Musical Scales

If, like me, you aren't a musician, this article might help in understanding musical scales


I recently purchased a Radio Shack MD501 MIDI keyboard to experiment with, and perhaps to learn a little about music and musical scales. The cost was only $40, and well worth it, it turns out. The keyboard has small keys, but 49 of them in four octaves, which is a useful range (a piano has six). What the keyboard calls "tones" are really voices. Voice 49, Synth Effect 2, seems most useful for our present purposes, since it produces a sustained tone that is nearly pure. Of course, the keyboard can sound like a piano, flute, trumpet, and so on, or at least try to. The phone jack did not work, so I took off the back and connected a counter and an oscilloscope across one of the speakers. The power supply is 9VDC 300 mA from a wall adapter, but the polarity on the barrel plug had to be reversed. Incorrect polarity won't hurt the keyboard, but it will not work. The center is +, the opposite of the usual polarity.

The first thing to do is to measure the frequency of the notes. The signals are not sine waves, so the counter may do funny things. There were problems at the lowest and highest frequencies, but in general my counter (which needs replacing) gave reasonable results. Middle c is at the centre of the keyboard, not surprisingly, and gives about 262 Hz. The correspondence between the notes and the keyboard can be deciphered from the pattern c*d*ef*g*a*b, which is repeated four times across the keyboard, black keys being represented by an *. The keyboard is tuned to a' = 440 Hz, which I think is called "concert pitch." The keyboard runs from C to c''', four octaves and an extra note, which should be 65 Hz to 1047 Hz. This covers the frequency range for most of the information in speech and music, though higher and lower tones may contribute to the quality of the sound. Note that the telephone passes 300 Hz to 3000 Hz. Harpsichords and spinets, with plucked strings, usually covered four octaves, but the compass of the piano is as much as eight octaves. The waveforms can also be observed with the oscilloscope, and their complexity will be obvious. Many notes have a strong second harmonic component. To test this, play the octave as well, and see if the beats are reasonably slow, which means that there is some phase coherence.

The absolute frequencies used for the scale depend on what is chosen for any one note. At a convention of German physicisists at Stuttgart in 1834, a' = 440 Hz (a' is the A above middle C) was recommended, and is the standard pitch today. The French Academy decreed a' = 435. If c' = 256 (a' = 426.67), all the C's would have frequencies that are powers of 2, and this pleased physicists for some reason. The pitch was lower in the 17th century, and has increased since. With a' = 440, the eight octaves covered by the piano extend from C,, = 16.35 Hz to c'''' = 3951.0 Hz, of which the middle four octaves, from 66 Hz to 990 Hz, are most used. Large and small letters, used with marks below or above, are used for notes in the different octaves, changing between B and C. B is followed by c, and b by c', for example.

The importance of the physical quantity frequency to the subjective quality pitch is obvious, but frequency is not all there is to pitch. Usually the pitch is that of the fundamental, or lowest frequency, of the sound, modified by its upper harmonics, which are integral multiples of the fundamental frequency, for many sources of sound. For our current purposes, we'll assume that pitch and frequency are about the same thing. A pitch interval is a ratio of frequencies, not a difference. The octave is a factor 2. If we divide an octave into 12 equal intervals, then the frequency ratio corresponding to one interval is the 12th root of 2, or 1.0595. The keyboard has 49 keys, so there are 48 intervals, if the interval between neighboring keys, white or black, is the same. Now, 1.059548 = 16. 65.438 Hz, which is the lowest note on the keyboard, times 16 is 1047 Hz, the highest note. We have an instrument that can produce pitches in fixed ratios, and wish to discover why that is so.

