Each point of the basilar membrane can be thought of as a band pass filter, allowing frequencies within a certain band to vibrate the membrane. The average critical bandwidth of a human ear is ~1mm. Frequency differences (between sounds) smaller than the critical bandwidth cause the perception of dissonance due to the instability of vibration within the basilar membrane. To write frequency in multiples of the critical bandwidth, x, we have:

x [mm] = speed of sound [mm/s] * 1/(frequency of sound) [s/cycle]

It has been found that for complex tones, the measure of dissonance between two tones is given by:

f(x) = 4*abs(x)*exp(1-4*abs(x))

where x is the frequency difference of the (fundamental frequency?) of the two tones in multiples of the critical frequency. f(x) = 0 corresponds to no dissonance.

Artificial spectra can be manufactured using this function. For example a note with partials at

225 Hz, 440 Hz, 615 Hz, 860 Hz

has corresponding x values of

And f(x) values of

This note alone (?) and played with a third note (below) is more consonant than manufacturing a spectra with partials at

220 Hz, 430 Hz, 602 Hz, 841 Hz

which has corresponding x values of

and f(x) values of

When these two notes are played against a third note with partials at

440 Hz, 860 Hz, 1203 Hz, 1683 Hz

the latter sounds dissonant, the former consonant.

Pierce (1966) suggested that almost any set of intervals can sound consonant by adjusting the harmonic content of the individual notes. Many scales have been constructed with the principles of consonant sounds. Pierce’s scale is an eight note octave scale constructed so that the partials of the notes either coincide or are separated by at least 1/8 of an octave.

In another example exploring consonance, Houtsma et. al. created four versions of a four part Bach chorale. The versions and their results are given below:

Ver. 1: exactly harmonic partials, amplitudes inversely proportional to harmonic number, exponential time decay. Equally tempered scale with semi-tones representing frequency ratios of the twelfth root of two.

Result 1: sounds normal

Ver. 2: partials of each note are stretched. scale stretched by same factor so that each semitone represents a frequency ratio of the twelfth root of 2.1.

Result 2: sounds consonant but weird

Ver. 3: partials are exactly harmonic, scale is stretched.

Result 3: out of tune.

Ver. 4: partials stretched, scale is equal temperament.

Result 4: out of tune.

How can these results be used for musical purposes? Perhaps there is an application in merging the music of different scales/genres. How do these results relate to the dynamic tuning? Also, what about the research indicating a psychological, physiological, or statistical reason for the Western scale to be judged as the most consonant scale? What do these results bring to bare on the discussion?

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