An interesting question, which musicians often overlook, is "what makes a chord sound good ?". In other words, why a particular combination of notes sounds better than another one. With "sounds better" I mean "is percieved as more desiderable" or "gives more pleasure to the listener"... choose the definition that best matches for you.

This article attempts to give at least some hints for answering this simple question.

By sound, we commonly mean the vibrations that travel through air and can be heard by humans. However, scientists and engineers use a wider definition of sound that includes low and high frequency vibrations in air that cannot be heard, and vibrations that travel through all forms of matter, gases, liquids and solids. The matter that supports the sound is called the medium. Sound propagates as waves of alternating pressure, causing local regions of compression and rarefaction. Particles in the medium are displaced by the wave and oscillate.

As a wave, sound is characterized by the properties of waves including frequency, wavelength, period, amplitude and velocity or speed.

The range of frequencies that humans can hear is approximately between 20 Hz and 20,000 Hz. This range is by definition the audible spectrum, but some people (particularly women) can hear above 20,000 Hz. This range varies by individual and generally shrinks with age, mostly in the upper part of the spectrum. The ear is most sensitive to frequencies around 3,500 Hz. Sound above 20,000 Hz is known as ultrasound; sound below 20 Hz as infrasound.

Variations in air pressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the experience called "sound". Most sound that people recognize as "musical" is dominated by periodic or regular vibrations rather than non-periodic ones (called a definite pitch), and we refer to the transmission mechanism as a "sound wave". In a very simple case, the sound of a sine wave, which is considered to be the most basic model of a sound waveform, causes the air pressure to increase and decrease in a regular fashion, and is heard as a very "pure" tone.

Pure tone at 1 Hz.

The rate at which the air pressure varies governs the frequency of the tone, which is also measured in oscillations per second, or hertz. Frequency is a primary determinate of the perceived pitch.

Pitches are often labeled using scientific pitch notation or some combination of a letter and a number representing a fundamental frequency. Human pitch perception is logarithmic with respect to fundamental frequency: the perceived distance between the pitches "220 Hz" and "440 Hz" is the same as the perceived distance between the pitches "440 Hz" and "880 Hz".

The distance between two pitches related by a frequency factor of 2 is called octave. A sound oscillating at 440 Hz (that is, 440 times per second) is called A (A4) and is being used as reference in the widely adopted musical pitch standard (American Standard). All the pitches obtained by multiplying or dividing 440 by 2 are still called A, just belonging to higher or lower octave.

Obviously there is an infinity of other pitches between two adiacent A. However we choose to use only some of them to compose music and the reason of this choice will be depicted later in this document.

Whenever two different pitches are played at the same time, their sound waves interact with each other - the highs and lows in the air pressure reinforce each other to produce a different sound wave. As a result, any given sound wave which is more complicated than a sine wave can, nonetheless, be modelled by many different sine waves of the appropriate frequencies and amplitudes (a frequency spectrum). In humans the hearing apparatus (composed of the ears and brain) can isolate these tones and hear them distinctly. When two or more tones are played at once, a single variation of air pressure at the ear "contains" the pitches of each, and the ear and/or brain isolate and decode them into distinct tones.

Octave: the sum of pure sounds at 1 Hz and 2 Hz. The overall wave still vibrates at 1 Hz.

There are mathematic instruments that are able to decompose complex sounds into combinations of pure sounds. One of such instruments is the Fourier Transform. The spectrum obtained by applying the Fourier Transform to the octave waveform in the figure above would have two peaks (Dirac's deltas) placed at 1 Hz and 2 Hz, that is it would decompose the complex waveform into its pure sinusoidal components.

Qualitative Fourier Transform of an octave interval.

The "natural" sounds usually aren't pure. The waveforms aren't necessairly sinusoidal but may be, for example, square or triangular. However, as long as the frequency of the waveform is well defined, the perceived pitch is the same. The perceived property that changes with the waveform is called timbre.

Non pure waveforms can be still decomposed in combinations of pure sounds by the means of the Fourier Transform. An interesting property of the FT is that the spectrum of a periodic signal (waveform) is made up of peaks that "sample up" the spectrum of the waveform obtained by truncating the original one to a single period. The detailed explaination of the Fourier Transform properties go beyond the scope of this document and can be found out by consulting a book about Signal Theory. It is sufficient to state that the peaks are placed at integer multiplies of the fundamental frequency f (so at 2f, 3f, 4f etc..). The peaks at 2f, 4f, 8f, 16f etc (powers of 2) are octaves while the others are just partial harmonics.

