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NOTE: for the purpose of this question, assume we are talking about 12-TET and assume that references to overtones in the harmonic series are being referred to in relationship to the 12-TET note that they have closest parity with.

In another post, there were many contributions that made reference to the harmonic series and also to the heavy influence 12-TET has had on our understanding of music theory and the language we use to describe it. If you believe that the principles of harmony have an origin in the natural universe and in the harmonic series, then it would seem that the tuning system we use today has permanently set us apart from that.

I have long believed that the overtone series of any given root note strongly affects which intervals will "fit" on top of it harmonically. David Cope, a professor of music at UC Santa Cruz refers to this concept as "interval strength". I have also heard a great deal of support for this concept when talking with professors of harmony and other musicians with graduate degrees in harmony from some of the top music programs in the world, so I believe that this idea is well-established in the music community, even though I haven't seen much research on the topic.

Approximation of overtone series

If we look at the harmonic series, we see that the overtones occur in an order that appears quite related to the consonance of pitches:

8, 5, 8, M3, 5, m7, 8, M2, M3, d5, 5, m6, m7, M7, 8, m2, M2, m3, M3, wikipedia

Notes

1) I have spelled all intervals enharmonically to represent the most frequently-used intervals in harmony.

2) I have spelled all intervals as closed intervals, regardless of how many octaves separate them from the fundamental.

3) I have not noted the approximations of how sharp or flat these overtones may be in comparison to their 12-TET counterpart.

Here's my four-part question:

1) To what extent does an overtone's first appearance in the overtone series affect the consonance of the 12-TET interval to which it corresponds?

2) If the overtone series is related to consonance as mentioned in #1, to what extent is this consonance reinforced by an overtone which appears repeatedly in the overtone series?

3) If the overtone series is related to consonance as mentioned in #1, to what extent does the inversion of one of the overtones correspond to consonance? For instance, the fifth is supported very well in the overtone series, but the fourth is not present, except as an overtone halfway between the fourth and the augmented fourth. However, the fourth seems to have a similar consonance as the fifth, perhaps because its own overtone series corresponds to the root.

4) I have been challenged before when I have said that the higher the overtone is, the less significant it is, because the overtones become less audible as they get higher and higher. Is it true or not true that overtones become less audible as they get higher? If so, doesn't this mean the lower overtones are of more importance? Has there been any research done on amplitude of various overtones (say, given a certain instrument such as guitar or piano) and the point at which that amplitude is so low that humans can't hear it?

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3 Answers 3

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We first need to categorize each interval, assign it a "consonance amount". That's the first problem we find. In the case of the fourth, for example, some consider it perfect consonance, and others consider it a dissonance, depending context (and who you ask).

For simplicity, let's define ours based on Wikipedia's:

1: Perfect consonances: unison, octave, fourth, fifth.
2: Imperfect consonances: major third, minor third, major sixth, minor sixth
3: Dissonances: minor second, major second, tritone, minor seventh, major seventh

1) To what extent does an overtone's first appearance in the overtone series affect the consonance of the 12-TET interval to which it corresponds?

The problem here with "which it corresponds" is that some harmonics will be not very close to its "corresponding" interval. The difference can be as big as 40-50 cent, which is the max difference before the harmonic is closer to another interval. It's more a "middle" than that particular interval at that point.

But let's ignore that. Here is the order in which the intervals appear in the harmonic series, the place in which they appear, and the difference between the interval and the harmonic in cents:

Interval   1st Appearance  Difference
----------------------------------------
root            1            0.0
fifth           3            1.9
major third     5           13.7
minor seventh   7           31.2
major second    9            3.9
tritone         11          48.7
minor sixth     13          40.5
major seventh   15          11.7
minor second    17           5.0
minor third     19           2.5
fourth          21          29.2
major sixth     27           5.9

The first two intervals that appear in the harmonic series are perfect consonances. The third one is an imperfect consonance. So far, so good. But here we reach a stop. The fourth, fifth, and sixth intervals are all dissonant, but they appear fairly early in the harmonic series (harmonic 7, 9, and 12 respectively). Also worth noting that the major sixth, an imperfect consonance, is the last interval to appear.

Other than the first 3 intervals that appear in the harmonic series (root, fifth, and major third), seems that there is no strong relationship. The order is all over the place in relation to their "consonance amount". To me it seems that the first appearance in the overtone series doesn't affect the consonance of the interval (considering the restrictions of the question: interval = closest interval to the harmonic and 12-TET).

As Bradd Szonye mentioned in the comments, it's worth diving into the interval relationship between harmonics. Here, as we go higher in the harmonic series, the interval between harmonics becomes smaller and smaller, quickly converging in one interval, starting with the minor third, as early as the 6th harmonic. So let's see the first 6 harmonics.

1-2 root
2-3 fifth
3-4 fourth
4-5 major third
5-6 minor third

Here we see the fourth and minor third. I don't know if "they appear early" has any significance here, given the few harmonics we can analyze before they converge in one interval more and more. What might be significant is the fact that they appear, and all of them are consonant. In fact, if we consider their inversions, those 6 harmonics cover all the perfect and imperfect consonances.

