So what exactly are the pitches of the harmonic series and how do we describe them?

We’ve already developed some tools with which we can begin to go about doing so. We know that a sound is a series of pressure waves, caused by a vibration, that oscillate with a certain frequency in hertz. When someone hears two or more pitches simultaneously, or in rapid succession, the perceived character of that sound is based on a ‘snap’ assessment of the difference between the two frequencies. In other words, we perceive a discrepancy between two frequencies and instantly translate that into an emotional and aesthetic response. The mathematical difference in frequency at the core of that response can be described using ratios.

Let’s start from the ground up. What is the simplest musical interval, and ratio, that we can come up with?

1:1

Simple. This much is the same as that much. This is equal to that. These two things are the same. Equal parts flour and equal parts water.

Rama perceives a sound at 200hz. He then hears another sound at 200hz. These two sounds stand in a 1:1 relationship with each other. They are the same pitch, otherwise known as a musical ‘unison’.

 

The first member of the harmonic series, known as the first partial, stands in a 1:1 relationship with itself, and will serve as a constant reference point as we ascend. It is the lowest pitch produced by a vibrating string, and it is therefore also known as the fundamental.

Just like vibrating strings, the human vocal chords produce audible overtones that correspond with the same harmonic series. Some of these pitches can be isolated and strengthened by changing the shape of the mouth, throat, and lips, and in Mongolia and Tuva in particular, this ability has been developed into a full fledged musical art-form. See this site for a primer on different types of “throat” or “overtone” singing. If you haven’t already, I recommend that you check out some recordings by groups like Huun-Huur-Tu and Alash (listen to this incredible performance). Khoomei is the basic style, while Sygyt produces a crystal clear whistling overtone, and Kargyraa is noteable for its production of undertones, pitches beneath the fundamental frequency.

It’s not exactly Tuvan throat singing, but do watch this scene from the classic film Baiju Bawra for a follow up to my previous post. First of all, it’s truly an amazing performance of a beautiful raag, Desi Todi, by two masters, Amir Khan and D. V. Paluskar. Secondly, be sure to watch the end to see Tansen and Baiju melt a slab of marble, break a tanpura string, and shatter a bowl of glass with the power of their voices!

In an episode of the television show Myth Busters, Jaime Vendera became the first documented person to shatter a wine glass with the power of his vocal chords alone. As you can see from the video below, he went on to practically perfect the art.

Based on what we’ve already covered in previous posts, it makes perfect sense that this is possible. The wine glass essentially acts in the same manner that the sympathetic strings of a sitar do. The glass has a natural resonant frequency, and when the same pitch is project into it, it reacts by ringing in sympathy. If the volume is loud enough, the glass may vibrate with such strength that it shatters. The Myth Busters episode is fun and I suggest you track it down somehow if you’re interested.

I’ve heard the story of an eminent Hindustani musician who performed the same feat by accident with a glass wall in a museum where he was performing. If the conditions were just right, hypothetically, it is possible! So how about bringing rain by playing the right melody? I’d like to see the Myth Busters episode on that…

Still don’t quite get the whole sound = wave thing? Check out the video below, which contains a demonstration of a classic physics experiment: the Ruben’s Tube. Flammable gas is circulated through a perforated tube and ignited. Then, music is projected through the tube, which creates points of higher or lower pressure, and causes the gas flames to mirror the properties of the sound. This one was constructed by Jeff Ryan at the University of Portland. A beautiful visual example of sound waves in action!

Photograph courtesy of Andrew Davidhazy – http://people.rit.edu/andpph/exhibit-6.html

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If you pluck a string and watch it closely, you may be able to see a pattern of oscillation in the shape of a standing wave. This pattern is not created by a single waveform however. Aside from the fundamental length, numerous subdivisions of the string also vibrate simultaneously, and these nodes and their simple mathematical progression have a profound influence on the way we perceive sound.

The smaller segments of the string also create audible tones as they oscillate and if you listen closely, you’ll hear that a single plucked string produces a combination of pitches. This can include, but is not limited to, the fundamental note, the first octave, the fifth above that, the second octave, the “major” third above that, the second octave of the fifth, and the very flat “minor” seventh.

The same tonal relationships correspond to what is known as the harmonic series and are some of the fundamental building blocks of music. They are produced by most vibrating strings and columns of air (like the vocal chords), and can be generated by taking simple fractional divisions of the string. The same proportions can be found by taking integer multiples of the fundamental frequency (in Hz) of the string (more on this dynamic later).

In one way or the other, these ubiquitous pitch relationships and their corresponding ratios have shaped the evolution of most musical tone systems. Below are the first seven harmonic divisions of a string, though this sequence theoretically extends to infinity.

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For a bit more info on string waves see:
http://www.acs.psu.edu/drussell/Demos/string/Fixed.html

Robert McLassus. Surface Waves.

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So what is a sound and how do we perceive it? Just good vibrations? Not quite.

A sound itself is actually a series of pressure waves traveling through a medium (gas, liquid or solid). These waves (or “impacts,” as the ancient Greeks sometimes described them) are caused by a source vibration, which is technically defined as an oscillation around a point of equilibrium. Sound waves, in turn, induce other things to vibrate, such as the human eardrum (tympanic membrane).

So basically, something vibrates causing waves of pressure to issue forth via colliding and rebounding molecules, which in turn causes the eardrum to vibrate, the frequency of which is interpreted by the human brain as sound and pitch.

The frequency of sound is measured in hertz (Hz), which is equal to the number of cycles per second. Human beings can usually perceive sounds between 20 and 20,000 Hz. The slower the frequency, the “lower” the perceived pitch, and the faster, the “higher.”

Another related and very important type of wave in music is that which is produced by a taut, vibrating string. This has an amazing relationship to the frequency of sound, and it’s where we’re headed in the next post. Until then, hang loose!

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Shalom Jacobovitz (photographer). 2010 Mavericks Competition.

Johann Wolfgang von Goethe. Farbenkreis zur Symbolisierung des menschlichen Geistes. 1809

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As a musician, I used to cringe at the thought that art might have anything to do with math (a subject I dreaded, and still dread for the most part). Music, I felt, was something to be perceived, experienced, and understood emotionally, not calculated and formulated.

After years of devotion to musical practice in one form or the other, I have to say, I was right.

I was also wrong.

My understanding of pitch began to transform when I started to learn what eventually became my principle instrument: the sitar. I threw myself experientially and emotionally headfirst into a musical system in which I found myself responsible, at every turn, for all aspects of the music, including the basic intonation of the scales. I quickly began to notice differences in the tuning of certain pitches; my ears were telling me things that my background in equal tempered systems never had.

Years later, the responsibility and task of determining the intonation of a complete musical system every time I sit down with my instrument, is what has ultimately led me to the perception of mathematical and scientific principles that are in accord with the aesthetic experience of music. Music arose out of the fundamental nature of sound, and the two are inherently interrelated.

A good analogy to musical pitch is the painter’s palette and the color wheel. From a set of basic primary colors, the artist can mix many intermediate shades, all part of a continuous gradation. Just like musical pitch, colors arise from our perception of certain periodic waveforms, and are perceived in relation to one another by the senses. In music, as in visual art, the plane on which we intuit these interrelationships is that of feeling, emotion and beauty.

This blog is devoted to exploring the palette of musical colors.