Music Revolution

Stravinsky's Prediction

With an extensive history of music already in place, one might wonder, what more could possibly be accomplished? Are there really any musical components left for radical exploration? Here is how Igor Stravinsky, one of music's most historically celebrated composers, decisively answered a similarly posed question:

Yes, pitch. I even risk a prediction that pitch will comprise the main difference between the “music of the future” and our music.
— Igor Stravinsky, Memories and Commentaries

Why would Stravinsky say that the transformation of pitch is the future of music?

If we look at the components of a single musical tone, whether it be from the voice or any instrument in the world, we see an endless series of pitches, many of which have still never been heard consciously in isolation, let alone heard in a musical or harmonic context. These sounds within one sound, or pitches inside one tone, which I will now refer to as partials, could be the music of the future that Stravinsky predicted. Partials work with largely different rules and principles than those that have already been established.

One reason for the discovery of additional partials, along with the expansion of what pitches are deemed acceptable, is modern technology. In Stravinsky's time, along with all the ages before, there was not an efficient way of isolating, and thereby hearing and discovering, complex partials. Only with relatively recent advancements in science and technology are we able to bring the more complex and higher partials into our listening experience.

With further advancements in technology, modern instruments can play nearly any combination of tones and partials that can be conceived. As in many areas of life, the modern era is a unique time for musicians to be alive due to the contemporary opportunities and changes that only this period has yet been able to provide. Certainly, a number of musical creators from the past would be envious of our current circumstances, for many of those prominent composers and players felt limited by the constraints of their generation. Even Arnold Schoenberg, a true innovator in his time, articulated:

We ought never to forget that the tempered system was only a truce, which should not last any longer than the imperfection of our instruments requires. I think, then, contrary to the point of view of those who take indolent pride in the attainments of others and hold our system to be the ultimate, the definitive musical system—contrary to that point of view, I think we stand only at the Beginning. We must go ahead!
— Arnold Schoenberg, Theory of Harmony

Perhaps we are entering that age where imperfection need not last any longer.

Theoretical Music - A Definition

           Those who have discussed the subject of music with myself recently know that I prefer to use the term "theoretical music" (TM) when describing musical functions. Naturally, this phrase is met with slight confusion, as I am currently unaware of its usage anywhere else.

Theoretical Music: 
            A division of music that uses cognitive processing, mathematics, and aspects of the scientific method to understand and explain musical phenomena.

            Along with the simple definition above, theoretical music attempts to examine music from alternate systems and perspectives. For example, approaching certain musical concepts in a mathematical, physical, or even philosophical way can provide new insights. It's not just intellectual comprehension that these multiple perspectives provide. They also present ways to enhance and progress the practical, creative musical landscape. See my post titled M=m for a more in-depth look at how mathematical operations support music, and vice versa.

            A theoretical musician studies how music works with a scientific and evidence-based, while at the same time open, mindset. She forms ideas from creative thinking, then tests and validates these theories through reasoning and experimentation. Curiosity is vital, and with the consistent practice of theoretical music, she continually questions and shapes the existing musical understanding found in modern society.

            Other terms that deal with explaining musical functions are often rigidly entrenched and settled through years of cultivation and tradition. Theoretical music offers a chance to doodle and muse about the workings of music with a "clean slate," where the ideas therein carry no offense to any establishment. In other words, TM is its own practice with unique skills and interests. Being a theoretical musician is as simple as applying and utilizing the definition above. If these principles already come naturally to you, it is surely already adding to our understanding of how music works. If this is the first time hearing about these ideas, welcome to the world of theoretical music!

Going Around in Circles (Of Fifths)

     Many ideas in the world are superficial constructions created to simplify and avoid the complexities of life. What's more, some of these ideas go largely unexplored and unquestioned. A concept in music known as the circle of fifths (COF) could qualify as one of these ideas. Indeed, a number of musical theorists elegantly simplify music into this single circle, and it has become a foundational symbol of most modern music and theory. While the COF has been useful to a degree, like most complexity-simplifying concepts, might it stifle deeper understanding?

      First off, what is the circle of fifths? It is a repeated cycle of one of the most, or if not the most, important musical intervals known as the "perfect fifth." Musicians throughout history have analyzed where, and how, fifths shed light on music. In modern music theory, when the fifth of every tone is taken consecutively, it circles back around to the same tone. At least, it seems that way. Curiously, it takes twelve successive fifths to complete a circle. Hence, the result is the twelve-tone scale found in most modern music. This, in theory, makes music more understandable and practical, because any musical pattern or song can then be taken through only twelve permutations (keys) before returning to square one.

