Theory and Practice of Piano Construction / With a Detailed, Practical Method for Tuning

CHAPTER V.
THE MUSICAL SCALE AND MUSICAL INTONATION.

We have now considered as much of the phenomena of musical sounds as may be considered to have a bearing upon the purpose of our investigations. We may then devote some space to the matter of the expression of musical ideas, and the intonation which has been devised in order to reduce the mental products of composers to the limitations of musical instruments. Music is expressed through the medium of a scale of tones, all of which bear definite relations to each other as to pitch. The “diatonic scale,” which is the foundation of musical intonation, is composed of a series of eight tones which are named after letters of the alphabet, the last tone having the same name as, and being the octave to, the first. The frequencies of these tones always bear the same ratios, one to another, whatever may be their positions within the compass of any instrument. Now, considering the frequency of the first tone to be unity, the frequencies of the others are in the following proportions:

If we now divide these proportionate numbers each by the other we have the proportionate intervals that separate them. Doing this, we have the following result:

Now, it will be observed that we have in the above table three different kinds of interval represented by the three ratios, ⁹⁄₈, ¹⁰⁄₉ and ¹⁶⁄₁₅. The first of these is called the major tone and the second the minor tone, while the third is known as the diatonic semitone. Following out these ratios, we may obtain the frequencies of any diatonic series. We shall choose the scale of which C 528 is the key-note. Its frequencies are as follows:

Knowing as we do the ratios and frequencies already calculated, it is obvious that we may similarly calculate the ratios and frequencies for the diatonic scale, of which any given tone is the tonic or key-note. Before doing this, however, it is well for us to remember that the diatonic scale is not adequate to all the requirements of music. Musicians have found it necessary to interpolate other sounds in between those which form the diatonic progression. The reason for this is that music, in order that it may have the greatest possible freedom of expression, must be written in a larger number of keys, and must contain more distinct sounds than the diatonic scale is able to afford. For these and other cognate reasons the chromatic scale was introduced. The addition of five chromatic semitones, obtained by taking the difference between a minor tone and a diatonic semitone, gives the chromatic scale thirteen semitones from key-note to octave. Unfortunately, however, the same number of keys upon the pianoforte cannot provide us with thirteen pure chromatic sounds in every key. This may be demonstrated as follows: The ratio of a chromatic semitone is ²⁵⁄₂₄. The sharp of C 528 is, therefore, 550. But in the diatonic scale of D (the major second in scale of C), C sharp has a frequency of 1113 ³⁄₄. The octave below this latter sound is the C sharp, which is one chromatic semitone above C 528. We know the frequency of the latter to be 550. The frequency of the octave below C sharp, 1113 ³⁄₄, ought, therefore, to be 550. But we know that the octave below any given note has a frequency that is one-half that of the given note. Now, one-half of 1113 ³⁄₄ is 556 ⁷⁄₈. Therefore, we see that there is a difference of 6 ⁷⁄₈ vibrations per second between the C sharp that is a chromatic semitone above C 528 and the C sharp that is the octave below the major seventh of the scale of D, and which ought to be the same sound, as it is in the same position on the key-board as the former. By carrying the same investigation further we are enabled to perceive that sounds of the same name are not identical when played in different keys, or, rather, that the same name does not imply that the sound so denoted means the same thing when it is considered in its relation to any tonic different to that to which it was first related. There is another difficulty also that confronts us in the problem of playing pure sounds upon the pianoforte; that instrument, as we know, does not provide us with different keys for the sharp of one sound and the flat of the sound next above it. There is a general belief that C sharp, for example, and D flat are identical. But this is not so. The flat of D is a chromatic semitone below that note, while the sharp of C is the same interval above the latter. By referring to our former calculations it will be seen that the chromatic semitone ratio is ²⁵⁄₂₄. The sharp of C is, therefore, obtained by multiplying the frequency of C by ²⁵⁄₂₄, and the flat of D is likewise evolved by an inverse process, namely, by dividing the frequency of D by the same ratio. This is equivalent to adding a chromatic semitone to C and subtracting the same from D. If we take the notes C and D from the scale of C 528, we have the frequencies of C and D as 528 and 594 respectively. Effecting the multiplication and division as above we see that C sharp has a frequency of 550, while that of D flat is 570 ⁶⁄₂₅. That is to say that these two notes differ by no less than 20 ⁶⁄₂₅ vibrations per second.

It thus becomes obvious that the expression of all the sounds within the compass of an octave, in such a manner that absolutely correct sounds in every key may be obtained, is a problem that calls for more sounds than are provided by the pianoforte. As a correct understanding of this most important subject is essential, a somewhat elaborate treatment of it will be given here. The reader who takes the pains to master the true inwardness of the problem of musical intonation will have an insight into the matter which few pianoforte makers or musicians possess.

“Just intonation” is the name given to that system whereby we are enabled to command the expression of all the sounds that are required to be heard within the compass of an octave in order that the degrees of each and every possible scale may be correctly and exactly rendered. It is not difficult to see that performers upon instruments which do not have fixed tones should have no difficulty in adjusting the intonation of every tone to correspond with the variations in pitch required by the different positions in the scale that such tones may occupy. Experiments have, in fact, been carried out with violinists and it has been shown that artists upon this instrument do naturally play the true diatonic and chromatic intervals when left to themselves and when not forced to adjust their intonation to that of fixed tone instruments.

