The Musical Scale1

The musical scale is based on octaves. Moving up one octave is defined as doubling the frequency. The octave appears to be an important musical interval in all cultures. You can listen to a .wav file of two notes one octave apart here [180 kb; requires patience]. To a human ear there is an obvious "sameness" about the two notes. But why do they sound similar? The ears can't know about the mathematical relationship. Probably this is connected with the way the brain processes sound, but as far as I know, no one knows for sure.

Three Western Musical Scales

The octave must then be divided into notes. This is where it gets complicated. There are two issues: (1) how many notes per octave, and (2) the "tuning," meaning the frequency ratios from note to note. Standard western scales have 12 notes per octave. There does not seem to be a consensus on the reason for this choice. Three of the possible tunings for the 12 notes are given below. The frequency change from one note to the next is called a "half-step."

Finally a reference tone is required - a "standard of pitch." For most western music "A4," the fourth A from the bottom of the piano keyboard is set to 440 Hz. The one-octave step in the .wav file above is from A4 to A5.

The "Just" tuning is based on ratios of small integers so that harmonics of complex tones will tend to coincide. This avoids beats that occur between two tones of nearly the same frequency. This tuning was used in Europe in the 1600's.

The "Pythagorean" tuning is based on increasing the frequency of a note 7 half-steps higher by a factor of 3/2. The notes at octaves above and below the new note are then also determined. The process is then iterated. This scale dates back to the Greek philosopher, and supposedly relates to the perfect harmony of the heavens. (Unfortunately, if the process is extended to create the original note, the frequency is different than the initial original note, upsetting the perfection).

The "Equal Temperament" tuning ratios are identical from one note to the next. That is, the frequency is increased by a factor of 21/12 for each step. For the other two tunings above, the frequency of the notes differs when the starting notes differ, so each key has a different set of frequencies. For the equal tempered scale the frequencies are the same for all keys.

With A4 at 440 Hz, the frequencies of three notes of a major chord are shown in the table below. The frequencies of the three tunings are quite close. So close that the largest differences are barely perceptible to the human ear.

 Tuning C4 E4 G4 Just 264 330 396 Pythagorean 260.741 330 391.11 Equal Temperament 261.625 329.63 392.00

A chord containing only the three fundamental tones sounds the same, to my ears, for the three tunings. But when harmonics are added, then the tunings sound different. With amplitudes (picked out of the air) of 1/3, 1/5, 1/7, and 1/9 for the first four harmonics, the sound of the chord can be heard here [328 kb, also takes patience], played in the order of Just, Pythagorean, and Equal Temperament. Another comparison between Just and Equal Temperament is provided by Prof. Bryan Suits.

A Little History

Jourdain states that a recreation of Egyptian flutes found in Pharaoh's tombs produce tones very similar to the modern scales. The Pythagorean tuning was used for almost 2000 years. The problem was that music only sounded good in the key from which the scale notes are derived. By the 17th century the equally tempered scale was being adopted.

At the time of Beethoven and before the reference note A4 was lower - around 420 Hz. Therefore all of the music written in that period is now really being played in a different key than for which it was composed! Some Stradivarius violins had to be reinforced to take the tighter strings a 440 Hz A4 requires. A higher tuning leads to a "brighter" sound, and tunings based on an A4 up to 465 Hz have been used in the 20th century.

An interesting discourse on the Just tuning and others, including Mesopotamian Tunings, can be found at the Light Bridge Music site.

Non-Western Scales

According to the Encarta encyclopedia, one of the most important duties of the first emperor of each new Chinese dynasty was to search out and establish that dynasty's true standard of pitch. Most Chinese music is based on the five-tone, or pentatonic, scale, but the seven-tone, or heptatonic, scale, is also used, often as an expansion of a basically pentatonic core. The pentatonic scale was much used in older music. The heptatonic scale is often encountered in northern Chinese folk music.

Arab melodies use tones half-way between western notes, leading to 24 notes. Scales of 22 steps are used in India. At the other extreme, Australian aborigines chant to a 2 note scale.

The distribution of notes within the octave also varies. Music of India theoretically offers 35 tunings. The tunings of the 5 notes of the gamelan music of Bali and Java are intentionally different for each orchestra, so that each has its own harmonic personality.

According to the Native American Flute Forum, American Indians use pentatonic scales, which "are the most widely used musical scales in the world. They are found in China, Tibet, Mongolia, Oceania, India, Russia, and Africa, in the folk songs and hymns of Europe and the United States, and among Native Americans." The site defines a pentatonic scale based on the following criteria:

• the scale must consist of five tones between the root tone and its octave
• there must be at least two half-steps and no more than three half-steps between adjacent tones of the scale, which means,
• given twelve half-steps in an octave, that the scale will have two tones separated by three half-steps and three tones separated by two half-steps
• the two intervals of three half-steps cannot be adjacent to each other.

The pentatonic scale was also used by Incas, and in Africa. Finally it is also used by western composers such as Debussy, and Dvorák.

Back to Music and the Human Ear

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1. Material in this section is based mainly on Chapter 11 in Signal's, Sound, and Sensation, by Hartmann, and on the book Music, the Brain, and Ecstasy, by Jourdain. Another interesting discussion of scales is given by Patel.