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The Nature of Interval is the result of research begun after a series of compositions I produced in the mid-70's which were based on overtone formations in a manner similar what has become known as “spectral composition”. This revision of the paper is a somewhat abbreviated version from nearly 20 years later. It details some surprising results and suggests a number of research paths. With luck, I'll have the chance to document thoughts and follow up investigations.
The Nature of Interval is the result of research begun after a series of compositions I produced in the mid-70's which were based on overtone formations in a manner similar what has become known as “spectral composition”. This revision of the paper is a somewhat abbreviated version from nearly 20 years later. It details some surprising results and suggests a number of research paths. With luck, I'll have the chance to document thoughts and follow up investigations.
The Nature of
Interval
Introduction
Interval lies at the
core of music. It is the most basic concept of all musical styles,
the most vital - perhaps the only - universal of musical systems and
cultures. We define style by the inclusion, usage, and function, and
even the exclusion of interval. It is the central if hidden topic of
all analysis. It is on the basis of interval that we identify the
specifics and background generalities of individual works, composers,
styles, eras, and cultures.
Interval is said to
be a measure of distance expressed as letter names, as pitches in
notation, or through one of several and various mathematical
expressions. Each interval has its own sound, immediately evident
even to those with no inkling of its measure. This sound, quality, or
effect is how we as musicians learn to recognize interval. Yet for
all our explanations of this most vital component our knowledge of
interval remains largely anecdotal. We learn that a sound is, say, a
perfect fifth simply because “anyone
can tell it is.”
The most generally
accepted model of interval is that of ratios between partials of the
overtone series. However unassailable as science, however accurate as
mathematics, this view has no relevance to the practice or perception
of music. With the exception of octaves of the fundamental, no
partials of the series duplicates the realities of equal temperament
and, therefor, the “intervals”
they produce among themselves are similarly skewed. To take but one
example, in the case of the apparent “major
seconds” formed by the
8th and 9th partials and the 9th and
10th, the first is too narrow compared to the
equally-tempered seconds, and the latter too wide.
When we have an
interval we are hearing not the interaction of disembodied partials
but two tones, each of which is the fundamental of its own
projected overtone series. It is the interaction of these complete
series, their conflict and coincidence, that are responsible for the
unique sound and specific effect of an interval. That the overtones
of this “composite
overtone series” do not
duplicate equal temperament is of no import, as they are not being
compared to sounding fundamentals but only to other overtones.
All partials of the
series are not created equal. Lower partials are projected with more
energy and are therefor more audible participants than higher
partials. Therefor any results that involves these lower partials are
considered to color the results more strongly. Second, although the
series is theoretically infinite, those partials above the 12th
contribute little to the overall effect. Even the 7-10 partials often
require special performance techniques such as sul ponticello -
playing by the bridge of stringed instruments - in order for them to
participate in the effect of the tone. In addition, the partials of
higher sounding fundamentals quickly approach the limit of human
hearing. Therefor, only those results that fall within the first 12
partials of the lower fundamental of the interval are considered.
Finally, results that have octave support in their own series are
seen as significant.
Even with these
caveats, the data still requires a certain amount of interpretation.
Some results will seem almost intuitive, others more arbitrary or
argumentative. The definition of conflict and coincidence need some
amplification in this respect.
Conflict occurs
when overtones of the composite series lie within a half or whole
tone distance of each other. Coincidence is self evident. Any other
relations are considered neutral. In effect this employs an approach
that accepts the traditional definitions of consonance and
dissonance. This may seem an example of circular reasoning,
accepting that whole and half step distances create conflict even
before their composite arrays have been examined. If we had never
heard fundamentals forming an interval this might be a valid
criticism. But we have a knowledge and experience of interval. Our
study is aimed at understanding the factors that contribute to the
unique sound of each interval.
The following
discussion accepts much of the common wisdom regarding interval, as
is evident in the organization of perfect and imperfect consonances
and dissonances. A modern fashion suggest that “tension”
and “relaxation”
are better adapted to the modernist and atonal use of interval and
that consonance and dissonance are purely stylistic terms.
