Maximal dissonance tuning: Difference between revisions

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A '''~~maximal~~ dissonance ~~tuning~~''' is ~~a tuning where the intervals between consecutive pitches are selected to maximize [[~~sensory dissonance~~]] according to a psychoacoustic model~~ of the ~~human~~ inner ear.

While musical [[dissonance]] depends on many complex factors including the listener's conditioning and cultural background, ''sensory dissonance'' or ''roughness'' is much more consistent among average human listeners (i.e. those not affected by conditions such as congenital amusia). In its simplest form, sensory dissonance happens when two sine tones are played simultaneously at roughly equal intensity, close enough in frequency that the basilar membrane has trouble distinguishing them but far apart enough that beating is not audible. This is purely about the physiology of the inner ear, and not about approximations to familiar intervals such as JI ratios.

~~There are several~~ published ~~formulas on which~~ a ~~maximal dissonance tuning can be based. One~~ of ~~the earliest (not necessarily the most accurate) was found by Kameoka & Kuriyagawa in 1969, who played two~~ sine ~~waves simultaneously at equal intensity for human listeners~~. They ~~computed based on experiments~~ that if the lower ~~sine wave~~ has frequency ''f'' Hz, ~~then~~ the upper frequency ~~that maximizes sensory dissonance~~ is ~~about~~ <math>D(f) = f + 2.27 f^{0.477}\ \text{Hz}</math>. The ratio <math>\frac{D(f)}{f}</math> changes slowly with absolute frequency. It is about 3.22 semitones at 100 Hz, 1.55 semitones at 440 Hz, and 1.03 semitones at 1000 Hz.

In 1969, Kameoka & Kuriyagawa published the results of a study where listeners were asked to rate the roughness of pairs of sine tones. They calculated an approximate formula that, given a sine tone frequency, produces another frequency above it that ostensibly maximizes the sensation of roughness. If each sine tone is played at a comfortable 57 dB SPL and the lower one has frequency ''f'' Hz, their analysis states that the upper sine tone of maximal frequency is given by <math>D(f) = f + 2.27 f^{0.477}\ \text{Hz}</math>. The ratio <math>\frac{D(f)}{f}</math> changes slowly with absolute frequency. It is about 3.22 semitones at 100 Hz, 1.55 semitones at 440 Hz, and 1.03 semitones at 1000 Hz.

From ~~the Kameoka & Kuriyagawa formula, a maximal dissonance tuning can be constructed. Start with a low frequency such as ''f'' = 20 Hz and iteratively compute~~ <math>D(f)</math>~~, <math>D(D(f))</math>, etc. Collect all these frequencies and~~ the ~~result is a maximal dissonance tuning~~. ~~As <math>\frac{D(f)}{f}</math> is frequency-dependent,~~ the ~~resulting tuning is not periodic~~ and ~~can't even be arbitrarily transposed, but small amounts won't upend the overall psychoacoustic effect~~.

From <math>D(f)</math> Kameoka & Kuriyagawa designed a somewhat ad hoc formula for measuring the roughness of any two sine waves played at roughly equal intensity. This was one of the earliest studies into quantification of roughness; later research has poked holes in their methodology and produced somewhat different results.

~~Other maximal dissonance tunings are possible, using different formulas based on assumptions~~ of ~~more complex timbres than sine waves. Simply iterating a function like~~ <math>D(f)</math> ~~makes~~ the ~~tuning only psychoacoustically interesting for~~ consecutive ~~scale steps~~. ~~It may be interesting to, for example, optimize every group of~~ ''n'' > ~~2 consecutive pitches~~ to ~~maximize dissonance~~.

Regardless of the accuracy of <math>D(f)</math>, it can be used to construct a strange tuning that maximizes the sensory dissonance between consecutive tones. Start with a low frequency such as ''f'' = 20 Hz and iteratively compute <math>D(f)</math>, <math>D(D(f))</math>, etc. Collect all these frequencies to produce a tuning. As <math>\frac{D(f)}{f}</math> is frequency-dependent, the resulting tuning is not [[periodic scale|periodic]] and can't even be arbitrarily transposed without ruining its nature (although small transpositions won't upend the overall psychoacoustic effect).

[[Category:~~Dissonance~~]]

More sophisticated approaches are possible for constructing tunings from sensory dissonance power laws. Simply iterating a function like <math>D(f)</math> makes the tuning only psychoacoustically interesting for consecutive scale steps. It may be interesting to, for example, optimize every group of ''n'' > 2 consecutive pitches to maximize roughness. <math>D(f)</math> also works on the assumption of sine wave tones, and can be expanded to more complex timbres.

