Music - The Language of (my) Soul
(Most recent change to this page: 12/27/97)
THE PHYSICS OF MUSICAL KEY
An Exploration
Preface
For years I had been curious about the relationships involved in music. Why do chords affect us the way they do, and why is it that a tune can be sung in one key (whatever key is), then transposed immediately into another, with a very obvious psychological effect? Why does changing one note in a three-note chord change the way it feels to us?While I was confident that this information was documented somewhere, I had pondered it, rather than researched it. I did ask several musician friends, but they didn’t seem to understand what I was getting at.
Little by little I collected facts, as well as some musical scores and instruction books. I graphed the frequencies of notes of different scales, and tried to see the relationships. For some reason it was important to me to work it out myself, rather than have the whole thing handed to me, although I would have snatched at somebody else’s explanation like a hungry animal.
The admittedly tentative result is here. Its imperfections are the result of assumptions I haven’t checked out yet, I suppose. My education is, undoubtedly, like that of most who are self-taught, full of holes. But this is basically correct, I believe, and I’ve had the satisfaction of solving the riddle.
The Basic Questions
Why do certain combinations of notes sound “right”? What is “key”?To talk about these things with any precision, it seems we have to talk physics. Not heavy-duty physics--and you don't have to know heavy duty math, either. I do refer to logarithms in order to illustrate (not to prove) the things I discovered. And I’ll define my terms as I go.
(Note: I've also written a little piece of fiction about this, for those who find their eyelids gettng heavy before they get to the end of this rather technical piece. Click here to start with the story.)
How to "See" Music
A musical “note” is a tone of a particular frequency.In a printed document, of course, we cannot provide examples that you can
hear. We are here using a standard graphic representation of sound waves to
illustrate our descriptions. The sound of a single tone is depicted as a sine
wave:
Figure 1. Sine wave of a pure tone
What we’re most interested in this description is the wave shape of a single cycle or two of a sound. A sine wave sounds smooth. An irregularly shaped wave has a distinctive sound that depends upon its shape, not as smooth as a sine wave. A square wave, that is produced by a rapidly switched back-and-forth electrical current, has a buzzing sound, depending upon its frequency. At a low-enough frequency, it is more like a clicking.
Every wave shape can be created by combining two or more sine waves. Even the square wave can be made by combining sine waves. Conversely, every wave shape can be "broken down" into component sine waves. Electronic synthesizers create the sounds of different musical instruments by combining sine waves (and sometimes square waves, which are very easy to make with electronic circuits).
With practice, our ears can often interpret complex waves as distinctive to certain musical instruments. A piano key will sound a "note" (a complex wave shape) that is different from the same note produced by a violin. A trained ear can even detect the combinations of sine waves in an irregularly shaped sound wave, especially if they are few in number.
What this document is about is precisely that ability we have—to recognize the component wave shapes in various combinations of tones. For it is rare that we hear music made up of simple sine waves. We’ll show the graphic forms of musical sounds so that you can see why certain tones have an affinity to each other when we hear them together—either in harmonies (played together), or in series, as in melody.
Some background:
1. If the vibrating part of an instrument, say a violin string, is producing a certain tone, it is vibrating at a certain frequency. Most instruments produce tones within a range of 16 vibrations per second, or 16 Hertz (abbreviated Hz), to 16,000 Hz, about ten octaves. The human voice is comfortable from around 100 Hz to 1000 Hz, three or four octaves.2. What makes a tone of a certain frequency a certain note is a matter of convention. Western musicians have standardized the scale by agreeing (mostly) that the vibration frequency of 440 Hz will be named “A”. (That’s the note the oboe plays to tune the orchestra.)
3. An octave is an interval between two notes whose frequencies have the relationship of 1 to 2. In other words, the frequency of the higher note is twice that of the lower one. For example, one octave above 440 Hz is 880 Hz, and an octave below it is 220 Hz. No matter where in the audible range one starts, this relationship is the same.
All notes in our system that are exact multiples of each other have the same letter name. In between, there are eleven more notes: A, Bb, B, C, C#, D, Eb, E, F, F#, G, G#, A. Each has a specific frequency (or pitch) relationship to the others, which we'll describe later.
