To understand musical consonance/dissonance, you must understand that consonance of simple harmonic ratios is an artifact of a much simpler underlying rule. The human hearing system does not analyze frequency ratios of individual notes, it examines the frequency domain clustering of partials of the sound as a whole.
If you listen to two sine waves of near identical frequency they sound consonant. Widen the frequency difference and they become dissonant. Further widen the frequency difference and they become consonant again. This was measured back in 1967 by R. Plomp and W. J. M. Levelt. The consonance of a musical harmony depends on the separation of the individual partials. We need a “critical bandwidth” of separation between frequencies to clearly distinguish them. You could think of dissonance as the unpleasant feeling of hearing different frequencies but failing to resolve them.
The majority of musical instruments used in Western classical music create sound by vibration constrained at two points, either the ends of a string or the ends of a column of air. Therefore the partials are all integer multiples [2] of the fundamental. It turns out that if these sounds are played together at small integer frequency ratios, the frequency of the partials align such that the quantity of dissonant, smaller than the “critical bandwidth”, frequency differences is at a local minimum.
However, percussion instruments are not constrained in this way, so cultures with a percussion focused musical tradition (eg. Indonesian gamelan music) developed alternative tuning systems better suited to the timbres of their instruments. Early electronic musicians, eg. Wendy Carlos, also noticed how the consonance of different tuning systems depended on the timbre of the notes.
As far as I am aware, the first person to mathematically formalize this relationship, and develop a method to generate arbitrary tuning systems for arbitrary timbres and vice-versa, was William Sethares [3]. He has a great webpage at http://sethares.engr.wisc.edu/ , with many audio examples. His book “Tuning Timbre Spectrum Scale” should be considered the most important book on music theory ever written because it generalizes all previous musical theories, and solves the problem of the exhaustion of harmonic novelty in music without having to resort to unlistenable crap like serialism.
And now we get to the link to the main article, and the reason why Sethare’s work was such a revelation to me. I shared a house with a music student for several years, and I became heavily involved in the classical music subculture. Back then I only knew of the Pythagorean ratio-based concept of harmony. I listened to a great variety of Western classical music, and attended several concerts. As my knowledge increased, I became disillusioned with pre-modern classical music, because each new composition began to sound like a reworking of something I had heard before. Traditional music theory simply didn’t have enough scope for novelty. I studied the works of Harry Partch, who pushed ratio-based music theory about as far as it can go, and I wasted a lot of time attempting to extend his theory, but I never felt I had reached a satisfactory conclusion.
Of course I was exposed to atonal composition via my musician friends, and my initial reaction was the same as almost everybody’s: I hated it. But both the obvious high status of this kind of music and my lack of knowledge of any alternative source of novelty slowly changed my preferences. I started listening to Second Viennese School composers and free jazz. The more I listened the more I liked it, and I gradually turned into an atonal music snob like my musician friends.
And then I left university and lost all contact with them. I forgot all about classical music for several years. When I listened to atonal music again I found I had reverted to my original preference. I’m now very certain the only reason I liked it was social signaling. I declared music to be dead and lost all interest in it.
When I later discovered Sethares’s work it shook my beliefs about music to the core. My whole atonal adventure was built on a mistake. We’re no longer limited by physical instruments and it’s really possible to compose music simultaneously strange and beautiful. I now promote Sethares’s work in the hope that more musicians will adopt it and create sometime great.
[1] R. Plomp and W. J. M. Levelt, “Tonal Consonance and Critical Bandwidth,” Journal of the Acoustical Society of America.38, 548-560 (1965).
[2] Approximately. Note that octaves on a piano are tuned slightly sharp, because piano strings are not simple mathematical abstractions, but have thickness and other properties such that they don’t produce perfectly harmonic sound.
[3] Sethares, W.A. (1993), Local consonance and the relationship between timbre and scale. Journal of the Acoustical Society of America, 94(1): 1218.
That’s fascinating, thank you! I will definitely check out Sethares’ work, as a music listener and an amateur composer.
It sounds very different for the type of music I have the most experience with (choral church music of various eras.)