The keyboard, or any musical instrument, is intended to produce sounds that are pleasing when heard in sequence, as in a melody, or at the same time, as in a chord. The pleasing pitches were long ago found to be described by small whole numbers, as in the lengths of the strings that produced them, or in their frequencies of vibration. Music was studied mathematically in the ancient world, and there were many such collections of pleasing pitches, generally called modes. We are a little poorer, only the major and minor modes having survived, and a few others in folk music. The major diatonic scale consists of notes with frequencies in the ratios 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 and 2 relative to a key note. The ratios between adjacent notes are 9/8, 10/9, 16/15, 9/8, 10/9, 9/8 and 16/15. These notes are called C, D, E, F, G, A, and B or do, re, mi, fa, sol, la, ti. This is called a diatonic scale with a keynote C. Such a scale can be formed with any note as keynote by multiplying by the same factors. We are so used to this scale that we can sing it off naturally, though it is a learned thing.

If we examine the intervals between successive notes, we find they are not equal. The interval between C and D, 9/8 = 1.125, is called a major tone, while the interval between D and E, 10/9 = 1.111, is called a minor tone. The following interval between E and F is only 16/15 = 1.067, and is called a semitone. If we call either a major or minor tone just a tone, then a semitone squared is only a little larger, 1.137, so rather loosely we can speak of tones and semitones, with a tone equal to two semitones (i.e., the square in frequency ratio). The major scale is then tone, tone, semitone, tone, tone, tone, semitone, and this sequence of intervals defines the scale.

Now suppose we want to start on G as the keynote. A and B give us tone, tone, and C is conveniently a semitone above B. D and E give us tone, tone--so far, so good--but next we need another tone, and F is only a semitone away. We need a note a semitone above F, called F# (F-sharp) in this scale. Now we need a semitone, and fortunately G is just a semitone above F#. We can play a scale of G if someone puts a note between F and G. Try this out on the keyboard. Play a scale of C on the white keys. Then begin at G and play the scale up to E. Instead of F, play the black F#, and then finish with G. Try it both ways, and notice the difference. So that the keyboard can make any key, that is, give a semitone where needed and a full tone where needed, we need to put a black key in the "middle" of each full tone on the keyboard. Black keys will not be needed between E and F, or between B and C, since they are already a half-tone apart. This means that we need 12 intervals in each octave, not 8, so the frequencies should differ by the 12th root of 2.

This also means that we make the semitone 1.0595 (instead of the correct 1.067), and force every tone to the ratio 1.1225 (instead of the correct 1.125 and 1.111). It appears that the discrepancies are too small to affect the concordance of the music, though there must be subtle differences in quality. The resulting 12-tone scale is called the chromatic or even-tempered scale. The diatonic scale is called just temperament. Playing music in a different key is called transposing, and can affect how the music sounds. Composers write music in certain keys for subtle effects, and the music must be played in those keys to realize these effects. Music must be raised or lowered in pitch to accomodate the ranges of singers, and full octave differences are too large. The chromatic scale makes pianos and keyboards that can play music in any key possible, but it is a compromise.

An equal-tempered keyboard is shown at the right. There are 12 keys in each octave, the notes differing by the same interval, 21/12, which is a semitone. These are divided into two groups. The 7 white keys make up a heptatonic scale approximating the diatonic scale. The 5 black keys consititute a pentatonic scale. Scottish and Irish music is very often pentatonic, and simple songs can easily be picked out on the black keys alone. The Chinese discovered a heptatonic scale, and even transposition, but most Chinese music has remained pentatonic.

Both Rayleigh and Lamb, in their texts on Sound, devote the first chapter to the relation of hearing and acoustics--that is, to the distinction between subjective and objective aspects--and include an explanation of musical scales. Both treatments are excellent, and Rayleigh's insights particularly so. Here, I will give briefly the theory behind the usual musical scales, as I understand it from this and other sources. I hadn't given the matter much thought before, though of course I have encountered music since my days in the church choir. It was always a mystery.

