Building Blocks of Music, Vol.

Building Blocks of Music, Vol. 2. Tone

When a sound has a repeated pattern between 20 Hz and 20,000 Hz, we use the term pitch or tone to describe the sound. I am trying to avoid excessive mathematics, although music is a mathematical art, so some mathematics will be necessary. I am going to ask you to remember back to the old Cartesian graph, with the horizontal X-axis and the vertical Y-axis.

On our graph, the X-axis represents time, moving constantly from left to right. The Y-axis represents the amount of energy of the wave (the amplitude). Any pitch may be graphed using this system. In an analog sound generator, whether it be a string, a vibrating column of air, or the diaphragm of a loudspeaker, the far points on the Y axis represent the vibrating thing, whether it be air, string or paper, at the points where the motion stops in one direction and goes the other direction. So, if a vibration moves in a pure pattern of up/down over time, the result is always a sine wave (that roly-poly paradigm of regularity). This sine wave is the purest expression of tone. A sine tone sounds excessively pure. In fact, because it is so rarely found in nature, it is sometimes difficult for us to grasp which pitch the tone is. We are used to dirtier sounds, with multiple layers of sound energy at different pitches and different energy levels.

The closest we have in the world of acoustic instruments to a sine wave is the flute or recorder. Even then, there can be harmonic energy as well as noise energy (but we will leave noise out until Vol 3). We tend to use language of purity to describe sounds that are close to the sine wave: crystalline, ringing, etc.

The sounds we associate with musical instruments, however, tend to have layers of pitches, known as harmonics. The lowest pitch is the fundamental. When we say that an oboe is playing A, nowadays we mean that the fundamental is vibrating at 440 Hz (cycles per second). Mozart would have assumed 432 Hz and Bach would have been comfortable with 415 Hz, but the ever-brightening sound of the orchestra is for later discussion. However, we do not expect all of the sound energy of the oboe to be vibrating at 440 Hz. In fact, above this fundamental is a whole range of sympathetic vibrations that follow a pattern known as the harmonic series.

The harmonic series is expressed as ratios of vibrations, and it is a constant. The fifth harmonic, for instance, has the same relationship to the fundamental whether that fundamental is A or E flat. When the harmonics are present in a certain pattern, the wave starts to look less like a sine tone and more like a saw blade. We call that sort of sound a sawtooth wave or a triangle wave. A tone with a triangle wave pattern is richer, and easier for us to hear as a tone than a sine wave. If we add energy to the higher harmonics, we get a square wave, which is a rather complex arrangement of harmonics.

Now, jumping ahead two volumes in the Building Block series, we will get to the interval, which describes the relationship between two pitches. We will see that each tone of the musical scale is based on the harmonic series. We will also see that the purest expression of the harmonic series is problematic in terms of real music, as the harmonics get a little out of whack, so that the logical progression of fifths, thirds, and octaves, do not come together. As a result we have a variety of tuning systems to reconcile this gap. When we look at temperament we will particularly pay attention to the tyranny of the equal 12 system, why it was accepted, and why it is about the worst sounding tuning possible.

Tone is related to timbre, or the quality of the sound, because the timbre is made of harmonics interacting with the fundamental, however, timbre generally refers to the stable relationship of the harmonics to all of the pitches of the set. An A (or la, as I prefer to use fixed do solfage (with do and ti, not ut and si, for my French and Italian readers) can have the same timbre as D (re), if the amount of sound energy is the same, relative to the fundamental, between the two. The significant difference is the frequency at which the fundamental vibrates.

When we get into intervals, we will discuss these ratios more in depth, but there is one that you need to know right away, and that is the interval of the octave. An octave is simply the doubling of a pitch, so if A=440, the next A will be 880. What we do to the notes in between is a matter for our discussion of temperament. So, if we look at the octaves between the standard points of audible sound we get: 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, and finally 20480, or about 10 octaves. When we discuss Stockhausen’s “Four Criteria of Electronic Music” and look how it can help us understand all music, we will visit this concept in depth.

When we look back at our Cartesian graph, we can determine how loud a sound is by the range between high and low points on the Y-axis. The louder the sound, the more energy is in the sound, and the more distance will be registered on the Y-axis. The pitch is not changed by the dynamic (loudness), but our perceptions can be, particularly in a sound with a lot of harmonic activity in it.

One final point of dynamics and pitch is that this analog is exactly what happens when the sound hits our eardrum. The points that the ear drum moves to in either direction are the points on the Y-axis. The amount of time between crests and troughs is the X-axis.

The next Building Block to consider is the effect of a distribution of sound without a regular pattern, which we call noise.

Posted by erik at July 3, 2003 7:49 AM | TrackBack