A great amount of musical thought examines consonance and dissonance.
Musical compositions play these two basic concepts against each other using timbre, melody, harmony, loudness, phasing, and articulation effects. Patterns of consonance and dissonance playing out through time, and reacting with memory and anticipation, create musical experience.
There's plenty of psychoacoustic data relating the measurable aspects of sound to what human ears can detect and understand. It generally states that humans ears can hear sounds in a certain frequency range and only if they are "loud enough" at each frequency. They can show how close two tones' frequencies can be before they are perceived as the same. As the the tones' interval widens, it passes through a period of "roughness" and then dissonant distinction, and occasionally, a resolution to harmony. The ability to recognize consonance is rather dependent of the timbre and loudness of the tones in question. But all things being equal, two tones are perceived as being consonant if their frequencies are related to each other as a simple whole number ratio. This simple ratio of frequencies means that, after a relatively short period of time, the two tones' pressure waveforms will maintain a constant phase relationship. In effect, the two tones merge into one tone.
Consider a raw sound source, with a series of pulses at a frequency "F0". Almost any repeating process can generate a series of pulses like this. It doesn't even have to be a pure pulse: often the high harmonic magnitudes are lower or missing.
If this pulse is run trough a tube or transmitted to a string of a certain length, a phenomenon called "comb filtering" occurs, where the frequencies of harmonics of the natural frequency of the resonator ("R0") are emphasized, while other frequencies are attenuated or suppressed. Thus, when the "excitation" frequency F0 is not related to the resonant frequency R0, the resulting wave, played through the resonator, is quieter. The resulting comb filtered sound will appear as a timbre of its own, taken from the energy in the excitation signal.
The harmonics of the resonator are determined by its lenght (and the speed of sound in the resonating medium). The shorter the resonator, the higher the resonant frequencies.
An equivalent phenomenon happens if you keep the resonator at a standard length and change the excitation frequency. As the two frequencies change, different harmonics will be emphasized.
If you make that excitation frequency low, the higher harmonics will be more audible.
And here is the magic part: if you feed an increasing excitation signal into three resonators, one at 4/3 of R0, R0, and 3/2 of R0, the harmonics of each resonator will be excited, and, if most of the harmonic content of the excitation signal is below the 7th harmonic, it creates as part of the series of excited, resonated frequencies, a familiar diatonic scale, each tube resonating in this order:
Cents Ratio 12TET interval +/- cents 0 1/1 I +0 203.91 9/8 II +3.91 386.31 5/4 III -13.69 498.04 4/3 IV -1.96 701.95 3/2 V +1.95 884.35 5/3 VI -15.65 1088.26 15/8 VII -11.74You can see that the unison and fifth are highly reenforced in this system. If you expand the audible harmonics up to the 10th harmonic, you get this expanded scale:
Cents Ratio 12TET interval +/- cents 0 1/1 I +0 203.91 9/8 II +3.91 266.87 7/6 IIIb -33.13 386.31 5/4 III -13.69 470.78 21/16 IV -29.22 498.04 4/3 IV -1.96 701.95 3/2 V +1.95 884.35 5/3 VI -15.65 905.86 27/16 VI +5.86 968.82 7/4 VIIb -31.18 1088.26 15/8 VII -11.74or, if you add resonators for the 5/4 and 4/5 to those of the 3/2 and 4/3, you get this dense scale:
Cents Ratio 12TET interval +/- cents 0 1/1 I +0 203.91 9/8 II +3.91 315.64 6/5 IIIb +15.64 386.31 5/4 III -13.69 498.04 4/3 IV -1.96 701.95 3/2 V +1.95 772.62 25/16 VIb -27.38 813.68 8/5 VIb +13.68 884.35 5/3 VI -15.65 1088.26 15/8 VII -11.74
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