Seattle Washington, 1986-2002
Please download the MIDI enabled version of the programs Glass Bead Game.exe, Electric Monochord.exe and Interval Viewer which were used to make the illustrations for this paper. Use them while reading the paper to hear the musical intervals described.
The files have been off the site for a while (my apologies!), but I've now changed servers and they should be available again. I've had a few reports of download trouble with these files, so feel free to e-mail me to request mailed copies if you need to.
MIDI performance may vary depending on your configuration. I have had good luck with most (but not all) machines.
--Norm
The author presents a graphical tool for the visualization of consonance and dissonance, based on an acoustical model of the interaction of musical tones.
In 1986, an early version of this paper appeared in the journal 1/1. I have revised it since then, added more graphics, and a couple of interactive programs which will let you hear the intervals and chords discussed here. I've also corrected some errors, and in general brought the paper up to date. Since page count is no longer an issue, I've added some introductory material as well--most musicians have not had the time to study musical acoustics in any depth, and the same can be said for most acousticians and the study of music. For further study, there is a considerable library of good reference works on musical acoustics, especially "Horns, Strings, and Harmony" and "Fundamentals of Musical Acoustics", both by Benade.
For more advanced reading, see the new book "Tuning, Timbre, Spectrum, Scale" by Sethares (Springer-Verlag), which focuses on the harmonic consequences of non-harmonic partials in musical tones, and is accompanied by an interesting demonstration CD.
This paper remains a brief introduction to the subject, and one that I hope will generate some discussion. I will continue to update sections as I am able-feel free to send questions and comments to me at sohl@sohl.com. Thanks for your interest!
Composers of every period have tried to expand their palette of compositional resources. This desire to continually re-invent music theory has shaped western music, and is one of the features that distinguishes our musical tradition. Composers in the 20th century have found themselves in a difficult position as they try to continue this effort-the compositional rules of the past have to a great degree played out, and new theories, techniques and technologies have yet to be proven completely valid (or emotionally satisfying) as compositional resources.
We are in the difficult position of developing new theories at the same time as trying to compose music. I like to think of our time as similar to that of the birth of polyphony, when composers such as Leonin and Perotin formalized an entirely new style of music based on the medieval improvisational art of descant. These composers discovered some of the basic elements of harmony and counterpoint and which still serve as the foundation of common- practice composition today. Starting from a strong philosophical tradition, these composers linked their theories to the absolutes of their time-theology, mathematics and astronomy, and it is only in the last century or so that this philosophical foundation has crumbled, and with it the basis for much of our music theory.
It's odd to think that in the middle ages, music was considered to be physical manifestation, almost a proof of spiritual truth. A medieval scholar using the monochord (a box strung with several strings tuned to the same pitch) could show how dividing the strings in small number ratios (1/2, 2/3, ¾, etc.) produced the basic musical intervals of the octave, fifth, and fourth. This was seen as evidence of the underlying pattern and order in the world of God's creation.
In our century, theorists have turned this logic around, using the collapse of historical theology and philosophy to discredit the idea of basic musical "absolutes" like consonance and dissonance. Ironically, modern studies of acoustics have given new evidence for the real existence of the same.
Current research seems to show that consonance and dissonance do exist as physical phenomena, and are to some degree constant across cultural boundaries. Most anyone, child or adult, will recognizes the octave, the fifth, and basic elements of rhythm without any particular musical training, just as they recognize simple visual patterns and colors. Not only do these musical patterns remain constant, but all of our popular music is based on them.
The challenge to the modern composer is not in using these elements-their use is obvious and perhaps too well understood. The challenge is in finding a new musical structures which are still based on these elements yet go somewhere new and interesting. The logical place to start the search for these new elements is by studying the basic elements of consonance, dissonance and rhythm.
In other words, I feel that compositional rules for harmony (and to a great extent melody and rhythm) can be deduced from a good basic understanding of consonance and dissonance. This seems like a reasonable place to start when looking for ways of expanding compositional resources for modern music. This paper is intended to provide some foundation for that search, and answer (as much as is possible) the following questions:
The next sections will discuss these subjects in greater depth.
