SIR JAMES JEANS
SCIENCE & MUSIC

CHAPTER V

HARMONY AND DISCORD

In the present book, we are dealing with subjects which lie partly within the province of science and partly within that of art, and the boundary between the two provinces is not always perfectly clear. If the question is debated as to whether the music of John Sebastian Bach is superior to that of his son Philipp Emanuel, science can bring nothing to the discussion. The question is purely one for artists, and it is quite conceivable, although perhaps rather improbable, that they may not be able to agree as to the answer. On the other hand, if the question is whether the music of either Bach is superior to that produced by a chorus of cats singing on the roof, there will be little doubt as to the answer. The artists will all agree, and science is able to explain to a large extent why they agree.

To say the same thing in another way, the aim of music is to weave the elementary sounds we have been discussing into combinations and sequences which give pleasure to the brain through the ear. As between two pieces of music both of which give pleasure in a high degree, only the artist can decide which gives more, but the scientist can explain why some give no pleasure at all. He cannot explain why we find Bach specially pleasurable, but he can explain why we find the cat music specially painful. And this brings us to the subject of the present chapter---why is it that some combinations of sounds are agreeable to the ear, while others are disagreeable?

Through Beats to Discord

On p. 49 we imagined two tuning-forks sounded together. The pitch of one was kept fixed at 261 vibrations to the second, while the pitch of the other started at 262, and was gradually raised. As the pitch rose beats were heard, for a time, but subsequently could no longer be distinguished as such. The sound of the combined tones began by being pleasant to the ear, but as the number of beats per second increased, it gradually became more unpleasant. The unpleasantness reached a maximum when there were about 23 beats to the second, and then began to decline. Brues, extending the experiment into the region in which beats can no longer be distinguished, finds that the decline is only slight, and that, broadly speaking, the unpleasantness remains at a fairly uniform level until the octave of frequency 522 is reached, at which point it suddenly disappears.

If the same experiment is performed with violin-strings, very different results are obtained. The unpleasantness no longer stays at a fairly uniform level, but fluctuates wildly. It almost vanishes at the interval of a major third, and again at the interval of a fourth, while it disappears completely at the intervals of the fifth and octave. At the exact points at which these minima of unpleasantness are reached, the frequency ratios of the variable to the fixed tone are found to have the simple values 5:4, 4:1 3:2 and 2:1.

Concord associated with Small Numbers

It is found to be a quite general law that two tones sound well together when the ratio of their frequencies can be expressed by the use of small numbers, and the smaller the numbers the better is the consonance. This will be clear from the following table, in which the intervals are arranged in order of increasing dissonance:

Interval

Frequency ratio

Largest number occurring in ratio
Unison

1 : 1

1
Octave

2 : 1

2
Fifth

3 : 2

3
Fourth

4 : 3

4
Major Third

5 : 4

5
Major Sixth

5 : 3

5
Minor Third

6 : 5

6
Minor Sixth

8 : 5

8
Second

9 : 8

9

and so on.

In brief, the farther we go from small numbers, the farther we go into the realms of discord. This was known to Pythagoras 2500 years ago; he was the first, so far as we know, to ask the question, "Why is consonance associated with the ratios of small numbers?" And although many attempts have been made to answer it, the question is not fully answered yet.

The central Pythagorean doctrine that "all nature consists of harmony arising out of number" provided of course the simplest of all answers, but only by building on an unproved metaphysical basis. An answer on equally uncertain foundations was given by the Chinese philosophers of the time of Confucius, who regarded the small numbers 1, 2, 3, 4 as the source of all perfection.

Euler's Theory of Harmony

In 1738 the mathematician Euler attempted an explanation on psychological lines, saying that the human mind delights in law and order, and so takes pleasure in discovering it in nature. The smaller the numbers required to express the ratio of two frequencies, the easier it is---such was his argument---to discover this law and order, and so the pleasanter it is to hear the sounds in question. Euler went so far as to propose a definite quantitative measure of the dissonance of a chord. His plan was to express the frequency ratio of the chord in question by the smallest numbers possible, and then to find the smallest number into which all these could be divided exactly. This last number, he thought, gave a measure of the dissonance of the chord. For example, the frequency ratio of the notes of the common chord C E G c' is 4 : 5 : 6 : 8. The measure of dissonance is accordingly 120, since this is the smallest number of which 4, 5, 6 and 8 are all factors.

It is easy to criticise this theory from all sides. In the first place it fails to explain the facts, since it assigns the same measure of dissonance, namely 120, to the chord of the seventh C E G B (with frequency ratios 8: 10: 12 : 15) as to the far less dissonant common chord. Again if we put one note, say E, out of tune by one per cent. of its frequency (about a sixth of a semitone) we increase Euler's measure of dissonance 100-fold; if we now reduce the out-of-tuneness to a tenth of this, we increase the measure of dissonance another tenfold. If one note is only infinitesimally out of tune, the measure of dissonance at once shoots up to infinity, which is a complete reductio ad absurdum. Finally, Euler's theory fails to explain why we enjoy hearing the common chord, with its 120 units of annoyance, when we could reduce the annoyance to 24 units by dropping E out of the chord, and could eliminate the annoyance altogether by sitting in silence. It must be admitted, however, that this is a defect of most theories of discord. Innumerable theories are ready to tell us the origin of the annoyance we feel on hearing a discord, but none even attempts to tell us the origin of the pleasure we feel on hearing harmony; indeed, ridiculous though it may seem, this latter remains one of the unsolved problems of music.

If we were compelled to attempt a solution, it would perhaps be somewhat on the following lines. The exercise of any of his faculties gives pleasure to a healthy being---otherwise he would never attempt crossword puzzles or mountain ascents---and the greater the use made of the faculty the greater the pleasure, at any rate within limits. We like to hear CG rather than C because the irritation produced by the very slight discordance of the notes is far less than the pleasure added by the hearing of the G. On the other hand, we do not enjoy hearing CC#, because the annoyance is so great that the balance swings in the opposite direction.

