The Citizen Scientist

Like many other amateur science enthusiasts, I like to play music. The other day I was playing my French horn with others in our local community band. As we encountered a new piece, the director made the offhand comment that the melody line, though unfamiliar to us, was an "inversion" of a well known melody. That led me to think about just what he meant. And one thing led to another.

First of all, what is an "inversion?" Is it just playing the notes backwards, from finish to start? Is it playing down in pitch when the written notes went up? Or is it something else? Then somebody mentioned the theory that minor chords in music are just inverted major keys. That sounded intriguing.

What is music, anyway? For the moment, close your eyes and throw away all your written music, your musical instruments and your preconceptions. Music is then what you hear people singing alone or in groups. Sometimes it is a single melody line and at other times it is rich and spontaneous harmony. Your ear will tell you that much of the style is a product of its culture of origin. American music sounds different from that of Africa or the Far East. But music has rhythm, melody and harmony no matter where it originates. Rhythm is the timing, melody the song line, and harmony the mix of sounds at any given point in time.

Experts in acoustics, the science of sound, tell us that pure notes (sine waves) sound rather plain compared to the more complex mixtures of notes that come naturally from a human voice or a musical instrument. And when a choir or orchestra plays well together, the sounds can be glorious! But what makes combinations of pure notes sound harmonious, or "good," rather than dissonant, or "bad?" The experts tell us that it has to do with the ratio of frequencies in the mix of notes. If the ratio of the frequencies (a measure of how high or low a note is) of two or more notes is equal to the ratio of relatively small whole numbers, then the notes tend to sound good together. If the ratio is equal to the ratio of larger whole numbers, then the harmony becomes "closer" and eventually dissonant.

The reason this matters is that we usually like our music to sound harmonious, or "good." Now a singer or a trombone player or a violinist can adjust a note to produce whatever frequency is necessary to achieve harmony with other musicians. However, with most instruments, the player has a choice of playing from a limited choice of actual notes. For example, a piano has 88 keys. If a pianist wants to play a note between two keys, he is simply out of luck. Most players of wind instruments, such as the trumpet, clarinet, flute and saxophone, have learned to make small adjustments up or down in the frequency of a given note by changing the way they blow air.

While music is what comes out of a singer's mouth, musicians had to come up with a way to write it down on paper. In Western culture, much of what people were singing conformed fairly well to a scale of twelve small intervals in an "octave." The highest note in the scale had a frequency twice that of the lowest note. These twelve intervals correspond to the black and white keys on a piano.

That sounds simple, but it is definitely not! Organ tuners and instrumentalists had to figure out how to tune their instruments so they sounded good, particularly when playing notes together (chords). It turned out that if the composer J. S. Bach, for example, tuned an organ to sound good for one piece of music, it could sound terrible for another. This is what led to the so-called equal temperament scale, where the frequency increases by a constant ratio from one note to the next. In order for twelve such equal multiplications to span a ratio of 2:1 in frequencies, an octave, the ratio has to be the twelfth root of two, or about 1.0595. Tuning an organ or piano by this method makes all pieces a little out of tune, but it allows all pieces to be equally out of tune!

This ratio of 1.0595 is almost magical in that it causes some of the notes of a scale to fall very close to small integer ratios that would allow the playing of (almost) harmonious chords. For example, Table 1 shows the the ratios for each note in the twelve interval scale:

NOTE	ACTUAL RATIO	NEARBY INTEGER RATIO
0	1.00	1:1
1	1.0595
2	1.1225	9:8
3	1.1892	6:5
4	1.2599	5:4
5	1.3348	4:3
6	1.4142	7:5
7	1.4984	3:2
8	1.5876	8:5
9	1.6819	5:3
10	1.7817	16:9
11	1.8877
12	2.0000	2:1

Table 1. The ratios for each note in the twelve interval scale. The ratios for notes 1 and 11 are intentionally left out, because the numbers would be ratios of large integers and would be ambiguous and hard to achieve accurately in reality. Chords using these ratios would be highly dissonant.

The "note" number 0 in Table 1 is any chosen starting point on the piano keyboard. As the number goes up, you count rightward on the keyboard. As the number goes down, you count to the left. Once you take twelve steps, you span an octave, from middle C to high C, for example.

If you label middle C on the piano as note 0, then notes to the right of it are numbered with positive whole numbers, and numbers to the left of it can be similarly numbered with negative whole numbers. Then any musical piece written for ordinary instruments could be thought of as combinations of notes, each with a positive or negative whole number label. Imagine a C major chord, with notes 0, 4, 7 on this chart (C,E and G in conventional music terms). You can see that the frequency ratios are 1:1, 5:4 and 3:2. Because these are fairly small whole numbers, the chord sounds good.

Now suppose we "invert" the chord by making all the positive labels negative and vice versa. We will then have a chord with labels 0, -4, -7. If we want to transform this chord "up an octave" and onto the existing limited table, we can add 12 to each label and get 12, 8, 5. Viewed in another way, the chord could also be listed in the opposite order of 5,8,12. While this chord still corresponds to small integer ratios, it is a different kind of chord. It is F minor. Therefore, we "inverted" C major chord and arrived at F minor. Interesting.

Suppose you invert a whole musical piece, one with all sorts of melodies and chords? Well, it turns out that harmonious chords transform into different harmonious chords and melodies take on an entirely different sound. You are unlikely to recognize any similarity to the original.

Thirteen-year old Ana Peterlin of Fairbanks, Alaska went a step further and composed her own musical duet called "Upside Down and Right Side Up." This retrograde inversion canon was written in such a way that one player read the music in the normal fashion, and the other player turned the page upside down and played from the same sheet of paper but from the end to the beginning. Imagine that! She analyzed her composition, and a similar one by Wolfgang A. Mozart, mathematically and presented her findings at a student science fair. Her project, and others, is described at http://www.sciencenewsforkids.org/articles/20040602/Feature1.asp.

Figure out a way to configure a keyboard or computer to invert music automatically as you play it.

Experiment with alternative methods of notation and with microtonal scales in which the steps in an octave are different from the usual twelve. You may find a way to transform familiar music into similar but barely recognizable microtonal pieces.