents or 720 cents, with great variations; indeed, tradition requires
that each gamelan ensemble have its own distinctive intonation and not
try to copy that of another ensemble.
In world musical traditions tending toward an equal or equable
heptatonic style of intonation, small fifths around 4/7 octave or 686
cents are often favored; while styles tending toward an equal or
equable pentatonic (e.g. the slendro modes of gamelan) may favor a
size around 3/5 octave or 720 cents.
In world traditions like that of Western Europe, where strongly
harmonic timbres (e.g. harpsichord) are favored for polyphonic music,
tunings at or quite near 4:3 and 3:2 may prevail; most regular
European and related keyboard temperaments stay within about 7 cents
of these ratios, whether in the narrow direction of tempering for the
fifths and the wide one for the fourths (the historical norm), or in
the opposite direction (as in some historical modified meantones for
certain fifths and fourths, and regularly in modern neomedieval
temperaments).
The interval of a fourth narrowed by a septimal comma (21:16, 471
cents), or a fifth enlarged by this comma (32:21, 729 cents) occurs,
for example, in just tunings based on the diatonic of Archytas with
steps of 9:8, 8:7, and 28:27. Pythagorean tunings with 12 or more
notes will produce a narrow fifth at 262144:177147 or 678 cents,
smaller than 3:2 by a Pythagorean comma at 531441:524288 or 23.46
cents, and likewise a wide fourth at 177147:131072 or 522 cents.
Pental just intonation in classic forms also produces narrow fifths
and wide fourths at 40:27 and 27:20 (680 and 520 cents), differing
from pure by the syntonic comma of 81:80 or 21.51 cents.
If we desire an integer ratio to gauge the neighborhood of the upper
end of our region, then 256:189 at 525 cents, a 3:2 fifth plus 64:63,
might serve this purpose.
Thus we have impressionistic ranges something like this:
Perfect fourth region
21:16 2/5 oct 4:3 3/7 oct 256:189
...__|------|--------------------|---------------|----------|__..
471 480 498 514 525
|-------------------------|-------------|-----------------------|
468? small or 491? middle 505? large or 528?
narrow wide
The situation for perfect fifths may be taken as generally similar:
Perfect fifth region
189:128 4/7 oct 3:2 3/5 oct 32:21
...__|------------|--------------|-------------------|------|__..
675 686 702 720 729
|-------------------------|-------------|-----------------------|
672? small or 695? middle 709? large or 732?
narrow wide
These ranges should not be taken as rigid or exhaustive, since, for
example, intervals rather larger than 732 cents may sometimes serve
musically as perfect fifths -- especially in some inharmonic timbres.
However, when treated flexibly, these concepts of general ranges for
perfect fourths and fifths nicely set the stage for considering some
especially intriguing categories we will meet below.
-----------------------------------------------
8. Tritonic intervals (a 6561:6272-plus region)
-----------------------------------------------
As an intuitive approach to defining a region of tritonic intervals
(also known as augmented fourths or diminished fifths), we may begin
with the concept of a fourth augmented or a fifth diminished by a
minor second or semitone of some kind (Section 5).
Another way of putting this is that a tritonic interval should have a
size somewhere between that of a 4:3 fourth at 498 cents and a 3:2
fifth at 702 cents, while differing from _either_ of these ratios by
at least a "semitone" of some kind.
If we consider the septimal 28:27 semitone or thirdtone of Archytas at
63 cents as close to the lower end of the minor second range, then the
tritonic region would cover a range equal to 9:8 or 204 cents (the
space between a 4:3 fourth and 3:2 fifth), less a 28:27 zone at either
end of this potential range.
To estimate the size of our tritonic region using integer ratios, we
might thus take 9:8 at 204 cents and subtract twice the size of a
28:27 semitone at 63 cents, that is 784:729 or about 125.92 cents,
leaving a range of 6561:6272 or 77.99 cents.
A simpler way of estimating this range is to consider that a 4:3
fourth plus a 28:27 semitone yields an interval of 112:81 or 561
cents; while a 3:2 fifth less 28:27 yields 81:56, or 639 cents. If we
round conveniently to 560-640 cents, this gives us a handy gauge for
the region. If we strive consistently to apply our earlier concept of
a semitone ranging down to about 60 cents, then we might have a
slightly wider range of 558-642 cents. Since borders are inherently
fuzzy, this is no great matter.
From a viewpoint of traditional European theory, an augmented fourth
or diminished fifth should more properly be defined as fourth plus a
_chromatic_ semitone, or a fifth less such a semitone, rather than by
means of a usual or diatonic semitone such as our septimal 28:27.
However, we can arrive at the same result as this theory by defining
an augmented fourth such as F-B as equal to the fifth (F-C) less a
diatonic semitone (B-C); and a diminished fifth such as B-F likewise
as a fourth (C-F) plus a diatonic semitone (B-C).
Thus the smaller septimal tritonic interval 112:81 at 561 cents may be
considered a diminished fifth, and the larger 81:56 at 639 cents an
augmented fourth.
We could reach the same result more conventionally by using the
septimal _chromatic_ semitone, equal to the difference of 9:8 and the
small 28:27 semitone, or 243:224 at about 141 cents -- actually by our
usual definition a type of neutral second (Section 6)! Defining a
septimal augmented fourth as 4:3 plus 243:224 (e.g. F-B as F-Bb plus
Bb-B), we arrive at 81:56; and defining a septimal diminished fifth as
3:2 less 243:224 (e.g. B-F as Bb-F less less Bb-B), we get 112:81.
This example illustrates how, while a tritonic interval as here
defined differs from either 4:3 or 3:2 by _at least_ a "semitone" with
a minimum size of 60 cents or so (Section 5), it may sometimes differ
from one of these ratios by a small semitone of around 60-75 cents,
and from the other by a small to middling neutral second, here 243:224
at about 141 cents.
If we take 560-640 cents as a convenient guide to the tritonic region,
then the septimal 112:81 and 81:56 at 561 and 639 cents are just
within its fuzzy borders. Close by are the simpler tridecimal or
13-based ratios of 18:13 and 13:9 at 563 and 637 cents, which may be
produced by the division of a 9:8 tone into a small diatonic semitone
at 27:26 or 65 cents, and a chromatic step equal to a 13:12 neutral
second at 139 cents. This division yields an augmented fourth equal to
4:3 plus 13:12, or 13:9; and a diminished fifth of 3:2 less 13:12, or
18:13. The augmented fourth and diminished fifth of 17-EDO at about
635 and 565 cents, nicely approximate these values.
Pental just intonation also yields some tritonic intervals in this
general neighborhood -- but this time with the diminished fifth as the
larger interval and the augmented fourth the smaller. Two pure 6:5
minor thirds, for example B-D-F, yield a diminished fifth of 36:25 or
631 cents, which differs from 3:2 by the small chromatic semitone of
25:24 or 71 cents. Tuned in similar fashion, the augmented fourth F-B
would exceed 4:3 by the same 25:24 semitone, yielding 25:18 at 569
cents. These tritonic ratios occur in 1/3-comma meantone.
In a classic pental just intonation context, the small 25:24 chromatic
semitone has its counterpart in the notably large diatonic step of
27:25 at 133 cents -- in many other contexts a small neutral second,
as has been noted (Section 5). Together, these steps add up to 9:8.
In the tempered system of 1/3-comma meantone, while the tritonic
ratios of 36:25 and 25:18 precisely obtain, both of these steps are
smaller, at about 64 cents for the small chromatic semitone and 126
cents for the large diatonic semitone -- the latter near our usual
border area between a large minor second and a small neutral second.
To this point, we have encountered pairs of tritonic ratios where the
small form differs from the large by more than 60 cents -- indeed, by
almost 78 cents for the septimal pair 112:81 and 81:56. Gradually,
however, we have been moving toward the central portion of the
tritonic region where these differences become less dramatic.
Medieval Near Eastern theory around 1300 features the tritonic ratio
of 56:39, or 626 cents, which could be described as an augmented
fourth equal to 4:3 plus a small 14:13 neutral second at 128 cents.
This ratio differs from 3:2 by a small diatonic semitone of 117:112 or
76 cents. The corresponding diminished fifth, equal to 3:2 less 14:13,
is 39:28 at 574 cents. These tritonic ratios, like 13:9 and 18:13,
could be described as tridecimal, since they are based on 13 as the
highest prime factor.
With 56:39 and 39:28, we still have a situation where tritonic ratios
are based on the division of a 9:8 tone into a small semitone plus a
neutral second, here 14:13. One way to define the "central" tritonic
subregion is to specify that intervals in this subregion should differ
from either 4:3 or 3:2 by some kind of "semitone" as we have defined
it in Section 5: that is, by at least 60 but no more than 125 cents,
to take conveniently rounded figures.
This criterion suggests a central subregion of about 577-623 cents.
In the small tritonic subregion of 560-577 cents, intervals differ
from 4:3 by a small semitone but from 3:2 by an interval larger than
125 cents fitting our concept of a neutral second (Section 6). In the
large subregion, intervals likewise differ from 3:2 by a small
semitone and from 4:3 by a small to middling neutral second.
Moving now into the central subregion, we encounter a pair of ratios
notable for their simplicity: the pental-septimal diminished fifth at
7:5 or 583 cents, and its complementary augmented fourth at 10:7 or
617 cents. The simplest integer ratio found in the tritonic region,
7:5 is also the classic mediant between 4:3 and 3:2.
