A few centuries ago, the main hypothesis as to what causes dissonance is 'complexity of ratios'. Basically, a simple ratio such as 3/2 will regularly and often have waveforms 'coincide' - the tops (or troughs) of the waves will recur in the same position relative to each other often. With more complicated ratios they will recur less often, and therefore 'keep the ear in torment'. Thus 2/1 will be more consonant than 3/2, 3/2 more consonant than 5/4, 5/4 more so than 9/8, 9/8 more so than 11/8, and so on. The exact way of measuring 'complexity' of a ratio differed between scholars that held that hypothesis, but higher prime factors were more complex than lower prime factors; however, 25/16 probably was comparable to, say, 13/8, despite 25/16 having no factors higher than 5, and 13 having 13 and 2 as its only factors. (25/16 = 52/24, 13/8 = 13/23).
Nowadays, the main hypothesis deals with overtones. Ultimately, for regular overtone series, the results will be similar to the previous hypothesis - simple ratios will mostly be more consonant than complex ones. We noted previously that tones tend to consist of a whole stack of frequencies.
If you were to generate sinusoidal tones (i.e. tones lacking all overtones), you'd find that only when they're fairly close to each other do they cause dissonance - anything wider than (roughly) a minor third never sounds dissonant, even solidly dissonant intervals like the tritone and the major seventh sound kind of neutral and uninteresting.
The reason for these is dissonance between overtones. The dissonance of two notes played together is basically the sum of the dissonances between the overtones. Thus, tones whose overtones often are within the critical bandwidth of each other are very dissonant, tones whose overtones often are closer than 10hz from each other or further than 6/5 from each other will be consonant.
Let us compare the overtones of A440, E660 and D#622. Each of these tones gets a column, so A440 and its overtones are under A, E660 and its overtones under E, etc. The overtones are ordered wrt pitch:
Looking at the table, we find that d# has way more "conflicts" with A than E does. Certainly, E will start having dissonances with A - heck, after a certain point, overtones will start clashing internally (which is why a note with very strong overtones far up the series will start coming off as noise!)
We would end up with something like this:
However, this also explains why the very complex ratios that lightly detuned intervals produce do not strike us as very dissonant. Clearly 659/440 is more complex than 3/2, (659 is a rather big prime compared to 3 and 2!) yet it's quite a tolerable detuning. The reason is that the overtones now come within the tolerable range just before the critical bandwidth, where they previously perfectly synched. Of course, the volume of each overtone has an impact on the perceived dissonance.
This model also predicts that instruments with unusual overtone series may cause dissonance in other intervals than more regular instruments do, and we find that this in fact is the case. One can even make intervals like the octave come out as very dissonant by picking weird timbres.
I have no methods for efficiently finding the least dissonant intervals for two tones with arbitrary weird overtones, so I won't give you that kind of mathematicking right now.
In addition, we must note that consonance and dissonance are somewhat subjective - we perceive them in part due to conditioning. However, it does seem the main thing we condition is 'where' the line is drawn - how many conflicting overtones we can tolerate. We also seem to fill in some missing overtones - so functionally a tritone will sound somewhat dissonant even if the overtones that would usually cause the dissonance now are missing. The dissonance will be less acute, however.
Finally, most human musics have both dissonant and consonant intervals - the interplay between consonance and dissonance, or tension and release - seems to be fairly common. However, due to the way intervals work, we easily get dissonant intervals as a byproduct once we have a sufficient number of consonant intervals - the cracks in which consonances can fit without trampling into other tones' critical bandwidths quickly shrink.
* A more accurate description would be "6/5 or 50hz, whichever is bigger"; it seems the critical bandwidth is wider in the lower registers. In fact, go low enough and octaves and the like get dissonant. This is why most music with more detailed timbres and such tend to leave a lot of space around the bass - you don't want to crowd the lower registers.
Nowadays, the main hypothesis deals with overtones. Ultimately, for regular overtone series, the results will be similar to the previous hypothesis - simple ratios will mostly be more consonant than complex ones. We noted previously that tones tend to consist of a whole stack of frequencies.