      While the COF makes sense conceptually, the subject in question is this: does a consecutive series of fifths actually make a perfect circle? Surprisingly, mathematics shows otherwise. A true fifth originates from the third harmonic of a tone, which is measured at 702 cents. If a cycle of twelve true fifths is taken when measured at 702 cents, it very nearly shuts into a circle. The key word in the last immediate sentence is: nearly. When fully calculated, it is off from a perfect circle by a mere 23.46 cents, an interval that baffled Greek musicians over two thousand years ago. This seemingly minute difference is so important that the infamous circle symbol associated with the fifth completely depends on it.

       Consequently, if the math doesn't hold up to support the circle of fifths theory, then what symbol might best represent the concept of a series of fifths? If a closer look into nature is taken, one eventually finds a perfect fit: the spiral. A spiral always remains open, as do successive fifths. A spiral can expand or contract, as can successive fifths. A spiral can theoretically extend indefinitely, as can successive fifths. The parallels are clear.

       Symbols are powerful due to humanity living largely visual-based lives. With a simple change of musical imagery, the very nature of music becomes dramatically more mysterious and complex. When the spiral starts to symbolize the musical fifth, society may begin to visualize music in its true, boundless form. Indeed, the spiral can lead musicians away from circles, and straight into the unknown, spiraling as deep as the imagination allows.

Spiral of Fifths

Spiral of Fifths

M = m

            The field of mathematics has been a source of monumental discovery and realization for over 3,000 years, especially when used in conjunction with another medium. Architecture, engineering, physics, and communication have all reached previously unknown heights when combined with mathematical thinking. Likewise, music has carried a tremendous amount of influence on humanity throughout the ages. It has especially stimulated art and creativity, while at the same time displaying the ability to transform emotions and shape behavior and personality. There are many aspects in the field of music that are still very mysterious. For instance, how is it that a simple combination of sounds from a piece of wood causes a person to cry, or dance, or both? Despite the rich history of these two fields, they seem in the public eye to have grown further apart over the years. In a recent TED talk, one professor frustratingly raised the question, "Why not admit there is a problem with mathematics and music?" (Formosa). With a little simple analysis of some congruent principles, one can see that there might not be a problem with math and music after all.

            One of the easiest shared concepts to understand between math and music is the idea of the octave, or doubling. In music, an octave is a musical tone seemingly the same as another, only higher or lower in pitch. In mathematics, the exponential function or, more specifically, doubling, functions in a highly similar manner. Just like the octave, any doubled number or multiple of two can easily be reduced, leaving the essence of the number intact. A musical octave is attained by dividing or multiplying a string or flow of air exactly by the number two. In other words, without the alternative musical terminology, one could say that the octave literally equals two!

            Like the octave, every other musical tone that exists can be defined using simple math terms. In the words of Harry Partch, a twentieth century pioneer of new music, "Tone is number, and since a tone in music is always heard in relation to one or several other tones­­­­­­­­­­­­­­­­­­­–actually heard or implied–we have at least two numbers to deal with: the number of the tone under consideration and the number of the tone heard or implied in relation to the first tone. Hence, the ratio" (76). Ratios, as Partch implies, contain valuable information about musical tones themselves: namely, they reveal the relationships of tones (intervals) through number interactions, and they relate the number of vibrations and cycles inherent inside all musical notes. Not only do mathematical ratios share scientific and intellectual information, but they also state the accurate physical measurements needed for tones and instruments. When the length of a string or sound hole are taken into consideration, ratios give the musician an exact dimension on where to place the fingers or frets. For example, to hear a sonic form of a mathematical ratio, simply pick a ratio of the total length of a string, say 3/4, measure it out with a ruler, and play. This successful experiment will render the whole musical spectrum of tonal relationships to be as simple as 1/1, 3/2, and 4/3.

            On a deeper level, math and music can work to explain the sonic and musical phenomena that escape conscious perception. This can be achieved through more research in the areas of string waves, string motion, the effects of music on the body and matter, and how sound distributes into the atmosphere. Studies on the effects of music on the body and neurology have lately been particularly prominent, as seen in highly successful books such as This is Your Brain on Music (Levitin). A study by the Academy of Finland has shown that music engages "wide networks in the brain, including areas responsible for motor actions, emotions, and creativity." These groundbreaking findings show what is possible when music, math, and technology work together. In addition, math and music have been instrumental in developing important theories in the field of physics, particularly wave theory. "Physics of music is really the physics of waves. We will concentrate on sound waves, but all waves behave in a similar way. Wave theory is probably the most important concept in physics and especially modern physics, much more so than projectile motion and classical mechanics" (Gibson). As shown, there have already been many breakthroughs from the empirical study of music, and perhaps the best findings are yet to come.

