Musical Concepts

Appendices: Part D – Musical Concepts (excerpt 2)

This harmonic interval, which Bekesy pointed out is the basis of music, language and speech and, by extrapolation, is also the basis of our DNA (not only human DNA, but all earthly DNA), corresponds to the twelve frequency ratios within an octave. Our present even-tempered (diatonic) scale has seven notes (C, D, E, F, G, A and B.), but the twelve steps (semitones), which represent the flats and sharps as the average of the frequency of the note above and below it. Therefore, the entire octave is divided into C, C#, D, D#, E, F, F#, G, G#, A, A# and B.

The reason for the twelve steps or semitones is explained in P͟s͟y͟c͟h͟o͟l͟o͟g͟y͟ ͟o͟f͟ ͟M͟u͟s͟i͟c͟: “It is found that on the whole, our present half-tone step is as small a step as the average of an unselected population can hear with reasonable accuracy, enjoy and reproduce in the flow of melody and harmony in actual music.”¹⁹

These twelve steps or ratios of successive notes to the tonic in the major mode are as follows:

N͟o͟t͟e͟s͟ R͟a͟t͟i͟o͟s͟
C:D 9:8
C:E 5:4
C:F 4:3
C:G 3:2
C:A 5:3
C:B 15:8
C¹ :C 2:1 (an octave)

In the A minor mode, the ratios are: 9:8 (A:B), 6:5 (A:C), 4:3 (A:D), 3:2 (A:E), 8:5 (A:F), 9:5 (A:G), and 2:1 (A:A) (an octave).

As demonstrated by Sir James Jeans, the frequency ratios between the twelve notes represent “a frequency ratio of 1.05946, since this is the twelfth root of 2.”²⁰ The square root of 2 is used because “the exact interval of 2 is spread equally over the twelve semitone intervals which make the octave.” 2:1 is C¹ :C, which represents an octave.²¹

“Frequency ratios within the octave are as follows (C=1):

C# = 1.05946
D = (1.05946)² = 1.1225
D# = (1.05946)³ = 1.1892
E = (1.05946)⁴ = 1.2599
F = (1.05946)⁵ = 1.3348
F# = (1.05946)⁶ = 1.4142
G = (1.05946)⁷ = 1.4983
G# = (1.05946)² = 1.5874
A = (1.05946)⁸ = 1.6818
A# = (1.05946)¹⁰ = 1.7818
B = (1.05946)¹¹ = 1.8877
C1 = (1.05946)¹² = 2.0000”²²

The octave corresponds to a 2 to 1 ratio frequency, just as hydrogen does to oxygen in the water molecule, H₂O, having the molecular shape of a right , whose = 90° and whose height/amplitude or hypotenuse = √2, so that the frequency doubles when you increase the octave, and it is a general law that the frequency is proportional to the square root of the tension [of a string] and that the period is exactly proportional to the length of the string, so that the frequency of vibration varies inversely as the length of the string. The frequency interval of 1.05, π/3, which corresponds to our interval of consciousness, is what both the melody and harmony are built upon, and it is also the ratio that determines the lunar and solar year in the eclipse cycle.²³

As pointed out in P͟s͟y͟c͟h͟o͟l͟o͟g͟y͟ ͟o͟f͟ ͟M͟u͟s͟i͟c͟, “While all music is objectively due to physical sound waves, we must bear in mind that we can never be directly aware of the rate of vibrations as such, for we hear it as musical pitch. This is one of the wondrous transformations from matter to mind.”²⁴

Then pitch (frequency) represents the 120 discrete tones which, as the rest of the electromagnetic spectrum, are quantized by musical instruments which “are inherently resonant systems and therefore respond to only certain frequencies.”²⁵

End Notes:

19 Psychology of Music, Seashore, p. 140
20 Science and Music, Jeans, p. 25
21 Ibid., p. 63
22 Ibid., p. 29
23 Sacred Geometry, Lawlor, p. 51
24 Psychology of Music, Seashore, p. 128
25 Music, Physics, and Engineering, Olson, p. 59