D'Alembert's Theory of Harmony

So far all theories of harmony had been either arithmetical or metaphysical. The first attempt at a physical theory of harmony originated with another mathematician, d'Alembert (1762), who admitted his indebtedness to some earlier speculations of Rameau (1721). Their theory was based on the fact that every fundamental tone heard in nature is accompanied by its second harmonic (the octave), by its third harmonic (the twelfth), and so on. The interval between the octave and twelfth being a fifth, they argued that it was "most consonant to the scheme of nature" that two notes a fifth apart should sound together, and so on.

Helmholtz's Theory of Harmony

Then Helmholtz (1862) developed a theory of consonance and dissonance in terms of beats---a theory which has been much discussed and criticised, but still holds the field to-day. We have already seen that C and C# sound badly together because they make unpleasant beats. In the case of wider intervals such as C and F# there are no beats to be heard, either pleasant or unpleasant, but Helmholtz asserted that C and F# sound badly together because certain of their harmonics (e.g. g' and F#) make unpleasant beats. On the other hand C and G sound well together because few of their harmonics beat badly:

C c' g' c" e" g" etc.
G g' d" g" b" d'" etc.

indeed many harmonics are common to both notes. On this theory the octave becomes the most perfect of all concords, since none of the harmonics can possibly beat worse than when one note is sounded alone. The theory is sometimes stated in the slightly different form that two notes sound well together when, and because, they have certain harmonics in common, but this form of statement overlooks the annoyance which may be introduced by such harmonics as are not possessed in common.

The theory explains at once why the dissonances of tuning-forks (p. 153) are so completely different from those of musical instruments---the tuning-forks have no upper harmonics to make beats with one another.

A few simple experiments with orchestral instruments, or at the keyboard of the organ, will convince us of the essential soundness of the theory. If we draw a flute-stop, and sound the chord C E G c', we hear no perceptible dissonance. If we now sound the same chord on a stop in which the harmonics are more developed than in the flute, the dissonance is more marked; the dissonance must have been introduced by the harmonics, since nothing else has been added which could have introduced it. It is noticeable on the diapason, and becomes unpleasant on the trumpet or clarinet. Finally it becomes intolerable on the mixture, a stop which consists of harmonics and nothing but harmonics; we shall, for instance, hear c" (the second harmonic of c') beating badly with b' (the third harmonic of E), and g" (the third harmonic of c') with g#" (the fifth harmonic of E). If we hold the chord C E G c', and add suitably chosen stops in succession, we shall hear the dissonance growing pari passu with the harmonic development.

It is the same in the orchestra; chords which sound well on the flutes or strings are impossible on oboes and clarinets. We understand the reason for this as soon as we notice the rich harmonics shewn in Plates VIII and IX.

Helmholtz attempted to test his theory by calculating the amount of dissonance it implied for different intervals. He first assumed a law, somewhat arbitrarily, for the amount of dissonance produced by the beats of two pure tones at an assigned distance apart, and was then able to calculate, by simple addition, the total dissonance produced by all the beats of all the harmonics of a pair of notes. This naturally depended on the proportions in which the different harmonics entered into each tone; Helmholtz assumed the proportion to be that of violin tone.

Fig. 53. The degree of dissonance, as calculated by Helmholtz, of two violin tones sounding together. The lower tone c' sounds continuously, while the upper tone moves gradually from c' to c".

The result of his calculation is shewn in fig. 53. One violin-string is supposed to sound c' continuously, while the pitch of the second ranges from c' to c". The degree of dissonance at any point is shewn by the height of the curve above the horizontal line in fig. 53. Obviously the main consonances and dissonances within the octave are reproduced with remarkable fidelity.

A still better test can be made by employing the exact experimental results of Brues to give the dissonance of pure tones, but the final result is much the same as that just given.

The Origin of the Musical Scale

With all this in our minds, let us try to imagine how the different musical scales may have come into being. No one seems to know precisely how music itself came into human life, but it probably was through either stringed instruments or wind instruments. Primitive man may have enjoyed the rhythm he could make by pounding sticks together, or even by beating primitive drums, which he may have used for marking time for dances or marches, but it seems likely, as already suggested, that he first discovered the pleasures of tonal music by hearing vibrating strings---perhaps the twang of his bowstring---or sounding pipes, such as the wind whistling over the top of a broken reed. Ancient drawings and reliefs shew him, in the infancy of civilisation, playing both on the lyre and on pan-pipes or syrynxes. At Ur Sir Leonard Woolley unearthed the remains of an eleven-stringed lyre, which proves that 5000 years ago or more, man had already passed from the enjoyment of a single musical sound to that of a succession of sounds. Two pictures of bands of Sumerian musicians of 4600 years ago are reproduced as the frontispiece of this book, and explain themselves. An Egyptian painting of about 2750 B.C. shews a complete orchestra of seven players, two of whom are playing on stringed instruments and three on wind instruments, while two in the middle seem to be engaged in clapping their hands as though to beat time---the discovery that the right number of conductors to an orchestra was one had yet to come, but mankind of 5000 years ago was at least acquainted with melody. He may even have been acquainted with harmony as well, although this is far from certain. For in the most primitive civilisation of all, music seems always to have been homophonic (one-part), as it still is to some extent among the Chinese, Indians, Turks, Arabs and even Greeks of to-day.

In time, however, the idea must have occurred to sing or play two or more notes at once---possibly because it was impossible for men and boys to sing together in the same pitch, or possibly because one-part music began to pall. Up to now, the exact pitches of the notes selected to form the scale had been almost a matter of indifference; from now on it was important that two or more notes of the scale could be sounded together without undue dissonance. Even to-day, many of the races which have not advanced beyond homophonic music---as for instance the Arabs, Persians and Javanese---use scales whose notes are not at all consonant; the dissonance is harmless because two notes are never heard together. On the other hand, even primitive races whose music is polyphonic use scales in which most intervals are consonant.

