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:
| C | D | E | F | G | A | B | C |
| 1 | 9⁄8 | 5⁄4 | 4⁄3 | 3⁄2 | 5⁄3 | 15⁄8 | 2⁄1 |
| 528 | 594 | 660 | 704 | 792 | 880 | 990 | 1056 |
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:
| C 528 | D 594 | E 660 | F 704 | G 792 | A 880 | B 990 | C 1056 |
| C-sharp 556 7⁄8 | D-sharp 626 3⁄64 | E-sharp 696 3⁄32 | F-sharp 742 1⁄2 | G-sharp 835 5⁄16 | A-sharp 928 1⁄8 | B-sharp 1044 9⁄64 | C-sharp 113 3⁄4 |
| D 594 | E 668 1⁄4 | F-sharp 742 1⁄2 | G 792 | A 881 | B 990 | C-sharp 1113 3⁄4 | D 1188 |
| E-flat 625 4⁄9 | F 703 45⁄72 | G 781 34⁄36 | A-flat 833 25⁄27 | B-flat 938 1⁄18 | C 1042 11⁄27 | D 1172 59⁄72 | E-flat 1250 8⁄9 |
| E 660 | F-sharp 742 1⁄2 | G-sharp 825 | A 880 | B 990 | C-sharp 1100 | D-sharp 1237 1⁄2 | E 1320 |
| F 704 | G 792 | A 880 | B-flat 938 2⁄3 | C 1056 | D 1173 1⁄3 | E 1320 | F 1408 |
| F-sharp 742 1⁄2 | G-sharp 835 5⁄16 | A-sharp 928 1⁄8 | B 990 | C-sharp 1113 3⁄4 | D-sharp 1237 1⁄3 | E-sharp 1392 3⁄15 | F-sharp 1492 |
| G 704 | A 792 | B 880 | C 938 | D 1056 | E 1173 | F-sharp 1320 | G 1408 |
| G-sharp 825 | A-sharp 928 1⁄8 | B-sharp 1031 3⁄4 | C-sharp 1100 | D-sharp 1237 1⁄2 | E-sharp 1375 | Fx 1546 1⁄8 | G-sharp 1650 |
| A 880 | B 990 | C-sharp 1100 | D 1173 1⁄3 | E 1320 | F-sharp 1466 2⁄3 | G-sharp 1650 | A 1760 |
| B-flat 938 2⁄3 | C 1056 | D 1173 1⁄6 | E-flat 1258 8⁄9 | F 1408 | G 1564 4⁄9 | A 1760 | B-flat 1877 1⁄3 |
| B 990 | C-sharp 1113 3⁄4 | D-sharp 1237 1⁄2 | E 1320 | F-sharp 1485 | G-sharp 1650 | A-sharp 1856 1⁄4 | B 1980 |
| C 528 | D 594 | E 660 | F 704 | G 792 | A 880 | B 990 | C 1056 |
| C-sharp 556 7⁄8 | D-sharp 626 3⁄64 | E-sharp 696 3⁄32 | F-sharp 742 1⁄2 | G-sharp 835 5⁄16 | A-sharp 928 1⁄8 | B-sharp 1044 9⁄64 | C-sharp 113 3⁄4 |
| D 594 | E 668 1⁄4 | F-sharp 742 1⁄2 | G 792 | A 881 | B 990 | C-sharp 1113 3⁄4 | D 1188 |
| E-flat 625 4⁄9 | F 703 45⁄72 | G 781 34⁄36 | A-flat 833 25⁄27 | B-flat 938 1⁄18 | C 1042 11⁄27 | D 1172 59⁄72 | E-flat 1250 8⁄9 |
| E 660 | F-sharp 742 1⁄2 | G-sharp 825 | A 880 | B 990 | C-sharp 1100 | D-sharp 1237 1⁄2 | E 1320 |
| F 704 | G 792 | A 880 | B-flat 938 2⁄3 | C 1056 | D 1173 1⁄3 | E 1320 | F 1408 |
| F-sharp 742 1⁄2 | G-sharp 835 5⁄16 | A-sharp 928 1⁄8 | B 990 | C-sharp 1113 3⁄4 | D-sharp 1237 1⁄3 | E-sharp 1392 3⁄15 | F-sharp 1492 |
| G 704 | A 792 | B 880 | C 938 | D 1056 | E 1173 | F-sharp 1320 | G 1408 |
| G-sharp 825 | A-sharp 928 1⁄8 | B-sharp 1031 3⁄4 | C-sharp 1100 | D-sharp 1237 1⁄2 | E-sharp 1375 | Fx 1546 1⁄8 | G-sharp 1650 |
| A 880 | B 990 | C-sharp 1100 | D 1173 1⁄3 | E 1320 | F-sharp 1466 2⁄3 | G-sharp 1650 | A 1760 |
| B-flat 938 2⁄3 | C 1056 | D 1173 1⁄6 | E-flat 1258 8⁄9 | F 1408 | G 1564 4⁄9 | A 1760 | B-flat 1877 1⁄3 |
| B 990 | C-sharp 1113 3⁄4 | D-sharp 1237 1⁄2 | E 1320 | F-sharp 1485 | G-sharp 1650 | A-sharp 1856 1⁄4 | B 1980 |