As in all the senses, the perception of sound is determined by mental processes. It is not simply a Fourier analysis of the aerial vibrations reaching our ears. The purpose of hearing is to inform us of our surroundings, not to record frequencies. The relation between the perceived tone and the physical stimulus is complex, but on an elementary level the simplest sensations of pitch are related to the frequency of sinusoidal vibrations, an observation known as Ohm's Law. This is all we will need here, leaving aside the many interesting auditory 'illusions' and paradoxes.

It has been anciently known that musical harmonies have something to do with the ratios of small integers. These are, in fact, frequency ratios, as established by Mersenne in the 17th century. The reason why notes with frequencies in these ratios, sounded sequentially or simultaneously, are harmonic is not precisely known. Helmholtz showed that it has something to do with the higher partials of the notes. When they are all in the same ratios of small numbers, then harmony results, otherwise dissonance. This is closely connected with Ohm's Law, and with the model of the cochlea (inner ear) as a resonant body. Nonlinearities of the ear, causing the creation of sum and difference tones, may also enter. Whatever the reason, the following ratios within the octave are perceived to be harmonic:

Some ratios apparently absent are actually there. For example, 5:2 is 2.5 or 2 x 1.25, an octave and a third. 3:1 is an octave and a fifth. All the ratios for the small integers up to 8 are present, except for those involving 7, which are absent. A musical scale is a collection of notes from which as many of these harmonic intervals can be realized as possible. The word interval is used for a particular frequency ratio. It is characteristic of our senses to recognize ratios, not arithmetic differences. In musical intervals, the intervals are compounded (multiplied), not added. Inverting an interval is subtracting it from an octave, which means dividing 2 by the ratio. This calculation with ratios is fundamental.

For example, if we consider the major third c and e (note that "third" comes from counting the three keys from c to e in Roman fashion, including start and finish), then its inversion is e to c', which is six keys, so this is a sixth, actually a major sixth. The inversion of the fifth c and g is g and c', a fourth. If you have a keyboard, play these inversions and you will discover that they sound almost alike!

We accept with little wonder that a note and its octave (and third harmonic or twelfth as well) sound about the same, but this is really a remarkable thing. There is no way in which one color could be similiar to its octave, even were the compass of vision sufficient. Helmholtz's concept of a resonant cochlea, however, gives a ready explanation, since the same oscillator will be excited by all the harmonics of a tone.

A minor third is c and d#, an interval of a tone and a half, while the major third is c to e, an interval of two tones. The inversion of the minor third is the major sixth. These intervals have quite a different feeling, but are also harmonic, though to most ears not as harmonic as the major intervals. In all these cases, the relations between the overtones are the important thing in music, and there is a very complex theory about what is pleasant and what is not, and what conventions must be followed in writing music. These "minor" intervals occur in a minor scale which is slightly different from the major scale we have been considering, but contains similar intervals.

The Greeks invented the diatonic scale with intervals that can be represented by - - u - - - u , where - is a full tone and u is a semitone. They permitted a scale to start on any one of these notes, with the result that there were seven modes. Note that what is meant is not that a diatonic scale was begun on each note, but that the same notes were used as they came in each scale. The diatonic scale of C was called the Lydian mode, what is now called C major. If we begin with D instead, the intervals are - u - - - u -, which is called the Phrygian mode. If this scale is transposed so that it begins at C, then the notes are C, D, Eb, F, G, A, Bb, C (the "b" stands for the "flat" symbol, which HTML does not support). This is now called the scale of C minor. By using the same intervals, a minor scale can be created for any keynote.

The scale beginning with E, and intervals u - - - u - -, was the Doric mode, followed by the Hypolydian, Ionic, Eolic and Mixolydian modes. The Doric and Hypolydian modes were unmelodic, but could still accompany voices. All these modes were donated to early church music; all but the Lydian and Phrygian have disappeared.