Consonance means "to sound together", and it occurs in music when two musical tones are played together at particular intervals. It was well known in the middle ages that simple consonant intervals could be played on strings by stopping one string at ½, 2/4, and ¾ of its full length . It took until the 1800's for a reasonable explanation of why this should be so. The German scientist Hermann Helmholtz published "On the Sensation of Tone" where he suggested that these consonant intervals were caused by the alignment of the harmonics of the 2 musical tones being produced by the strings.2
Consonance is often now defined as the result of the synchronization of the partials of two or more different musical tones. Even though parts of Helmholtz's theory have been disproved, his general theory of harmonic interaction remains a useful and empirically sensible model for practical musicians, and a common starting point for current research in musical perception. There is still no entirely satisfying physiological explanation for the perception of consonance and dissonance.
There are an amazing variety of musical tones. A violin sounds very different from a tuba, and both sound very little like a flute. One major way that they differ is in their "spectrum". Most all noises are made up of many different frequencies, or partials, and these partials can be graphed to form a spectrum. The spectrum of a musical tone differs from random noise in that its "spectrum" for the most part falls into what a mathematician would call a harmonic series.
Spectrum of a musical tone.
In the preceding chart, the vertical axis of the graph is pitch (marked in octaves of c) and the horizontal axis is loudness of the harmonic partial.
Without going into any great depth ("Horns, Strings, and Harmony" by Benade is a good introduction to the subject) this structure gives musical tones the property of repeating over time, which allows our ears to perceive the noise as having a distinct pitch.
How is consonance related to harmonic partials? Let's return to the monochord for a moment. Imagine that we have two strings tuned to the same pitch. It should be obvious that all the significant partials of these two musical tones all align, and the two strings sound in tune.
Two tones at the same pitch.
If we slide the bridge of one string slightly higher, the strings begin to go out of tune, and the partials no longer align. This produces unpleasant "interference tones" or beats, which increase in speed as the two pitches move apart.
If this just continued to get worse as the distance between the tones increased, there would be no such thing as music! Fortunately, that is not the case. As the two musical tones get further apart, different harmonic partials come into alignment-the 4th partial of one tone aligns with the 5th partial of the other tone (creating the interval of a major third.).
Moving on, then the 3rd aligns with the 4th for the interval of the fourth.
The 2nd aligns with the 3rd, giving the interval of the perfect 5th.
and finally the 1st aligns with the 2nd to give the interval of the octave.
There can be many more partials in a musical tone than is shown here, and many of them (to a limit of 11) can be seen in the image below. Many of the strongest of these intervals are used as the basis of the 12 tone scale common to western music. It is important to remember that these are not the only consonances to be found in nature, and that the intervals of the equal-tempered scale are, in fact an approximation of these consonances, a compromise scale needed to make the mechanical instruments of the past possible. Today we have the ability to completely control musical instruments (and the associated spectrums of musical sound) and we may be nearing a time when these compromises are no longer needed.
Interactions of two musical tones, one fixed and one moving over the span of an octave.
A musical scale made up of these pure consonances, with no compromises is often referred to as a "just" scale. There is nothing particularly modern about this scale-it crops up throughout history and all over the world. The number of intervals found in a just scale is limited only by the number of harmonic partials included in the calculation.
(1) Intervals of a just scale with a harmonic limit of 6
(The white diamond on the left represents the note middle C, and the gray diamond on the right represents c at the octave. the black tick marks represent 12 tone equal temperament, and the red marks above the steps of a just scale.)
A just scale of a certain limit, for example the senarius (a limit of six), can be produced by physical harmonic interaction and graphed as shown in Figure 1. What this represents is a graph of two tones, one moving through the span of an octave against a tone held fixed at 1/1. As differing harmonics of the two tones match with each other a consonance occurs, and is marked with a diamond. The size of the diamond is inversely proportional to the highest of the interacting harmonics. This scale, with a limit of 6, contains all the consonances commonly used in western music.