Ironically, champions of these terms scrupulously avoid such
imprecise and narrative referential vocabulary in all other topics.
Any number of factors can contribute to a more tense or relaxed
atmosphere; rhythm, tempo, dynamics, instrumentation, linear
direction, structural contrast, compositional procedure, pitch
context, and interval. In terms of interval alone, it is true that
some combinations are more tense, some more relaxed. These
perceptions are the direct result of the use of consonant and
dissonant intervals. Regardless of compositional style or
philosophic fashion, interval retains its unique sound. Tension is
the effect; dissonant intervals are the cause. To substitute the
effect for the cause is like renaming the rhino viruses as “the
sniffles.”
The Arrays
The
self-identifying stability of the unison and octaves is the result
of the perfect coincidence of each partial of the upper or added
series to the lower. The unison coincides at every placement , while
the octave shows coincidence on every other.
Both arrays exhibit
apparent conflict in the upper reaches, against the 7th
through 12th partials of the lower series. (All examples
employ C as the lower series.) As these conflicts exist even in the
series of a single tone it leads us to consider whether a pitch can
exhibit conflict with itself.
At first glance and
based on experience, one might be tempted to discount the effect of
these upper conflicts. We have learned to accept a single pitch as
self-defining and stable and as these higher partials possess little
energy, they seldom affect our perception. Yet special techniques
can imbue a pitch with a more “conflicted”
almost unstable sound, such as performing sul ponticello - bowing or
striking strings near the bridge. This particular effect emphasizes
the upper partials, providing the upper partials more energy with
which to project their conflict. It is undoubtedly due to the manner
in which different instruments distribute energy to their upper
partials that sets a disparity in the ear that allows us to identify
different instruments playing a unison. It is the agreement among
the partials of all instruments of a section of strings that allows
us to hear them as a unit.
In the case of the
unison or a single pitch this conflict may remain merely incipient
but it is a vital component of the octave. As the partials of the
octave above are in perfect coincidence at every placement with
every other partial of the lower, without the conflict of the upper
register it is very likely that the upper tone would be totally
absorbed in and hidden by the lower. Indeed, this is the effect of
some moments in the literature or at least particular performances.
The perfect fifth
CG coincides throughout the range on G, except for the first -
fundamental - instance. (It is this point of the array exclusively
that ratios most clearly describe.) The stable effect is produced by
the extra energy with which the coincident partials are projected.
The first conflict occurs at CdE (partials of C are shown in upper
case, as well as those that coincide with C) and there is complete
conflict in the upper range of C versus the midrange of G.
In fact, the very
nature of the series is in conflict with itself in this range as we
saw with the unison. It should not be surprising then, that all
intervals exhibit complete conflict in this register. Even so,
distinctions can be made. As lower partials possess more energy than
higher ones, the b will not project its conflict as strongly as
would fundamentals. Of course, under the right conditions this
conflict can be heightened.
The conflict in the
lower register is more significant. The 4th overtone of
G, d, is in conflict with the 4th and 5th of
C, C and E. As has been noted, without some significant conflict the
upper pitch is easily subsumed into the lower. This exact sequence
of pitches is repeated an octave higher, where there is now
coincidence on D. The octave recurrence with coincidence on D may
function to soften the conflict of the lower register.
The perfect 4th
has always been regarded as mutable in its effect, consonant and
stable in some situations or styles, dissonant and unstable in
other. The composite array of this interval does much to explain
this dichotomy. While there are three points of coincidence, 2 are
on octaves of C and one on the highest partial G, none involve the
fundamental of F. In fact, the octaves and all partials of the
fundamental, except the first, are in conflict at every point
throughout the range. Most significantly, the first conflict occurs
between the 2nd partial of F and 3rd of C,
much lower and therefor more influential than the first conflict in
the perfect 5th array.
The P4 and P5 are
considered to be an inversional pair, retaining identity whether
expressed as one interval or the other. Yet the arrays identify a
clear and essential difference between the two, a difference that is
in keeping with our experience and is evidenced in caveats
surrounding the use of the P4. This has profound implications for
one of the most basic concepts of interval, that of the identity of
intervals under inversion.