The octave, the simplest and most perfect consonance of all, must have been discovered at a very early stage; it is fundamental in the music of all peoples, even the most rudimentary. The early Greeks seem to have employed no other concord in their music, although they were certainly acquainted with others. Aristotle tells us that the voices of men and boys formed an octave in singing, and asks " Why is only the consonance of the octave sung, for this alone is played on the lyre?" He suggests that other consonances were not in favour because "both tones are concealed, one by the other", and compares part-music to many speakers "who are saying the same thing at the same time, when we should understand a single speaker better", which seems to suggest that he did not possess a true polyphonic ear.

Nevertheless, the time was bound to come when incessant movement in octaves was found sterile and uninteresting---witness the scholastic prohibition of consecutive octaves, which was subsequently extended to consecutive fifths also. We can imagine our primitive musician discovering the consonance of the fifth c-g, possibly first introduced, as some have thought, in the form of a descending fourth, c'-g or f-c. This, with the octave, would give a set of four strings c f g c', of which any two except f g could be played together without creating unpleasantness. Nicomaeus tells us that down to the time of Orpheus, lyres were tuned to sound these notes.

The possessor of such a lyre would still have no great variety of tones for melody, so that we can imagine him increasing the number of his strings, and planning that each new string should create no new unpleasantness, or at any rate as little as possible, when sounded together with the already existing strings. On a lyre sounding c f g c', two of the strings c, c' can be sounded with either of two new tones f and g without creating discord; the other two strings f and g can be sounded with only one, since c and c' introduce the same new tone. Our pioneer might try to give another possibility of concord to g by introducing its fifth d. The problem then repeats itself, and to find a new and pleasant companion to d, he introduces a. So he goes on and gets a succession of notes F-C-G-D-A-E-B etc., each of which can be sounded in perfect harmony with either the note preceding it or the note succeeding it in the sequence. But there is always the trouble that the first and last note have only one agreeable companion, and so we can imagine him pressing on until finally, when he has gone far enough, he finds his sequence repeating itself. His notes no longer form a straight-line sequence but a circle of notes, which can be arranged like the numbers on the face of a clock, as in fig. 54, so that each string has now two notes with which it can sound in perfect harmony, the one in front and the one behind.

Fig. 54. The clock-face of twelve notes---the twelve semitones of the octave. Each note sounds the harmony of a fifth when played with either of the notes next to it.

It need hardly be said that the foregoing account has been purely fictitious; if for no other reason, because there was no single primitive man, but vast numbers of tribes and peoples who developed music independently, and the in the most varied surroundings. But all were striving for the same goal, and the principles which guided them---to choose pleasant noises rather than unpleasant, consonances rather than dissonances---must have been precisely those which we have imagined guiding our fictitious primitive man, so that they were led to much the same result as he, and this with a unanimity which is remarkable. They exhibit enormous differences in their language, customs, clothes, modes of life and so forth, but all who have advanced beyond homophonic music have, if not precisely the same musical scale, at least scales which are all built on the same principle.

The main differences are found in the numbers of notes which form the scale. By stopping at different places in the sequence F-C-G-D-A-...,(1) we obtain the various scales which have figured in the musics of practically all those races which have advanced beyond the one-part music of primitive man.

The first three notes of the sequence C, F and G formed the main tones of the scale of ancient Greece. If we proceed as far as five notes C, D, F, G, A we have the pentatonic scale in which a considerable amount of Chinese and ancient Scottish music is written, as well as much of the music of primitive peoples in Southern Asia, East Africa and elsewhere; transpose it a semitone up, and we have the scale provided by the black keys of the piano---hence the fact, beloved of school-children, that many Scottish melodies, "Auld Lang Syne ", etc., can be played without touching the white keys at all, and that almost any sequence of notes strummed on the black keys sounds like a Scottish melody. On taking the first seven notes, we have the ordinary diatonic scale, which seems to have been introduced into Greece in the middle of the sixth century B.C., was standardised by Pythagoras, and has remained the normal scale for western music ever since. The beginnings of this scale cannot be traced. Garstang found two Egyptian flutes, the date of which cannot be later than about 2000 B.C.; these gave the seven-note scale C D E F# G A B, which is identical with the Syntolydian scale of ancient Greece.

Finally, the full clock-face of twelve notes supplies the complete chromatic scale of modern music.

The Problem of Temperament

This last scale is affected by a serious complication. The tones which give the best concord after the octave have their frequencies in the exact ratio 3 2, or 1.5, as is evident from their constituting the second and third harmonics of the same fundamental note. Thus, for perfect concord, each step of one "hour" on the clock-face would increase the frequency by a factor 1 .5, and twelve such 12 steps would increase it by a factor of (1.5)12, of which the value is 129.75. We have just said that these twelve steps bring us back to the c seven octaves above the c from which we started. But we now see that they do not bring us back exactly; the frequency of this last c is only 128 times that of our starting-point, so that our twelve steps slightly overshoot the mark, and bring us to a note whose frequency is greater than that of c by a factor of 1.0136; this interval is commonly known as the "comma of Pythagoras", and is rather less than a quarter of a semitone.

To put the same thing in another way, we have just identified the frequency ratio 1.5 with the interval of a fifth, although our table (p. 25) gave the value as 1.4983. The difference is only small---1.13 parts in a thousand---but by the time we have taken the twelve steps needed to pass completely round the clock-face, it has been multiplied twelvefold into the difference of 13.6 parts in a thousand, which represents the aforesaid difference in pitch of almost a quarter of a semitone. When this is allowed for, the true clock-face is that shewn in fig. 55; it extends to infinity in both directions, and all simplicity has disappeared.

Fig. 55. The form assumed by the clock-face of fig. 54, when true fifths are used. There is now an endless series of notes, such notes as B#, C and Dbb being of different pitch.

