Hermann von Helmholtz (1821-1894)
The following intervals that commonly occur in traditional tonality are defined in terms of ratios.
An A major scale can be derived by multiplying the frequency of the tonic, in this case A at 440 Hz, by whole number ratios, as shown below: [ H.L.F. Helmholtz, "Sensations of a Tone" (New York, Dover Publications Inc., Revised Edition, 1954) p. 15]
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The A Major Scale in Ratios
The same A major scale may be constructed in scale steps using only major and minor seconds, as shown below:
When tuning by ratios, it should be kept in mind that scales are derived from chord progressions, not vice-versa. The frequency set above, given by the brilliant 19th century German physicist-chemist-psychologist H.L.F. Helmholtz, is perhaps the most common tuning for the major scale, but is by no means the only one. Modulation is one reason that fixed values must be "unfixed" in order to accommodate the new theory.
Another point to note is that most pieces of traditional tonal music, even short and seemingly simple pieces by Mozart and Haydn, cannot be translated without modification from traditional notation into ratios. Chromatic passages are especially difficult to recast into this new framework.