The imperfect
intervals also show results that are at odds with traditional
notions of inversion. The common view is that minor intervals invert
to major, and the sum of the original interval and its inversion
equal 9. (*In set theory the interval class and its inversion, each
a measure of half-steps instead of letter names, equals 12.)
Therefor the m3 and M6 are seen as linked as are the M3 and m6. The
arrays suggest otherwise.
If we compare the
thirds we find more significant conflict in the lower register of
the m3 array. The m3 coincides at four placements, none of which
duplicate either fundamental. The M3 has only one low coincidence
but it is an octave of the E. Its other three common partials are in
the conflicted incipient (upper) register. There are a preponderance
of “neutral”
placements, especially in the lower registers. This allows each
fundamental some range in which to establish itself without
conflict.
The 6ths
similarly show one array having a single lower placed conflict, and
the other having 2 significant conflicts. One shows lower
coincidence than the other. Surprisingly, it is the M6 that compares
most closely with the M3 and the m6 with the m3. Other
correspondences reinforce this pairing. Both major intervals are
coincident first and lowest on E. The upper fundamentals of the
minor intervals, Eb and Ab, project octaves that are in half step
conflict with that of the lower C fundamental: Eb vs. E and D, Ab
vs. G]
The seconds and
sevenths conform more closely to the traditional inversional model.
The inversions of the seconds essentially duplicate the conflicts at
higher placements. That is, where the major second produces whole
tone conflict from the first placements on., the m7 produces whole
tone conflict beginning against the octave C. Yet, there is also an
essential difference between the second and seventh as well, again
one that is confirmed by perception and experience. Where the 2nds
conflict at every placement the sevenths allow the first and third
partials of C to project themselves with no conflict. This result
explains the oft cited perception that widely spaced dissonance
somewhat soften the effect.
The remaining
interval, the tritone has been considered in some systems to be the
most dissonant of all. It may be the most destabilizing in some
tonal contexts, it is true, but its natural occurrence in that same
tonal system as the 4th and 7th scale degrees
is necessary to establish a key. In post-tonal styles its
description echoes that of the P4; more restless under some
circumstances than others.
The tritone array
clearly identifies all these descriptions as fictions based on
stylistic use. In fact, the tritone produces only moderate conflict.
It has as much in common with the minor imperfects as the major
2nd/minor 7th. Like the minor 6th it produces no conflict
against the lowest 2 C's.
Like all imperfect intervals it conflicts in the next 2 placements.
In fact, the tritone seems to lie well directly between the effect
of the imperfects and dissonances.
III Implications
and Applications
The Myth of
Inversion and the Gradation of Dissonances
The composite
overtone arrays are a visible reflection of reality. As such, they
impact every level of the art of music. In some cases they confirm
and explain perceptions. In some they suggest revisions to our
thinking. In others they clarify, providing us with tools for
greater compositional control.
Register,
spacing and the arrays
The overtone arrays
confirm some common wisdom, clarifies perception and challenges some
long held beliefs. For one, they provide a clear explanation of the
effect spacing and register has on interval., Widely spaced
intervals are known to present a different effect than the same
interval placed in close position. Dissonant intervals are said to
be less sharp, consonant intervals disembodied, and widely separated
lines appear more independent and self-involved. This is simply due
to the fact that the lower pitch is given an expanded range in which
to establish itself with no conflict and that the upper tone will
project fewer partials within the range in which partials affect our
perception. For instance placing a higher fundamental pitch in the
area of the 8th overtone of a lower, the third octave of
the array, will essentially remove all possibility of either
conflict or coincidence in upper partials as the 2nd
overtone of the upper pitch occurs above the highest effective
partial of the lower pitch. Conversely, closely spaced intervals are
perceived as more dense, linked or related due to the rich
environment in more resonant, influential registers.
Intervals, whether
close or widely spaced, are also affected by register. Generally,
the higher the tone the greater the possibility that even the
partials of the mid-range will be beyond the limit of human hearing.
When 2 pitches are projected in high register the conflict and
coincidence of the interval may be too high to be of any influence.