[[Category:Consonance and dissonance]]

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-A '''maximal dissonance tuning''' is a tuning where the intervals between consecutive pitches are selected to maximize [[sensory dissonance]] according to a psychoacoustic model of the human inner ear.
+While musical [[dissonance]] depends on many complex factors including the listener's conditioning and cultural background, ''sensory dissonance'' or ''roughness'' is much more consistent among average human listeners (i.e. those not affected by conditions such as congenital amusia). In its simplest form, sensory dissonance happens when two sine tones are played simultaneously at roughly equal intensity, close enough in frequency that the basilar membrane has trouble distinguishing them but far apart enough that beating is not audible. This is purely about the physiology of the inner ear, and not about approximations to familiar intervals such as JI ratios.
-There are several published formulas on which a maximal dissonance tuning can be based. One of the earliest (not necessarily the most accurate) was found by Kameoka & Kuriyagawa in 1969, who played two sine waves simultaneously at equal intensity for human listeners. They computed based on experiments that if the lower sine wave has frequency ''f'' Hz, then the upper frequency that maximizes sensory dissonance is about <math>D(f) = f + 2.27 f^{0.477}\ \text{Hz}</math>. The ratio <math>\frac{D(f)}{f}</math> changes slowly with absolute frequency. It is about 3.22 semitones at 100 Hz, 1.55 semitones at 440 Hz, and 1.03 semitones at 1000 Hz.
+In 1969, Kameoka & Kuriyagawa published the results of a study where listeners were asked to rate the roughness of pairs of sine tones. They calculated an approximate formula that, given a sine tone frequency, produces another frequency above it that ostensibly maximizes the sensation of roughness. If each sine tone is played at a comfortable 57 dB SPL and the lower one has frequency ''f'' Hz, their analysis states that the upper sine tone of maximal frequency is given by <math>D(f) = f + 2.27 f^{0.477}\ \text{Hz}</math>. The ratio <math>\frac{D(f)}{f}</math> changes slowly with absolute frequency. It is about 3.22 semitones at 100 Hz, 1.55 semitones at 440 Hz, and 1.03 semitones at 1000 Hz.
-From the Kameoka & Kuriyagawa formula, a maximal dissonance tuning can be constructed. Start with a low frequency such as ''f'' = 20 Hz and iteratively compute <math>D(f)</math>, <math>D(D(f))</math>, etc. Collect all these frequencies and the result is a maximal dissonance tuning. As <math>\frac{D(f)}{f}</math> is frequency-dependent, the resulting tuning is not periodic and can't even be arbitrarily transposed, but small amounts won't upend the overall psychoacoustic effect.
+From <math>D(f)</math> Kameoka & Kuriyagawa designed a somewhat ad hoc formula for measuring the roughness of any two sine waves played at roughly equal intensity. This was one of the earliest studies into quantification of roughness; later research has poked holes in their methodology and produced somewhat different results.
-Other maximal dissonance tunings are possible, using different formulas based on assumptions of more complex timbres than sine waves. Simply iterating a function like <math>D(f)</math> makes the tuning only psychoacoustically interesting for consecutive scale steps. It may be interesting to, for example, optimize every group of ''n'' > 2 consecutive pitches to maximize dissonance.
+Regardless of the accuracy of <math>D(f)</math>, it can be used to construct a strange tuning that maximizes the sensory dissonance between consecutive tones. Start with a low frequency such as ''f'' = 20 Hz and iteratively compute <math>D(f)</math>, <math>D(D(f))</math>, etc. Collect all these frequencies to produce a tuning. As <math>\frac{D(f)}{f}</math> is frequency-dependent, the resulting tuning is not [[periodic scale|periodic]] and can't even be arbitrarily transposed without ruining its nature (although small transpositions won't upend the overall psychoacoustic effect).
-[[Category:Dissonance]]
+More sophisticated approaches are possible for constructing tunings from sensory dissonance power laws. Simply iterating a function like <math>D(f)</math> makes the tuning only psychoacoustically interesting for consecutive scale steps. It may be interesting to, for example, optimize every group of ''n'' > 2 consecutive pitches to maximize roughness. <math>D(f)</math> also works on the assumption of sine wave tones, and can be expanded to more complex timbres.
+[[Category:Consonance and dissonance]]