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Encyclopedia of Microtonal Music Theory

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[Joe Monzo]

The relative consonance/dissonance of a musical interval.

Rather than use the above phrase (as Partch did), Monzo has adopted the single term sonance, because he agrees with the assertion - made by both Schoenberg and Partch, among others - that rather than describing two diametrically-opposed sensations, consonance and dissonance refer instead to the opposite poles of a single continuum of sensation. (An early and influential expression of this idea was presented by Helmholtz - see below.)

Monzo's theory of sonance actually holds that there are two separate continua of sensation, based on the ratio perceived by the listener to be that between the frequencies of the two tones in the interval:

  1. one, determined by the values of the prime-factors of that ratio,
  2. the other, determined by the values of the exponents of those factors.

Sonance is directly proportional to both of these values; thus, dissonance increases (and consonance simultaneously decreases) as both the prime-factors and the values of the exponents of those factors become larger. This idea was expressed earlier by Ben Johnston and others; the earliest reference to it which known by the authors is The true character of modern music, written in 1764 by the mathematician Leonhard Euler. Harmonic lattice diagrams are a graphical representation of this theory of sonance.

The perceptual basis of Monzo's theory thus allows that the actual tuning of the interval may be a ratio with far higher primes or exponents, or in fact may not be rational at all (as in the case of temperaments), but that the listener will, at least to some extent, interpret or understand that interval as a rational one with the smallest prime-factors and exponents recognized by his aural and/or music-theoretical experience.

Recent speculation among tuning theorists (mid-1999) has raised the idea that consonance and dissonance may actually be two separate and not mutually-exclusive dimensions of sonance. Monzo has extrapolated this to the idea that each prime factor may in fact be responsible for a separate dimension of sonance that does not necessarily exclude any of the others.

It is also important to note that sonance is usually determined not merely as an auditory phenomenon, but rather as a result of musical context, highly dependent on the style of a particular composer or era. Many tuning theorists have recently (1999) come to the consensus that the term "accordance" (describing the continuum from concordance to discordance) should be used for the former, restricting "sonance" for the latter.

The concept of sonance goes back to Benedetti.

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[John Chalmers, Yahoo tuning message]

I recommend we distinguish between "sensory consonance" (aka roughness, sonance etc.) and "contextual consonance" as Tenney does in his History of Consonance and Dissonance.

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[Hermann Helmholtz, On the Sensations of Tone, p 226 and 227]

Consonance is a continuous, dissonance an intermittent sensation of tone.

... We have found that from the most perfect consonance to the most decided dissonance there is a continuous series of degrees, of combinations of sound, which continually increase in roughness, so that there cannot be any sharp line drawn between consonance and dissonance, and the distinction would therefore seem to be merely arbitrary.

(Immediately after this quote, Helmholtz devotes three pages to a discussion of Euler's theories of consonance and dissonance.)

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