In a typical just intonation context, a 7:5 diminished fifth
(e.g. B-F) resolves to a pure 5:4 major third (e.g. C-E) by stepwise
contrary motion, the upper voice descending by a small diatonic
semitone of 21:20 (84 cents), and the lower ascending by a usual
pental diatonic semitone of 16:15 (112 cents). In the division of a
9:8 tone, the 21:20 semitone is complemented by a larger chromatic
semitone of 15:14 or 119 cents. From 3:2 less 15:14 we get the 7:5
diminished fifth; and from 4:3 plus 15:14, the 10:7 augmented fourth.
Pythagorean intonation yields rather similar tritonic intervals. The
augmented fourth is equal to 4:3 plus a chromatic semitone of
2187:2048 or 114 cents, or 729:512 at 612 cents; and the smaller
diminished fifth at 1024:729 to 3:2 less 2187:2048, or 588 cents.
In the division of the 9:8 tone, the rather large chromatic semitone
of 2187:2048 is complemented by the 256:243 diatonic semitone at 90
cents. This rather compact and incisive semitone makes available a
very effective resolution by oblique motion from an augmented fourth
at 729:512 to a pure 3:2 fifth, for example G-C# with the lower voice
remaining stationary and the upper ascending by a 90-cent semitone to
the fifth G-D, as happens in the Montpellier version of Perotin's
organum _Alleluia posui adjutorium_ from around 1200.
Along with the more polarized tritonic ratios of 36:25 and 25:18 that
we have already encountered, pental just intonation also yields the
forms within our central subregion of 45:32 and 64:45, respectively a
smaller augmented fourth at 590 cents and a larger diminished fifth at
610 cents. These forms are based on the division of a 9:8 tone into a
diatonic semitone at 16:15 or 112 cents, and a chromatic semitone at
135:128 or 92 cents. Thus 4:3 plus 135:128 yields the 45:32 augmented
fourth, and 3:2 less 135:128 the 64:45 diminished fifth.
Two ratios yet more centrally located within the region are 17:12 at
603 cents and 24:17 at 597 cents, based on the near-equal division of
the 9:8 tone into semitones of 18:17 at 99 cents and 17:16 at 105
cents. Depending on the context, either 18:17 or 17:16 might be
regarded as the diatonic semitone, and the other as chromatic. If we
arbitrarily take the smaller 18:17 as diatonic, for example, then the
17:16 chromatic semitone plus 4:3 would yield a 17:12 augmented
fourth, and 3:2 less 17:16 a smaller 24:17 diminished fifth. We may
refer to these ratios, following Scala, as septendecimal, or 17-based.
Some tempered systems have a tritone equal to precisely 600 cents or
half of a 2:1 octave, which may be used as either diminished fifth or
augmented fourth. This interval will be found in any equal division of
the 2:1 octave (EDO) with a number of steps evenly divisible by two,
with 12-EDO and 22-EDO as familiar examples. Irregular temperaments
such as 12-note modified meantone systems may feature this form of
tritone along with other sizes.
The following diagram may give an overview of the tritonic region and
some of the ratios by which it is populated. Indeed the ratios we have
discussed provide a rich enough population that the region is here
divided into two mirrorlike halves, with the "axis of symmetry" at 600
cents:
Tritonic region (~112:81-81:56)
pental-
septimal pentalA septimal Pyth pentalB
112:81 18:13 25:18 39:28 7:5 729:512 45:32 24:17
....|-----|-------|-------|----------|-------|-------|--------|---->
561 563 569 574 583 588 590 597
|------------------------------|----------------------------------->
560? small 577? middle 600
pental-
pentalB Pyth septimal pentalA septimal
17:12 64:45 1024:792 10:7 56:39 36:25 13:9 81:56
<---|--------|----=---|--------|---------|-------|-------|------|..
603 610 612 617 626 631 637 639
<-------------------------------------|---------------------------|
600 middle 623? large 640?
Starting at the left of the first portion at 560 cents, and at the
right of the second portion at 640 cents, one encounters the pairs of
tritonic intervals considered above in the order we have discussed
them: first the small 112:81 and large 81:56, then 18:13 and 13:9, and
so on until we reach 24:17 and 17:12 near 600 cents. This is only a
sampling of the many shades found in just and tempered systems.
------------------------------------------------
9. Interseptimal intervals (49:48-minus regions)
------------------------------------------------
We now come to the intriguing categories of intervals which may be
called _interseptimal_, since they occupy regions intermediate between
two septimal ratios such as 8:7 and 7:6, or 12:7 and 7:4. We will
begin with these two regions, with ranges of about 240-260 cents and
940-960 cents.
Intervals of these sizes routinely and beautifully adorn many world
musics, for example in the slendro tunings of gamelan. From the
perspective of at least one world musical tradition, the Western
European, however, they may have more of an unconventional beauty
which captures the imagination by exploring the space intervening
between two more familiar interval categories.
-----------------------------------------------------------
9.1. Regions between large major and small minor categories
-----------------------------------------------------------
Thus the region of 240-260 cents intervenes between the large or
septimal major second at 8:7 (231 cents), and the small or septimal
minor third at 7:6 (267 cents). The ratios 8:7 and 7:6 differ by a
factor of 49:48, or 36 cents, which is the potential size of this
interseptimal region. However, since we wish to allow 8:7 some "sphere
of influence" including slightly larger intervals, and likewise with
7:6 in regard to slightly smaller intervals, a rounded 240-260 cents
may roughly represent the range with an "interseptimal" quality
somewhat distinct from either of these simpler ratios.
Similarly, we find an interseptimal region between the large 12:7
major sixth at 933 cents and the small 7:4 minor seventh at 969 cents;
a rounded range of 940-960 cents might represent this distinct
region.
These regions are of a type we might call "major-minor," since they
are situated in the space between a large major and small minor
category -- the space between 8:7 and 7:6, or 12:7 and 7:4. The other
type, which we will explore in the next subsection, is found between a
large major and a perfect category, or a perfect and small minor one.
Looking first at the 240-260 cent region between 8:7 and 7:6, we find
that the complex integer ratio of 147:128 at a rounded 240 cents is
one possible guide to the lower border area. This is equal to 8:7 plus
a small interval of 1029:1024 or 8.43 cents. The 147:128 ratio is very
close to an even 1/5 octave or 240 cents, one possible point of
departure for the subtly unequal division of the octave typical of
slendro.
From a neomedieval European-oriented perspective, a 240-cent interval
is large enough that, while still often serving as a large melodic or
vertical major second, it can also serve in some contexts as a very
small and intriguing kind of "minor third," for example contracting to
a unison with one voice moving by a small tone of about 180-192 cents
and the other by a step of about 50-60 cents which serves as a
semitone, although it may be rather smaller than 60 cents we have set
as the minimum size for a "usual" semitone.
One theoretical point of interest is the Nobly Intoned or NI mediant
between 8:7 and 7:6 at around 344 cents. The complex integer of
1152:1001 at 243 cents, found in the Zephyr-24 tuning based on Erv
Wilson's eikosany concept (with factors of 1-3-7-9-11-13), nicely
approximates this NI value, as does the 242-cent interval found in
extended versions of Zarlino's 2/7-comma meantone (regular 192-cent
tone plus 50-cent enharmonic diesis).
The classic mediant of 8:7 and 7:6 is 15:13 at 248 cents, very closely
approximated in 29-EDO. As we continue our journey through the region,
another of the simpler ratios we encounter is 22:19 at 254 cents.
Not too far from the upper border area of the region around 260 cents
is 297:256 at 257 cents. An interval of around this size seems to me
largely interchangeable with 7:6, and yet may have its own distinct
charm and shading -- as also with sizes in the more immediate
neighborhood of 250 cents.
Here is an overview of the region showing these landmarks:
1/5 oct or CMed 8:7-7:6
147:128 1152:1001 NI 15:13 22:19 297:256
..|----------|-----|------------|---------------|-------|-------|..
240? 243 244 248 255 257 260?
Over much of this region, an interval in a vertical context may act as
_either_ a very large or "ultramajor" second or a very small or
"ultraminor" third; toward the upper end, intervals may lean more in a
"thirdlike" direction. Often the musical impression may depend on the
context: an interseptimal interval around 250 cents may sound like a
major second if it expands stepwise to a fourth, and like a minor
third if it contracts stepwise to a unison. This creative ambiguity is
one of the delights of neomedieval style.
The region of 940-960 cents between 12:7 and 7:4 is in many ways
similar: here, in a neomedieval style based largely on 13th-14th
century European polyphony, an interval may often act either as an
"ultramajor sixth" expanding to an octave, or an "ultraminor seventh"
contracting to a fifth. Indeed, some interpretations of Marchettus of
Padua (1318), who calls for a very large major sixth in directed
progressions expanding from this interval to the octave, suggest a
size of around 950 cents; an alternate reading calls for something
like 12:7 (933 cents), for example.
With the region between 8:7 and 7:6, the Noble Intonation value at
about 244 cents was located closer to the more complex 8:7. Here it is
likewise, at about 943 cents, located closer to the more complex 12:7
(933 cents) than to the simpler 7:4 (969 cents). This value is very
nicely approximated in 14-EDO. The classic mediant of these septimal
ratios is 19:11, at 946 cents. Another ratio of interest is 26:15, at
952 cents, or its near-just equivalent in 29-EDO, also close to the
geometric mean of 3:2 and 2:1 at around 951 cents.
