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Musical scales

A musical scale is a set of notes used by musicians to play music. Musical scales are not defined by science, though in some ways we can understand why a scale may be the way it is through the application of science.

Musical scales have been known and studied since before the time of Pythagorus (ca. 550 BCE). A detailed list of many experimental measurements of scales, as measured in the mid- to late 1800's, is contained in the book by Helmholtz.

 


You will note that the most "pleasing" musical intervals above are those which have a frequency ratio of relatively small integers. Some authors have slightly different ratios for some of these intervals, and the Just scale actually defines more notes than we usually use. For example, the "augmented fourth" and "diminished fifth," which are assumed to be the same in the table, are actually not the same.

The set of 12 notes above (plus all notes related by octaves) form the chromatic scale. The Pentatonic (5-note) scales are formed using a subset of five of these notes. The common western scales include seven of these notes, and Chords are formed using combinations of these notes.

As an example, the chart below shows the frequencies of the notes (in Hz) for C Major, starting on middle C (C4), for just and equal temperament. For the purposes of this chart, it is assumed that C4 = 261.63 Hz is used for both (this gives A4 = 440 Hz for the equal tempered scale).

 


Note

Just Scale
Equal
Temperament
Difference
C4 261.63 261.63 0
C4# 272.54 277.18 +4.64
D4 294.33 293.66 -0.67
E4b 313.96 311.13 -2.84
E4 327.03 329.63 +2.60
F4 348.83 349.23 +0.40
F4# 367.92 369.99 +2.07
G4 392.44 392.00 -0.44
A4b 418.60 415.30 -3.30
A4 436.05 440.00 +3.94
B4b 470.93 466.16 -4.77
B4 490.55 493.88 +3.33
C5 523.25 523.25 0

Since your ear can easily hear a difference of less than 1 Hz for sustained notes, differences of several Hz can be quite significant!

IntervalRatio to Fundamental
Just Scale
Ratio to Fundamental
Equal Temperament
Unison 1.0000 1.0000
Minor Second 25/24 = 1.0417 1.05946
Major Second 9/8 = 1.1250 1.12246
Minor Third 6/5 = 1.2000 1.18921
Major Third 5/4 = 1.2500 1.25992
Fourth 4/3 = 1.3333 1.33483
Diminished Fifth 45/32 = 1.4063 1.41421
Fifth 3/2 = 1.5000 1.49831
Minor Sixth 8/5 = 1.6000 1.58740
Major Sixth 5/3 = 1.6667 1.68179
Minor Seventh 9/5 = 1.8000 1.78180
Major Seventh 15/8 = 1.8750 1.88775
Octave 2.0000 2.0000