These pitch relationships are what this article is about, for they create
the feeling we get when we hear them in combination.
Now, let’s dig:
As you can very easily perceive, two notes that are an octave apart somehow sound like the same note, at least to Western ears. If you play them together, there’s a pronounced affinity between them. Physically, it’s obvious—on every other vibration of the higher note, both are pushing and pulling on our eardrum at precisely the same time, and so they reinforce each other. In between, they cancel each other. They are clearly “in tune”. It’s the most basic tonal relationship in Western music.
Figure 2. Wave forms of two tones at 2:1 ratio
Notes that push and pull together at some regular interval sound pleasing to us. With the octave, it’s every other vibration of the higher, beating with every vibration of the lower . We can even see this regularity in the resultant wave form:
Figure 3. Resultant wave form of tones at 2:1 ratio
What happens when the lower note beats with every third beat of the higher? To be sure, it is also recognized by our ears, and our ears are somehow pleased. To a decreasing degree, the same thing happens with “every-fourth” note, “every-fifth”, “every-sixth”, and so on.
Here are some actual frequencies:
What is remarkable is that the octave and the “every-third” note also sound “right” together. The octave beats two times for every three of the “every-third” note. That makes a more complex physical effect on our ears, but it is still a regularly repeating pattern that we can readily perceive, both graphically and in our ear.440 - fundamental frequency 880 - octave 1320 - “every-third” note 1760 - “every-fourth” note (This is the octave of 880!) 2200 - “every-fifth” note 2840 - “every-sixth” note
Figure 4. Resultant wave form of tones at 2:3 ratio
Looking at our series above, we see that "every-fourth" note is the octave of the octave. What about the next one? It relates to the octave of the octave by a ratio of 5:4. The next pair--"every-sixth" and "every-fifth"--relate as 6:5.
Figure 5. Resultant wave form of tones at 6:5 ratio
The fragment shown here illustrates a time span of less than a hundredth of a second. Even so, our ears detect this pattern, and depending upon our training, we can often tell the difference between two differing patterns. Many musicians can readily identify the pattern of a particular chord.
There’s more to the physics of music. That violin string that we plucked in the beginning of this discussion—it doesn’t simply vibrate at one frequency. It does that most strongly, but at the same time it vibrates in halves, each half producing a note of double the frequency of the whole—the octave! Since the tension is the same, and the cross section and the material, the distance the half can travel is much less, so the amplitude is less. And the string vibrates in thirds as well, and fourths, and fifths—each at less and less amplitude.
You might surmise that the “thirds” note is identical to the “every-third” note we discussed earlier, and the “fourths” is identical to the “every-fourth”, and so on. That’s why a plucked string doesn’t sound like an electronic tone of the same frequency: it is simultaneously vibrating at a number of frequencies. (The instrument itself resonates (that is, it vibrates in sympathy) to reinforce some of those “harmonics” and suppress others.)
How we use it
So far, we’ve found that how frequently two notes “beat together” has something to do with the way they sound to us. Let’s look at how notes are set up in traditional music. We’ll use the piano, because notes are fixed there.The piano has twelve notes to the octave, equally spaced (“equal temperament”). The name “octave” comes from the eight notes out of the twelve that are used together to form a scale in a particular key. The white keys on the piano constitute the notes of the C Major (or C Natural) key. The other notes (black keys) were added so that a scale of notes could be moved up or down in pitch while retaining the same inter-note relationship. (More on this later).
Equal spacing of notes does not mean an equal number of Hertz between notes. This can be readily understood in relation to the octave itself, since doubling the frequency each octave means that the distance between keeps doubling, too. All the notes have the same relationship. They are spaced in pitch (frequency of vibration) so that each is the same multiple (1.0595649) above the one below, and so that they each have a corresponding octave note with the same name, in the next octave. In other words, A1 has an octave note A2, with double its frequency; B1 has a B2 at double its frequency, C1 has a C2, and so on.
