NUMERICAL COMPARISON OF THE DIATONIC SCALE WITH THE TEMPERED SCALE.
The following table, comparing vibration numbers of the diatonic scale with those of the tempered, shows the difference in the two scales, existing between the thirds, fifths and other intervals.
Notice that the difference is but slight in the lowest octave used which is shown on the left; but taking the scale four octaves higher, shown on the right, the difference becomes more striking.
|DIATONIC.|TEMPERED.| |DIATONIC.|TEMPERED.| C|32. |32. |C|512. |512. | D|36. |35.92 |D|576. |574.70 | E|40. |40.32 |E|640. |645.08 | F|42.66 |42.71 |F|682.66 |683.44 | G|48. |47.95 |G|768. |767.13 | A|53.33 |53.82 |A|853.33 |861.08 | B|60. |60.41 |B|960. |966.53 | C|64. |64. |C|1024. |1024. |
Following this paragraph we give a reference table in which the numbers are given for four consecutive octaves, calculated for the system of equal temperament. Each column represents an octave. The first two columns cover the tones of the two octaves used in setting the temperament by our system.
TABLE OF VIBRATIONS PER SECOND.
C |128. |256. |512. |1024. | C[#] |135.61 |271.22 |542.44 |1084.89 | D |143.68 |287.35 |574.70 |1149.40 | D[#] |152.22 |304.44 |608.87 |1217.75 | E |161.27 |322.54 |645.08 |1290.16 | F |170.86 |341.72 |683.44 |1366.87 | F[#] |181.02 |362.04 |724.08 |1448.15 | G |191.78 |383.57 |767.13 |1534.27 | G[#] |203.19 |406.37 |812.75 |1625.50 | A |215.27 |430.54 |861.08 |1722.16 | A[#] |228.07 |456.14 |912.28 |1824.56 | B |241.63 |483.26 |966.53 |1933.06 | C |256. |512. |1024. |2048. |
Much interesting and valuable exercise may be derived from the investigation of this table by figuring out what certain intervals would be if exact, and then comparing them with the figures shown in this tempered scale. To do this, select two notes and ascertain what interval the higher forms to the lower; then, by the fraction in the table below corresponding to that interval, multiply the vibration number of the lower note.
EXAMPLE.—Say we select the first C, 128, and the G in the same column. We know this to be an interval of a perfect fifth. Referring to the table below, we find that the vibration of the fifth is 3/2 of, or 3/2 times, that of its fundamental; so we simply multiply this fraction by the vibration number of C, which is 128, and this gives 192 as the exact fifth. Now, on referring to the above table of equal temperament, we find this G quoted a little less (flatter), viz., 191.78. To find a fourth from any note, multiply its number by 4/3, a major third, by 5/4, and so on as per table below.
TABLE SHOWING RELATIVE VIBRATION OF INTERVALS BY IMPROPER FRACTIONS.