The Pythagorean Scale

Various ways have been suggested for avoiding this complication. When Pythagoras standardised the musical scale in mathematical terms, he did not encounter it at all, because he did not think of his series of notes as forming a closed circle. He assigned to his C exactly 1 1/2 times the frequency [Actually he was unacquainted with the concept of "frequency" and spoke in terms of wave-lengths. I have expressed his ideas in more modern language.] of F, to his G exactly 1 1/2 times the frequency of C, and so on, thus arriving at a scale with the ratios shewn in the following table:

Pythagorean frequency ratio

Pythagorean interval

Equal temperament frequency ratio
C = 1.0000  

1.0000
 

Tone
 
D 9/8 = 1.1250  

1.1225
 

Tone
 
E 81/64 = 1.2656  

1.2599
 

Hemitone
 
F 4/3 = 1.3333  

1.3348
 

Tone
 
G 3/2 = 1.5000  

1.4983
 

Tone
 
A 27/16 = 1.6875  

1.6818
 

Tone
 
B 243/128 = 1.8984  

1.8877
 

Hemitone
 
C = 2.0000  

2.0000

From the way in which the scale is formed, it follows, as a matter of pure arithmetic, that the intervals C D, D E, F G, G A and A B must all be exactly equal, with a frequency ratio of 9:8. Pythagoras described each of these intervals as a "tone", and was left with the two smaller intervals E F and B C, each of which is represented by the more complicated frequency ratio of 256:243, or 1.0535. Pythagoras called such an interval a "hemitone". It is distinctly less than either the half of a Pythagorean tone or the modern semitone, the frequency ratios being

Pythagorean hemitone = 1.0535,
Half of Pythagorean tone = 1.0606,
Equal temperament semitone = 1.0595

Thus the Pythagorean octave was made up of five equal tones and of two equal hemitones, which were rather less than half-tones.

The scale was perfect and complete---so far as it went. In addition to the concord of the octave, it contained no fewer than four fifths and five fourths, a greater wealth of concords than can be attained from any other selection of eight notes.

The scale could of course be extended indefinitely in either direction by a process of trespassing into neighbouring octaves. On the other hand, the severity of Greek taste resulted in melodies being restricted to a compass of an octave---and frequently even of a fourth---so as to employ only the best and most agreeable registers of the human voice, so that a trespass into an upper octave involved a corresponding curtailment of the lower octave. The normal eight-stringed lyre might begin at any note of the scale, but it would end at the same note in the octave above. As there were seven choices possible for the lowest note--- d, e, f, g, a and b---this could be done in seven ways, which were referred to as "modes". These were as follows:

Range Ancient Greek name Scale (beginning with c) Glarean's ecclesiastical name
c-c' Lydian c, d, e, f, g, a, b, c' Ionian
g-g' Ionian* c, d, e, f, g, a, bb, c' Myxolydian
d-d' Phrygian c, d, eb, f, g, a, bb, c' Dorian
a-a' Aeolian** c, d, eb, f, g, ab, bb, c' Aeolian
e-e' Dorian c, db, eb, f, g, ab, bb, c' Phrygian
b-b' Myxolydian c, db, eb, f, gb, ab, bb, c' Locrian
f-f' Syntolydian c, d, e, f#, g, a, b, c' Lydian
*Or Hypophrygian when in another pitch.
** Or Hyperdorian when in another pitch.

Mediaeval music, and ecclesiastical music in particular, took over the Greek modes. originally only four were recognised as "authentic", namely, d-d', e-e', f-f' and g-g', these having been approved by Ambrose, Bishop of Milan, in the fourth century. Then Pope Gregory the Great added four more, which were known as Plagal. Finally, in the sixteenth century Glarean (Dodecachordon, Basle, 1547) distinguished twelve modes, and assigned Greek names to them, although many of his identifications with the ancient modes were incorrect. Some of these twelve modes had never been used at all, having been found unsatisfactory from the outset, while others fell into disuse in the course of time. Then, sometime in the seventeenth century, musicians began to find only two modes entirely satisfactory; they became known as the major CDEFGABC and the minor ABCDEFGA.

A collection of notes played in succession does not of itself constitute a melody which can awaken our musical imagination; to satisfy modern musical feeling, there must be a further element, which we describe as tonality. Our musical thought does not wish to wander indifferently all over the scale; it remains associated always with one particular note, the tonic or key note, which we somehow think of as giving a fixed and central point. Just as the traveller thinks of each point of his journey in terms of its distance from his home, so we moderns think of each note of a melody in terms of its interval from the key note. The skilful composer contrives to make us conscious of the key note from the very beginning of his music, and keeps our minds conscious of its position through all the notes that are played. In general, for instance, we expect the music---or at least the bass of it---to end on the key note, just as the traveller expects his journey to end at his home; we refuse to accept any other ending place as final. Even ancient Greek music had a sort of key note---the tone of the middle string of the lyre; Aristotle tells us that "All good melodies often employ the tone of the middle string, and good composers often come upon it, and if they leave it, recur to it again; but this is not the case with any other tone."

At first the composer could give variety to his music by writing in many different modes, but as the number of available modes decreased, he found it necessary to impart variety in other ways. The modern musician not only writes his music in a great variety of keys, but also, to maintain the interest at a high level, finds it necessary to change frequently from one key to another in the same piece. He may begin in the key of C, using as his scale the sequence of notes

CDEFGABC,

and may very soon change to the key of G, in which his scale consists of the notes

CDEF#GABC.

We can now see the first great objection to the Pythagorean scale from the standpoint of modem music---it is impossible to modulate from one key to another because the scale contains no F#, and indeed no semitones at all. And it was impossible to create them by halving the whole tone intervals, because two hemitones did not make a tone.

A second, and hardly less weighty objection, remains. The only numbers which enter into the frequency ratios of the Pythagorean scale are 2 and 3, with their powers 22 = 4, 23 = 8, 32 = 9, etc.; the numbers 5, 7, 11 do not occur at all. But the frequency ratios of a note and its various harmonics are represented by the complete sequence of numbers 2, 3, 4, 5, 6, 7, . . ., so that most of these harmonics are not represented by notes on the Pythagorean scale at all. And Helmholtz's theory of dissonance makes it clear that the pleasurable consonances are the harmonics, and not the notes of the Pythagorean scale.