It is a curious question just how far the "sphere of influence" of 7:4
might reach out to embrace somewhat smaller intervals. Thus the fine
958-cent interval of Zarlino's 2/7-comma (e.g. Bb-G#) comes within our
interseptimal region, although it could also be taken as a rather
heavily tempered representation of 7:4. Here 960 cents, or 4/5 octave,
is arbitrary taken as around the upper border of the region.
CMed 2/7-comma
12:7-7:4 meantone
NI 19:11 26:15 (e.g. Bb-G#) 4/5 oct
..|----------|--------|-------------|-------------|-----------|..
940? 943 946 952 958 960?
This region, like that of around 240-260 cents, is very typical of the
slendro tunings of gamelan.
An alternate term for an interval of around 250 cents would be a
"hemifourth," equal to about half of a fourth of some kind as defined
in Section 7 (around 468-528 cents). Thus the 940-960 cent region
could be described as that of "an octave less hemifourth," or possibly
of "a fifth plus hemifourth." Such terms might be attractive, for
example, in describing the Soft Diatonic of Aristoxenos, which as
interpreted by John Chalmers includes a step of half a fourth or 250
cents.
Whereas neutral or "interpental" intervals (Sections 3, 6) occur in
regions between a large minor and small major category as typified by
pental ratios -- for example, neutral thirds between 6:5 and 5:4 --
interseptimal intervals of the two regions we have just considered
occur in the space intervening between a large major and small minor
category as typified by septimal ratios: 8:7 and 7:6, or 12:7 and 7:4.
An advantage of distinguishing between "neutral" intervals and
"interseptimal" ones is that some tuning systems may feature intervals
of one type but not the other. Thus George Secor's 17-tone
well-temperament offers a wealth of neutral intervals, but no
interseptimal ones around 250 or 950 cents (or 450 and 750 cents, as
we are about to consider). In contrast, his 29-tone High Tolerance
Temperament (HTT-29) offers an abundance of intervals in all of these
categories.
-------------------------------------------
9.2. Major-perfect or perfect-minor regions
-------------------------------------------
A related type of interseptimal region occurs at around 440-468 cents,
between the large or septimal major third at 9:7 (435 cents) and the
narrow fourth at 21:16 (471 cents); and likewise at around 732-760
cents, between the wide fifth at 32:21 (729 cents) and the small or
septimal minor sixth at 14:9 (765 cents).
These regions, like those around 240-260 and 940-960 cents, are very
commonly represented in gamelan music, for example. They are also
represented as premiere resources in a tuning system such as 24-EDO or
29-EDO which features sizes at or around 250, 450, 750, and 950 cents.
Especially with certain types of strongly harmonic or customized
inharmonic timbres, the regions of 440-468 and 728-760 cents can have
distinctive qualities because of their rather close proximity to the
simple ratio of a 4:3 fourth or 3:2 fifth. This attraction is most
evident for the portions of these regions nearest to 4:3 or 3:2, say
458-468 cents and 732-745 cents. Depending on the choice of timbre and
style, the musical consequences may be undesirable or highly desirable.
Thus in a 1/4-comma meantone tuning used in a 16th-century European
fashion, the interseptimal intervals at around 462 cents (Eb-G#) and
738 cents (G#-Eb) are mostly regarded as "Wolf" fourths or fifths,
"howling" and musically "mistuned" counterparts of usual fourths and
fifths close to 4:3 and 3:2. Here both the intense beating which
occurs in a strongly harmonic timbre (e.g. organ or harpsichord), and
the categorical "strangeness" of an interval somewhere between a major
third and perfect fourth, or perfect fifth and minor sixth, may have
contributed to a cognitively as well as acoustically dissonant
impression.
In contrast, the very similar interval sizes available in 13-EDO at
around 462 and 738 cents can in certain customized inharmonic timbres
serve in a neomedieval context as stable "fourths" and "fifths" -- and
the 462-cent "fourth" also as a very large cadential major third
expanding to a 738-cent fifth! This practice might be taken as a
tribute to the large gravitational spheres of 4:3 and 3:2.
Conversely, intervals in the lower portion of the 9:7-21:16 region or
the upper portion of the 32:21-14:9 region, say 440-455 and 745-760
cents, may have qualities rather like those of large septimal major
thirds around 9:7 or small septimal minor sixths around 14:9, with the
former in a neomedieval context often expanding stepwise to a fifth
and the latter to an octave, for example.
Looking first at the region between the large 9:7 major third and the
narrow 21:16 fourth, we find that this interseptimal zone has a size
equal to the 49:48 difference between these two ratios, or 36 cents,
less a few cents at either end to allow these ratios their "spheres of
influence": thus about 440-468 cents.
Here a rounded 440 cents is taken as marking a fuzzy border area where
we are moving from widely tempered variants of 9:7 to intervals with a
more distinct character. Thus the 442-cent interval of 19-EDO, or the
443-cent interval of 46-EDO, might be regarded either as a rather
imprecise version of 9:7, or as having its own interseptimal flavor.
By around 22:17 or 446 cents, we may have entered more distinctively
interseptimal space.
While 9:7 and 21:16 are the septimal ratios bounding this region, it
may be best as a guide to Noble Intonation (NI) to seek a Phi-weighted
mediant between 9:7 and the simpler ratio of 4:3, yielding an NI value
of around 448 cents, or not far from 22:17. This is in theory one
measure of the area of maximum complexity between 9:7 and 4:3, and it
is a very pleasant and effective one in neomedieval contexts.
The classic mediant of 9:7 and 4:3 is 13:10 at 454 cents, an interval
available in virtually just form in 29-EDO, and also in Zarlino's
2/7-comma meantone, for example at Eb-G#. It has been proposed that
the style of vocal tuning described by Marchettus of Padua in 1318
calls for an enlarged major third of around this size in a directed
progression expanding to a fifth -- although a more moderate
interpretation prefers something around the septimal 9:7. In any
event, an interval at 13:10 or so acting as an "ultramajor third"
expanding to a fifth can be musically at once striking and beautifully
effective.
Much larger than this, and we move more into the peripheral sphere of
influence of the fourth. The classic mediant between 9:7 and 21:16 at
30:23, or about 460 cents, is one possible guide to this transition.
At 462 cents we encounter the interval of 4:3 less a 49:48 diesis (as
we shall term many intervals in the range of about 33-60 cents), or
64:49, with the augmented third or small fourth of 1/4-comma meantone
(e.g. Eb-G#), or the small fourth of 13-EDO, very close to this.
At 17:13 or around 464 cents we are getting closer to the upper border
of the region, and the 467-cent interval found in 72-EDO and the
version of George Secor's Miracle temperament based on it might be
considered as a variant on the 21:16 fourth.
The following diagram may give an overview of some of these points,
with the reader's patience asked for the multiple labels attached to
certain rounded values in cents:
2/7-comma/
CMed 1/4-comma/
19- 46- 9:7-4:3 9:7-4:3 13-EDO/
EDO EDO 22:17 NI 13:10 30:23 64:49 72-EDO
..|-----|---|-------|------|---------|--------|------|------|---|..
440? 442 443 446 448 454 460 462 467 468?
The region of 732-760 cents between 32:21 and 14:9 is rather similar,
with the especially strong "gravitational attraction" of the 3:2 fifth
often exerting its sway over the lower portion of the region. At about
738 cents we encounter the ratio of 49:32, equal to 3:2 plus the 49:48
diesis. This interval can have a very striking character in certain
septimal tunings, including one of Robert Walker's, where the 49:48
step actually serves as a kind of very small semitone or "ultraminor
second." In such a setting, it might be perceived as a very small
"minor sixth" contrasting with a 3:2 fifth. In the different context
of 13-EDO, as mentioned, an interval of almost this identical size can
serve as a "xentonal" equivalent of 3:2 in a neomedieval setting. This
is also the size of an interval such as G#-Eb in 1/4-comma meantone.
In gamelan, interval sizes in the neighborhood of 740 cents may occur
in both slendro and pelog tunings, with much variability in size:
intervals which would generally be described here as wide fifths,
small minor sixths, or denizens of the intervening interseptimal
region seem all congenial to the style, with the choice between these
varying from ensemble to ensemble.
By 20:13, or 746 cents, we may be moving more into the domain of a
"sixthlike" impression, with an interval such as G#-Eb in Zarlino's
2/7-comma meantone at almost precisely this size. Such an "ultraminor
sixth," much like a more conventional septimal minor sixth at 14:9,
might in a neomedieval context expand stepwise to an octave, or
resolve by oblique motion to a fifth with the upper voice descending
by a small step of about 50 cents (in context, a kind of "ultraminor
second").
The classic mediant of 3:2 and 14:9 is 17:11, or 754 cents, an
intriguing form of very small "minor sixth": from here on we more and
more closely approach the upper border of the region, or the lower
border of the 14:9 neighborhood, here placed at a rounded 760 cents.
In this upper portion of the region are found the 46-EDO interval of
757 cents and the 19-EDO interval of 758 cents.
1/4-comma/ 2/7-comma/ CMed
13-EDO/ 29-EDO 3:2-14:9 46- 19-
49:32 20:13 17:11 EDO EDO
..|-----------|-----------|-----------------|---------|---|-----|..
732? 738 746 754 757 758? 760?