Lower pitches, on the other hand, will project more of their
partials in the range of human auditory response with enough energy
to influence the sound. This accounts for the thick or muddy effect
of even the most consonant intervals in the lower pitches.
The Myth of
Inversion
This may appear at
first glance a surprising result, yet we have always been aware of
the problem of the P4, consonant under certain circumstances only.
Schoenberg may have intuited non-invertiblilty when he said that
inversion itself is a transforming operation. Even so, the concept
of identity under inversion is so ingrained, so prevalent, so
demonstrable throughout the literature that proof and explanation is
necessary.
The proof is in the
ear and relatively easy to demonstrate. We are taught to accept the
m6 as the inversion of the M3. We learn to distinguish these
intervals by ear-training, often by finding melodies that contain
the interval in question. “My
Bonnie” opens with a
major sixth. “MY Bonnie”
in minor teaches us the minor 6th. Compare this interval
with the first 2 notes of “Kumbaya”,
a M3. It would take willful manipulation and a very disciplined ear
to hear the two examples as reflections of one another.
The traditional
inversional pairs are only true for systems that restrict pitch
choice by some structure or system, such as membership within
“keys.”
If we look for a certain interval above a key member, the key itself
provides only one choice. In free chromaticism, atonality, etc,
traditional inversion must be seen as a transformational device
based on its changed effect and the choice to emphasis pitch rather
than the effect of interval. This is most clearly a vital concern
for the imperfect consonances and the P 4/5/ When all tones of the
chromatic are available we can choose that interval that most
closely retains the effect, linking by quality - m3 to m6 and M3 to
M6 - rather than the retention of pitch and the resultant change of
effect.
This clearly
defines the pitch bias of traditional views of interval: it is not
the interval that is important, it is the re-spacing of pitch. When
we choose to retain pitch identity under inversion we lose
intervallic identity. In that interval is the basis of music, a
theory that strips inversion of pitch definition and considers
interval first is not only possible but vital to the control and
focus of chromatic styles.
The Gradation
of Dissonance
Before we take up
this topic we must return to the very terms consonance and
dissonance. The prefixes - con and dis - appended to “sonance”
are translated as sounding together and sounding apart. Further we
have accepted a stylistic example as part of that definition, that
consonances are members of traditional triadic tonal harmonies, the
triads, and dissonances are all “non-chord
tones.” If we disavow
tertial chord structure as the only basis for chord construction, if
we allow any of the full panoply of intervals to take part in
chordal construction, or if we focus entirely on linear procedures
either contrapuntal or serial, then what defines a non-chord tone?
If we divorce
ourselves from this stylistic view and focus instead on the
composite overtone arrays we might clarify the definition. Now
“sounding together”
points to those arrays in which there is more significant or
strongly weighted coincidence. “Sounding
apart” or dissonance
then, refers to arrays in vital conflict.
In such a view the
order of increasing conflict is unison/octave, P5, M imperfects, m
imperfects, P4, tritone, m7/M2, M7/m2. The P4 remains the “kicker.”
In pitch-based intervallic use it will function as the inversion of
the P5 with the usual caveats. In terms of conflict it more properly
occurs between the imperfect consonances and the tritone.
The Effect of
Interval on Complex Structures
In Basic Atonal
Counterpoint (SAF Publications; rev 2001) I detail the
use of the gradation of dissonance in categorizing and controlling
mixed intervallic atonal harmonic structures. Each interval was
given a numerical expression that corresponds to its placement in
the intervallic ordering. I chose to measure
dissonance/tension/conflict and so assigned the greater dissonances
higher numbers. In this system each contiguous interval is noted and
added to all other contiguous intervals to produce an “intervallic
quotient.”
For instance, the
ordered collection C Db A B would produce a quotient based on the
minor second, minor sixth and major second. Reordering the pitches
as Db C B A would produce a more “dissonant”
quotient consisting of two contiguous major 7ths and a minor 7th.
