Tuned Sounds

A sound with a perceivable frequency (pitch) is a periodic function. The shape of that waveform (and its change in time) is what gives a sound its unique character. It's how we recognize different voices and different instruments.

In the 19th century, while working on the problem of heat flow, Jean-Baptiste Joseph Fourier noticed that any periodic function (such as, in our case, a tuned note) can be represented by a series (perhaps infinite) of simpler periodic functions, such as a sine or cosine. This is a useful insight in our analysis of why sound sounds the way that it does.

This utility is perhaps easiest to explain if we examine a specific example, and for this I'll chose a vibrating string (but the situation is the same for many sound generators, such as a struck bar or the resonating column of air that you will find in many instruments).

When a physical impulse is imparted to a fixed string (by plucking, bowing, etc.) is begins vibrating at a frequency that is determined by the string's length and diameter, as well as it's tension:

If that were the end of it all strings would sound more-or-less the same, because they would all have the same waveform. But that isn't the end of it. Other vibratory modes will also be excited, and given that the ends of the string are fixed (and they modes arise at the same tension and string diameter), these vibratory modes will be at a integral multiple of this fundamental tone. So, for example, the first four harmonics would look something like:

If the fundamental (the blue line) were 100 Hz, the second harmonic (the green line) would be at 200 Hz, the third at 300 Hz, and the 4th at 400 Hz. Thus the harmonics themselves exist in simple, whole-number ratios. Through Fourier analysis any waveform can be analyzed, and it is found that the timbre of the sound relates to the relative strength (and to extent phase) of these harmonics.

Every tuned sound that we hear is a superimposition of tones which are related to each other by simple ratios. This is true from a mathematical perspective, and there is also evidence that our ears and brains actually work in this way from a physiological perspective. Our ears, for example, seem to perform something of a spectral analysis, as different parts are sensitive to different frequencies.

Given the fact that all pitched sounds that we hear already encode simple ratios of tones, perhaps it isn't too surprising that this colors our perception or multiple tones (harmony). It is possible that we have evolved to apprehend these ratios, because in a basic way this is a fundamental property of sound.

The Harmonic Scale

Extending the discussion of the harmonic nature of sound, it is possible to use the harmonic series to generate the notes of a scale. If, for example, we take the first 10 harmonics of a tone, we end up with the following series:

Remember, though, that an octave is a factor of two, and we perceive notes separated by octaves as somehow the same note. Therefore it is common in musicology to normalize such an interval, taking each note and either multiplying of dividing by a power of two to make the note fall within a single octave (or, numerically, making it fall between the value of 1 and 2). In doing this the above series would be transformed to the following:

The \(\frac{9}{8}\) interval, for example, comes from the penultimate note, 9, when scaled to the octave:

(in other words, its the ratio 9 scaled down by three octaves).

It is also possible -- and common -- to reference the ratios to a harmonic different than the fundamental. So, for example, if we were interested in referencing things to the third harmonic instead of the first, our original series would be:

Dividing each interval by three and then normalizing to the octave yields:

For the scale documented here I took a long harmonic series -- 30 harmonics, referenced to the fourth harmonic. I used a long series so there would be many potential notes from which to choose the scale (the method of selection will be described below). This scale has 16 degrees (including the unison and octave):

The issue with the series is that, while it is harmonically complex, it is harmonically very complex. If we examine all the distinct intervals within an octave (in the same way that we did for the even tempered scale above): we end up with a large set of distinct intervals:

That is a lot of harmonic complexity!

The way of handling the complexity it to choose a subset of notes from this scale in he same way that the standard major and minor scales are a chosen subset of the 12 chromatic tones of the standard tuning. But to do this we have to have a way of estimating which intervals may be consonant or dissonant.

Consonance and Dissonance

In order to characterize the feel of a scale or mode, it is useful to define a metric that measures an interval's consonance. And while musicologist have done formal work in this, I decided to create some simple metrics that proxy for the fact that "the ratio of simple whole numbers sounds consonant." Much more complicated (and better defined) metrics are possible, but I wanted to use a simple metric as a proof of concept.

The metric I chose for this scale is to take all of the distinct intervals within the scale and to sum up the numerators and denominators of all intervals. The lower the number the smaller the composite numbers for the intervals, and the more consonant the scale is rated.

It is also useful to visualize the consonance of a given scale. For this we can create a heat map that compares every degree to every other degree. For the harmonic scale given above, this follows:

Each matrix position is the interval between two degrees, the diagonal being each note played against itself. The label is the denominator of the interval as expressed in lowest terms (the thought being that lower numbers are more consonant).

To create the mode I chose seven tones (eight including the octave) from the scale and calculated the consonance matrix. As it turns out there are 3003 different ways to choose those tones from the full scale, and to evaluate each scale requires the evaluation of 64 separate intervals. Overall 192,192 intervals had to be created and calculated to find the scale that minimized the metric function. This is a lot of computation, but is fairly trivial for modern computing technology (the combinatorial analysis takes less than a minute on a reasonably powered modern computer).

The Simplest Expression

The above parameters describe how I originally came across the scale, but they don't represent the simplest formulation. The scale is also found in the harmonic scale constructed from the 1st to 21st harmonic:

• First harmonic: 1
• Last harmonic: 21

which yields, when normalized:

When the same metric is applied, the scale results:

Harmonic Complexity

If you recall, the standard 12 tone Equal Division of the Octave has a small set of distinct intervals:

If we examine the intervals in this scale, it is a slightly larger set:

Some of these intervals are consonant, whereas some (for example \(\frac{21}{20}\)) will sound somewhat dissonant.

What Does it Sound Like?

The following widget plays two clips: a small riff in the standard major scale in 12-EDO, and the same riff played in the scale described above:

At first the scale sounds a little foreign, but in working with it I've found that the ear quickly adapts and the scale subsequently starts to make musical sense.

It is also interesting to compare this scale numerically to a commonly used scale, such as the 12-EDO major scale. In the following table this is done, with the first note of the two scales pegged to A = 220 Hz:

We can also examine the scale's relationship to the (harmonic) minor. This is even further removed:

(A cent is an equal division of the octave too, with 1200 comprising a single octave.)

In the comparison to the major scale, the largest delta is about 69 cents, which is the better part of a semitone (which is defined as 100 cents in 12-EDO).