In various musical styles, these interseptimal regions typified by
intervals of around 450 or 750 cents happily combine with those of
around 250 and 950 cents that we have considered. Thus in gamelan, all
of these ranges are common.
In neomedieval styles, also, interseptimal intervals of the
"major-minor" type (240-260 or 940-960 cents) nicely complement those
of the "major-perfect" or "perfect-minor" type (440-468 or 732-760
cents).
For example, using the rounded values of 24-EDO for convenience, a
sonority of 0-700-950 cents is rather ambiguous: the outer 950-cent
interval might be taken as an "ultramajor sixth" seeking expansion to
the octave or an "ultraminor seventh" inviting contraction to a
fifth. The upper 250-cent interval, likewise, might be an "ultramajor
second" expanding to a fourth or an "ultraminor third" contracting to
a unison.
While precisely this creative ambiguity may be the desired effect,
with either resolution open, adding a fourth voice can resolve the
ambiguity. Thus 0-450-700-950 cents, with the 450-cent "ultramajor
third" adding both excitement and definition, suggests the
interpretation of 950 cents as a very large major sixth and 250 cents
as a very large major second, with all three intervals resolving by
stepwise expansion to the fifth, octave, and fourth respectively.
In contrast, 0-250-700-950 cents, with the added 250-cent interval
above the lowest voice as an "ultraminor third," suggests that the
identical interval between the highest voices should likewise be taken
as a very small minor third, and the outer 950 cents as an "ultraminor
seventh," with these thirds contracting to unisons and the seventh to
a fifth.
One purpose of the term "interseptimal" is to describe these intervals
in a relatively objective way based on their location along the
spectrum, whether they represent the routinely flowing intonational
currents of a gamelan ensemble, for example, or the less familiar
effects of some other world musical settings where superlatives like
"ultraminor" or "ultramajor" might be more in order.
While giving these interseptimal regions a choice place on the
continuum, we should also remember that in various world musical
traditions they may blend quite unassumingly with what are here
considered distinct categories, as in the free gamelan use of
intervals we might here consider "fifths," "minor sixths," or
somewhere in between.
----------------------------------------------------
10. Superfourths and subfifths (49:48-minus regions)
----------------------------------------------------
We now come to the regions of superfourths and subfifths, most
versatile and intriguing intervals for whose names I am much indebted
to David Keenan. As here defined, superfourths are found in the region
between a wide perfect fourth at around 256:189 or 525 cents and a
small septimal tritonic interval at 112:81 or 561 cents -- say around
528-560 cents. Likewise subfifths populate a region situated between
the large septimal tritone at 81:56 or 639 cents and the narrow fifth
at 189:128 or 675 cents; allowing our usual leeway at each end of the
range, around 640-672 cents.
Especially familiar examples are the 11:8 superfourth at 551 cents,
and the 16:11 subfifth at 649 cents, with many other shadings also in
evidence, as we shall see.
Superfourths and subfifths have affinities to either neutral or
interseptimal types of intervals, and might be considered as an
instance of either, but are perhaps best considered in their own
special category. In gamelan music, they are often favored along with
interseptimal and neutral intervals alike, the latter especially in
pelog. In various Near Eastern music, neutral steps and intervals
interact with others such as the 9:8 tone to generate superfourth or
subfifth relationships, as with the 11:8 or 16:11 intervals found
within Zalzal's tuning according to al-Farabi.
From a certain theoretical viewpoint, the affinity between
superfourths or subfifths and interseptimal categories is especially
compelling. If we take a 4:3 fourth and subtract an interval up to
around a 64:63 comma at 27 cents or a tad more, say around 30 cents,
we have some kind of narrow fourth, ranging down to 21:16 (471 cents)
or a bit smaller. If we subtract a small semitone of around 28:27,
then we have a large major third at around 9:7 (435 cents). If we
instead subtract an interval somewhere in the range of 30-60 cents, a
_diesis_ as we shall term it, then an interseptimal interval somewhere
in the range between 9:7 and 21:16 results.
Similarly, if we begin again with 4:3 and add some kind of comma up to
64:63 or so, we have a wide fourth at up to around 256:189 (525 cents)
or a bit larger. If we add a small semitone around 28:27, we have a
small septimal tritone or diminished fifth at around 112:81 (561
cents). Should we add a diesis somewhere in the 30-60 cent range,
however, then we have a superfourth of some flavor.
The same logic would apply for subfifths. Adding a diesis of 30-60
cents to a 3:2 fifth produces an interseptimal flavor somewhere
between the wide 32:21 fifth and the small 14:9 minor sixth.
Subtracting 30-60 cents from 3:2 produces some flavor of subfifth
siituated between the large septimal tritone or augmented fourth at
81:56 (639 cents) and the narrow fifth at 189:128 (675 cents).
A certain affinity between these categories is also suggested by the
intonational practices of gamelan. Just as either slendro or pelog
often favors intervals at around 445-480 cents, which using our
terminology would be either narrow fourths or interseptimals in the
9:7-21:16 range, so pelog especially often favors intervals around
515-545 cents, which might be styled either wide fourths or
superfourths. This association might lead one to think of the two
categories as kindred -- and likewise with intervals around 15-50
cents from 3:2 in either the wide or narrow direction.
While this last example illustrates a pleasant association between
these categories, it can also serve as a useful caution on the
limitations of any generalized interval mapping scheme. Following the
scheme presented here, we might classify a pelog interval of 515 cents
as a "wide fourth" very close to 3/7 octave, but 535 cents as a
"small superfourth." What is not clear, however, is that a Balinese or
Javanese musician would take these sizes as representing two distinct
categories, rather than shadings of a single basic category. Any
generalized scheme, including this one, should always supplement
rather than substitute for the categories and patterns of a specific
musical tradition.
Douglas Leedy has made a very cogent case that superfourths and
subfifths as here styled (following Keenan) might be considered
members of the same family as neutral thirds and sixths, for example.
If we take an 11:9 neutral third at 347 cents and add a 9:8 tone, we
arrive at an 11:8 superfourth at 551 cents. Similarly, one might add,
subtracting a 9:8 tone from an 18:11 neutral sixth at 853 cents yields
a 16:11 subfifth at 649 cents.
Situations of this kind often arise in Near Eastern maqam and dastgah
traditions, and would also occur in certain ancient Greek modes based,
for example, on the Equable Diatonic tetrachord of Ptolemy with its
division of the 4:3 fourth into string ratios of 12:11:10:9.
Regarding superfourths and subfifths as belonging to their own
category leaves open an appreciation of their co-occurence with
neutral and/or interseptimal intervals (as here described) in various
world musics, and also some conceptual affinities between these
categories.
As here defined, the region of superfourths or of subfifths spans a
range similar to that of an interseptimal region, for reasons already
touched upon in the comparison above. To derive a superfourth from a
4:3 fourth, for example, it must be widened by somewhat more than a
septimal comma at 64:63 or 27 cents, say 30 cents, but less than a
small or septimal semitone at 28:27 or 63 cents, which would yield a
small septimal tritone at 112:81 or 561 cents. The difference between
28:27 and 64:63 is 49:48 or 36 cents. In practice, the range is a bit
smaller, say 528-560 cents, or about 32 cents.
Starting from 528 cents, we soon encounter the interesting ratio of
49:36 or 533 cents, with 36-EDO and multiples providing a very close
approximation. This very small superfourth is equal to 4:3 plus a
septimal diesis at 49:48, or 3:2 less the very large septimal
chromatic step of 54:49 at 168 cents, near the top of our neutral
second range (125-170 cents), and equal to the difference between a
small 7:6 minor third and a large 9:7 major third. Possibly this small
superfourth is large enough to have a degree of "independence" from
the region of perfect fourths because it differs from 4:3 by 49:48, a
diesis step more melodically discrete than a smaller interval of the
comma type (up to about 30-33 cents).
Another small superfourth is 15:11 at 537 cents, equal to 3:2 less
Ptolemy's large neutral second at 11:10 or 165 cents, or 4:3 plus a
small 45:44 diesis at 39 cents. As the studies of John Chalmers have
suggested, 15:11 would arise in some ancient Greek modes based on the
Equable Diatonic of Ptolemy with its 11:10 steps. This superfourth is
very closely approximated by the 538-cent equivalent in 29-EDO.
Possibly one of the first superfourths written in the Western European
tradition of composition occurs in Fabio Colonna's treatise of 1618 on
his harpsichord with 31 notes per octave, _La Sambuca Lincea_, where
the upper voice starts at a fourth from a middle voice and then rises
a diesis, about a fifth of a tone, to form a superfourth. If the
instrument is tuned in 1/4-comma meantone, and we take to diesis to be
the regular one of 128:125 or 41 cents, then this interval would be
equal to the slightly wide meantone fourth at 503 cents plus the
diesis, or 544 cents. If the octave were divided into 31 precisely
equal parts, then the fourth would remain at around 503 cents but the
diesis be slightly smaller, at around 39 cents, for a 542-cent
superfourth.
At 545 cents, we encounter a superfourth which, in systemic terms, is
the regular diminished fifth of 22-EDO (e.g. B-F), equal to the
somewhat wide fifth at 709 cents less the large chromatic step at 164
cents, very close to Ptolemy's 11:10. This superfourth is also equal
to the 22-EDO fourth at 491 cents plus the regular diatonic semitone
(or literally diatonic quartertone, since four make a major second) at
not quite 55 cents. Although considerably smaller than a usual tritone
as defined above (starting around 560 cents), it can quite happily in
this context be called both a regular 22-EDO diminished fifth and a
superfourth.