If you were to generate sinusoidal tones (i.e. tones lacking all overtones), you'd find that only when they're fairly close to each other do they cause dissonance - anything wider than (roughly) a minor third never sounds dissonant, even solidly dissonant intervals like the tritone and the major seventh sound kind of neutral and uninteresting.
Music File Hosting - Audio Hosting - differencetone_2sources
an example of two sinusoidal tones a bit too close to each other to be dissonant
Upload Music - Play Audio - Sinusoidal Dissonance
two sinusoidal tones that are further apart yet close enough to be dissonant
We introduce the concept of 'critical bandwidth', which is a range of intervals that by their very nature are dissonant - roughly, this corresponds to 'about ten hertz off' to 'about 6/5' (roughly A-C, the minor third)*. So, for A440, any frequency in the rough area of 450hz to 528hz will be dissonant. But any musician knows that there are intervals wider than the minor third that are dissonant - for instance, the major seventh (A-G#) and the tritone (A-D#).
The reason for these is dissonance between overtones. The dissonance of two notes played together is basically the sum of the dissonances between the overtones. Thus, tones whose overtones often are within the critical bandwidth of each other are very dissonant, tones whose overtones often are closer than 10hz from each other or further than 6/5 from each other will be consonant.
Let us compare the overtones of A440, E660 and D#622. Each of these tones gets a column, so A440 and its overtones are under A, E660 and its overtones under E, etc. The overtones are ordered wrt pitch:
A sample of how overtones serve to 'create' the timbre of a tone may serve to help at this point:
A E D# A 440 D# 622 E 660 A 880 d# 1244 e 1320 1320 a 1760 a# 1866 b 1980 c 2200 d# 2488 e 2640 2640 g* 3080 3110 g# 3300 . . .
Play Music - Audio Hosting - Overtones
Looking at the table, we find that d# has way more "conflicts" with A than E does. Certainly, E will start having dissonances with A - heck, after a certain point, overtones will start clashing internally (which is why a note with very strong overtones far up the series will start coming off as noise!)
We would end up with something like this:
Music File Hosting - Listen Audio Files - Tritonus by overtonesvs. this:
Upload Music - Audio Hosting - Fifth in Overtones
However, this also explains why the very complex ratios that lightly detuned intervals produce do not strike us as very dissonant. Clearly 659/440 is more complex than 3/2, (659 is a rather big prime compared to 3 and 2!) yet it's quite a tolerable detuning. The reason is that the overtones now come within the tolerable range just before the critical bandwidth, where they previously perfectly synched. Of course, the volume of each overtone has an impact on the perceived dissonance.
This model also predicts that instruments with unusual overtone series may cause dissonance in other intervals than more regular instruments do, and we find that this in fact is the case. One can even make intervals like the octave come out as very dissonant by picking weird timbres.
I have no methods for efficiently finding the least dissonant intervals for two tones with arbitrary weird overtones, so I won't give you that kind of mathematicking right now.
In addition, we must note that consonance and dissonance are somewhat subjective - we perceive them in part due to conditioning. However, it does seem the main thing we condition is 'where' the line is drawn - how many conflicting overtones we can tolerate. We also seem to fill in some missing overtones - so functionally a tritone will sound somewhat dissonant even if the overtones that would usually cause the dissonance now are missing. The dissonance will be less acute, however.
Finally, most human musics have both dissonant and consonant intervals - the interplay between consonance and dissonance, or tension and release - seems to be fairly common. However, due to the way intervals work, we easily get dissonant intervals as a byproduct once we have a sufficient number of consonant intervals - the cracks in which consonances can fit without trampling into other tones' critical bandwidths quickly shrink.
* A more accurate description would be "6/5 or 50hz, whichever is bigger"; it seems the critical bandwidth is wider in the lower registers. In fact, go low enough and octaves and the like get dissonant. This is why most music with more detailed timbres and such tend to leave a lot of space around the bass - you don't want to crowd the lower registers.
No comments:
Post a Comment