To take the simplest instance, the fifth harmonic of C has five times the frequency of the fundamental. But the nearest note on the Pythagorean scale, e", has a frequency 87/16 or 5.06 times that of the fundamental C, and so is about a fifth of a semitone out of tune with the fifth harmonic of C. If we sound C, this latter insists on sounding anyhow, for we have seen (p. 83) that the natural harmonics alone are forced by resonance, and makes considerable discord with the Pythagorean e". Thus the note that our ear wants to hear sounding with C is not the Pythagorean e but the harmonic e.

It will be clear from what has already been said that these complications are absolutely fundamental; they arise out of the laws of arithmetic, which the musician is completely powerless to alter. If we visited another planet, we should find the same laws there as on earth. Here, 312 is very nearly equal to 219, which means that 12 fifths are very nearly equal to 7 octaves, and the same would be true there, so that if the inhabitants were at about the same musical level as we are, we might expect to find them employing the same diatonic scale as ourselves. But, there as here, complications would arise from the two numbers not being exactly equal; they like ourselves would have a "comma of Pythagoras", and like our own musicians might have devoted a vast amount of thought to minimising its baneful effects.

The Mean-Tone Scale

A solution which prevailed for many centuries led to a scale known as the "mean-tone" scale. Four steps from C on our clock-face, C-G-D-A-E, bring us to E, a third above C. We have already seen that the frequency ratio between. E and C is 5.06 on the Pythagorean scale, whereas the pleasurable ratio is 5.00. The bearings of the mean-tone system were laid by diminishing each of the four steps C-G, G-D, D-A and A-E equally by such an amount as gave the exact frequency ratio 5.00 to the interval C-E. Each of these steps was accordingly represented by a frequency ratio of 4th root of 5 or 1.49527, in place of the ratio 1.5 which characterises the exact fifth. The two quantities differ by about 3.15 parts in 1000. Each "hour" of the clock-face was accordingly in error by one 3.15 parts in a thousand, and the accumulation of twelve such errors amounted to 38 parts in a thousand, which is well over three-fifths of a semitone. When the scale was laid out as in fig. 56, the interval G #-Eb proved to be one of 7.395 equal temperament semitones, which is three-eights of a semitone more than the exact fifth (7.020 equal temperament semitones) ---it was accordingly known as the quinte-de-loup or "wolf-fifth", wolves being howling animals.

Fig. 56. The clock-face on the mean-tone scale.

Howling effects such as this could only be kept out of the music by carefully choosing the key in which a composition was written. By making slight departures from the mean-tone system, it was found possible to tune the notes so that music played in one key should sound harmonious, while that in a few other and nearly related keys did not sound too bad. For the rest, musicians simply had to avoid writing or playing in the more remote keys; they were virtually limited to three sharps or two flats, unless indeed instruments were specially arranged for them. Organs were sometimes built in which two black keys were interpolated between D and E, one sounding D# and the other Eb; and other notes were often treated in the same way. For instance, the organ which was built for the Foundling Hospital by Thomas Parker in the year 1768, "upon the new principle invented by the late Doctor Smith (Master of Trinity College, Cambridge)", contained devices for replacing the C#, G#, Eb, Bb pipes by others sounding Db, Ab, D# and A#, the scheme accordingly being that shewn in fig. 57.

The general principles of the mean-tone system were foreshadowed by Schlick (Spiegel der Orgelmacher und Organisten, 1511), who suggested tuning the fifths F C, C G, G D, D A "as flat as the ear could endure", so that the third F A should "sound decent". In its more precise form the system seems to have been the invention of a blind Spanish musician Francis Salinas who lived the greater part of his life in Italy, and described the exact mean-tone system in his De Musica Libri Septem (1577). Gradually, but only very gradually, it was superseded by the system of "equal temperament", which had been proposed at an even earlier date, 1482, by another Spaniard, Bartolo Rames.

Fig. 57. The clock-face on the mean-tone scale, with four extra notes added, to make it possible to play in several keys.

The Equal-Temperament Scale

In this system the "comma of Pythagoras" is distributed equally over the twelve intervals which make up the circle on the clock-face. As the comma is about a quarter of a semitone, this involves flattening each interval of a fifth by about a forty-eighth of a semitone. Or to be more precise, since the twelve steps round the clock-face are to represent an interval of exactly seven octaves, and so a frequency ratio of 128:1, each step must represent a frequency ratio of the twelfth root of 128 or 1.4983. All semitones are now equal, and, as already explained on p. 25, each represents precisely the same frequency ratio, 1.05946. Although these frequency ratios had been correctly calculated by the French mathematician Mersenne, and published in his Harmonie Universelle as far back as 1636, the system does not seem to have been employed in practice until late in the seventeenth century, when it began to be used in North Germany. The first occasion on which we hear of its use is in the famous organ which Arp Schnitger built for S. Jacobi at Hamburg in 1688-92; this is said to have been tuned by the builder to something which at least approached to equal temperament. J. S. Bach subsequently advocated the system; not only were his own clavichord and harpsichord tuned to it, but he wrote the well-known "forty-eight" (Wohltemperiertes Klavier) to prove that it enables compositions in all keys to be played without disagreeable discords. Yet even he was unable to convert contemporary organ-builders to the new system, and there seems to be no doubt that the organs of his day were usually tuned to the mean-tone system. This doubtless explains why Bach seldom wanders into the more remote keys in his organ works, in striking contrast to his compositions for the clavichord.

After the death of J. S. Bach, his son Philipp Emanuel Bach started an active campaign in favour of equal-temperament tuning, but its adoption was slow, and especially so in England; it was not until about the middle of the nineteenth century that English pianos began to be tuned to equal temperament, and not a single one of the English organs shewn in the Great Exhibition of 1851 was so tuned.