At 551 cents, we have the cardinal landmark of 11:8, the simplest
ratio in the region and also, interestingly, the classic mediant
between 4:3 and the 7:5 diminished fifth. it is equal to 3:2 less a
12:11 middle neutral second, and thus arises both in ancient Greek
modes based on Ptolemy's Equable Diatonic (12:11:10:9 tetrachord), and
in Zalzal's scale as described by al-Farabi with its 12:11 steps. The
11:8 superfourth is also equal to a 4:3 fourth plus the large diesis
of 33:32 or 53 cents, a large enough step to serve under some
circumstances as a kind of small semitone. The 550-cent superfourth of
24-EDO and multiples provides an excellent approximation.
Extended versions of Zarlino's 2/7-comma meantone also offer a good
approximation of 11:8 by adding the meantone diesis of about 50 cents
to the somewhat widened fourth at 504 cents, yielding a 554-cent
superfourth. In 13-EDO, we also find an interval of this rounded size.
At around 558 cents, near the upper border of the region, we have
another form of superfourth (or small tritone) occurring in Zalzal's
scale: 243:176, equal to 3:2 less the smaller 88:81 middle neutral
second of this tuning at 143 cents. If we choose to regard the minimal
size for a usual "tritone" as 4:3 plus a semitone at the rounded size
of 60 cents, then 243:176 would be considered a very small tritone,
since it differs from 4:3 by a large diesis or small semitone of
729:704, or 60.41 cents. If we prefer a rounded minimal size of 560
cents, then it is a very large superfourth. Since borders are fuzzy,
the question seems somewhat moot.
CMed 2/7-
36-EDO/ 29- 31- 1/4- 22- 24- 4:3-7:5 comma
49:36 15:11 EDO EDO comma EDO EDO 11:8 13-EDO 243:176
..|------|------|----|-----|---|-----|------|-----|------|-------|....
528? 533 537 538 542 544 545 550 551 554 558 560?
The realm of subfifths is similarly diverse in its shadings, ranging
from about 640 to 672 cents. In the Persian dastgah tradition, for
example, it is very common in a dastgah or family of related modes
such as the popular Shur for the fifth step of a mode to have two
forms: a regular form at around 3:2, plus an inflected form lowered by
a _koron_, a small interval which may range from about 30 to 70 cents.
If values in the range of 30-60 cents were subtracted from a 3:2
fifth, then some flavor of subfifth would result. Indeed the Persian
tuning of the fretted tar as suggested by Dariush Anooshfar based on
often complex integer ratios, and available in the Scala scale archive
(persian.scl), has some subfifths at 3200/2187 or 659 cents. The
tuning described by Hormoz Farhat has subfifths at 655 and 660 cents.
Gamelan tunings abound with intervals in this range, as also with
superfourths. They may also occur in a tuning style such as that of
the xylophone tuning favored by the Chopi people of southern
Mozambique, where fifths may average around 4/7 octave or 686 cents,
but with considerable variation in both directions, so that some are
small enough to come within our subfifth region. This variety of
"equable heptatonic" tuning, with melodic steps tending toward an
average size of about 1/7 octave or 171 cents, will be considered at
more length below.
Here is a diagram showing some landmarks of this subfifth region.
13- 24- 22- 9-EDO
352:243 EDO 16:11 EDO EDO 22:15 72:49
..----|--------|-----|----|---------|------------|---------|----..
640? 642 646 649 650 655 663 667 672?
If we take the region as starting around 640 cents, then just wider
than this is the 352:243 subfifth at 642 cents found in Zalzal's
scale. We quickly move into the nearer neighborhood of 16:11 at 649
cents, with either the 13-EDO interval at 646 cents or 24-EDO at 650
cents, for example, providing a fine approximation of this subfifth
with the simplest integer ratio.
In a neomedieval style, subfifths around 16:11 are often substituted
for regular fifths close to 3:2 in minor seventh sonorities such as
1:1-13:11-16:11-39:22 (0-289-649-991 cents) or tempered variations.
Here it is common for the lower two voices as usual to contract from
minor third to unison, and the outer ones from minor seventh to fifth;
if the lower voice rises by a semitone, for example, with the one
immediately above it and the highest descending by a tone to bring
about these resolutions, then the voice at 16:11 ascends by a neutral
second to arrive at a unison with the highest. Here the subfifth is
closely allied with neutral intervals, both vertical and melodic.
In 22-EDO, the 655-cent subfifth is the regular tritone or augmented
fourth (e.g. F-B), equal to a 709-cent fifth less the regular semitone
of almost 55 cents. This interval lends itself to a very effective
neomedieval resolution by oblique motion, for example from F-B to F-C,
with the lower voice stationery and the upper ascending by the very
incisive 55-cent step B-C. This is a kind of modern variation on the
classic idiom of Perotin's era around 1200 mentioned in the discussion
of tritonic intervals (Section 8).
In 11-EDO, a subset of 22-EDO, this same 655-cent interval is used in
certain "xentonal" styles as a kind of curious equivalent for a 3:2
fifth, with customized timbres designed to maximize a "fifthlike"
impression. Given that this "xenofifth" is about 47 cents narrow of a
pure 3:2, the resemblance is necessarily partial, with one listener
hearing a kind of synthesis between a large tritone and a fifth.
The 22:15 subfifth at 663 cents is notable for its use in a just
gamelan tuning of Lou Harrison, with intervals of around 660-670 cents
often reported in Javanese or Balinese pelog tunings, as well as
others in nearby portions of the spectrum.
The 667-cent subfifth found in 9-EDO and its multiples is reported
sometimes used in certain timbres as a very small perfect fifth,
likely somewhat more firmly than its 11-EDO counterpart at 655 cents.
By this point we are approaching the region of small fifths, here
estimated to have its fuzzy lower border, also the fuzzy upper border
for subfifths, at around 672 cents.
Superfourths and subfifths thus occur in various world musical
traditions and tuning systems. They may be found in association with a
rich use of neutral intervals, as in medieval and current Near Eastern
traditions; or with the use of interseptimal and sometimes also
neutral intervals, as in certain pelog tunings of gamelan.
While superfourths and subfifths are available in some tuning systems
such as extended meantones as supplementary or "special effects"
intervals, in others such as 22-EDO when used as a regular neomedieval
temperament and some 17-note circulating systems they occur as usual
diminished fifths or augmented fourths, for example. In certain
17-note circles, we might diminished fifths of around 550-575 cents
and augmented fourths of around 625-650 cents, thus spanning a range
from the superfourth or subfifth region to that of small or large
tritonic intervals as defined above (Section 8). Since these diverse
sizes at different locations in the circle are regarded as musically
interchangeable, this is an instructive example of how theoretical
divisions of the interval spectrum should not trump the more flexible
realities of musical practice which often disregard such borders.
-------------------------------------------------------------
11. Small intervals: commas, dieses, and "ultraminor seconds"
-------------------------------------------------------------
To this point, while we have often brought intervals smaller than a
usual "semitone" into play when defining or exploring other
categories, we have not yet made these types of intervals our main
focus.
We shall do so now, considering the many-featured terrain to be found
and surveyed between a pure unison and what we have termed a small
usual semitone at the septimal 28:27 (63 cents) or the slightly
narrower size of a rounded 60 cents.
By tradition, this territory may be divided into two general
categories: "commas" or the like ranging from the smallest discernible
intervals to around 30-33 cents or so; and "dieses" here taken as
ranging from around 30-33 cents to our small form of "usual" semitone
at 60 cents.
One imperfect but helpful concept is that notes at a comma of some
kind apart may often be perceived as two "versions" of the same note
or step; while notes connected by the step of a diesis are heard as
more clearly "different" or "discrete" in their identity, whether that
difference is perceived as a kind of small semitone, or as something
quite distinct from a semitone.
The first situation of a diesis used as a very small semitone is
common in neomedieval styles, where steps of around 50-60 cents are
routinely used in forming regular modes as well as in special
cadential progressions. The second, of a diesis used as "something
else again" and quite distinct from a semitone, occurs par excellence
in Nicola Vicentino's 16th-century enharmonic style where meantone
dieses of somewhere around 35-41 cents are used as special melodic
"fifthtone" steps.
Here we shall begin with the smaller intervals or commas, then moving
to the larger dieses and "ultraminor seconds," while emphasizing that
this is a very partial and incomplete survey not meant to substitute
for various more comprehensive lists and discussions.
Pyth
352:351 pental 531441: septimal 41-
364:363 896:891 144:143 72-EDO 81:80 524288 64:63 EDO
|------|-------|-------|--------|--------|------|---------|-----|...|
0 5 10 12 17 22 23 27 29 30?
A small comma often coming into play in neomedieval tunings is 352:351
(about 4.925 cents), the difference between the middle minor third at
the classic mediant of 7:6 and 6:5, 13:11 (289 cents), and the
slightly larger 32:27 Pythagorean minor third (294 cents) from three
pure fifths or fourths. This comma is also the difference between a
Pythagorean major third at 81:64 or 408 cents and a slightly larger
33:26 third at 413 cents, the latter plus 13:11 yielding a pure 3:2.
A related comma minutely smaller is 364:363 (4.763 cents), the
difference between a 33:26 major third and the classic mediant of 5:4
and 9:7 at 14:11 (418 cents), the simplest ratio for a middle major
third. This 364:363 comma is also the difference between 13:11 and the
slightly smaller 33:28 minor third at 284 cents, the fifth complement
of 14:11.