            After shortly examining the striking similarities between math and music, it is possible to find a new perspective on both fields. While music, presently perceived as a "right-brain" activity, and math, perceived as a "left-brain" activity, stand seemingly worlds apart in the public eye, they are in reality like the right and left hemispheres of the brain: part of a cohesive whole that work brilliantly together. In the end, math and music have always been part of a common goal: to understand, discover, and connect with existence more fully. When these two fields, of which separately have accomplished awesome feats for humanity, come together, their impact will be multiplied by two, or in other words, their impact will go up an octave. Same thing, right?



Works Cited

Formosa, Dan. "Why Not Admit There is a Problem With Math and Music? Dan Formosa at TEDxDrexelU." Online video presentation. YouTube. YouTube, 9 Jun 2012. 31 March 2016.

Partch, Harry. Genesis of a Music. 2nd Ed. New York: Da Capo Press, Inc., 1974. Print.

Levitin, Daniel J. This is Your Brain on Music: TheScience of a Human Obsession. New York: Penguin, 2006. Print.

Suomen Akatemia (Academy of Finland). "Listening to music lights up the whole brain." ScienceDaily. ScienceDaily, 6 December 2011. <>.

Gibson, George N. "Why Learn Physics Through Music?" Uconn. Uconn, n.d. Web. 31 March 2016.

The Aural Singularity

            Does the infinite really exist? For now, the constraints of perception may keep humanity from reaching a definitive answer to this all-encompassing question. Restricted knowledge of concepts is inherent, but this does not stop many minds around the world from attempting to grasp the anomaly of infinity. One fascinating way to experience the infinite is through a particular field in which it is overwhelmingly present. These boundless fields consist of numbers and mathematics, color, gravitational singularities, perhaps universes (Kaku), time itself, and an old cliché, human stupidity. Curiously, most of these divisions have been the source for great leaps of discovery. This paper briefly examines another way of experiencing and working with the infinite: the way of music.

            Consider the following very simple thought experiment: if one number is selected from the vastness of mathematics, then divided in half continuously, is there ever a limit reached? It is a concept such as this that leads one straight into infinity. Likewise, if a musical interval of, say, 100 cents, is continuously divided in half, no limit can be reached. The same endless divisions can be said for the string of an instrument. While this is of course impractical for many modern, popular instruments, perhaps instruments in the near future could make this a practical capability, in the same way that calculators are capable of generating a staggering amount of numbers.

            Every time a string vibrates or a musical tone sounds, a series of harmonics (or their close cousins, partials and overtones) instantly sounds at the same time. When harmonics are taken into account, the infinite nature of music perhaps becomes more apparent. Though the harmonic series is musical at heart, mathematicians have conceivably studied it in further detail throughout the years, and they have come to the conclusion that it is a sequence of numbers, or frequencies, that indeed extends into infinity (Weisstein). With this realization, keep in mind when listening to, or playing, music, that a representation of infinity is happening with every tone, and every rhythm, and every chord struck. Unfortunately, somewhere down the line, the harmonics become lost above or below the range of human perception. Luckily, however, any harmonic interval can be rescued back into conscious perception by mathematically locating the harmonic, then dividing it until it falls into hearing range.

            Self-expression is thought of as a major goal among musicians and artists alike. In an infinite music space, self-expression could be fine-tuned in the most intricate fashion, and individuality would have the chance of becoming commonplace. Think about this: there are nearly countless ways an individual looks at, and feels about, the world. In fact, it is likely safe to say that no one in the history of humanity has been, in totality, the exact same as another. This is also true for musical tones and rhythms. With no two tones being alike, players and composers would be able to capture the most detailed sonic form of their feelings and visions. Take an example: if any note from a well-known piece of music, like Beethoven's "Ode to Joy", was altered by a mere 20 cents, it would diverge the atmosphere and feeling for the listener, albeit subtle. It is these faint differences that could bring significant assistance to those who are searching to find their own voice.

            A brief look at the nature of key musical components has revealed music's plausible connection to the infinite and, consequently, everything. According to legendary saxophonist, Charlie Parker, "They teach you there's a boundary line to music. But, man, there's no boundary line to art." When the boundary line is removed from the limitless ocean of music, the art will flow like never before. Once music becomes recognized as a new member of the infinity alliance, then musicians everywhere can break away from a small planet, and go exploring out into the aural cosmos.



Works Cited

Kaku, Michio. Parallel Worlds: A Journey Through Creation, Higher Dimensions, and the Future of the Cosmos. New York: Knopf Doubleday Publishing Group, 2006. Print.

 Weisstein, Eric W. "Series." Mathworld. Wolfram Research, n.d. Web. 2 March 2016.

 Parker, Charlie. "About Charlie 'Yardbird' Parker - Quotes." CMG Worldwide. N.p., n.d. Web. 1 March 2016.