(2) Intervals of a just scale with a harmonic limit of 7
If we increase the limit to 7, we get a couple of new intervals, including the interval of the flattened minor 7th familiar from blues and jazz. This interval lies outside of the 12 tone scale, and is used only by performers who have instruments that are not locked into a fixed tuning (slide guitar and voice, for example).
(3) Intervals of a just scale with a harmonic limit of 11
It should be obvious that calculations that include more harmonics will create scales with many more intervals than the common 12 tone scale can accommodate. Not all of these intervals are really useful, since they are often produced by weak partials and often occur in parts of the scale where they end up masked by dissonances created by other stronger partials (more on that later!). Still, there are situations where some of these intervals are very valuable, and can provide considerable new resources for composition.
(4) Chords generated by interactions of the 1st. 6 harmonic partials
Each axis of the graph represents the scale generated by one tone moving against a drone, as described above. Every possible combination of these tones--(one drone and two moving tones) within the span of the octave is represented as a point somewhere on the graph. The red lines mark places where the drone and one moving tone form a consonance. The diamonds mark a place where all three tones form consonances with each other.
In order to graph the interactions of more than two tones, additional axes can be added to the graph. For all possible three-note chords with a common bass tone, and again a harmonic limit of six, the graph in Figure 4 is produced.
Any possible combination of three tones within the specified range will be found as a point somewhere on this chart. A grouping of three tones is defined as a chord if all three members of the group share consonant relationships with each other. Obviously, the notes from the two axes share consonance with the bass tone, but in addition, they must be consonant with each other.1
(5) Chords generated by interactions of 7 harmonic partials
If the harmonic limit is expanded to seven (see Figure 5), we have, in addition to the previous chords, an incomplete seventh chord, and several other new chords unique to Just Intonation. Increasing the limit causes more chords to be displayed. Of course there is no reason to stop at 7:
(6) Chords generated by interactions of 17 harmonic partials
So far, we have been able to think of all points in this graph as representing the set of all chords sharing a common bass tone (1/1). This graph helps to define consonant chords, and can also be used to compose music involving two moving parts against a drone. It can also be used without a drone, to plot two tones playing in counterpoint, if we accept the just scale melodically as well as harmonically. In this case, the 1/1 is not sounded, except as a melodic interval, but it remains as the tonal center of the piece.
There s not enough space in one article to expand the graph much further, but one important aspect of chord structure can be added, that of inversions of chords with common roots.
Illustrate
If the graph is expanded to cover two octaves, with 1/1 at the center (now the root, not just the bass tone),second inversions will fall in the lower right as well as the upper left, since these two parts of the graph are mirror images of each other. Inversions of other chords which share a common element with the center tone can also be found.
Illustrate
The system can be expanded to represent four tones, that is three notes over a drone, by adding a third axis to the graph. This means either constructing a three-dimensional model, or as is more practical, printing a series of graphs representing the planes of the third axis. It should be remembered that all tones must share consonance with the drone or keynote, as well as with each other. This set of graphs can represent all possible consonant combinations of four tones with a particular harmonic limit. Most of these chords involved some form of doubling, but some contain four unique tones.
As was mentioned above, the 1/1 may act as a drone of even not sound, and simply imply the tonality of the piece. In this case the set of graphs can be used to describe the fully consonant chords of any scale degree in just intonation. As anyone who has tried to play conventional chords against a drone will know, certain common chords clash vividly - obviously these chords will not be marked as consonant in these graphs. This has some serious implications since one of these chords is the major triad to the step 3/2, the V chord, which is typically necessary for cadences. The V-I progression is perhaps the most powerful chord progression in music theory, so it might seem that something is wrong here. The problem lies not with the model, but in the assumption that the V must belong to the same tonality as the I. This cannot be true, since the V contains a tone (call it B natural in the graph of the 3/2) which fills within the region of greatest possible dissonance surrounding the 1/1.
In monophonic writing in the Western tradition, intervals this close to the unison are used to develop a strong drive to that unison. The tension of a strong dissonance near perfect consonance, drives the melody toward union. This explains part of the drive in the V-I progression. The tonality seems to change to allow the V chord in the same way a secondary dominant momentarily shifts the tonality of a piece of conventional harmony.