To accomplish this relationship, and provide equal spacing between adjacent notes, we have to use logarithms. The logarithm of 440 is 2.6434527. The logarithm of 880 is 2.9444827. The difference is .30103 (don’t give up yet, this will get back to music in a moment). To divide the difference between A1 and A2 into 12 equal steps, we simply divide that .30103 into twelve parts, or .0250858, and add this to the logarithm of 440 (2.6434527 + .0250858 = 2.6675385) which is the logarithm of 465.09164, which happens to be the frequency of the next note in our series. (Unfortunately, it’s not called B, but B flat). We can add .0250858 to its logarithm for the next step, and so on up the scale, until we arrive at 880 Hz.
What we’re doing at this point, of course, is describing what the frequencies are in the musical scale. The names of the notes involved are A (remember the oboe), B-flat, B, C, C-sharp, D, E-flat, E, F, F-sharp, G, and G-sharp. As the piano is tuned for equal temperament, we could as easily call B-flat A-sharp instead. It’s a half tone (“semitone”) between A and B.
A 440.000 Bb 466.164 B 493.883 C 523.252 C# 554.364 D 587.330 Eb 622.253 E 659.254 F 698.456 F# 739.989 G 783.990 G# 830.608 A 880.000
The note which has a 4:3 ratio would vibrate at 586.667. The “D” vibrates at 587.330—also an error of one-tenth of 1%. To get a 5:4 ratio, we need 550 Hz. The closest we have in our scale is “C#”, 554.364—an error of .8%
To get a 6:5 ratio, we need 528 Hz. The closest we have in our scale is “C”, 523.252—an error of .9%
We can continue for a little while. A 5:3 ratio gives us 733.333, within .9% of “F#”, vibrating at 739.989 Hz.
A 7:6 ratio would be 513.333. That’s in between “B” and “C”, and not too helpful. But it’s also stretching the power of perception, for most of us, anyway. To a trained musician, it would sound "out of tune."
Within a single octave, we have six notes with a very close acoustic relationship between them and the lowest (fundamental) note. And remember that the same relationship exists no matter where you start!
We obviously need all the notes anyway, just to maintain that extensibility, even though to create a “pleasing” tune or chord we won’t use all of them. Which notes one uses for a particular piece of music depends upon where one starts. You may have noticed that most ordinary music begins on a particular note, with all the other notes having some special affinity to that first one. And often, the piece returns to that first note at the end. Would you believe it’s called a “key note”?
If we were to select notes for a particular piece, we’d usually start with the “best” ones available—the ones with the closest beat ratios. In our selection above, the 3:2, the 4:3, the 5:4 and the 5:3 give us a pretty good range of notes:
A C# D E F# A
That process may sound arbitrary, but the fact remains that the acoustics don’t lend themselves to our desire to have this system work exactly, beginning with any note we choose, up and down the scale. We have to compromise, and accept what sounds “right”. People have been playing by ear for a long time, and the mathematical relationships have been known only relatively recently . The art came before the science.
Anyway, we’ve chosen these notes:
A B 2 semitones apart C# 2 semitones apart D 1 semitone apart E 2 semitones apart F# 2 semitones apart G# 2 semitones apart A 1 semitone apart
C D 2 semitones apart E 2 semitones apart F 1 semitone apart G 2 semitones apart A 2 semitones apart B 2 semitones apart C 1 semitone apart
What happened to the sharps and flats? Well, the easiest way to explain it is to look at the piano keyboard. Remember we said that there are twelve equally spaced notes in an octave?
Figure 6. Portion of a standard keyboard
Now, do the same thing starting at A, from the previous list. Same numbers. Two different sets of notes, beginning from different starting points but containing exactly the same pattern of intervals between notes.
These sets of notes are called diatonic scales, or simply “scales,” like our full set (called the chromatic scale), and identified by the fundamental in each series. The two scales listed above are the A Major and C Major scales. (This doesn’t mean that in composing a piece of music one cannot use any other notes than those listed. But these notes form the basis for traditional composition, using what we’ve called the ones that go together best from the full selection. If someone extends the selection to include other notes, they should have their reasons. Picasso didn’t stick to “normal” perspective in his paintings, either.)
The C Major scale is the most-used scale, because it can be represented on the musical staff without sharps or flats. The standardized usage of these scales led to the term “key” which, for our purposes, can mean the same thing as “scale.”
