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What Is the Sound Frequency of the Musical Scale?

Throughout history, there have been a myriad of ways to tune an instrument. Europeans were not the only ones to tune to a scale. In fact, the modern idea of equal temperament tuning may come from the Chinese music theorist Zhu Zaiyu in 1584. Today, we use a different type of equal temperament than described in the 16th century. Although these tuning types overlapped in usage, they can be seen as a progression from Pythagorean tuning.

  1. Pythagorean Tuning

    • The first known musical scale was attributed to Pythagoras. By dividing a string two-thirds at a time, he created a "pure fifth." It is a "pure fifth" because the string vibrates at 3/2 of the whole string. The "pure fifth" can also be described as the ratio 3:2. The eleven "pure fifths" and one "wolf" (an imperfect fifth that sounded out of tune) make up the 12-tone scale used in Pythagorean tuning.

    Well-tempered

    • There are numerous ways to well-temper tune an instrument, however, the basic idea was to eliminate all the "wolf" intervals by basing each scale on a pure octave. Because each scale had its own schematics to make pure octaves, the scales sounded very different. "Key color" refers to the unique sound of each scale in well-tempered tuning.

      Johannes Sebastian Bach's "The Well-Tempered Clavier" (1722) is a series of piano compositions for each of the 12 major and 12 minor keys.

    Equal Temperament

    • In equal tempered tuning, each note in the 12-tone scale is tuned to 440 Hz, also known as A4 or A440. Each octave is pure and the 12 tones in between are separated by equal intervals---identical minor seconds. Therefore, when starting at A440, an octave above would be A880 and an octave below would be A220. However, the minor second is not the same as it is in Pythagorean tuning.

      Although the theory of equal temperament dates back to the 16th century, it only became the standard in European music during the mid-18th century.

    Ratios of Vibrations

    • The ratio of vibrations in Pythagorean and well-tempered tuning are:

      Interval Ratio
      Octave 2
      Fifth 3/2
      Fourth 4/3
      Major 3rd 5/4
      Minor 3rd 6/5
      Major 6th 5/3
      Minor 6th 8/5

      The ratio of vibrations in equal temperament are:

      Interval Ratio
      Unison 1.000
      Minor Second 1.05946
      Major Second 1.12246
      Minor Third 1.18921
      Major Third 1.25992
      Fourth 1.33483
      Fifth 1.49831
      Minor Sixth 1.58740
      Major Sixth 1.68179
      Minor Seventh 1.78180
      Major Seventh 1.88775
      Octave 2.0000

    Sound Frequency in Hertz

    • The following are the hertz frequencies of each note in equal tempered tuning starting with C:

      Note Hz
      C4 261.63
      C4# 277.18
      D4 293.66
      E4b 311.13
      E4 329.63
      F4 349.23
      F4# 369.99
      G4 392.00
      A4b 415.30
      A4 440.00
      B4b 466.16
      B4 490.55
      C5 523.25

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