Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the ear drum in perpetual torment, bending in two different directions in order to yield to the ever-discordant impulses.Galileo Galilei, 1638
This page is an interactive demonstration of the human physiological perception of musical dissonance - according to recent thinking. If you have a browser that supports it (Chrome, Safari), you should be able to hear sounds while you play with the settings, by turning on the Sound switch under the "Sound" tab, above. You can then get started by clicking on points of the curve above, to add notes.
To start, there is a single note set to 440Hz - at the 'zero' position of the graph above. The graph shows an approximate scale of human-perceived dissonance when adding similar but differently pitched notes. The x-axis measures pitch in cents - hundredths of a semitone - and the vertical grey lines mark off semitones. The dips (local minima) of the graph are indicated by dots. Clicking on these will add notes that should sound consonant, or at least "minimally dissonant" compared to nearby pitches. The curve will update to reflect the new, combined sound. You can also click on the curve at any other point (e.g. dissonance maxima) to see what other tunings and collections of notes sound like. You can add, delete or edit notes using the notes tab. Slider values can by fine-tuned by clicking in the track to the left and right of the slider 'handle'.
There are more options for the basic note sound, under "Sound". Here you can change the pitch of the root note, the number of harmonic partials in each note, the dropoff (how quickly the partials of a note get quieter as they get higher), and also there is an option to use only odd partials - as in the sound of a square or triangular wave.
This app was built with the intention of exploring theories of dissonance perception advanced by Helmholtz (1895), Plomp & Levelt (1965), Sethares (1993), Vassilakis (2005) and others. The roughness calculation used here is that of Vassilakis. The app was inspired by a question on the Microtonal Music and Theory Facebook group and this Stack Exchange question and accepted answer. If you have any questions or comments please contact me.