The equal-temperament system is now in universal use for keyed instruments, and has the great advantage that music can be played equally well in all keys. On the other hand its defects are many. The most obvious is that of all the seventy-eight intervals that lie within the range of a single octave, not a single one is in perfect tune; every one could be improved if there were not the others to think about. The pianist and organist accept this accumulation of lesser evils in order to escape the major evils of badly discordant intervals. But the violinist and singer are under no such necessity; as each interval comes along, they can make it what they like, and so naturally tend to make it that which gives most pleasure to the ear. Observations shew that the intervals which such performers produce when they are left to themselves differ greatly from those they produce when accompanied by an instrument tuned to equal temperament.

Just Intonation

An attempt to standardise the former intervals has led to the introduction of yet a further system, known as the system of "just intonation". This is limited to one single key, and aims at making the intervals as accordant as possible with both one another and with the harmonics of the key note and of the closely related tones.

The ratios chosen are shewn in the following table:

It will be seen that most of the frequency ratios can be expressed in terms of comparatively small numbers, indicating consonant harmonies.

On the other hand the whole tones are not all equal, some, known as major tones, having a frequency ratio of 9/8 (1.125), while others, known as minor tones, have a frequency ratio of only 10/9 (1.111). The two semitones have the same frequency ratio of 16/15, but this is more than half the frequency ratio of any full tones, since (16/15)2 = 1.138. The second column of the following table shews the intervals of just intonation for the scale of C; while adjacent columns give the harmonics of C. G, D, F and A on this scale, and also the intervals on the meantone and equal-temperament scales.

Frequency ratios in scale

The table shews that, on the scale of just intonation, C, G and A are true harmonics of F; E, G, D of C; and B and D of G.

The frequencies given in the second column are those which would actually be played by a violinist playing in the key of C. If, however, his music modulates to the key of G, his A will no longer have 5/3 times the frequency of C, but 9 times the frequency of G, and so 27 times the frequency of C; the frequency of his A changes from 1.667 to 1.687 times that of C. Thus the pitches of his notes are not fixed, but vary with the key in which he happens to be playing at the moment.

The classical observations on this subject are due to Delezenne and Helmholtz. The latter wrote:

That performers of the highest rank do really play in just intonation has been directly proved by the very interesting and exact results of Delezenne. This observer determined the individual notes of the major scale as it was played by distinguished violinists and violoncellists, by means of an accurately gauged string, and found that these players produced correctly perfect Thirds and Sixths. I was fortunate enough to have an opportunity of making similar observations by means of my harmonium on Herr Joachim. He tuned his violin exactly with the g d a e of my instrument. I then requested him to play the scale, and immediately he had played the Third or Sixth, I gave the corresponding note on the harmonium. By means of beats it was easy to determine that this distinguished musician used b1 and not b as the major Third to g, and el not e as the Sixth.

Key Characteristics

On an instrument tuned to equal temperament, the semitones are all equal, so that the scales which represent the different keys differ only in pitch. They are completely similar in all other respects, the frequency ratios being the same in all. We can verify this by making a gramophone record of a chromatic octave C C#, D, D#, E, ... C played on a piano or other instrument tuned to equal temperament, and running it through the gramophone at 1.05946 times the speed at which it was taken. Then C becomes C# exactly, the C# becomes D exactly, and so on, so that what we hear is exactly the chromatic octave of C#.

On an instrument which is not tuned to equal temperament, the semitones are not all equal, so that a musician whose ear was infinitely sensitive would say: "This is not the chromatic octave of C# that I hear; it is the octave of C played a semitone too high." It follows that in every system of tuning other than equal temperament, each scale has its own special characteristic quality; we do not pass from one scale to another by a mere uniform change of pitch.

Some regard it as a defect of equal-temperament tuning that it obliterates the different characteristic qualities of the various keys. How serious this defect is will, of course, depend on how far these differences, if indeed they can be perceived at all, contribute to the interest or enjoyment of our music.

In the original Greek modes, the octave was divided into its seven intervals by steps which varied greatly from one mode to another, with the result that the characteristic qualities of the various modes were unmistakable and could be recognised at once. Plato tells us, for instance, that the Lydian mode (our modern major mode!) was specially associated with sorrow; it and the closely associated Ionian mode, which only differed from it in bb replacing b, were also the modes of softness, relaxation, self-indulgence, and even drunkenness. The Dorian and Phrygian modes on the other hand were---so he tells us---associated with courage, the military spirit, temperance and endurance. Because of this association, Plato would have permitted only the Dorian and Phrygian modes to be employed in his ideal republic, the Lydian and Ionian modes being prohibited.

The only modes which are in general use in modern music are the Greek Lydian (major mode) and Aeolian (minor mode), and their characteristic qualities are still easily recognisable. There was a time when the church frowned on the major mode as being too sensual for ecclesiastical music, but to-day we associate the major mode primarily with strength, virility, gaiety., and even frivolity, while the minor mode suggests sadness, seriousness and profundity; indeed, because of these associations, such expressions as "in a minor key" are in common use as part of the English language.

The differences between the various major keys are far more subtle than those which differentiated the various Greek modes, or those which produce the differences between the major and the minor keys in modern music. Instead of depending on the difference between whole tones and semitones, they depend at most on the difference between major whole tones and minor whole tones (p. 177).

When we compare two scales in major keys with one another, we find that, unless the tuning is that of equal temperament, the octave is still divided into its seven intervals by slightly different steps, and the question is whether this slight difference is perceptible to the trained musical ear, and if so, whether it has an appreciable influence on the emotional qualities of the music.

Many musicians, including Berlioz, Schumann and Beethoven, seem to have believed that both questions must be answered in the affirmative. We find Beethoven writing of B minor as a "schwarze tonart", describing Klopstock as "always maestoso---Db major", changing the key of a song in an effort to make it sound amoroso in place of barbaresco, and so forth.