Adding these two commas yields 896:891 (9.688 cents), the difference
between the Pythagorean 81:64 and 14:11, or likewise 32:27 and 33:28.
In regular neomedieval temperaments where fifths are about two cents
wide of a pure 3:2, a chain of four such fifths will come close to
dispersing the full 896:891 comma, so that usual major thirds are
quite close to 14:11. A chain of three fifths up or fourths down
disperses an amount slightly in excess of the 352:351 comma, so that
usual minor thirds are likewise close to 13:11.
A comma of 144:143 or 12.044 cents defines the difference between the
neutral seconds at 12:11 or 151 cents and 13:12 or 139 cents, and
interestingly is available as a direct melodic step in tuning systems
such as Zephyr-24 based on Erv Wilson's 1-3-7-9-11-13 eikosany. This
small comma thus also marks, for example, the difference between the
large tritone or augmented fourth at 13:9 (637 cents) and the 16:11
subfifth (649 cents), equal respectively to 4:3 plus 13:12 or 12:11.
This fascinating comma also seems to approximate the difference
between a Nobly Intoned (NI) major third at around 422-423 cents, the
evident region of maximum complexity between 5:4 and 9:7, and the pure
septimal ratio of 9:7 at 435 cents. In tuning systems like Zephyr-24,
both of these flavors of active major thirds are thus available.
The step of 1/72 octave or 16-2/3 cents available in 72-EDO is another
interesting size of "comma" which in context can represent various
integer ratios. It is also used by contemporary composers such as
Julia Werntz as a melodic "microinterval" in its own right, without
any deliberate regard to approximating intervals based on small
integers, for example.
We now come to the "big three" among the commas, which might be known
by this phrase both because they are relatively large, and because
they are the ones likely most often mentioned. The pental or syntonic
comma of 81:80 (21.506 cents) is equal to difference between a small
or pental major third at 5:4 (386 cents) and a Pythagorean third at
81:64; and likewise between a Pythagorean minor third at 32:27 and a
large or pental minor third at 6:5. In meantone temperaments, this is
the comma dispersed by chains of somewhat narrowed fifths (or widened
fourths). Thus 1/4-comma produces pure 5:4 major thirds; 1/3-comma,
pure 6:5 minor thirds; and 2/7-comma, equally impure approximations of
5:4 and 6:5 alike (both at 1/7 comma narrow, or about 3.072 cents).
The syntonic or pental comma is also known as the comma of Didymis,
named after a Greek musician describing the use of the 5:4 third.
The Pythagorean comma at 531441:524288 or 23.460 cents, is equal to
the difference between 12 pure 3:2 fifths and 7 octaves at 2:1. This
comma must be dispersed in 12-note circulating temperaments, such as
well-temperaments or modified meantone systems, where all 12 fifths
are "reasonably close" to 3:2 (typically within about 7 cents for
Western European styles often realized in strongly harmonic timbres.)
The septimal comma or comma of Archytas at 64:63 or 27.264 cents marks
the difference between the Pythagorean 81:64 middle major third and
the large septimal major third at 9:7 (408 and 435 cents); and
likewise between the Pythagorean 32:27 minor third at 294 cents and
the small 7:6 minor third at 267 cents.
A slightly larger interval, near the upper end of the "comma" range,
is the 41-EDO step of 29.268 cents. Exactly where the line should be
drawn between a "comma" and a larger "diesis" is unclear, and in any
event it is a fuzzy border; but somewhere around 30-33 cents seems
about right.
To sum up on our sampling of commas, we might note that the smaller
ones like 352:351, 364:363, and their sum of 896:891 serve to mark
subtle differences of shading, as between middle major thirds at the
Pythagorean 81:64 or slightly larger 33:26 or 14:11; and likewise
middle minor thirds at the Pythagorean 32:27 or the slightly smaller
13:11 or 33:28 popular in neomedieval tuning systems. These commas can
be dispersed in regular temperaments by a gentle widening of the fifth
on the order of 2 cents, bringing about some new colors without
compromising too seriously the medieval idea of pure fifths and
fourths.
The syntonic or Didymic comma at 81:80 or 22 cents, and the septimal
or Archytan comma at 64:63 or 27 cents, are considerably larger, and
thus call for a considerably greater compromise of the fifths if they
are to be dispersed in a regular system such as a Renaissance European
meantone for 81:80; or 22-EDO for 64:63, where a regular major third
is very close to the large septimal ratio of 9:7.
Some modified meantone systems seek a curious balance betwen the
syntonic, septimal, and Pythagorean commas. Most of the fifths are
tempered narrow by around 1/4 or 2/7 syntonic comma, but a chain of a
few in the remote portion of the 12-note circle are tuned about
equally wide, thus producing pental-flavor intervals in the nearer
portion of the circle, a few septimal-flavor intervals in the most
remote portion, and at the same time balancing out the Pythagorean
comma so that all 12 fifths are within about 7 cents or pure and the
system circulates.
While even the largest commas, approaching the range of 30-33 cents,
may often be heard more as two "versions" of "the same step," there
are special situations where these intervals can take on more
discretely melodic properties.
For example, taking a given note and the pure 3:2 fifth above it as a
sustained drone, let us add two notes at a 21:16 narrow fourth, and at
63:32 or a 3:2 fifth above 21:16 (471 and 1173 cents) -- with these
added notes then moving to a simple 4:3 fourth and 2:1 octave (498 and
1200 cents), and thus making the 64:63 or 27-cent step more tangible.
Above our drone we could also have two moveable voices starting at
21:16 and 7:4 (471 and 969 cents), and moving up 64:63 to 4:3 and 16:9
(498 and 996 cents). LaMonte Young and his colleagues are renowned for
this type of melodic use of the septimal comma, which can also be
approximated in temperaments with reasonable equivalents of these
ratios.
Moving now to large intervals or "dieses," the lower zone of
demarcation remains uncertain, but as suggested might be placed around
30-33 cents.
2/7-
smaller larger comma
1/4- 1/4-comma Pyth 24- 22-
36-EDO comma 49:48 31-EDO 29-EDO diesis 36:35 EDO 33:32 EDO 91:88
|....|------|-----|-----|-------|---------|------|----|----|---|----|..|
30 33 35 36 39 41 45 49 50 53 55 58 60?
The 36-EDO step of 33-1/3 cents might be considered a very large comma
or a very small diesis, depending on the musical context; this
interval may be near the transitional zone between the two categories.
An interval that nicely epitomizes a small diesis is the 34.990-cent
step found in 1/4-comma meantone, which together with the larger
128:125 diesis at about 41 cents divides a 76-cent small or chromatic
semitone (e.g. C-C#) into two nearly equal melodic intervals. These
small steps play a premier role in the enharmonic or "fifthtone" music
of such Manneristic composers as Nicola Vicentino and Fabio Colonna.
Their harpsichord or organ tunings of 31 fifthtone steps to the
octave, if realized in 1/4-comma, would thus produce diesis steps of
35 and 41 cents, a tuning quite successful in practice although not
the only possibility.
The 49:48 diesis at 35.697 cents, which we have met frequently in this
exploration of the interval spectrum, represents the different for
example between the large major second at 8:7 and the small minor
third at 7:6; or the large major sixth at 12:7 and the small minor
seventh at 7:4. In tunings such as that of Robert Walker having steps
above the modal final or note or repose such as 8:7 and 7:6, or 12:7
and 7:4, this 49:48 or 36-cent step between the major second and minor
third degrees, or major sixth and minor seventh degrees, can indeed
play the role of a very small "semitone" or "ultraminor second."
The 31-EDO diesis at 38.710 cents divides the small or chromatic
semitone into precisely steps, and is another possibility for the
interpretation of fifthtone music by Manneristic composers such as
Vicentino and Colonna. In 1666, at any rate, Lemme Rossi demonstrated
the very small mathematical distinction between this tuning and
1/4-comma, and suggested that Vicentino's tuning favors the former
arrangement, championed by Christian Huygens in 1691.
In 1/4-comma meantone, as already mentioned, a large diesis or
fifthtone at 128:125 or 41.059 cents complements the smaller 35-cent
fifthtone in making up a 76-cent chromatic semitone. Interestingly,
29-EDO has an almost identical diesis at 41.379 cents. While the
1/4-comma or 31-EDO dieses as typically used in a Manneristic
enharmonic or "fifthtone" style have an effect quite distinct from
that of a "semitone," the 41-cent fifthtone of 29-EDO is sometimes
used in effect as a very small semitone in certain neomedieval
cadential progressions involving interseptimal intervals. Thus a
sonority with the 29-EDO "ultramajor third" at 455 cents and
"ultramajor sixth" at 952 cents (close to 13:10 and 26:15) may resolve
with these intervals expanding respectively to fifth and octave, the
lower voice descending by a usual 207-cent tone and the upper ones
ascending by 41-cent dieses.
The Pythagorean diesis of about 45.112 cents is equal to half of a
usual 256:243 semitone at around 90 cents; the term "diesis" is also
sometimes used for this semitone itself, and "diaschisma" for this
division of it into two equal parts. Note that "diaschisma" can also
refer to a ratio of 2048:2025 or about 19.553 cents, a later
usage. This type of division was likely used in some ancient Greek
tunings in the enharmonic genus where a semitone or some kind is
divided into two more or less equal parts, and is also mentioned as an
added feature for a sophisticated organ by Ugolino of Orvieto writing
around 1425-1440. After advocating a Pythagorean tuning with 17 notes
per octave (Gb-A#), he adds that the diatonic semitones E-F and B-C
(at 256:243 or 90 cents) might be divided into two approximately equal
parts after the manner of the ancients.