Illustrate
The graph above has been expanded to cover two octaves, and shows all chords with a span of two octaves. One interesting result of a physical definition of consonance is that the octave can no longer be considered invariant. Scale structure is similar across octaves, but more sparse as the distance between the drone tone and the moving tone increases, which creates more potential compositional resources.
We can add another axis to the four-tone graph to represent either a fifth tone or a tonal center with four tones moving. This is the construct necessary to fully represent four-part writing. The complexity of a four-dimensional graph limits its usefulness as a physical model but in the application of computer-aided composition, those restrictions need not apply. The ease of calculation and generation of chords and scales, and the extendibility of the system to any number of tones makes this method particularly suited for sophisticated computer-aided modeling, analysis, and composition.
Contrary motion, or counterpoint between the two moving tones can be seen as diagonal motion from lower right to upper left, where parallel motion runs contrary to that. Similar motion runs between these extremes.
This last experiment leaves us on theoretically shaky ground. Because the model of consonance is based on the actual physical interaction of specific sounds, it can only apply directly to the perception of consonance between simultaneous tones, to the vertical aspect of harmony, and not to horizontal structure or melody. We normally assume that the same rules apply to musical intervals when used melodically as when they are used harmonically. It is often possible to think of melodic structure in simpler terms however, since the human ear's resolution of pitch is much more accurate in terms of harmony than melody.
It is not uncommon to find historical references to strictly harmonic just intonation. For example the debates earlier this century concerning intonation in the playing of string quarters (chap. 9, Intervals, Scales, and Temperaments by Lloyd and Boyle). What concerned these writers was a harmonic Just Intonation, all intervals in tune with each other but within the melodic structure of the twelve-tone scale. A similar idea is advocated by Harold Waage in "The Intelligent Keyboard" (1/1, Vol. 1, No. 4).
The use of the just scale melodically does have some advantages. This creative use of dissonance as well as consonance seems to give music a strong sense of power and movement, and this is one case where it is useful to free melodic movement from the strict scale generated by a drone, or allow modulation of the tonal center. In analysis of conventional music based on this model of tonality, a much greater role must be assigned to these transitory modulations. This idea is not entirely foreign to modern conventional music theory, but its application has never been widespread.
Still, extremely complex and interesting polyphony can be written within a strict definition of tonality. There are other forms of V chords within the strict chart which, although they would be considered incomplete in traditional music, might still perform a dominant function. An example is the V7 lacking the third.
Dissonance is often thought of as the absence of consonance, and in a general way, this is true enough, but there is more to the structure of dissonance which may be of value to a composer.
Rhythm can be thought of as a similar phenomena to consonance as shown in the following experiment: Record a drummer playing a simple series of ¼ notes. As that recording is played back, artificially speed it up. As the beats pass the speed of 30 or so a second, the resulting sound will be perceived as a musical pitch. When the rate reaches 440 a second, the sound of concert A will result. Now repeat that experiment, with the drummer playing a pattern of 2 beats to 3 beats (two on three). When that pattern is played back at a higher rate of speed, the sound of the interval of the 5th is heard! This is not to say that rhythm and consonance are identical phenomena-they are not, but they are similar in some very interesting ways.
Calculating ratios, note names and pitches. <<to be completed>>
Calculating beats <<to be completed>>
Footnotes
1 It is interesting to note that although this graph is based on the physical structure of sound, there is no problem in generating the minor triad or other chords which have presented problems for previous "physical" theories (for example, see Tone, Levarie and Levy, page 189). In Figure 2, note that the major triad "M3", the minor triad "m3", and first and second inversions "1st" and "2nd" all appear.
2 Anyone can perform the following simple experiment to discover consonant intervals: Make a "monochord" by stretching two similar rubber bands over a cardboard box. Place pencils under the bands at each end of one side of the box, so that the bands are held above the box about ¼ inch. Adjust the tension of the bands so that the two "strings" of the monochord produce the same note. Now slide a pencil under only one string, so that the pitch moves up and down.
Illustrate
Last modified: 03/17/07
Reset 4/14/02