The "simple frequency ratio" theory of consonance is one of the oldest hypotheses in western musical thought. Some version of this theory is also the most commonly held view by ordinary musicians and music theorists -- mixed with some notion of the importance of culture.
The frequency ratio hypothesis is called into question by a simple but vivid demonstration. Using pure tones, play a just-tuned major third (ratio of 4:5) between the pitches C4 and E4. Now play a just minor second (ratio of 11:12) between C4 and C#4. * Listeners are virtually unanimous in judging the major third as "more consonant" or "less dissonant" than the minor second. So far, this is consistent with the notion that simpler ratios are more consonant than complex ratios.
Now, repeat the task with the pitches Ab2 and C3 (just major third)
and A#2 and B2 (just minor second):
Note that for this demonstration to work, it is essential to use pure-tones only. The effect is destroyed by complex tones (for reasons that are explained by other theories of dissonance, such as the tonotopic theory).
Although this demonstration has been known for more than 40 years, it is not well-known to musicians or music scholars. Only a few musicians have access to sine-tone generators, and fewer yet are disposed to try this demonstration.
[*] Audio distortion will ruin the effect, so it is best to
reproduce these sounds through good quality loudspeakers
at a relatively low volume.
For convenience, here are some helpful frequency values:
C4 (concert pitch)
C#4 (just minor 2nd above C4)
E4 (just major 3rd above C4)
Ab2 (concert pitch)
A#2 (concert pitch)
C3 (just major 3rd above Ab2)
B2 (just minor 2nd above A#2)