It seems entirely fitting that the masterwork on the physiological foundations of music theory by one of the pioneers of electrodynamics should have an electronic version. When the book was first published in 1863, Alexander Graham Bell was 15 years old and 14 years from inventing the telephone (although Bell is praised in one of the translator’s idiosyncratic footnotes for his ability to tap out nine distinct vowel sounds with his finger against his teeth). The oscilloscope, which Helmholtz could have used to visualize waveforms was 35 years away from invention, while the frequency analyzer, which he could have used to investigate the harmonic content of a tone, and the electronic synthesizer on which he might have replayed it were both a century in the future.
To acquire the waveform of a bowed string, Helmholtz uses a ‘vibration microscope’. This consists of a lens which vibrates vertically, driven by a tuning fork, through which the observer views a piece of starch attached to a string which vibrates horizontally. The afterimage in the observer’s visual cortex forms a lissajous figure, which he can sketch and by comparison with lissajous figures of known functions extract the waveform.
To break down the harmonic content of tones, Helmholtz plugs resonators into his ears to amplify overtone frequencies and notes down the apparent strength of the resonator’s output using musical loudness terms. To play specific frequencies, he has tuning forks and acoustic sirens, while to synthesize tones with given overtone structure, he builds an apparatus using excited tuning forks to generate the tones and resonators with adjustable openings to control their amplitude.
When I first read Helmholtz’s “On the Sensations of Tone” in the early 1980’s, my computer had 8K of onboard memory, and a cassette drive for storage, but I dreamt of an electronic version of the book, where you could listen to the auditory phenomena, and where the ingenious experiments described within could be reproduced or simulated. I needed an environment where the printed material could be faithfully presented and easily read, but where the sensations of the tones could be accurately conveyed. Modern web browsers gave me the former, and their implementation, in the last couple of years, of the sophisticated Web Audio API gave me the latter, so now the tools are all in place for the electronic version I dreamed of forty years ago.
The English version of “On the Sensations of Tone” can be viewed as a collaboration between Helmholtz and his translator Alexander Ellis. Our electronic version is a collaboration between myself and Danny Aley at Saltire Software, supported by Hannah Kemper, Dao Hoang and Chris McCormick, along with teams of interns working over 3 summers: Urara Kaneko, Andrea Nguyen, Jalen Chrysos, Duncan Soiffer, Padmini Bhagavatula, Julia Gilley, Kokoa Kaneko.
As you read this electronic version, you will be able to listen to a variety of different types of audio content. Some of the effects Helmholtz is eliciting are subtle, and will be best appreciated with a reasonable quality audio output. Best to use headphones or decent speakers, and not rely on the speakers built into your laptop. In one or two places, there is the opportunity to give audio input. Again a better quality mic and a quieter background environment would lead to a better experience.
There are a number of different types of audio element in the book. The following table lists them, and gives you a link to an example of the form to let you try it.
| Chord from written notes | Chapter 12 p. 212 |
| Chord displayed on staff | Chapter 12 p. 215 |
| Sequences of notes (scales) | Chapter 14 p. 257 |
| Frequencies | Chapter 1 p. 17 |
| Tune snippets. Clicking on the staff plays the tune. | Chapter 14 p. 261 |
| Keyboards with special tunings | Chapter 10 p. 185 |
| Amplitudes of harmonics | Chapter 5 p. 79 |
| Apps | Chapter 1 p. 9 |
One thing to note is that Helmholtz uses the musical staff in two entirely different ways. In some examples, he intends note lengths to be taken to indicate timing, and the whole piece played as a tune. In other cases he uses note length to convey some other information, quite unrelated to timing. In these cases it is meaningless to play the entire piece of music as written. We instead give you the ability to click on the individual notes or note groupings.
Some tips on using the electronic book follow:
The Info Panel has three elements:
The Waveform Audio Display gives you a picture of the current audio as a waveform. It is flat if no audio is output and adapts quickly to new input.
The Frequency Audio Display is a picture of which frequencies are present in the waveform. As this involves an averaging process, there is a slight lag before it adapts to new input.
The Page Navigation Panel shows what page you are on, and lets you step ahead or back by page or by chapter.
Reading Helmholtz, I come away humbled by the quality of the results and the basis for theorization achieved with the relatively primitive experimental means at his disposal. I hesitate to imagine the creative use he’d have made of the toolbox we’re given by the Web Audio API and hope our efforts don’t fall too far short.
Philip Todd, Brightwood OR 2021