In this case, we do see a strong relationship. Dissonances do not appear in the intervals between harmonics (between n harmonic and n + 1) in the pre-convergence range.

2) If the overtone series is related to consonance as mentioned in #1, to what extent is this consonance reinforced by an overtone which appears repeatedly in the overtone series?

We concluded that they are not strongly related, but let's see the data anyway.

First we need to decide how many harmonics we will consider. If you go deep enough, you'll find that several consecutive harmonics will "belong" (be closer to) the same interval, so maybe it's a good idea to stop before that happens. Also, the higher the harmonic, the lower the amplitude (in general, consider a sawtooth wave for simplicity), so higher harmonics will not have as much as an impact as the first ones.

There is no pretty way to avoid this issue. Tritone appears consecutively in harmonics 22 and 23, and minor sixth appears consecutively in harmonics 25 and 26, but major sixth doesn't appear until harmonic 27, so limiting the analysis to harmonic 22 would exclude it. But that's what we are going to do to avoid consecutive appearances.

Here are the intervals and the frequency in which they appear in the harmonic series, first 22 harmonics:

root            5.0
fifth           3.0
major third     3.0
minor seventh   2.0
major second    2.0
tritone         2.0
minor sixth     1.0
major seventh   1.0
minor second    1.0
minor third     1.0
fourth          1.0
major sixth     0.0

Again, root, fifth, and major third are in the top, and again we see dissonances appearing early. Again, we see consonances in the last places. There doesn't seem to be a strong relationship here either, with the exceptions (again) of root, fifth, and major third.

3) If the overtone series is related to consonance as mentioned in #1, to what extent does the inversion of one of the overtones correspond to consonance?

It shouldn't make a difference in our current categorization. All the inverses correspond to an interval in the same category.

Perfect consonances:
P5 - P5

Imperfect consonances:
M3 - m6
m3 - M6

Dissonances:
tri - tri
m2  - M7
M2  - m7

4) I have been challenged before when I have said that the higher the overtone is, the less significant it is, because the overtones become less audible as they get higher and higher. Is it true or not true that overtones become less audible as they get higher?

It depends. You can synthesize a sound with any harmonic ratio you can think of. But yes, less exotic sounds will have that tendency, the higher the harmonic the lower the amplitude It's not a constant, this tendency is commonly broken: some higher harmonics might have a higher amplitude than some lower harmonics. Some lower harmonics might not even be present, as some instruments cancel or attenuate specific harmonics.

It's a tendency, not a constant, there are many exceptions. That's maybe why you were challenged about it.

Alto sax:

Alto Sax

Nylon guitar:

Nylon guitar

Trumpet:

Trumpet

Snare drum:

Snare drum

If so, doesn't this mean the lower overtones are of more importance?

The less amplitude an harmonic has, the less it adds to the sound. So, higher amplitude harmonics add more to the sound, they are "more important". As lower frequency harmonics tend to have more amplitude, there will also be a tendency of lower frequency harmonics being "more important", but it's not necessarily the case, as lower frequency harmonics not always have higher amplitude than other higher frequency harmonics.

It would be more accurate to say "higher amplitude harmonics are more important".

Has there been any research done on amplitude of various overtones (say, given a certain instrument such as guitar or piano) and the point at which that amplitude is so low that humans can't hear it?

Interesting question. Don't know of any study in particular, but you can do it yourself.

Use a very steep high pass (low cut) filter and increase the cutoff frequency until you can't perceive a sound. That's where you stopped hearing the harmonic series at that specific amplitude. Now check the cutoff frequency and the fundamental of the sound you filtered. Divide the cutoff frequency by the fundamental and that's roughly how many harmonics you were able to hear.

That doesn't deal with masking and other dynamics, though. Maybe another experiment you can make is the opposite. Use a very steep low pass (high cut) and decrease the cutoff frequency until you perceive that the sound is changing. If you don't perceive a change, it means that those harmonics (if any) aren't adding something to the sound (at least to your ears). When you perceive the smallest change, do the same: divide the cutoff frequency by the fundamental and that's roughly how many harmonics were "important" enough for you to hear a change.

You can then check the amplitude of those harmonics around the cutoff frequency in relation with the fundamental with an harmonic analyzer.

I wonder if the two experiments yield similar results.