To understand musical consonance/dissonance, you must understand that consonance of simple harmonic ratios is an artifact of a much simpler underlying rule. The human hearing system does not analyze frequency ratios of individual notes, it examines the frequency domain clustering of partials of the sound as a whole.
If you listen to two sine waves of near identical frequency they sound consonant. Widen the frequency difference and they become dissonant. Further widen the frequency difference and they become consonant again. This was measured back in 1967 by R. Plomp and W. J. M. Levelt. The consonance of a musical harmony depends on the separation of the individual partials. We need a “critical bandwidth” of separation between frequencies to clearly distinguish them. You could think of dissonance as the unpleasant feeling of hearing different frequencies but failing to resolve them.
The majority of musical instruments used in Western classical music create sound by vibration constrained at two points, either the ends of a string or the ends of a column of air. Therefore the partials are all integer multiples [2] of the fundamental. It turns out that if these sounds are played together at small integer frequency ratios, the frequency of the partials align such that the quantity of dissonant, smaller than the “critical bandwidth”, frequency differences is at a local minimum.
However, percussion instruments are not constrained in this way, so cultures with a percussion focused musical tradition (eg. Indonesian gamelan music) developed alternative tuning systems better suited to the timbres of their instruments. Early electronic musicians, eg. Wendy Carlos, also noticed how the consonance of different tuning systems depended on the timbre of the notes.
As far as I am aware, the first person to mathematically formalize this relationship, and develop a method to generate arbitrary tuning systems for arbitrary timbres and vice-versa, was William Sethares [3]. He has a great webpage at http://sethares.engr.wisc.edu/ , with many audio examples. His book “Tuning Timbre Spectrum Scale” should be considered the most important book on music theory ever written because it generalizes all previous musical theories, and solves the problem of the exhaustion of harmonic novelty in music without having to resort to unlistenable crap like serialism.
And now we get to the link to the main article, and the reason why Sethare’s work was such a revelation to me. I shared a house with a music student for several years, and I became heavily involved in the classical music subculture. Back then I only knew of the Pythagorean ratio-based concept of harmony. I listened to a great variety of Western classical music, and attended several concerts. As my knowledge increased, I became disillusioned with pre-modern classical music, because each new composition began to sound like a reworking of something I had heard before. Traditional music theory simply didn’t have enough scope for novelty. I studied the works of Harry Partch, who pushed ratio-based music theory about as far as it can go, and I wasted a lot of time attempting to extend his theory, but I never felt I had reached a satisfactory conclusion.
Of course I was exposed to atonal composition via my musician friends, and my initial reaction was the same as almost everybody’s: I hated it. But both the obvious high status of this kind of music and my lack of knowledge of any alternative source of novelty slowly changed my preferences. I started listening to Second Viennese School composers and free jazz. The more I listened the more I liked it, and I gradually turned into an atonal music snob like my musician friends.
And then I left university and lost all contact with them. I forgot all about classical music for several years. When I listened to atonal music again I found I had reverted to my original preference. I’m now very certain the only reason I liked it was social signaling. I declared music to be dead and lost all interest in it.
When I later discovered Sethares’s work it shook my beliefs about music to the core. My whole atonal adventure was built on a mistake. We’re no longer limited by physical instruments and it’s really possible to compose music simultaneously strange and beautiful. I now promote Sethares’s work in the hope that more musicians will adopt it and create sometime great.
[1] R. Plomp and W. J. M. Levelt, “Tonal Consonance and Critical Bandwidth,” Journal of the Acoustical Society of America.38, 548-560 (1965). [2] Approximately. Note that octaves on a piano are tuned slightly sharp, because piano strings are not simple mathematical abstractions, but have thickness and other properties such that they don’t produce perfectly harmonic sound. [3] Sethares, W.A. (1993), Local consonance and the relationship between timbre and scale. Journal of the Acoustical Society of America, 94(1): 1218.
Interesting! Examples 2 to 5 from here were particularly mindblowing. Thanks for the link!
That’s fascinating, thank you! I will definitely check out Sethares’ work, as a music listener and an amateur composer. It sounds very different for the type of music I have the most experience with (choral church music of various eras.)