The 36:35 diesis at 48.770 cents is found in ancient Greek enharmonic
tunings, and marks the difference between a small pental major third
at 5:4 and its large septimal counterpart at 9:7 (386 and 435 cents);
and likewise a small septimal minor third at 7:6 and its large pental
counterpart at 6:5 (267 and 316 cents), etc.
At around 50 cents, we enter a range where dieses may more readily or
often serve as "ultraminor seconds" in a variety of tuning systems;
Ivor Darreg has suggested that around 50-55 cents might mark a
transition from "microtonal" intervals to intervals with a more
familiar "semitone" quality.
Two tuning systems notable for steps of around 50 cents are 24-EDO,
where precisely this size of quartertone obtains, and 2/7-comma at
50.276 cents. In a regular form of extended 2/7-comma, or a modified
form such as Zest-24, the versatile 50-cent step can be used now as a
Vicentino-like meantone diesis (although other features of a 1/4-comma
or 31-EDO division do _not_ obtain), now as a small neomedieval
cadential semitone in cadences involving interseptimal intervals --
and also, in Zest-24, as a very "narrowly tempered" variation on a
usual 28:27 semitone at 63 cents in certain septimal modes.
The 33:32 diesis at 53.273 cents represents the difference between a
12:11 neutral second at 151 cents and a 9:8 tone at 204 cents; or a
4:3 fourth and a superfourth at 11:8 (498 and 551 cents). I recall the
opinion of one Turkish music theorist that this is the smallest
recognized "semitone" in some Turkish styles, which might reflect a
perception similar to that of Ivor Darreg. It is narrower than the
small 28:27 semitone by 896:891 or about 10 cents, so that in a tuning
system like Zephyr-24 both 28:27 and 33:32 occur and may be used more
or less interchangeably.
In 22-EDO, the regular diatonic semitone (or literally quartertone) at
54.545 cents serves in effect as a tempered form of 28:27 in a
septimal context, for example as the difference between the regular
near-9:7 major third at 436 cents and the perfect fourth at 491 cents;
or the large major sixth at 927 cents and the smallish minor seventh
at 982 cents, the latter about 13 cents wide of 7:4 but still close
enough to be nicely within its "septimal suburbs." Some people find
this small a regular diatonic semitone quite pleasant, and others more
problematic.
A large diesis closely approaching the realm of usual small semitones
is 91:88 at 58.036 cents, narrower than the septimal 28:27 by only a
comma of 352:351 or about 5 cents. This step occurs in Zephyr-24, for
example, and is almost exactly approximated by the interval of 58.090
cents in George Secor's HTT-29 and some derivative variants. The
"quasi-diesis" of the Peppermint-24 system, based on the regular
Wilson/Pepper temperament with fifths at about 704.096 cents, is a
very comparable 58.680 cents. In a neomedieval style, these 58-59 cent
steps invite use much like the slightly larger 28:27.
By this point we are near the border area between "dieses" and more
usual small semitones, possibly located somewhere around 60
cents. George Secor has suggested that one very attractive range for
melodical optimal semitones may be found around 60-80 cents, with the
28:27 semitone or thirdtone of Archytas as a fine example.
Having quickly surveyed the interval region of 0-60 cents, we should
note that especially with inharmonic timbres such as those used in
Javanese or Balinese gamelan, the smaller or "comma-like" intervals of
this region can very effectively play the role of complex and
shimmering unisons. Thus in Zephyr-24, the intervals of 144:143 or 12
cents, 78:77 or 22 cents, and 64:63 or 27 cents (the first and last
considered above) very nicely serve this purpose in a gamelan style.
-----------------------------------------------------
11.1. Ultramajor sevenths and the "perioctave" region
-----------------------------------------------------
Our exploration of the territory between a pure 1:1 unison and a usual
small semitone at about 60 cents or large suggests a similar survey of
the region between a large major seventh at around 1140 cents and a
2:1 octave. Here 1140 cents or so, a slightly wide tempering of the
septimal major seventh at 27:14 or 1137 cents, seems a likely place
for the lower border of this region. For the sake of symmetry, the
region of 1140-1200 cents is shown in the first portion, and the
region of 1200-1250 cents on the other side of 2:1 in the second
portion.
88- 22-
cET EDO 64:33 25-EDO 96:49 63:32 2/7-comma 143:72 2:1
|----|----|-----|------|-----------|------|-------|--------|-------|>
1140? 1144 1145 1147 1152 1164 1173 1180 1188 1200
2:1 288:143 128:63 88-cET
<|--------|------------------|---------|.........................|
1200 1212 1227 1232 1250?
Starting at 1140 cents, we soon a very interesting xentonal interval:
the 1144 cents of Gary Morrison's 88-cET, where steps of 88 cents are
used in an equal temperament without repetition at the 2:1 octave.
This interval, while it might play the role of a very large and active
major seventh in a more conventional setting, has been optimized for
an aurally "consonant" or blending quality in certain timbres designed
by William Sethares, who showcases Morrison's tuning as part of his
_Xentonality_ CD.
In 22-EDO, we find a regular major seventh at 1245 cents -- an octave
less the small minor or "ultraminor" second at 55 cents -- which lends
itself, for example, to a very effective oblique resolution ascending
by this small semitone to the 2:1 octave, a la Leonin or Perotin in
the style of Gothic Europe using the Pythagorean major seventh and
semitone at 1110 cents and 90 cents.
The interval of 64:33 at 1147 cents, equal to a 2:1 octave less the
diesis or very small semitone at 33:32, can likewise serve as an
"ultramajor seventh," more or less interchangeable with the septimal
27:14, as may happen in Zephyr-24 (which includes all these ratios).
In 25-EDO, the ultramajor seventh available at 1152 cents is an
especially striking example of this neomedieval effect, resolving to
an octave via the step of 1/25 octave or 48 cents, an "ultraminor
second" indeed! This seems to me a distinctively 21st-century sound
with 12th-13th century roots.
The ratio of 96:49, or 1164 cents, is equal to an octave less a
septimal diesis at 49:48. It occurs, for example, in Robert Walker's
earlier noted tuning if one plays the interval formed by two notes
located at a 7:6 minor third and 16:7 major ninth (8:7 plus a 2:1
octave) above the modal center. I find that this interval can indeed
have the effect of an "ultramajor seventh" if it resolves obliquely to
the octave via a 49:48 or 36-step step in the upper voice; and has a
very curious impression, possibly something between a major seventh
and a kind of tempered octave, when it contracts conjunctly to a
fifth, each voice moving by an 8:7 step.
Moving further along the spectrum, at around 1267-1270 cents we enter
the domain of intervals which may be defined as an octave less some
comma of up to 30-33 cents. The ratio of 63:32 (an octave less 64:63)
or 1173 cents can, as mentioned above in the discussion of that
septimal comma, be used very tellingly to make 64:63 at 27 cents
sound as a discrete melodic step, although somewhat different from
a "semitone."
In certain gamelan-like timbres, for example, an interval of around
63:32 can be used as a complex octave with a pleasantly shimmering
sound.
This kind of treatment is also attractive for ratios closer to 2:1
such as 143:72 at 1188 cents in Zephyr-24, a small octave narrower
than 1200 cents by the characteristic comma of 144:143 or 12 cents
discussed above.
The second portion of the diagram above, starting at 1200 cents, shows
a few possible ratios for wide octaves in gamelan-like situations, for
example. Thus, in Zephyr-24, 288:143 at 1212 cents is wider than 3:2
by 144:143; and 128:63 at 1227 cents by the septimal comma at 64:63.
A memorable instance of a wide octave is the 1232-cent interval found
in Gary Morrison's 88-cET, and optimized for aural consonance by
William Sethares as part of his artful exploration of this system.
Indeed, Sethares has designed timbres where a ratio as large as 21:10,
or 1284 cents (a middle 21:20 or pental-septimal semitone plus 2:1),
has "octave-like" qualities.
Jacky Ligon has suggested that narrow or wide octaves treated
musically in a role analogous to 2:1, more or less, might typically
range about 50 cents on either side of 1200 cents. Here, as the octave
variations favored for instruments such as xylophones or metallophones
in many world musics, and also the computerized timbres of Sethares,
suggest, it is difficult to draw boundaries but most edifying even
briefly, as here, to survey some of the terrain.
--------------------------------------------
12. The equable heptatonic and its heartland
--------------------------------------------
To conclude our survey of the interval spectrum and its regions, we
now come to a special kind of region which has a fairly clear
heartland located at around 160-182 cents, but no clear borders: what
is here termed the "equable heptatonic."
The basic pattern of an equable heptatonic style of intonation is to
divide the octave into seven not-too-unequal steps which thus average
at around 1/7 octave or 171 cents. In one variety of equable
heptatonic tuning reported as common in Thailand and in some African
traditions, for example, these steps are often consistently within a
relatively few cents of 171 cents, or a precise 7-EDO. In other
varieties, however, along with some steps in this "heartland" of
around 160-182 cents, say, there are others which may range from
middle neutral seconds on the lower end to around a 9:8 tone on the
upper end, say around 140-200 cents.
