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:
| 1. | The sound C | = 528 |  | Appears in the scales of |  | C, F, G, B-flat. |
| 2. | “ C | = 521 11⁄54 | E-flat | |||
| 3. | “ C-sharp | = 556 7⁄8 | D, B, F-sharp, C-sharp | |||
| 4. | “ C-sharp | = 550 | A, E, G-sharp | |||
| 5. | “ D | = 594 | C-G | |||
| 6. | “ D | = 586 2⁄3 | A, F, B-flat, E-flat | |||
| 7. | “ D-sharp | = 618 3⁄4 | E, B, F-sharp, G-sharp | |||
| 8. | “ D-sharp | = 626 31⁄64 | C-sharp. | |||
| 9. | “ E-flat | = 625 4⁄9 | B-flat | |||
| 10. | “ E | = 660 | C, G, A, E, B-flat | |||
| 11. | “ E | = 668 1⁄4 | D | |||
| 12. | “ E-sharp | = 696 3⁄32 | F-sharp, C-sharp | |||
| 13. | “ E-sharp | = 687 1⁄2 | G-sharp. | |||
| 14. | “ F | = 704 | C, F, B-flat | |||
| 15. | “ F-sharp | = 742 1⁄2 | G, D, E, B, F-sharp, C-sharp | |||
| 16. | “ F-sharp | = 753 1⁄3 | A | |||
| 17. | “ G | = 792 | C, D, F, G | |||
| 18. | “ G | = 782 4⁄13 | B-flat | |||
| 19. | “ G | = 781 34⁄36 | E-flat | |||
| 20. | “ Fx | = 773 6⁄16 | G-sharp | |||
| 21. | “ G-sharp | = 825 | A, E, B, G-sharp | |||
| 22. | “ G-sharp | = 835 5⁄16 | F-sharp, C-sharp | |||
| 23. | “ A-flat | = 833 25⁄27 | E-flat | |||
| 24. | “ A | = 880 | C, E, F, A | |||
| 25. | “ A | = 881 | D | |||
| 26. | “ A | = 891 | G | |||
| 27. | “ A-sharp | = 928 1⁄8 | B, F-sharp, C-sharp, G-sharp | |||
| 28. | “ B-flat | = 938 2⁄3 | F, B-flat, E-flat | |||
| 29. | “ B | = 990 | C, G, D, A, E, B, F-sharp | |||
| 30. | “ B-sharp | = 1031 1⁄4 | G-sharp | |||
| 31. | “ B-sharp | = 1044 8⁄64 | C-sharp |
| The sound | Appears in the scales of | ||
|---|---|---|---|
| 1. | C | = 528 | C, F, G, B-flat. |
| 2. | C | = 521 11⁄54 | E-flat |
| 3. | C-sharp | = 556 7⁄8 | D, B, F-sharp, C-sharp |
| 4. | C-sharp | = 550 | A, E, G-sharp |
| 5. | D | = 594 | C-G |
| 6. | D | = 586 2⁄3 | A, F, B-flat, E-flat |
| 7. | D-sharp | = 618 3⁄4 | E, B, F-sharp, G-sharp |
| 8. | D-sharp | = 626 31⁄64 | C-sharp. |
| 9. | E-flat | = 625 4⁄9 | B-flat |
| 10. | E | = 660 | C, G, A, E, B-flat |
| 11. | E | = 668 1⁄4 | D |
| 12. | E-sharp | = 696 3⁄32 | F-sharp, C-sharp |
| 13. | E-sharp | = 687 1⁄2 | G-sharp. |
| 14. | F | = 704 | C, F, B-flat |
| 15. | F-sharp | = 742 1⁄2 | G, D, E, B, F-sharp, C-sharp |
| 16. | F-sharp | = 753 1⁄3 | A |
| 17. | G | = 792 | C, D, F, G |
| 18. | G | = 782 4⁄13 | B-flat |
| 19. | G | = 781 34⁄36 | E-flat |
| 20. | Fx | = 773 6⁄16 | G-sharp |
| 21. | G-sharp | = 825 | A, E, B, G-sharp |
| 22. | G-sharp | = 835 5⁄16 | F-sharp, C-sharp |
| 23. | A-flat | = 833 25⁄27 | E-flat |
| 24. | A | = 880 | C, E, F, A |
| 25. | A | = 881 | D |
| 26. | A | = 891 | G |
| 27. | A-sharp | = 928 1⁄8 | B, F-sharp, C-sharp, G-sharp |
| 28. | B-flat | = 938 2⁄3 | F, B-flat, E-flat |
| 29. | B | = 990 | C, G, D, A, E, B, F-sharp |
| 30. | B-sharp | = 1031 1⁄4 | G-sharp |
| 31. | B-sharp | = 1044 8⁄64 | C-sharp |
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:
| A-flat 833 25⁄27 | B-flat 938 26⁄316 | C 1042 73⁄236 | D-flat 1111 73⁄81 | E-flat 1250 8⁄9 | F 1389 21⁄52 | G 1563 132⁄216 | A-flat 1666 50⁄54 |
| D-flat 555 154⁄162 | E-flat 624 640⁄1290 | F 694 365⁄648 | G-flat 741 130⁄486 | A-flat 833 150⁄162 | B-flat 926 284⁄486 | C 1042 528⁄1296 | D-flat 1111 146⁄152 |
| G-flat 741 130⁄486 | A-flat 823 3600⁄3888 | B-flat 926 1136⁄1944 | C-flat 988 520⁄1458 | D-flat 1111 308⁄456 | E-flat 1285 650⁄1458 | F 1389 1408⁄3088 | G-flat 1482 260⁄486 |
By examining the last table the reader will perceive that we have obtained fourteen new sounds. They are shown graphically in this manner: