As designers, we are always looking for connection between geometries, proportion, and the built environment.
So what does music have to do with it?
Musical frequencies are usually talked about in terms of musical notation. In the west, we use a standard 12-tone scale (13 if you include the octave of the first note) from A to G with a handful of flat and sharp notes in between. This notation, however, corresponds to a parabolic scale of geometric relationship in frequency and wavelengths of sound.
That may sound like gibberish, we know– so let’s start with the basics.
Sound is a pulsation of waves.
Any given note sounds the way it does because of the length of its wave, and correspondingly, the frequency at which that wave repeats each second (the other half of the equation is how the ear processes the sound waves, but that post is for another day). In any given space, sound travels at the same speed no matter the length of the wave, therefore notes with long waves necessarily have a lower frequency, and sounds with very short waves have high frequencies, because they can repeat many more times per second. Sounds with short wavelengths and high frequencies are perceived as high, treble notes, while the sounds with longer wavelengths and low frequencies are deep bass notes.
Visualizing harmonic relationships is easiest in terms of wavelength because unlike frequencies, wavelengths don’t require time to pass to register the relationships between notes—to be sure, they are the same relationships, only inverted. Wavelengths are also easy to visualize, because they are actual lengths, and they are measured in centimeters and meters. Anyone who has built a table or shelf has dealt with the same scale of measurements that makes music. The only difference, perhaps, is that for our ears to register sound as musical, is a strict adherence to proportion.
Have you ever wondered what an octave is? Why does a low A note sound like a higher one? The answer: simple geometry.
The following waves are all representative of A notes. As you can see, their wavelengths and frequency are simple multiplications and divisions of 2 of each other. Each octave, going up the scale, has half the wavelength and double the frequency. When these notes are played together they ring out sweetly because their waves interlock into a simple pattern.
Because the wavelengths of notes double each octave, the scale of frequencies and wavelength is parabolic. What looks like a straight line in musical notation, actually represents a parabolic scale of harmonic proportions. There is more geometry in music than just from one octave to another: Wavelengths of notes in the same key, relate to each other with simple fractions.
The image on the right shows depicts three notes that make a major triad, such as C, E, and G. Because their wavelengths are simple fractions of each other: 5/4, 3/2, 6/5, they combine into a tightly interlocking pattern that registers to our ears as a pleasing chord. Harmonics in music then are ratios of wavelengths, and can be measured in the same units that architects measure centimeters and meters, inches and feet.
Doesn’t this just apply to western music? What about music from other cultures?
One of the amazing things about music is its ability to connect people, and remind us that we are of the same nature. Throughout the world, peoples have come to the same conclusion in terms of what harmonic ratios sound best. While the actual notes used in the west are different from those in Asia, the harmonic relationships between them are shared. The sonic proportions of these notes have attracted the human ear for as long as we know. Examples of use of pentatonic scales include Celtic folk music, Hungarian folk music, West African music, African-American spirituals, Gospel music, American folk music, Jazz, American blues music, the music of ancient Greece, kulintang, Native American music, melodies of Korea, Laos, Thailand, Malaysia, Japan, China, Vietnam, the Philippines, and the Afro-Caribbean tradition, as well as Western Impressionistic composers. Pythagoras was perhaps the first to codify the notes into geometric proportions. The pentatonic scale is simple and concordant, whose relations are easily perceptible.
You probably would rather just hear it, though.
This “Circle of Fifths” shows perfect fifth notes adjacent to each other. Any 5 adjacent notes make one pentatonic scale. As you can see, the pattern eventually gets back to where it started. The pentatonic scale can be found by inscribing a pentagram into a circular octave.
Now for the mysterious.
If all this math has robbed you of the ethereal beauty of music, don’t despair! We aren’t just peddling through fractions, but approaching a natural phenomenon. If you are familiar with the golden ratio, or the Fibonacci sequence, you’re probably familiar with the golden spiral. But the golden ratio is also heavily associated with the pentagram, which contains several iterations of this sacred ratio: 1.618…
This is a great video that talks about the mathematics in music as well as the pentagram and golden geometry.
The pentagram happens to be a useful tool in finding the pentatonic scale on a circle of notes, which in and of itself is not mind-blowing, but intriguing enough to do a little more digging into the relationships contained in the scale.
This table shows all the relationships between the notes on the pentatonic scale:
|
G |
A |
B |
D |
E |
|
|
G |
1 |
8/9 |
4/5 |
2/3 |
3/5 |
|
A |
9/8 |
1 |
9/10 |
3/4 |
27/40 |
|
B |
5/4 |
10/9 |
1 |
5/6 |
3/4 |
|
D |
3/2 |
4/3 |
6/5 |
1 |
9/10 |
|
E |
5/3 |
40/27 |
4/3 |
10/9 |
1 |
The ratios between the notes B, G, and E are integers on the Fibonacci sequence, particularly, 3, 5, and 8. The E note is 3/5ths of the G note, the G is 5/5ths of itself, and the B note is 8/5ths of it (or 2/5th, 4/5th, 16/5th etc)). These notes are an E minor chord. This golden geometry found in music is not dissimilar from the golden geometry we find in nature: this sequence is found everywhere from the shapes of galaxies and sea shells, to the patterns in pine cones and plant growth. That we not only perceive it as visually pleasing, but also find these relationships sonically beautiful is astonishing.
This brings us to a broader note. Music is geometric and mathematical, but that doesn’t mean it’s unnatural.
Music is a great bridge between what we think of as nature and what we consider mathematics. Our ears don’t need to be told when notes sound right, and musicians certainly don’t think of music in terms of ratios of wavelengths, but for the same reason we find beauty in spiraling sea shells or the patterns in plant leaves, we love the sound of music. Behind all of nature is mathematical guidance encouraging fruitful growth.
By following nature, and the math that governs it, the design process can work to reflect the forces that we grow out of and appreciate innately. Through design we can consciously pursue our own nature, and reflect upon how we unconsciously perceive the world. Here is another great link that depicts what actual sound waves look like.