Music and the Making of Modern Science

250 Chapter 16

surface areas of our central bodies excite, seems more qualified than any other body to

open new vistas in the investigation about the nature and properties of matter. In particular,

the wavelengths of the first four hydrogen lines excite and arrest attention. ”^13 Where Stoney

had preferred to study a simpler spectrum , Balmer chose the simplicity and fundamentality

of the element , despite its more complex spectrum.

Seemingly aware of Stoney ’ s conclusions about hydrogen, Balmer notes that “ the rela-

tionships of these wavelengths allow themselves to be expressed with surprising accuracy

through small numbers. Thus, the wavelength of the red hydrogen line is to the violet as

8 to 5; that of the red to the blue-green as 27 to 20 and that of the blue-green to the violet

as 32 to 27. ”^14 Though in many texts Balmer is depicted as having guessed his results by

mere trial and error, he explains that they came from analogies with sound:

This circumstance must necessarily recall analogous relations in acoustics, and one believes the

vibrations of the same spectral lines of a substance ought to be understood almost as overtones of

the same characteristic fundamental tone [ Grundton ]. Yet all attempts to find such a fundamental

tone for hydrogen, for example, have not turned out satisfactorily. One would have come with such

a calculation to such large numbers that would not have yielded thereby a clearer insight. For

instance, one gets for the first, second, and fourth hydrogen lines the same fundamental tone of

which the second line represents the twenty-seventh multiple. With every newly included line the

sought-for fundamental tone will be represented with quite important augmented wavelengths.

Nevertheless, the idea suggested itself that there should be a simple formula with whose help the

wavelengths of the four indicated hydrogen lines could be represented.^15

From the hypothesis that hydrogen wavelengths represent overtones and using Å ngstr ö m ’ s

values for its four visible wavelengths, Balmer deduced that the fundamental wavelength

(corresponding to the Grundton ) is 3,645 × 10 – 7 mm = 3,645 Å , “ the fundamental number

[ Grundzahl ] of hydrogen, ” from which he then deduces the observed wavelengths expressed

as 3 645

2

,,Å 22

m

mn−

⎛

⎝⎜

⎞

⎠⎟

where m , n are integers. Though Balmer does not detail the “ various

grounds on which it is likely ” that the wavelengths should take this form, we can infer

from what he has already said that he derived it from his general idea of overtones applied

to the ratios between Å ngstr ö m ’ s wavelength values.

Based on this assumption, Balmer sought a common fundamental number as their

Grundton but then must have realized that none of the observed lines gave this number

directly, nor could he infer the wavelengths from the usual simple formulas for overtones

in terms of the Pythagorean ratios for octave, fifth, and so forth. Even so, his musical

assumption directed him to find a common number that, multiplied by simple fractions,

would give the observed Å ngstr ö m wavelengths, which his following description confirms:

“ the wavelengths of the first four hydrogen lines are given if the fundamental number h

= 3,645 [ Å ] is multiplied sequentially by the coefficients 9/5, 4/3, 25/21, and 9/8. Clearly

these four coefficients form no lawful sequence, but as soon as one multiplies the first and

the last by 4 the lawfulness comes forward and the coefficients maintain as numerator the