In order to show with accuracy the total number of different sounds that are required to produce “just intonation” in every possible key the reader is invited to consider the following table, which shows the smallest possible number of sounds that will give the true diatonic intervals in twelve keys. The first note in each row is the key-note and the last the octave thereto. The frequencies of those key-notes that are not represented in the first scale (that of C) have been calculated as follows:

• The key-note to scale of B-flat is the perfect fourth to key-note of scale F.
• The key-note to scale of E-flat is the perfect fourth to key-note of scale B-flat
• The key-note to scale of F-sharp is the octave below major seventh of scale G.
• The key-note to scale of G-sharp is the octave below major seventh of scale A.
• The key-note to scale of C-sharp is the octave below major seventh of scale D.

We therefore have the following results:

In order that the different sounds may more easily be separated, they have been collated in linear progression, together with their frequencies and the scales in which they or their octaves appear:

Thus we see that thirty-one different sounds are required to give the true diatonic intervals in only twelve keys. But it is not necessary to remind the reader that there are more keys than these used in music. We have, in fact, not yet considered the keys of A flat, D flat and G flat. The frequencies of the keynotes of these scales have been calculated as follows:

• A-flat is the perfect fourth to E-flat, which as calculated above = 625 therefore A-flat = 833 ²⁵⁄₂₇.
• D-flat is the perfect fourth to A-flat, which as calculated above = 833 ²⁵⁄₂₇ therefore D-flat = 555 ¹⁵⁴⁄₁₆₂.
• G-flat is the perfect fourth to D-flat, which as calculated above = 555 ¹⁵⁴⁄₁₆₂ therefore G-flat = 741 ¹³⁰⁄₄₈₆.

We are therefore able to construct these following additional scales:

By examining the last table the reader will perceive that we have obtained fourteen new sounds. They are shown graphically in this manner:

• In the scale of A-flat the new sounds are B-flat, C, D-flat, F and G.
• In the scale of D-flat the new sounds are E-flat, F, G-flat, and A-flat.
• In the scale of G-flat the new sounds are A-flat, C-flat, D-flat, E-flat and F.

None of these sounds had been obtained in the scales given before and, consequently, we have to consider that there are fourteen more sounds to be added to the thirty-one that we have already found.

The above calculations would suffice to provide us with the diatonic intervals in all the keys that are used in music. Harmony demands, however, certain other intervals. These are minor thirds, minor sevenths, dominant sevenths and minor sixths. Accordingly, if we desire to probe the matter of just intonation to its depths, we must calculate the sounds that are required to make up these intervals in such scales as are now without them. Examining the tables already prepared, we find that there are wanting the following members:

• Minor thirds to the key-notes of the scales C, D, E-flat, F, G, B-flat, A-flat, D-flat, G-flat.
• Minor sixths to the key-notes of the scale C, E-flat, B-flat, A-flat, G-flat, and D-flat.
• Dominant sevenths to the key-notes of the scales E-flat, F and B-flat.
• Minor sevenths to the key-notes of the scales A-flat, D-flat, and G-flat.

We shall have no difficulty in calculating the frequencies of the required notes by the same processes that we have followed heretofore.

The result of these calculations may now be collated and summarized. We find that there are no less than sixty-six separate sounds required for the production of the necessary intervals in all the possible scales. These sounds are thus classified:

Now the obvious conclusion to be drawn from this analysis is that the true sounds of the just musical scales are very different from any that we hear upon the pianoforte. Indeed, we may properly carry the reasoning a step further. If the expression of all the degrees of the true musical scales requires this formidable array of sounds, then surely, the sounds that are produced upon the piano are not all of the required true sounds, but are totally unlike any of them. For it is evident that if the sixty-six true sounds within the compass of an octave have to be reduced to the thirteen that are found upon the pianoforte, the process of compression to which the former must be subjected will force the latter into the position of so many compromises. In fact, with the exception of the standard tone from which all calculations and all tuning must start, and its octaves, there is no tone upon the piano, as it is now tuned, which is identical with any sound of the justly tuned scale. The process to which we have alluded, and which is necessary to secure to the piano and all other instruments with fixed tones the ability to perform music in all keys which are desired for the proper expression of the composers’ ideas, is called temperament. Upon the skill and cunning with which this compromise with natural laws is effected depends the whole beauty of, and the whole of our pleasure in, music as we are accustomed to hear it. It would be vain to pretend that tempered intonation is preferable to that which is pure and just, but it is equally vain and foolish to decry the accepted system of temperament until the mechanical skill of manufacturers of musical instruments and the taste of performers have risen to the point of appreciating the beauties of pure intonation and of devising mechanical means of attaining it. Until that time arrives we must fain be content to accept what we have and make the best of it. There have, of course, been attempts to provide instruments that could be used to give the pure intervals in every key, but they have been invariably failures. Most of them have been forced to depend upon tempered intonation to a certain extent, while others have been mechanically impossible.

In any case we must remember that the pianoforte, as at present constructed and played, depends entirely upon an equally tempered intonation. So strongly has the pianoforte entrenched itself in popular favor, indeed, that music and tempered intonation have become, to most people, exactly synonymous. It is proper that we should be able to draw true distinctions, however, as the practical work of piano building ought to be largely guided by the considerations induced from the necessity and fact of temperament.