Re-ordering a
single set for more or less conflict is relatively intuitive. Less
so is the comparison of the specific ordering of 2 dissimilar
collections or sets of dissimilar size. For instance, the structures
C Ab G C# D# B and C B F C# G F# contain m6, M7, t, M2. M7 and M7,
t, M6, t, M7, respectively. Merely noting these resultant intervals
as Howard Hanson does in Harmonic Materials of Modern Music,
is not sufficient to identify the effect of interval on the
combinations of these two structures. Both contain 4 dissonances and
one consonance. The M7's, two for each, may cancel out , as does one
tritone. Now what is the effect of the intervals remaining, m6 and
M2 vs. M6 and tritone.
If the intervals
are given the weights, or factors, shown in the following example
the two structures can be compared on the basis of their dissonance
quotients. This is a traditional gradation that accepts inversion,
as it was employed in Basic Atonal Counterpoint. An
alternative series that takes into account the results of the
composite arrays would differ only in that the major imperfect 3/6
would be linked, as would the minor 3/6 and that the P4 would be
placed between m3/6 and tritone. All values would be then adjusted.
P4/5=1 M3/m6=2
m3/M6=3 Tritone=4 M2/m7=5 M2/M7=6
Factoring the m6 as
2, the third in line of gradation, and M2 as 5 yields a combined
value of 7. The M6 (2) plus tritone (4) yields 6. Based on a
comparison of non-reciprocal intervals alone it is clear that the
first benefits from more conflict that the second. An examination
that includes entries for every contiguous interval gives a more
nuanced result, and is better suited to compare structures built of
different numbers of pitches.
The complete value
of the first structure is 2+6+4+5+6=23 and the second 6+4+2+4+6=22.
When viewed as the result of five intervals the distance which is
numerically the same as arrived at above is seen to be a much
slighter difference; 23/5=4.6, 22/5=4.4.
Constructions of
different size can be compared in this manner, simply by dividing
the result by the number of intervals in the structure. C Eb B Bb
(3+2+6=11/3=3.67) can then be compared to either of the above to
identify that this structure will affect a significant change of
tension, although still relatively subtle, if used in a progression.
Related
Topics
We have noted that
the overtone arrays explain the effect of spacing and register. A
further study might apply this information to instrumental
performance. Each instrument has its own signature sound that is the
result of the suppression of some partials and the projection of
others. For instance, a unison produced by two instruments - one
suppressing the equal numbered partials, the other suppressing the
odd - will have a different effect than two that both suppress the
same partials.
Further, specific
techniques and types of attack,, even dynamics, can affect the
composite overtone structure,. A study that investigates this
application of the arrays can bring a degree of certainty to both
the teaching and application of instrumentation and orchestration.
Once upon a time
theorists postulated that the overtone series displayed a
progression that could possibly account for the development of
Western musical styles. The progression of intervals formed by
successive overtones from perfect to triadic and 7th
chord structures, to whole and half step collections seemed to
duplicate the development of organum, the harmonic common practice
system, and even post-tonal construction. It was predicted that
micro-tonal music would be the next stage of development. This has
yet come to pass as a common aspect of training or practice, unless
one includes the inability of many performer to tune with any degree
of accuracy, or those styles where mere approximation of pitch is
acceptable.
Yet, one cannot
help but be struck by the apparent similarity between some composite
overtone structures and tonal - if not atonal - practice. For
instance, the perfect 5th array places the “dominant
7th”
of the upper fundamental G against tones of the lower C in a manner
that very much suggests the voice leading of G7 to C. The tritone
array produces coincidence on upper partials that combine with the
fundamentals to form a French 6th - C E F# A#. If the
composite array affects our perception of interval because of the
presence of coincidence and conflict, might it not also have
influenced music in other ways, as well ?
Others may find
applications for ethnic studies, the physics of music, the design of
instruments or concert halls. Electronic music seems uniquely
adapted to explore and benefit from this material. In the long run,
the overtone arrays have implications for all. It is hoped that
others will employ these results, to further our understandings and
abilities, and to bring us closer to a true vision of the invisible
art of music.