The scientific Helmholtz appears to have held similar views; he wrote:

There is a decidedly different character in different keys on pianofortes and bowed instruments. C major and the adjacent Db major have different aspects. The difference is not caused by a difference of absolute pitch, as can easily be verified by comparing two instruments which are tuned to different pitches. If Db on one instrument has the same pitch as C on the other, the C major still retains its brighter and stronger character on both, and the Db its soft, veiled harmonious quality.

To-day many musicians claim to hear the different characteristics very clearly, and associate them with the emotional quality of the music. They will tell us that music played in the "open" key of C major---with neither flats nor sharps in the key signature---sounds strong and virile; played in the key of G, with one sharp, it sounds brighter and lighter; in D, with two sharps, even more so; and so on. Every additional sharp in the key signature is supposed to add to the brightness and sparkle of the music, while every flat contributes softness, pensiveness, and even melancholy. Some writers go into greater detail. Here, for example, is part of an arrangement suggested by Ernst Pauer and quoted in the English translation of Helmholtz's Tonempfindungen:

C major Pure, certain, decisive; expressive of innocence, powerful resolve, manly earnestness and deep religious feeling.
Db major Fullness of tone, sonority and euphony.
E major Joy, magnificence, splendour; brightest and most powerful key.
E minor Grief, mournfulness, restlessness.
F major Peace, joy, light, passing regret, religious sentiment
F minor Harrowing, melancholy
F# major Brilliant, very clear.
Gb major Softness, richness.

It is clear that even if these qualities had ever been associated with the various keys, they must all be lost in equal temperament, in which, to take the most obvious instance, the key of F# major (six sharps) is absolutely identical with that of Gb major (six flats). Yet musicians who claim to find the association in music played by the orchestra claim also to hear it on the pianoforte, and have expended much ingenuity in maintaining that the keys retain their alleged characters and distinctive quality even on the pianoforte. Helmholtz, for instance, argued that the operation of striking a short black key must necessarily differ mechanically from that of striking the white keys with their longer leverage, and suggested that this may cause the required difference. It cannot be denied that it might make some difference, but it would be a most amazing coincidence if it made precisely the difference needed, so that the different lengths of the black and white keys on the pianoforte gave just the same characteristics to music played on the pianoforte as the deviations from equal temperament give to the same music when played by an orchestra.

Such a coincidence is, indeed, so utterly improbable that it seems safe to rule it out, and to assert that, in the case of pianoforte music at least, the special qualities of individual keys exist only in the imagination of the hearer, and possibly sometimes in that of the composer also, who may have chosen the key of a particular composition so as to fit in with his preconceived ideas of its emotional characteristics. In confirmation of this it may be remarked that pianoforte pieces retain their emotional qualities when played on a pianola, on which the mechanical difference between white and black keys disappears entirely.

The case of orchestral music cannot be dismissed so easily, but an obvious judgment may reasonably be based on the circumstance that those who claim to hear differences in the orchestra claim to hear precisely the same differences on the piano, although the equal temperament tuning makes it impossible that they should occur; and we may feel confirmed in our judgment by the circumstance that these differences are not always the same with all hearers. We have already noticed how Plato associated our modern key of C major with sorrow, weakness and self-indulgence, while Helmholtz associates it with brightness and strength, and Pauer with purity, innocence, manliness, and other virtues. And Helmholtz, in the passage just quoted, describes Db major as soft, veiled and harmonious, while Beethoven associated it with maestoso qualities, and Pauer's list tells us that it has fullness of tone, sonority and euphony.

All this suggests that the whole matter is one of subjective imagination, possibly based in the first instance on association of ideas. An obvious chain of associated ideas starts from sharps in the key-signature, and runs through sharpening of pitch to high notes and bright, joyous music; another runs from flats through flattening of pitch to deep-pitched notes, with their depression and seriousness. Obviously this does not explain everything; there may also be association with well-known pieces of music.

The power of subjective imagination seems to be very strong. Some hearers even claim to find emotional qualities in individual notes---here is a list from Curwen's Standard Course of Lessons and Exercises in the Tonic Sol-fa Method (1872):

Do (key-note) strong, firm.
Re rousing, hopeful.
Mi steady, calm.
Fa desolate, awe-inspiring.
So grand, bright.
La sad, weeping.
Ti piercing, sensitive.

We cannot but be reminded of the Beethoven enthusiast who claimed that a single chord, nay even a single semiquaver, of his favourite master contained more emotional quality than all the music of Bach added together.

In whatever way we answer these various questions it remains true that the introduction of equal temperament tuning has resulted in much of the music of the earlier masters not being heard tonally as it was intended to be heard---as for instance the vocal and organ works of Bach and Handel, and the clavichord and harpsichord works of Handel, all of which were written for the mean-tone system.

Our discussion will have made it clear that there is no perfect system of intonation, and that no scale can be devised which is suited for all instruments. In an orchestra we may hear the brass playing in harmonics, the strings in just intonation, or perhaps a compromise between this and equal temperament, and the organ, harp and piano in equal temperament. Yet we seldom feel that anything is wrong, except perhaps in a pianoforte concerto, where the conflict arises in its acutest form. It was not always so. Dr Robert Smith, writing in 1759, described equal temperament as "that inharmonious system, of 12 hemitones, which produces a harmony extremely coarse and disagreeable", and even in 1852 Helmholtz wrote:

When I go from my justly-intoned harmonium to a grand pianoforte, every note of the latter sounds false and disturbing.. . . On the organ, it is considered inevitable that, when the mixture stops are played in full chords, a hellish row must ensue, and organists have submitted to their fate. Now this is mainly due to equal temperament, because every chord furnishes at once both equally-tempered and justly-intoned fifths and thirds, and the result is a restless blurred confusion of sounds.

While we cannot deny the general truth of this, we hardly feel so critical to-day. Perhaps our ears are more tolerant than those of our ancestors. just as we have learned to tolerate and even enjoy harmonies which they found unbearable, so we may have learned to enjoy imperfectly tuned intervals which they heard only as a "hellish row".

