Scales: General Comments

A (musical note)

Designation by octave

Common scales beginning on A

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.

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).

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.
Major scales in different temperaments

Common Western Musical Scales

The spacing between adjacent notes on the chromatic scale is referred to as a half step. The number of half steps between adjacent notes for the common musical scales used in western music are:

* Major: 2-2-1-2-2-2-1
* Natural Minor: 2-1-2-2-1-2-2
* Harmonic Minor: 2-1-2-2-1-3-1

which are all 7-note (heptatonic) scales.

The major and natural minor scales occur in pairs which share the same set of notes, but start in a different place. For example, if you stick to the white keys on the piano and start your scale on C, then it is the C major scale. If you start on A, it is the A minor scale. Since there are seven possible starting notes, you get seven possible "modes." The Greeks gave them all names.

Modes Using Notes ABCDEFG

(Note: in 16th century Europe, Glarean of Basle assigned names to the modes using many of these same Greek names. While Glarean's assignments are in more common use today, they do not match those of the ancient Greeks. See Jean's book, page 168, or Helmholtz's book, pages 245 & 269 for more info on the different assignments)

Changing the starting note can give a different feel to the music and can be applied to any scale. For example, if you start the Harmonic minor scale on its fifth note you get intervals of (1-3-1-2-1-2-2) which is sometimes called the "spanish gypsy scale."

Other scales include the five note pentatonic scales and the six note blues and whole-tone scales.

If you want to try these on a keyboard but don't have any experience, refer to this figure:
keyboard figure

Chords - Frequency Ratios
A chord is three or more different notes played together. Michael Keith (see ref below) computed that for the equal tempered scale there are "351 essentially different chords."

The interval between adjacent notes on the chromatic scale is referred to as a half step. The number of half steps between adjacent notes in common three and four note chords, and their frequency ratios, are shown in the table below.

The fundamental beat frequency associated with a chord can be determined by looking at the repeat period - that is, for the frequency ratios given above (which are reduced to the lowest possible integer values), the repeat period for the major chord is 4 times the period of the lowest note in the chord. For the 7th, it is 20 times that of the lowest note. Since f = 1/T, the fundamental beat frequency for the major chord is 1/4th the frequency of the lowest note, and for the 7th, it is 1/20th the frequency of the lowest note. If you listen carefully, you can hear the beat frequency as an additional unplayed note.

What makes a chord sound consonant or dissonant depends upon human physiology and psychology. One "rule" is based on work by Helmholtz and relies on "overlapping harmonics." A nice explanation is contained in the article by Jan Wild listed below. Basically, for each pair of notes in the chord, find the lowest harmonics which match. If it is the 8th or less in every case, the chord is consonant. For example, the major triad has frequency radios of 4:5:6. The harmonics of the lowest note are then 4, 8, 16, 20, 24, etc. and the harmonics of the second are 5, 10, 15, 20, 25, etc. The fifth of the lower matches with the fourth of the upper so this interval should be consonant. (i.e. they are both less than the 9th harmonic). One gets a similar result for 4 and 6, and 5 and 6. Hence, the entire major triad is consonant.

If you try to use the rule of eight and the equal tempered scale, you will have to consider harmonics which "almost match" since none of them, except the octaves, will ever exactly match.

Based on this "rule of 8" the "nice" three note chords which start on middle C are:

plus those where the notes are related by octaves. These are (in order) C-minor, Ab-major, C-major, A-minor, F-minor, and F-major chords.

Chord progressions are the basis of most western music. If a tune stays in one key, then the basic chords are triads starting on each of the different notes of the scale. Since these chords are often expressed in terms of the root of the chord, this fact is not always clear. Many tunes will use just the three chords based on the fundamental (I), the fourth (IV) and the fifth (V). Sometimes a fourth note is added which is a third above the highest note of the triad. This gives a 7th chord. When a minor third is used it is a note which is not actually part of the original scale, but it sounds cool.

Simple Triads for C Major scale

* The roman numeral indicates the starting note of the triad, and upper and lower case signify whether it's a major or minor triad respectively. The b minor triad is B, D, and F#. When played with B, D, and F it is referred to as a "diminished" chord.

Particularly common is the I-IV-V progression. For some simple chord progressions and examples of tunes which use them, see, for example, Olav Torvund's site.

The semitone is always dissonant. Michael Keith concludes that the only 30 of the possible 351 chords have no semitone intervals, and that in fact there are only 12 which are "musically distinct."


Some references:

J. Wild, "The computation behind consonance and dissonance," Interdisciplinary Science Reviews, Vol 27, No. 4, p 299 (2002).

H. Helmholtz, "On the sensation of tone as a physiological basis for the theory of music," translated by A. J. Ellis, (Dover, NY, 1954).

Michael Keith, "From Polychords to Polya: Adventures in Musical Combinatorics," Vinculum Press, Princeton, N.J., 1991, ISBN 0-9630097-0-2.


Pentatonic Scales
Pentatonic ("5-note") scales appear in many cultures, and even within some modern Western music (Such as Blues, Jazz, and some Country Music).

There are two common pentatonic scales.

The intervals for the two scales are:

* Major: 2-2-3-2-3
* Minor: 3-2-2-3-2

where the numbers above represents the number of half-steps between the notes. You can, of course, create other interesting 5-note scales.

On a keyboard, you can play such a pentatonic scale by using only the black keys. Start on D# for the minor scale, F# for the major scale.

The frequency ratios for these notes relative to the first note in the scale are: