Is there a way to calculate consonance/dissonance?

[Qaiphyx]

I don't understand chords well for the most part etc, but frequencies/ratios I understand. Isn't there a set way to calculate these two things utilizing ratios?

 

[M Quack]

An octave is a doubling of frequency. Each half note corresponds to a factor of 2^(1/12).

The usual reference frequency is 440 Hz for the note A above middle C

http://en.wikipedia.org/wiki/A440_(pitch_standard)

From this you can calculate the frequency ratios for common chords.

 

[jtbell]

How do you quantify "consonance" and "dissonance"? I don't remember ever seeing them ranked on a numeric scale.

 

[M Quack]

Wikipedia has this to say about major chords: http://en.wikipedia.org/wiki/Major_chord

A major triad can also be described as a major third interval with a minor third interval on top or as a root note, a note 4 semitones higher than the root, and a note 7 semitones higher than the root.

If the root has frequency f,
then the major third has frequency f*(2^(1/12))^4 = f*(2^(1/3)),
and the "perfect fifth" has frequency f*(2^(1/12))^7

apparently that gives frequency ratios of 4:5:6 - I have not checked if these are exact numbers or approximate.

 

[Qaiphyx]

Let me ask this differently, if I know my primary fundamental/primary octave/frequency, says its 200. How can I find frequencies that are dissonant? Do I have to scale through and just find them by ear?

Sorry for the confusion I am a bit new to all this.

 

[M Quack]

http://en.wikipedia.org/wiki/Consonance_and_dissonance

Wikipedia claims that consonant tones have simple, integer ratios.

Why that sounds more or less pleasing to our ears is subject to debate - probably in a philosophy forum. I don't think that physics can give an answer to that.

 

[DrGreg]

apparently that gives frequency ratios of 4:5:6

This is correct if your musical instrument has been tuned with just intonation. The advantage of just intonation is that the consonance is perfect. The disadvantage is that you have to retune your instrument every time you play a tune in a different key.

If the root has frequency f,
then the major third has frequency f*(2^(1/12))^4 = f*(2^(1/3)),
and the "perfect fifth" has frequency f*(2^(1/12))^7

This is correct if your musical instrument has been tuned with equal temperament. This is a compromise which gives you a good approximation to consonance in all keys.

 

[Mr Vibrating]

There's a refreshingly simple mechanic to all of this:

- all factors of two of a given frequency (AKA 'fundamental') within discriminable range are resolved as being 'equivalent' to it - this is the magical, almost inexplicable percept of sameness between octaves. What's actually 'the same' between these tones of such wildly varying pitch? Nothing, of course.. This paradox is resolved by realising that we assign this property of parity to maximally simple frequency relationships.

In other words, 'consonance' and 'dissonance' are but degrees of this equivalence, and difference from it - or inequivalence.

'Consonance' has a maximum value, peaking at maximum simplicity; 2f. In other words, we're processing frequency relationships in terms of their relative complexity.

And, specifically, we're measuring this complexity in terms of complexity of the network activity in the associated resolving nuclei - a 2f relationship resolves every other cycle. The next most simple relationship, the 'perfect fifth' or 3f, resolves every 3rd cycle (ie. three cycles of the higher freq for each single cycle of the lower one). And so on; 5f, 6f etc. etc.; as we get to progressively higher ratios, more and more neurons are corralled into resolving their temporal integration window, and they become less harmonious.

In short, if a freq ratio can be resolved as any factor of two between the two freqs, then they're maximally simple and thus assigned parity (octave equivalence). For 3f, less so. 4f is equivalent again (2 x 2f), but 5f is even more inequivalent (dissonant) than 3f.

The harmonic series can be produced in a number of different ways, but from this angle it seems pretty obvious we're using recursive subdivision by two - that is, take a string (or monochord), halve its length, and the ratio between the new note and the full chord length is one octave.

Halve that distance again (so now a quarter of the original) and the relationship of the new note to the original is a perfect fifth. Repeat again and we have another octave. Next we get a Third, then another octave, and so forth. Keep repeating and this simple function reproduces all intervals of the harmonic series.

A simple biological architecture that might accomplish such recursive subdivision would be recirculation of input signals through a feedback / feedforward network with a 2f modulation. Miller et al have shown just such an effect between auditory cortex neurons and their thalmic afferents - although one can only speculate if this indeed affords us some kind of affinity for the series. What seems unequivocal however is this principle of maximal consonance correlating to maximal simplicity. Consonance has an elementary form, in this mysterious and perplexing sensation of parity between maximally-simple frequency ratios, and by which dissonance is, by default, defined in terms of relative complexity. Again, there is really nought but equivalence, and degrees of inequivalence in relation to it...

 

[Mr Vibrating]

NB there's a popular myth that consonance and dissonance are culturally subjective.

I think my explanation above should put the lie to this common misconception; 2f equivalence is an axiomatic universal for all sufficiently-complex multicellular lifeforms.

Indeed, if we ever encounter little green men, then provided they're also multicellular, they'll likewise be subject to 2f equivelance, and thus divide the resulting bandwidths into progressively less-equivalent intervals.

As a final thought, you can now probably appreciate that if we're to crack hard AI, and build a machine that can feel music the way we do, we'll first need a machine that 'hears' 2f intervals as equivalent too...