Tunings of both varieties may be found in the Scala scale archive, for
example. Thus one Burmese tuning reported by von Hornbostel
(burma3.scl) has a range of about 163-182 cents, all within the
equable heptatonic "heartland"; and a different tuning (burma.scl) a
range of about 154-193 cents, also encompassing what we might call the
realms of middle-to-large neutral seconds and small tones. Likewise,
while the xylophone tunings, for example, of some African traditions
may have steps consistently in the heartland range, South African
composer Kevin Volans reports a scale (volans.scl) with steps ranging
from 140 to 200 cents. A Chopi xylophone tuning from southern
Mozambique as reported by Andrew Tracy similarly appears to have steps
with a range of around 144-199 cents.
Since in Section 4 we defined the range of the major second region as
about 180-240 cents, and in Section 6 a neutral second region of
around 125-170 cents, our heartland of the equable heptatonic nicely
situates the intermediate range of 170-180 cents in a rightful home of
its own. Here is one possible map of this heartland:
CMed
11:10-10:9 55- 256: 65536:
11:10 54:49 7-EDO 21:19 EDO 231 59049 10:9
<|------------|--------|---------|--------|------|-----|-----|-----|>
160 165 168 171 173 175 178 180 182
The lower portion of this range coincides with the domain of large
neutral seconds (around 160-170 cents). Thus starting from 160 cents,
we soon encounter Ptolemy's 11:10 at 165 cents, and then the septimal
chromatic step at 54:49 or 168 cents (the difference between a small
7:6 minor third and large 9:7 major third, at 267 and 435 cents, for
example). At 171 cents, or more precisely 171.429 cents (171-3/7
cents), we have the 7-EDO step dividing the octave into 7 equal
parts.
A ratio quite close to this is 21:19 at 173 cents, the classic mediant
of the 11:10 neutral second and the small 10:9 tone.
Interestingly, 55-EDO, a tuning system close to 1/6-comma meantone
sometimes proposed in 18th-century Europe as a useful model for
intonation, has a 175-cent step which makes available a fine equable
heptatonic division of the octave with six steps of 175 cents and one
smaller step of 153 cents.
The ratio of 256:231 or 178 cents occurs in Zephyr-24. Slightly larger
is the complex Pythagorean ratio of 65536:59049, equal to twice the
diatonic semitone of 256:243 or 90 cents, or 180 cents, and more
precisely to around 180.450 cents.
This interval is located at around our lower border of 180 cents for a
small major second or tone, a category exemplified by the pental ratio
of 10:9 or 182 cents.
While equable heptatonic tuning systems have sometimes been referred
to above rather freely as "scales," they are often used as gamuts or
collections of notes from which pentatonic modes, for example, may be
selected. In Southeast Asian traditions, pieces may shift within this
gamut from one modal pattern or center to another, a process known by
the ancient Greek term of _metabole_ orginally used to describe an
analogous change from one genus or modal center to another.
In defining a region of minor sevenths at around 960-1025 cents
(Section 4), and of neutral sevenths at around 1030-1075 cents, we
likewise left a small gap at 1025-1030 cents which can now find its
own home in an equitable heptatonic heartland region centered at the
reference point of 6/7 octave or 1029 cents.
59049: 55-
9:5 32768 EDO 38:21 7-EDO 49:27 20:11 51:28 31:17
<|-------|-------------|-------|------|-------|------|------|------|>
1018 1020 1025 1027 1029 1032 1035 1038 1040
Here we begin with the large or pental minor seventh at 9:5 or 1018
cents, with the slightly larger Pythagorean interval of 59049:32768
or 1020 cents (an augmented sixth, e.g. Eb-C#, equal precisely to five
9:8 or middle tones) in close proximity. At 1025 cents, we encounter
the 55-EDO interval of 1025 cents. With 38:21 at 1027 cents, we are in
the immediate vicinity of 6/7 octave at 1029 cents,
Moving to intervals slightly larger than this 7-EDO size, we have
49:27 at 1032 cents, a complex septimal ratio which plays a very
prominent role in the "aaron" scale designed by Gene Ward Smith in
honor of composer and performer Aaron K. Johnson (available in the
Scala archive as pipedum_12k.scl).
By around 20:11, we are entering a portion of the equable heptatonic
territory which is also a typical domain for small neutral or
supraminor sevenths. The ratio of 51:28 or 1038 cents is equal to the
small 17:14 neutral third, emblematic of its subregion (Section 3),
plus a 3:2 fifth. Our diagram concludes at 1040 cents, with 31:17 an
integer ratio of almost precisely this size (1040.080 cents).
As with the equable heptatonic "heartland" around 171 cents, so with
that around 1029 cents, actual tunings in this style in different
world musical traditions may either remain within these precincts or
also include considerably smaller or larger intervals. The picture
becomes yet more rich and intricate when we consider that octaves
somewhat smaller, or larger, than 2:1 at 1200 cents may be used in
some of these traditions.
Some tuning frequencies for a xylophone of the Chopi people of
Southern Mozambique, as reported by Andrew Tracy, show octaves with a
range of around 1174-1223 cents. Intervals of a single step have a
range of about 144-199 cents, and those in the general vicinity of
6/7 octave a range of around 1003-1056 cents.
Such a brief discussion of the equable heptatonic can only begin to
suggest the diversity of world musical traditions and tunings fitting
this general and often quite flexible approach. If this section has
done so, highlighting "heartland" ranges of the interval spectrum
especially characteristic of this approach while emphasizing how
adjoining regions of the spectrum are often embraced in equable
heptatonic styles of intonation, then it has served its purpose.
--------------------------------------------------------
13. Summary of suggested interval regions and subregions
--------------------------------------------------------
The following table attempts to summarize the interval categories
proposed above by surveying the spectrum from a 1:1 unison to a 2:1
octave. Section numbers are included to show where each category
receives its main discussion.
Note that this table presents a crude although convenient overview
because, for "the sake of simplicity," it overlooks some basic and
vital points presented in the text. Thus a "comma" of up to 30 cents
or so might often serve as a kind of complex unison in a style such as
gamelan, and likewise an interval in the range of 1170-1200 cents,
for example, as a complex octave.
As developed in the text (Section 11), the line between a "comma" and
a "diesis" might be drawn at a fuzzy border region somewhere around
30-33 cents; this table uses the rounded value of 30 cents.
Pure unison (1:1) 0 cents
Commas 0-30 cents (Section 11)
Dieses 30-60 cents (Section 11)
Minor seconds 60-125 cents (Section 5)
small 60-80 cents
middle 80-100 cents
large 100-125 cents
Neutral seconds 125-170 cents (Section 6)
small 125-135 cents
middle 135-160 cents
large 160-170 cents
Equable heptatonic
(heartland range) 160-182 cents (Section 12)
Major seconds
(or tones) 180-240 cents (Section 4)
small 180-200 cents
middle 200-220 cents
large 220-240 cents
Interseptimal
(Maj2-min3) 240-260 cents (Section 9)
Minor thirds 260-330 cents (Section 2)
small 260-280 cents
middle 280-300 cents
large 300-330 cents
Neutral thirds 330-372 cents (Section 3)
small 330-342 cents
middle 342-360 cents
large 360-372 cents
Major thirds 372-440 cents (Section 2)
small 372-400 cents
middle 400-423 cents
large 423-440 cents
Interseptimal 440-468 cents (Section 9)
(Maj3-4)
Perfect fourths 468-528 cents (Section 7)
small 468-491 cents
middle 491-505 cents
large 505-523 cents
Superfourths 528-560 cents (Section 10)
Tritonic region 560-640 cents (Section 8)
small 560-577 cents
middle 577-623 cents
large 623-640 cents
Subfifths 640-672 cents (Section 10)
Perfect fifths 672-732 cents (Section 7)
small 672-695 cents
middle 695-709 cents
large 709-732 cents
Interseptimal 732-760 cents (Section 9)
(5-min6)
Minor sixths 760-828 cents (Section 2)
small 760-777 cents
middle 777-800 cents
large 800-828 cents
Neutral sixths 828-870 cents (Section 3)
small 828-840 cents
middle 840-858 cents
large 858-870 cents
Major sixths 870-940 cents (Section 2)
small 870-900 cents
middle 900-920 cents
large 920-940 cents
Interseptimal 940-960 cents (Section 9)
(Maj6-min7)
Minor sevenths 960-1025 cents (Section 4)
small 960-987 cents
middle 987-1000 cents
large 1000-1025 cents
Equable heptatonic 1018-1040 cents (Section 12)
(heartland range)
Neutral sevenths 1030-1075 cents (Section 6)
small 1030-1043 cents
middle 1043-1065 cents
large 1065-1075 cents
Major sevenths 1075-1140 cents (Section 5)
small 1075-1100 cents
middle 1100-1120 cents
large 1120-1140 cents
Octave less diesis 1140-1170 cents (Section 11)
Octave less comma 1170-1200 cents (Section 11)
Pure octave (2:1) 1200 cents
--------------
14. Conclusion
--------------
This paper has sought to sketch out an approach to interval spectrum
regions which recognizes the musical beauty and value of each region
or subregion, and encourages the further exploration of this spectrum
whose multicolored aural hues provide many diverse palettes for the
musics of the world.
Margo Schulter
Sacramento, California, USA
mschulter@calweb.com
26 July 2010