65.4064 130.9605
C
C
G C
E
G
C
C G C
E G Bb
C D E F#
G
32.7032 130.8128
---------------------------------------------------------------------------------------------------------------------------------------------------------
43.6535 97.9988 195.9976 293.9964 391.9925
---------------------------------------------------------------------------------------------------------------------------------------------------------
43.6535 97.9988 195.9976 293.9964 391.9925
G
G D
G B
D F
G
C
C G C
E G Bb
C D E
F# G
8.1096 196.2192 294.3288 392.4384
------------------------------------------------------------------------------------------------------
43.6535 130.9605 261.921 392.8815
8.1096 196.2192 294.3288 392.4384
------------------------------------------------------------------------------------------------------
43.6535 130.9605 261.921 392.8815
F
F C
F A
C Eb F
G
C
C G C
E G Bb C
D E F#
G
130.8128 261.6256 392.4384
------------------------------------------------------------------------------------------------------
41.2034 164.8136 288.4238 329.6272 370.8306
130.8128 261.6256 392.4384
------------------------------------------------------------------------------------------------------
41.2034 164.8136 288.4238 329.6272 370.8306
E
E B
E G# B
D E F#
C
C G C E
G Bb
C D E
F# G
163.516 294.3288 327.032 359.7352
---------------------------------------------------------------------------------------------------------
38.8909 194.4545 233.3454 388.909
Eb Eb Bb Eb G Bb Db Eb F G
163.516 294.3288 327.032 359.7352
---------------------------------------------------------------------------------------------------------
38.8909 194.4545 233.3454 388.909
Eb Eb Bb Eb G Bb Db Eb F G
C
C G C
E G Bb
C D E F# G
196.2192 228 .9224 392.4384
-------------------------------------------------------------------------------------------------------------
55 165 165.495 330 385 386.155
196.2192 228 .9224 392.4384
-------------------------------------------------------------------------------------------------------------
55 165 165.495 330 385 386.155
A
A
E
A
C# E
G
C
C G C
E G Bb
C D E
F# G
163.516 327.032 392.4384
-------------------------------------------------------------------------------------------------------------
51.9131 259.5655 363.3917
163.516 327.032 392.4384
-------------------------------------------------------------------------------------------------------------
51.9131 259.5655 363.3917
Ab
Ab Eb
Ab
C
Eb Gb
Ab
C
C G C
E G Bb
C D E
F# G
261.6256 359.7352
------------------------------------------------------------------------------------------------------------
36.7081 256.9567 293.6648 330.3729 367.081
261.6256 359.7352
------------------------------------------------------------------------------------------------------------
36.7081 256.9567 293.6648 330.3729 367.081
D
D A
D F# A
C D E
F# G#
C
C G C
E G Bb
C D E
F# G
261.6256 294.3288 327.032 359.7352
-------------------------------------------------------------------------------------------------------------
58.2704 233.0816 291.352
261.6256 294.3288 327.032 359.7352
-------------------------------------------------------------------------------------------------------------
58.2704 233.0816 291.352
Bb
Bb F
Bb
D
F Ab
C
C G C
E G Bb
C D E
F# G
228.9224 294.3288
------------------------------------------------------------------------------------------------------------
34.6478 381.1258
228.9224 294.3288
------------------------------------------------------------------------------------------------------------
34.6478 381.1258
Db
Db Ab Db
F Ab Cb
Db Eb F
G
C
C G C
E G Bb
C D E
F# G
392.4384
------------------------------------------------------------------------------------------------------------
392.4384
------------------------------------------------------------------------------------------------------------
61.7354
370.4124
B
B F#
B
D# F#
C
C G
C E G
Bb C D E
F# G
359.7352
------------------------------------------------------------------------------------------------------------
46.2493 231.2465 323.7451 369.9944
359.7352
------------------------------------------------------------------------------------------------------------
46.2493 231.2465 323.7451 369.9944
F#
F# C#
F# A#
C# E
F# G#
C
C G
C E G
Bb
C D E
F# G
228.9224 327.03 359.1352
-------------------------------------------------------------------------------------------------------------
©
copyright S. A. Funicelli,
2003; all rights reserved.228.9224 327.03 359.1352
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