In pentatonic Celtic music, the intervals are - - -u - -u, or C, D, E, G, A, C. The fourth (F) and seventh (B) with respect to the keynote are missing from this scale. Curiously, these are just the intervals between the black keys, so a Celtic scale can be played on them alone, as we have said. The Highland bagpipe plays only nine pentatonic notes. Five different modes can be constructed by starting on each of the five notes as keynotes, just as the Greeks did with the heptatonic scale. At least four of these modes have been used. It is very common for early musical traditions to reject the semitone. On the other hand, even smaller intervals, such as the quarter-tone, have been introduced, but have been rejected in Greek and European music.

A common chord consists of a note, its major third, and its fifth. The frequencies are in the ratio 1:5/4:3/2, or 4:5:6. The fifth above the note is called the dominant, and the fifth below the note (2/3 the frequency) is called the subdominant. To construct a diatonic scale based on a keynote, we find the common chords on the note, its dominant, and its subdominant. Any notes outside the octave we transpose to lie within the octave by multiplying or dividing by 2. When this is done, we find seven notes, labelled C (the keynote, 1), D (9/8), E (5/4), F (4/3), G (3/2), A (5/3), B (15/8). The next note is C in the next octave, and so on above and below. An interval is named by counting the notes from C = 1. That is, G is the fifth note, and C:G is a fifth. The second and the seventh are not considered harmonic. We see that a musical scale is a collection of pitches, arranged in a monotonic sequence of frequency, that are usually harmonic and pleasant when sounded sequentially or simultaneously in the performance of music. Musical scales in general, however, do not depend on harmony, which is a late development, though current musical theory is dominated by it.

The four strings of the violin are tuned to g, d', a' and e". We note that the interval between each string is a fifth, so that if one string is tuned to the correct pitch, say a' = 440, the others can then easily be tuned to their proper pitches by ear. A violin plays with just temperament, unless the player purposely pulls the notes a little sharp or flat to agree with the piano.

The frequency ratios 1:2:3 are given by the keynote, octave and twelfth (a third beyond the octave). The ratios 2:3:4 are the keynote, fifth and octave (as can be found by dividing by 2). 3:4:5 is the same as 1:4/3:5/3 or keynote, fourth and major sixth. All these chords, as simultaneous sounding of more than two notes is called, are harmonic. There is also a great deal of theory respecting chords.

The diatonic major third is an interval of 9/8 plus 10/9, or 90/72 = 5/4 = 1.2500. The equal-tempered third is an interval of 1.2600. The sharpness of the equal-tempered third is now familiar and accepted, though sensitive ears once deplored the difference. The difference between the intervals 9/8 and 10/9 is (9/8)/(10/9) = 81/80 = 1.0125, a ratio that appears frequently in discussions of temperament, and which is called a comma. The equal-tempered chromatic scale ignores differences of a comma.

A very interesting experiment that shows the distinction between diatonic and equal-tempered scales can be performed with the keyboard. Play the major triad CEG, and note how smooth and pleasing its consonance is. CE is a major third, EG a minor third, and CG is a fifth, so it contains all the consonant intervals when inversions are considered. Indeed, as we have seen it can form the basis for constructing any diatonic scale. The black key between G and A can be called G# or Ab, and they are the same note on the equal-tempered chromatic scale. The notes E and G# have the frequency ratio 5/4 in just intonation, so that EG# is a major third. The third is easily demonstrated by playing the two notes. CE remains a major third, while CG# is a minor sixth. All of these intervals are consonant, so that CEG# should be a pleasant chord when played. Indeed, this is true on an instrument with just temperament. However, the interval EAb has the frequency ratio 32/25 = 1.280, and is decidedly dissonant. When it is played on the keyboard, we get an interval of 1.260, closer to the major third 5/4 = 1.250 instead, which is pleasant. However, now play the chord CEAb--it will be found to be quite dissonant, and rather unpleasant. It is the higher partials of the C that cause the dissonance. The chord CEG# is best avoided with equal temperament, since our ears detect the fraud.

