Harmonic Series Interference

Question

I have been analysing intervals to look at the frequency difference between harmonics.

Before my analysis I was under the impression that harmonics for 'consonant' sounding intervals contained more harmonics that lined up - as is the case with the octave.

I analysed the tritone and as expected many of the harmonics fall close to each other causing interference and 'dissonance'.

Now I am looking at the perfect fifth (C3-G3 in this case). I was expecting some harmonics to be shared between these notes. Instead it seems that some almost line up but to not exactly.

For example the third harmonic of C (392.438) and the second harmonic of G2 (391.995) - a difference of .44. It is my understanding that this would create interference between these partials every 2.27 seconds.

I have tried recording guitar and bass playing this interval if i isolate this area I think I can hear some interference however i can't tell.

Is this correct? Is this small amount of beating between these partials just adding to the character of this interval?

Thank you!

Answer 1

This is called "12-tone equal temperament", and it involves dividing the span from our fundamental note to its first harmonic – i.e. from a fundamental frequency to double that frequency – into 12 equal ratios. So from one note to the next we have a ratio x such that if we took 12 of them (that is, if we considered x to the power 12) we would get the ratio 2; so x is the 12th root of 2. G is the 7th note of the 12, so corresponds to multiplying the fundamental frequency by a ratio of (12-th root of 2)-to the power of 7. Since any n-th root which is not a whole number is irrational, no integer multiple of the fundamental frequency can equal an integer multiple of any of our other notes – and as you perhaps know, harmonics are given by integer multiples of base frequencies.

An alternative to equal temperament is what's known as Just Intonation; where we in fact define the note G to be the note given by the 3rd harmonic of C. This obviously has better properties for exactly the reason you have observed, however ultimately generates an unlimited number of distinct notes.

Answer 2

In general your understanding is correct. The perfect 5th has quite a few of the harmonics of each note lining up and where they don't they are a 5th apart, which is perceived as a combination of two distinct notes. In contrast the whole tone has nothing lining up all, at least to the 8th harmonic, and there are several that are close.

There are a few concepts that are missing from your description. One is the critical band theory. If the frequency difference (or ratio) is large enough the combination is not muddy. This, by the way, also explains why the perception of dissonance is somewhat frequency dependent. That is to say a third or minor third will be judged dissonant in the bass register but not in soprano.

The next is that the alignment of harmonics is more exact in Just tuning, which is based on the harmonic sequence. Fretted instruments and the piano are closer to equal tempered tuning which does NOT follow the harmonic sequence. Hence the harmonics of intervals in this tuning will not be perfectly aligned. Many people with perfect pitch or very good relative pitch (after years of training) can hear the dissonance in 12TET intervals. I knew a man with perfect pitch who could not listen to equal tempered music due to the dissonance.

Answer 3

This is a fascinating area and the answer you are looking for is subtle because it depends on the conjunction of the spectra of partials of the two notes being played at once.

The definitive text in this area is Sethares’ Tuning Timbre Spectrum Scale. He derives many interesting results from looking at the relationship between timbre (i.e. the harmonic series of a specific note played on a particular instrument) and the mucial scale being used, including non 12 tone tempered scales.

It’s a long book but quite accessible. In particular do check out the example recordings where, for example, he makes the octave sound dissonant, or the tritone consonant, by manipulating the timbre of the instrument being used.