Physics of Music - Notes

Scales: Just vs Equal Temperament    (and related topics)

The "Just Scale" (sometimes referred to as "harmonic tuning" or "Helmholtz's scale") occurs naturally as a result of the overtone series for simple systems such as vibrating strings or air columns. All the notes in the scale are related by rational numbers. Unfortunately, with Just tuning, the tuning depends on the scale you are using - the tuning for C Major is not the same as for D Major, for example. Just tuning is often used by ensembles (such as for choral or orchestra works) as the players match pitch with each other "by ear."

The "equal tempered scale" was developed for keyboard instruments, such as the piano, so that they could be played equally well (or badly) in any key. It is a compromise tuning scheme. The equal tempered system uses a constant frequency multiple between the notes of the chromatic scale. Hence, playing in any key sounds equally good (or bad, depending on your point of view).

There are other temperaments which have been put forth over the years, such as the Pythagorean scale, the Mean-tone scale, and the Werckmeister scale. For more information on these you might consult "The Physics of Sound," by R. E. Berg and D. G. Stork (Prentice Hall, NJ, 1995). For an interesting discussion about the historical development of the equal tempered scale, you might read "How Equal Temperament Ruined Harmony," by Ross W. Duffin (W.W. Norton & Co., NY, 2007). For a very complete list of historical temperaments, see the book by Owen Jorgensen listed at the bottom of this page. A table showing a comparison of one meantone temperament with equal temperament can be found here.

The table below shows the frequency ratios for notes tuned in the Just and Equal temperament scales. For the equal temperament scale, the frequency of each note in the chromatic scale is related to the frequency of the notes next to it by a factor of the twelfth root of 2 (1.0594630944....). For the Just scale, the notes are related to the fundamental by rational numbers and the semitones are not equally spaced. The most pleasing sounds to the ear are usually combinations of notes related by ratios of small integers, such as the fifth (3/2) or third (5/4). The Just scale is constructed based on the octave and an attempt to have as many of these "nice" intervals as possible. In contrast, one can create scales in other ways, such as a scale based on the fifth only.

Interval Ratio 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

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).


Just Scale
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!

Listen to the difference:
The first second of this WAV file contains a major triad starting on F# (F# - A# - C#) using the Just scale appropriate for C Major. The last part of the file contains the same triad but using the Just scale appropriate for F# Major. (This is one of the worst case situations).
Tuning Shift WAV file.

Here's another example to test your ears. The following WAV file has two "players" playing a C major scale. One of the players is using the Just Scale, the other the Equal Tempered scale. Both start on exactly the same pitch. See if you can hear the notes where the pitches are different by listening for the beats.
Major scales in different temperaments

Equal Tempered Scale - Table of frequencies

For a detailed list of historical temperaments see:
"Tuning: containing the perfection of eighteenth-century temperament, the lost art of nineteenth-century temperament, and the science of equal temperament, complete with instructions for aural and electronic tuning," by Owen H. Jorgensen, Michigan State University Press, East Lansing, MI 1991. (This is an expensive reference book -- you might wish to look for a copy in your library rather than in a book store).

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