Very Qualitative Fourier Transform of a periodic triangle wave

Thus when the original sound sources are perfectly periodic, the note consists of several related sine waves (which mathematically add to each other) called the fundamental and the harmonics, partials, or overtones. The sounds have harmonic frequency spectra. The lowest frequency present is the fundamental, and is the frequency that the entire wave vibrates at. The overtones vibrate faster than the fundamental, but must vibrate at integer multiples of the fundamental frequency in order for the total wave to be exactly the same each cycle. Real instruments are close to periodic, but the frequencies of the overtones are slightly imperfect, so the shape of the wave changes slightly over time.

Finally, the combination of non perfectly periodic instruments, non sinusoidal sources and complex interactions with the transmitting medium (air) lets us perceive a large set of frequencies when a single note is played. The spectrum of such a sound is usually still made up of peaks but in certain cases may even end up being continous.

The spectrum of a playing violin.
The frequency is on the Y axis while X axis rappresents time.
The bright lines along the bottom are fundamental sounds while the upper ones are harmonics and overtones.

If two pure notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave will still be periodic with a short period, and the combination will sound consonant. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave will repeat thrice and the 200 Hz wave will repeat twice. Note that the total wave repeats at 100 Hz, but there is not actually a 100 Hz sinusoidal component present.

Perfect fifth: the sum of pure tones at 1 Hz and 1.5 Hz. The wave repeats at 0.5 Hz.

Additionally, when considering real instruments, the notes will have their harmonics and partials. It ends up that when the two notes played have simple fraction frequency ratios then they also will have many of the same partials. For instance, a note with a fundamental frequency of 200 Hz will have harmonics at (200,) 400, 600, 800, 1000, 1200, ... A note with fundamental frequency of 300 Hz will have harmonics at (300,) 600, 900, 1200, 1500, ... The two notes have the harmonics 600 and 1200 in common, and more will coincide further up the series. The combination of composite waves with short fundamental frequencies and shared or closely related partials is what causes the sensation of harmony.

The spectrum of a violin playing first the fundamental note and then a perfect fifth above it.
The shared partials are highlighted by white lines.

For several (historic and physic) reasons we choose a set of pitches between two octaves to form our contemporary music. The notes choosen turn out to relatiely simple frequency ratios when played together and thus sound "naturally" consonant.

The natural scale is attributed to the Greek philosopher Aristoxenus Tarentinus. After fixing the frequency of the first note - the C of the scale - the frequencies of the other notes are determined from the ratios indicated in the following table. On the last C the following octave begins and the operation can be repeated. The following table shows the ratios between the frequencies of all the notes of the scale and the fixed frequency of the first note of the scale.

The interval between C and G is our perfect fifth as depicted above.

There is one major problem with the natural scale and the just intonation derived from it. The ratio of the frequencies of two notes which differ for one tone is not always the same. Consequently a certain melody cannot be played starting from a random note of the scale. For instance, a melody starting with the two notes C and D (ratio 9/8) cannot be transposed one tone higher, since the ratio of the frequencies of E and of D is very near ((5/4)/(9/8) = 10/9), but not equal to 9/8.

To obviate this inconveniency, we today use the so-called equal temperament, which constitutes the compromise adopted in modern western music. Earlier western music used other compromises.

Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). The best known example of such a system is twelve-tone equal temperament which is nowadays used in most Western music. Other equal temperaments do exist (some music has been written in 19-TET and 31-TET for example, Arabian and eastern styled music is based on a 24-tone equal temperament).

Mathematically, each successive pitch is related to the previous pitch by a factor of the twelfth root of 2. That is, the ratio between the frequencies of any two successive pitches is 1.05946309436. This is also the value of the ratio of the widths of two consecutive frets on modern guitars. The twelfth fret divides the string in two exact halves (octave).

The first "usable" pitch above the reference A is thus 440 * 1.05946309436 = 466.1637615184.

12-TET (Tone Equal Temperament) allows the use of integer notation and modulo 12, and this allows for proofs concerning pitch.

The following table shows the values of the intervals of 12 TET, along with one interval from just intonation that each approximates, and the percentage by which they differ:

Obviously the 12-TET leads to "non perfectly simple" frequency ratios when notes are played together and thus leads to slightly dissonant chords. However, the dissonances aren't easily audible.

Above, a perfect fifth, Below, a 12-TET fifth

Above, a perfect fifth, Below, a 12-TET fifth