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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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[Joe Monzo, Yahoo Tuning List, Sun, 29 Nov 1998 06:17:37 -0800]

Well, regardless of what relationships the notes actually have (i.e., even if they're irrational proportions as in an equal-temperament), we are always perceiving harmonic relationships between simultaneous tones - that's simply a function of the musical notes having periodicity.

Prime factors in these ratios may be very high, but may be close enough in pitch to ratios of lower prime numbers, and the musical context may supply enough of an implication supporting these lower primes, that we will interpret the higher-prime ratio as one with lower prime factors, which would probably aid in our understanding the relationship by making it simpler.

Or it may be the case that a higher prime in any given instance may provide a more distinctive or unique quality (affect) than that provided by an interpretation which favors lower prime factors but higher exponents. This may help to make the note blend better or stand out better, depending on context.

{As an exaggerated example:

On the one hand, considered melodically, a "major 3rd" of 34 [= 81/64 = 4.08 Semitones] is merely another note in the cycle of powers of 3, but 51 [= 5/4 = 3.86 Semitones] is a powerful new identity, a new odd- or prime-base, 5. It would provide a whole new set of intervallic relationships which provide affective information which differs from that of the intervals in the 3-Limit system. In this sense, 51 would stand out, which would make it good as an emphasized melodic note. On the other hand, considered harmonically, in an accented triad of 64:81:96 [= 1/1 : 81/64 : 3/2] the 34 would stand out as a dissonance, whereas in a triad tuned to 4:5:6 [= 64:80:96 = 1/1 : 5/4 : 3/2], 51 would blend right in as the low-prime and low-exponent 5-identity of the 1/1-otonality. So in this sense, 34 would be the choice for a note that raises tension.

- this is a good example of how musical context determines our perception of the sounds.

Similarly, 5-limit JI gives us at least two "minor 7ths", which are both obviously related (according to my ears anyway) to the other 5-limit ratios, but 71 [= 7/4 = 9.69 Semitones] is a sound unlike any 5-limit ratio, and I would venture to argue that 3252 [= 225/128 = 9.77 Semitones] would most often be perceived as a slightly sharp 71, again, depending on context.}

This aspect ties in with the commonly-accepted idea that sonance is proportional to differing size of both prime factor and exponent. It's not necessarily that the ratio is able to be interpreted more simply, or more consonantly (or less dissonantly) - that is, simply by degree of sonance - but rather that the different interpretations also permit of different types of sonance.

The important point is that these interpretations are always fluid and changing, completely dynamic. And because the infinite quality of numbers permits an equally infinite spectrum of rational interpretation, it becomes very difficult, without computer frequency analysis (and probably even with it), to ascertain precisely what is the pitch of a certain note.

(I believe that it was Schoenberg's recognition of this, and his frustration at his inability to organize his harmonic ideas numerically, thru sheer plenitude of possibility, that led him to so strongly accept the 12-EQ system as a practical compromise.[*] I'm quite certain that he had multifarious rational implications of his harmonies firmly in mind while composing, whatever those implications were; for this reason I find him one of the most interesting of theorists and of composers, plus a lot of his stuff sounds great - just to prove that I can still like good ol' 12, too.)

In fact, modern research is showing that musical sounds are not quite as periodic or regular or easily-definable as was once thought. To me, this provides further weight to the idea that we are perceiving rational relationships which are much more complicated than has previously been described, the main complication being the fact that time is an essential dimension in which music must be perceived, and the sounds are changing all the time.

(Indeed, it is often the case that when sounds do not change enough to suit our appetite for stimulation, we find the music boring - as in "electronic- sounding" timbres with little nuance, or very consonant slow-moving JI music, etc. - altho both of these can of course produce good music if used well! [the big hit "Dah dah dah" by a German group in the 80s, with keyboard part played on a tiny cheap Casio, and used very recently in a TV commercial, for the former; the music of La Monte Young for the latter]

[*] Someone please dig out the old Tuning Digest from the spring of this year where I talked about Schoenberg stating that the possibilities were infinite, and citing book and page number.

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