Are all pure tones that are separated by intervals of greater than a minor third equally consonant?

Question

In this Quora answer

Why do certain musical notes sound good together?

the answerer claims

it is important to distinguish between pure tones, that is, simple sine waves, and real tones, such as those produced by a musical instrument or human voice, which in fact contain a (mostly) harmonic series of overtones of varying amplitudes. Psychoacoustic experiments on untrained listeners involving the perceived consonance of pairs of pure tones yields a surprising result. Consonance decreases rapidly as the pitch interval increases from zero (that is, a single tone), as one might expect from our experience with real tones, reaches a nadir at about one semitone, and then reaches nearly 100% again near a minor third. However, consonance is not perceived to decrease again as the interval is increased. That is, all pure tones that are separated by intervals of greater than a minor third are equally consonant.

I googled a bit and could not find evidence for this claim. It is an interesting claim and I would like to read more about the reasoning behind this.

He says

Why might this be? Johnston suggests that the answer may have something to do with the bandwidth of the frequency-tuned cochlear cells that detect sound in the inner ear. Two tones separated by larger than the bandwidth of a cochlear cell do not interfere in the ear at the site of transduction, whereas tones within the bandwidth do.

But again, I am unsure why a google search did not turn up more information. He cites a book and, if necessary, I will track it down at my library and read it, but I decided to ask here first to see if anyone has further information before resorting to that.

My Question:

Are all pure tones that are separated by intervals of greater than a minor third are equally consonant? Why?

Answer 1

The answer is no, according to William Sethares's Tuning, Timbre, Spectrum, Scale. See Fig 3.8 on Page 47, also reproduced as Figure 1 at http://www.acousticslab.org/learnmoresra/moremodel.html

This figure shows that at lower frequencies (100Hz), significantly more than a single octave is required for dissonance to die down between two pure tones (sinusoidal). Only above 1000Hz is the statement true.

Actually, looking at Kameoka & Kuriyagawa, Consonance theory part I: consonance of dyads, which Sethares claims as evidence, the curves seem to be off, and should be expanded horizontally. So when Sethares claims, "such curves have become widely accepted", that apparently shouldn't mean that they are in any way accurate.

But at least, Kameoka & Kuriyagawa's data also supports the answer: No, you need more than an octave. But after that the statement is roughly true.

Answer 2

This question gets at the how, but right now I don't have a good explanation of the why.

To my ear: yes they are. For frequency ratios above ~1.25 you just hear two tones "on top of" one another. Each is identifiable as a separate entity. It is only at the lower ratios that they merge together into a single, more dissonant, sound.

I tested this by constructing a PureData patch, using two osc~ objects, the first at 440Hz, the other tunable from 440Hz-880Hz. The PureData source code follows:

#N canvas 1920 0 1918 1049 10;
#X obj 305 531 dac~;
#X obj 365 319 osc~ 440;
#X obj 401 43 hsl 128 15 0 1200 0 0 empty empty empty -2 -8 0 10 -262144
-1 -1 0 1;
#X obj 412 189 pow 2 0;
#X obj 395 125 t b f;
#X obj 361 88 / 1200;
#X floatatom 391 272 5 0 0 0 - - -;
#X floatatom 570 262 5 0 0 0 - - -;
#X obj 291 441 *~ 0.49;
#X obj 393 162 f 2;
#X obj 166 230 osc~ 440;
#X obj 372 238 * 440;
#X connect 1 0 8 0;
#X connect 2 0 5 0;
#X connect 3 0 7 0;
#X connect 3 0 11 0;
#X connect 4 0 9 0;
#X connect 4 1 3 1;
#X connect 5 0 4 0;
#X connect 6 0 1 0;
#X connect 8 0 0 0;
#X connect 8 0 0 1;
#X connect 9 0 3 0;
#X connect 10 0 8 0;
#X connect 11 0 6 0;

Answer 3

If true, a possible explanation is that, since the tones are pure, there is no beating between upper harmonics of the tones. Dissonance is certainly more noticeable when there is beating between harmonics. If harmonic-rich 200 Hz and 293 Hz frequencies are played simultaneously, the 3rd harmonic of the 200 Hz tone will be 600 Hz while the second harmonic of the 293 Hz tone will be 586 Hz. There will be beating between these harmonics at a rate of 14 beats per second, the difference between 586 and 600. This will make the interval sound dissonant. If the 200 and 293 Hz tones are pure, there will be no beating since the upper harmonics will not be present. Whether this will sound equally consonant to an interval with pure frequencies of 200 and 300 Hz, I don't know, but the pure tone interval of 200 and 293 will certainly be less dissonant than the harmonically-rich interval of the same frequencies.