the Nature of Interval
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


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
       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
      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
      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
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
          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
          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
   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
             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
  Db                  Db              Ab         Db              F            Ab             Cb             Db            Eb            F                      G
C                    C                G            C              E             G             Bb           C             D              E               F#          G
              61.7354                                                                                                                                                                         370.4124
                 B                              B                             F#                         B                            D#                    F#
C                     C                G            C              E             G              Bb          C             D             E               F#          G
           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.