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2  
Excellent answer! One comment regarding point 1: While the perfect fourth and minor third show up very late in the harmonic series as partials, they show up much earlier as inversions. We usually define a just perfect fourth (4:3) as the interval from the just fifth (second partial) to the next octave (third partial). Likewise, the just minor third (6:5) is the interval from the major third to the next fifth. –  Bradd Szonye Aug 1 at 19:42
    
@BraddSzonye It doesn't seem to be the case of the minor third. The inverse of minor third is the major sixth, and both appear late in the harmonic series. Am I missing something? –  JCPedroza Aug 1 at 21:10
    
Sorry for any confusion – I meant the other common sense of “inversion,” as in “not starting with the fundamental.” The minor third is usually defined as the interval from major third to fifth, rather than an interval directly from unison/octave. –  Bradd Szonye Aug 1 at 21:14
    
@JCPedroza If you take two notes forming a minor 3rd, so their ratio is 1:6/5, then sixth harmonic at 6 of the lower (corresponding to a musical octave+5th), matches the fifth harmonic of the higher note (corresponding to a musical 3rd). –  Dave Aug 1 at 21:15
1  
@BraddSzonye I'll try to make the same dynamic with the intervals between harmonics too then. Will update the answer in a bit, thanks for the input. –  JCPedroza Aug 1 at 22:16

For your specific points:

1) For most/all* instruments, the amplitude of the partials decreases as the partial number (i.e. frequency goes up). Thus, for almost all musical sounds there is "more of" the (first) octave and the fifth and less of the higher harmonics. So adding a second sound whose fundamental is at one of the lower harmonics reinforces something that is "already there", i.e. pretty easily percievable. Adding a new note whose fundamental is at one of the higher partials "brings out something new" -- i.e. less consonant.

Consider the just-intonated intervals in order of increasing denominator:


  unison    |    1/1 
  perf. 5th |    3/2 
  perf. 4th |    4/3
  maj.  6th |    5/3
  maj.  3rd |    5/4
[ min.  7th*|    7/4      (7th harmonic) ]
  min.  3rd |    6/5
  min.  6th |    8/5
  maj.  2nd |    9/8
  maj.  7th |   15/8      (5-limit JI)
  min.  7th |   16/9      (5-limit JI)
  min.  2nd |   16/15     

If you ignore the pitch related to the 7th harmonic as a special case, these are pretty much in order of degree of consonance. Consider the harmonics on top of each of these pitches, e.g. for the perf. fifth it will be 3/2,3,9/2,6..., every other harmonic of this note is already present in the overtone series of the fundamental (which, in this representation has frequencies of 1,2,3,4,...). Similarly, one out of every three of the overtones of the perfect 4th, and the major 6ths will already be present in the over tone series of the fundamental. In this sense, adding a note at a consonant interval is reinforcing harmonic content that is already present in the fundamental to a greater degree than notes at dissonant intervals would.

There is also a practical problem for non-synthesized instruments: the partials are not exactly integer multiples of the fundamental; these inharmonious arise due to the physical characteristics and limitations of the instrument. Thus trying to match a given high partial with another note produced a a different instrument will likely miss, and you'll end up with two frequencies that are close to one another but slightly different, and thus hear them beating against one another, leading to perceived dissonance.

2) Formally, there are an infinite number of each partial; however as mentioned above, the amplitude of the higher ones is smaller than the amplitudes of the higher ones; thus the first appearance is the most important indicator of how significant a given partial is.

3) In this context a way of thinking about inversions is that "the overtones of the higher note match up with overtones of the lower one", take the perfect 4th G-C -- the partial corresponding to a musical fifth (at a frequency 3 times the original) of the fundamental C matches up with one of the octave partials of the G.

4) For most/all* musical instruments the amplitudes of the higher overtones is much lower than the lower overtones; here I'm referring to a note held at steady state. The transient initial attack, if analyzed by itself, may show a variety of high-frequency content that is not related to the steady state note. A search on "frequency spectrum violin", for example, yielded a series of pages (some of which) include the power spectrum of a violin as measured, thus with a little searching you should be able to find representative information on any instrument of your choice. Almost any book on musical acoustics, e.g. Music Physics and Engineering (H. Olson), will probably also have similar spectral plots.

An important source of dissonance is the physical beating between pairs of closely spaced frequencies; the links in this wikipedia entry on the physiological basis of consonance should provide a pointer into the research in this area. Another important point is that (except for synthesized sounds) there are inharmonicities in the overtones of a musical note -- the overtones are not exactly integer multiples of the fundamental. So what happens when you play two notes whose musical interval corresponds to one of the higher partials: they (and all of their partials) tend to miss one another by a small amount; this leads to perceptable amplitude beating, and this is perceived of as dissonance.

I haven't yet worked all the way through it but The Harmonic Experience by W.A. Mathieu covers the aspects of how do the just-intonated harmonic relationships integral to the physics of sound relate to/with modern equal temperament practice.

  • I can't think of a counter example where the amplitudes of the partials doesn't go down as the frequency goes up. I'm sure someone, somewhere, has synthesized sounds with this property, but even most standard synth patches have this feature.
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These are all great answers. One little wrinkle: when the fundamental tone is very low -- i.e., in the subbass register (20-60 Hz), we often hear much more of the higher overtones than the fundamental or first partial. The human ear's sensitivity falls off dramatically as frequency drops below about 200 Hz.

When you play the lowest note on a grand piano (approx 27 Hz), most of what you're hearing is upper partials.

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