When it comes to rediscovering music, sometimes it is beneficial to think outside of the box.

In our previous lesson we saw that deep inside every music note is a beautiful and increasingly complex order known as the harmonic series. This hidden order gives rise to a set of sound phenomena: a fundamental that vibrates at a given frequency, and its set of complementary and enriching overtones that vibrate at higher multiples of the fundamental. In practice, it's really only useful to work with the first 7-8 harmonics. Remarkably, the first 6 harmonics spell out a major triad, and you can hear it by singing just one note.

If there is anything that I may impress upon you before we move on to the musical intervals, it is this: that each time we write a note on paper, we are writing the fundamental tone that comes with its own set of harmonious overtones.

Why is it important for musicians to appreciate the harmonic series?

The first few answers to the question might seem obvious to those who have been performing music for a while, but when we are discussing fundamentals for the purpose of deepening our relationship with sound it's important to consider them.

1. You will improve your intonation. Not just knowing, but experiencing how the major third sounds in nature will result in better ensemble playing.

2. You will learn about the universal relations of the tones and improve your knowledge of harmony. The fact that the fifth appears so audibly above every sounding note gives this interval a much richer meaning when it appears in music. You can then work with or against that fact to create or resolve tension in a piece.

3. You develop the practice of deeply listening to the totality of sound, which will shape your entire approach to music making.

Let us now pose another question: how are figures 1 and 2 related?

Figure 1. Rhythm usually has no place in the discussion of harmony—let's change that for the time being. The half and quarter note relation and the quarter and eighth note relation represent a lived 2:1 relation.

Figure 2. Listen to the harmonic series here: https://alexanderchen.github.io/harmonics/

Music: The Aesthetic Enigma

I mentioned briefly in my last post that music is different from any other art because it's the only one that has no model in our physical reality. The dancer, the painter, the sculptor, the architect, the designer—each artist adds meaning to the world by transforming its symbols and building upon them. It's easy to see how they each can find models for their art in our physical reality. But can we say the same about music? What physical model did Bruckner use when he constructed his symphonies?

Music doesn't have a model in the physical world, it simply refers to itself. Music is self-referent. We seem to derive meaning in music through repetition.

This is a profound fact that we can trace all the way back to the very definition of a musical tone, and, the patterns that arise from the natural motion of string vibration.

The Exercise: Discovering the Double

For today's exercise, we are arranging the tones of the harmonic series into a graph that makes its organizing principle easy to see. The purpose of this exercise is to give a greater appreciation for the relationships that naturally appear in sound, making music possible.

To learn how to create the graph, simply watch the video at the top of this blog post and work alongside it, before reading the discussion below. You can also download a version of the graph that you can easily fill out by clicking here.

To really benefit from this exercise, be sure to try creating this graph for yourself first before reading the discussion below. I promise it is easier than it may seem at first glance.

The Overtone Graph Discussion

This graph organizes the tones of the harmonic series in their ascending order. The graph follows two principles: (1) Octaves of the fundamental are written on the left and right sides of the page and (2) all octaves of the same note are stacked on top of each other. The graph can really be drawn out of intuition—all you need to start is a blank page and follow those two guiding principles, writing the names of the notes as they appear in the harmonic order.

What are the patterns that it reveals? By stacking the octaves of the notes as they appear throughout the harmonic series, we start to witness several remarkable patterns.

In each new octave of the series, there will be one new tone that emerges between two tones in the previous octave.

Also, check out how the octaves of each note relate to their harmonic number. It is such that the higher octave is twice the harmonic number of the previous octave. For example, the 5th harmonic is an E, and the next E in the harmonic series is the 10th harmonic, and then the E after that it is the 20th harmonic, and then the 40th, and so on.

This graph reveals that the harmonic series is drenched in the principle of the double.

Musical Consequences

Think back to figures 1 and 2. How are they related?

They are related because in both cases, we can live the double. The octave represents a 2:1 relation, a doubling of frequency, of the number of vibrations contained within a pulse of 1 second. The fact that we can experience the octave, the great organizer of scales, in the same way that we can live a doubling of rhythm is what makes music completely unique. No other art has the double at its core the way music does. We could try to imagine, what is the double of blue? In music, from octaves to rhythms, whether conscious of it or not, the double, and its generative properties, is always clearly experienced and can be found at the heart of the musical experience.

Some examples of overtone inspired music

Bartok’s Concerto for Orchestra (uses a scale inspired by the overtone series)

Bruckner Symphony No 7, first movement melody in the celli

The music of Kaija Saariaho, and other composers of spectral music. https://www.youtube.com/watch?v=HZ5Ow0thsco

The Intervals

Day #3 is going to be our deep dive into the musical intervals! Now that we have finally conquered the world of sound, or at least have discovered some paths to go about doing that, our next topic will be about the transformation of one sound into another.

Discovering the Double: Harmonic Series Analysis