- 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.
| Key-notes— | ||||||||
|---|---|---|---|---|---|---|---|---|
| C 528 | D 594 | E-flat 625 4⁄9 | F 704 | G 792 | B-flat 938 2⁄3 | A-flat 833 25⁄27 | D-flat 555 146⁄152 | G-flat 741 124⁄486 |
| Minor thirds—6⁄5 Ratio | ||||||||
| E-flat 633 3⁄5 | F 712 4⁄5 | G-flat 750 4⁄55 | A-flat 844 4⁄5 | B-flat 950 2⁄5 | D-flat 1125 11⁄15 | C-flat 1000 106⁄135 | F-flat 667 66⁄810 | B double flat 889 1330⁄2430 |
| Minor sixths—8⁄5 Ratio | ||||||||
| A-flat 841 4⁄5 | C-flat 1000 32⁄45 | G-flat 1501 13⁄15 | F-flat 667 38⁄276 | B double flat 889 358⁄810 | E double flat 593 10⁄2400 | |||
| Dominant sevenths—16⁄9 Ratio | ||||||||
| D-flat 1111 80⁄81 | E-flat 1251 5⁄9 | A-flat 1668 20⁄27 | ||||||
| Minor sevenths—9⁄5 Ratio | ||||||||
| G-flat 741 64⁄243 | C-flat 988 359⁄810 | F-flat 658 2908⁄4374 | ||||||
| Key-notes— | ||||||||
|---|---|---|---|---|---|---|---|---|
| C 528 | D 594 | E-flat 625 4⁄9 | F 704 | G 792 | B-flat 938 2⁄3 | A-flat 833 25⁄27 | D-flat 555 146⁄152 | G-flat 741 124⁄486 |
| Minor thirds—6⁄5 Ratio | ||||||||
| E-flat 633 3⁄5 | F 712 4⁄5 | G-flat 750 4⁄55 | A-flat 844 4⁄5 | B-flat 950 2⁄5 | D-flat 1125 11⁄15 | C-flat 1000 106⁄135 | F-flat 667 66⁄810 | B double flat 889 1330⁄2430 |
| Minor sixths—8⁄5 Ratio | ||||||||
| A-flat 841 4⁄5 | C-flat 1000 32⁄45 | G-flat 1501 13⁄15 | F-flat 667 38⁄276 | B double flat 889 358⁄810 | E double flat 593 10⁄2400 | |||
| Dominant sevenths—16⁄9 Ratio | ||||||||
| D-flat 1111 80⁄81 | E-flat 1251 5⁄9 | A-flat 1668 20⁄27 | ||||||
| Minor sevenths—9⁄5 Ratio | ||||||||
| G-flat 741 64⁄243 | C-flat 988 359⁄810 | F-flat 658 2908⁄4374 | ||||||
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:
| Different sounds in twelve diatonic scales | 31 |
| Sounds wanting to complete the diatonic scales of A-flat, D-flat, G-flat | 14 |
| Minor thirds wanting in scales of C, E-flat, F, G, B-flat | 6 |
| Minor sixths wanting in scales of C, E-flat, and B-flat | 3 |
| Dominant sevenths wanting in scales of E-flat, F and B-flat | 3 |
| Minor thirds wanting in scales of A-flat, D-flat and G-flat | 3 |
| Minor sixths wanting in scales of A-flat, D-flat and G-flat | 3 |
| Minor sevenths wanting in scales of A-flat, D-flat and G-flat | 3 |
| Total number of sounds in an octave | 66 |
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.