Note Diatonic Equal Temperament
C 1.000 1.000
C# - 1.060
D 1.125 1.122
D# - 1.189
E 1.250 1.260
F 1.333 1.335
F# - 1.414
G 1.500 1.498
G# - 1.587
A 1.667 1.682
A# - 1.782
B 1.875 1.888
The equally-tempered intervals are generally regarded as satisfactory, since the ear seems to be insensitive to small differences in frequency. Any beats produced by small differences in frequency are below the limit of perception, and this also seems to help, especially when the notes are played rapidly and not held. The interval between the diatonic and equally-tempered notes is not greater than 0.99. Since the intervals are all equal, you can begin a diatonic scale on any note, and there will always be notes available to complete your scale. This is the great advantage of equal temperament for instruments that produce fixed frequencies, such as pianos and organs, and even those with finger holes. The voice and strings (without frets) can adjust to an exact diatonic scale. The errors in frequency will be on different notes when different keynotes are used, and this may have some effect on the music.

The notes are represented as shown in the Figure in the traditional musical notation. The lines and spaces represent the notes in the diatonic scale. When a different keynote than C is taken, sharps and flats are written beside the clef (the S or C at the beginnings of the lines) to show that these other notes are to be sounded instead. On the piano, these notes are played with the black keys. In our example above of the key of G, we found that F# was to be played, not F. Therefore, a # is placed on the line representing F (the top one). Whenever you see a # in this position, in means that the key of G is in use. When a note is to be played that is not in the particular diatonic scale, it is distinguised by a 'natural' symbol to tell one to ignore the sharp or flat. This is getting perilously close to the outer limits of my knowledge, and is only for the nonmusical curious. There is, of course, a large amount of theory and lore in musical notation that goes far beyond the mere specification of pitch.

Harmonic music, where more than one part sounds simultaneously with others in harmonic relationships, is so familiar to us that it is remarkable that it is a rather recent development, only dating back to Palestrina (1524-1594), and is of modern European origin, though based, of course, on music from classical times. Most musical traditions are homophonic, with only one voice or possibly voices an octave apart, depending entirely on sequential harmony. Voices and simple instruments naturally produce this kind of music, which was extremely well-developed in ancient music, and gave rise to musical scales. In early medieval times, polyphonic music came into fashion, in which more than one voice sang independent melodies that fit together in very complex ways. Rounds, fugues and madrigals are descendants of polyphony, but rather simple ones. Harmonic music was apparently developed to substitute for polyphony with untrained voices, such as church congregations, at the Reformation. Until this time, organs were used mainly as complex pitch pipes to steady choir voices, but afterwards the development of harmony released a great surge of musical creativity, and music was composed that was simply to be heard, not sung or to be a mere accompaniment to recitation. At the present time, popular music seems to have sunk back into a repetitive homophony!

All this is getting perilously close to the outer limits of my knowledge, and I hope I have not said anything absolutely wrong. There is, of course, a large amount of theory and lore in musical notation that goes far beyond the mere specification of pitch, but an understanding of musical scales is an essential part of general education. Buying and playing an inexpensive electronic keyboard is a very rewarding activity.

References:

  1. H. Lamb, The Dynamical Theory of Sound, 2nd ed. (London: Edward Arnold and Co., 1925) Introduction. Horace Lamb is better known for his equally excellent text on Hydrodynamics.
  2. H. von Helmholtz, On the Sensations of Tone (New York: Dover, 1954) The Ellis translation with notes.
  3. Lord Rayleigh, The Theory of Sound, 2nd. ed. (London: Macmillan, 1926, 2 vols.). The first edition was dated 1877, the second 1894. Chapter I. Rayleigh is unsurpassed as an expositor of physics.
  4. A. E. E. McKenzie, Sound (Cambridge: Cambridge U.P., 1963). There is an exellent chapter on musical scales.
  5. Sir W. Bragg, The World of Sound (New York: Dover, 1968). Text of the Royal Institution Christmas Lectures for Young People, 1919.


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Composed by J. B. Calvert
Created 30 June 2000
Last revised 9 July 2005