Every second-grader learns to sing in the C Major key—“do re mi fa so la ti do.” And most people get pretty good at recognizing the pattern of relative pitches and intervals, in spite of the fact that those notes are not evenly distributed through the octave. Many people can “transpose” to a different major key and accurately sing the “do re mi fa so la ti do,” if given only the bottom “do.”
The major scales are not the only possible combinations of notes in a series. Remember, we compromised on a few that we chose. Suppose we simply used the white keys, beginning with “A”.
A B 2 semitones apart C 1 semitone apart D 2 semitones apart E 2 semitones apart F 2 semitones apart G 2 semitones apart A 1 semitone apart
The series of white-key notes beginning at C is called a C Major. If it begins at A it is A minor. One of the basic conventions of Western music is that this “key note” is the most important note, other notes in a piece being selected for their relationship to it.
There are, of course, twelve different starting places for each type (major and minor). Much music has been written in all twenty-four keys although, since those with fewer sharps or flats required are easier to read, it wouldn’t surprise me that those are more popular.
We’ll look at some more wave forms to better understand the differences between major and minor keys. One way to see this is in harmony—the playing of several notes simultaneously, as chords.
The standard chord in any key is the first note, the third note, and the fifth note in the scale. It’s called a triad. For example, in C Major, the notes are C, E, and G. Between C and E are four semitones; between E and G are three. In A Major, the notes are A, C# and E, and the intervals between them are the same.
In the A minor scale, the notes are A, C and E. That means that the third note in the scale is lower in pitch by a semitone (C# to C). The interval between the bottom note and the top one remains the same, so the interval between the middle note of the chord and the top note is a semitone greater than in a major scale.
Here’s what the waveforms look like for a major triad and a minor triad :
Figure 7. Waveform of a major triad
Figure 8. Waveform of a minor triad
:
Figure 9. Waveform of a harmonic triad
Conclusion
Discovering these relationships over the course of twenty years or so has not, alas, helped me learn to make music. It did give me a greater appreciation for musicians and composers and their products. I’ve never stopped feeling awe at the sound of a hundred or so people, all different shapes and sizes, undoubtedly all different in temperament and dreams, synchronize their respective instruments to the degree necessary to play a Ravel or Shostakovich orchestral piece, as though they comprised a single organism. And, especially after my own feeble attempts to master a guitar, I have an equal amazement in listening to a violinist traverse the solo passages in Tchaikovsky’s concerto.Still, my “research” has changed music for me, from “magic” to
enormous eloquence. And it has made this awareness something of my own, rather
than something I simply read. I’d like to think that it’s akin to that
feeling a musician must have when the music comes out clear and true.
Donald Skiff
December 27, 1997
Postscript:
What I didn't know, in the beginning of this search, was how much simpler it would have been had I started with philosophy, rather than music itself. I might have found the following:
Among [the disciples of Pythagoras] a preoccupation with music and mathematics played an important role. Here it was that Pythagoras was said to have made the famous discovery that vibrating strings under equal tension sound together in harmony if their lengths are in a simple numerical ratio. The mathematical structure, namely the numerical ratio as a source of harmony, was certainly one of the most momentous discoveries in the history of mankind. The harmonious concord of two strings yields a beautiful sound. Owing to the discomfort caused by beat-effects, the human ear finds dissonance disturbing, but consonance, the peace of harmony, it finds beautiful. Thus the mathematical relation was also the source of beauty.
—Werner Heisenberg, physicist, in "Science and the Beautiful," reprinted in Quantum Questions, ed. by Ken Wilber, Shambhala Press, 1985
Although many names of musicians are recorded in ancient sources, none played a more important role in the development of Greek musical thought than the mathematician and philosopher Pythagoras of Samos (6th-5th century BC). According to legend, Pythagoras, by divine guidance, discovered the mathematical rationale of musical consonance from the weights of hammers used by smiths. He is thus given credit for discovering that the interval of an octave is rooted in the ratio 2:1, that of the fifth in 3:2, that of the fourth in 4:3, and that of the whole tone in 9:8. Followers of Pythagoras applied these ratios to lengths of a string on an instrument called a canon, or monochord, and thereby were able to determine mathematically the intonation of an entire musical system.
—Calvin Bower, from the Internet
Post-Postscript: Click to see "An Addition to 'Music: The Language of (My) Soul'”
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