The Music of the Future

Earlier in the present chapter we let our fancy roam to the extent of imagining that we were visiting another planet, on which musical development had reached about the same level as on our own. As the laws of arithmetic would be the same on this planet as on earth., we conjectured that the inhabitants might quite possibly have arrived at the same musical scale as our own, the octave being divided into twelve equal, or approximately equal, divisions.

If we are prepared to take a further flight of fancy, let us imagine that we visit a planet on which music has developed to a far higher level than our own, or, if we take an optimistic view of the future of our race, let us imagine that we revisit our own planet some thousands of years hence. What kind of music shall we find and, in particular, what scale will be in use?

The simplest, although not necessarily the correct, conjecture is that the music of the future will be like that of the present, but intensified---as it is now. only more so. To see what is implied in this, we must read our histories of music and imagine that those tendencies which have moulded music into its present form persist, and mould it still further in the same direction.

One tendency is typified in the history of consecutive fifths. Harmonies which have seemed venturesome and perhaps ugly to one generation seem natural and beautiful to the next, but are destined through continued repetition to seem obvious and tedious to generations yet to come. The sated ear for ever demands new harmonies which it will fast learn to tolerate, and then dismiss as threadbare and uninteresting.

Thus we find a long succession of musicians, Palestrina, J. S. Bach, Beethoven, Liszt, Wagner, Debussy---each of whom broke new ground, and most of whom were regarded as revolutionaries in their day---and innumerable other modern composers, introducing chords which after being thought perilously discordant at first, have now passed into the common language of music, and are heard with pleasure by our modern ears. Not only so, but the indisputable dissonances of equal temperament no longer distress us in the way that they seem, from the quotations given above, to have distressed our more fastidious predecessors.

One way of picturing the future is to imagine our posterity becoming more and more tolerant of dissonance as time goes on. If they ever attain to a stage in which all possible combinations of notes in the present scale are heard as tolerable but boring concords, further progress will only be possible by music enlarging its territory---by adding more notes to the scale. There is already a tendency to experiment with split semitones---quarter-tones---although up to the present it can hardly be said to have met with overwhelming success.

This brings us to a second tendency in musical history---a tendency continually to enlarge the scale. This has been in turn pentatonic (5-note), heptatonic (7-note) and chromatic (12-note). Has it reached its final resting place in the 12-interval division of the octave, or will the subdivision still continue?

We have already seen that the question is one for the arithmetician. Without forgetting the proverbial dangers of prophecy, we may be fairly sure that the laws of arithmetic will not alter, and that the natural harmonics will not change their position---a million years hence, as now, their frequencies will be 2,3,4,. . . times that of the fundamental. And, unless the physiological quality of our ears changes appreciably, we may assume that we shall always obtain our basic pleasure from chords whose frequency ratios can be expressed by the smallest of numbers. Because of this., it seems likely that the present fifth, with the simplest frequency ratio of all, 3 : 2, and the major third, with the next simplest frequency ratio 5 : 4, will figure largely in the music of the future. Before we attempt a conjecture about the musical scale of the future, it is worth seeing how far the subdivision of the octave would have to be extended., to provide a scale richer and purer in this respect than our present scale.

More Complex Scales

We have already seen that the complexities of the present scale centre in the fact that 12 fifths are not exactly equal to 7 octaves. Let us first examine whether we can replace the numbers 12 and 7 in this statement by others which will reduce the degree of inexactness. Using the method of continued fractions, we find that the following are increasingly good approximations to the ratio of the intervals of a fifth and an octave:

12 fifths =

7 + 0.019 octaves

7 octaves +
1/4 semitone

41 fifths =

24 – 0.016 octaves

24 octaves –
1/5 semitone

53 fifths =

31 + 0.003 octaves

31 octaves +
1/28 semitone

306 fifths =

179 – 0.0014 octaves

79 octaves –
1/60 semitone

Each of these approximations is the best that can be obtained without extending the scale beyond the number of notes it contemplates, so that if the only problem was that of reducing the comma of Pythagoras to a minimum, the logical stopping-places would be at 12, 41, 53 and 306 notes to the octave. This is, however, far from being the whole problem: we want a scale which is rich in 5:4 consonances (major thirds) as well as in 3: 2 consonances. Now in the various scales just mentioned, the number of notes which constitute the exact 5:4 consonance are found to be 3.86, 13.20, 17.06 and 98.51 respectively. The only scale which is even as good as the present 12-note scale in this respect is the 53-note scale. On this the present "fifths" are replaced by intervals Of 31 notes, the tuning being almost perfect, while the present "major thirds" are replaced by intervals of 17 notes, these being flat by only a seventieth part of a present semitone.

So far as is known, a 53-note scale was first proposed in Europe in the seventeenth-century by Nicolas Mercator, Danish mathematician and astronomer, who, according to Yasser, found it mentioned in the writings of a Chinese theorist, King-Fang, of the second century B.C. In the middle of the last century, two harmoniums with 53 notes to the octave were built, one for Mr R. H. M. Bosanquet of London, and one by Mr J. P. White of Springfield, Mass., but neither seems to have been regarded as more than a curiosity.

We have already seen that the present 12-note scale has its roots embedded very deeply in the unalterable properties of numbers; we now find that music will have to go very far before finding a better scale. But a 53-note scale would give far purer harmonics than the present scale, and we can imagine future ages finding it worthy of adoption, in spite of all its added complexities---especially if mechanical devices replace human fingers in the performance of music. For, in the last resort, our limited scales have their origin in the limitation of our hands.

Yet, if ever music becomes independent of the human hand, may not the race then elect to use a continuous scale in which every interval can be made perfect---as with the unaccompanied violin of to-day?


Footnote

1. There is no theoretical reason for starting with F rather than with any other note of the scale. F has been selected merely in order to keep off the black notes of the piano for as long as possible.


Chapter Six

Table of Contents