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When the tones C, E and G are played, C is perceived as the root. In the case of this major triad, the root is easily identifiable. However, for many chords, there is no identifiable root. For example, the tones C, D and E do not indicate any tone as the root tone.

The ways to identify the root of a chord are fairly well understood. Whenever a fifth interval is present, the bottom tone must be the root. (For example C,F -> F is root). Whenever minor or major third intervals are stacked, the bottom most tone is the root (For example A,C,D,F -> D is root).

Is there any mathematical / physical theory that describes why chords have roots? It would seem that based on the general qualitative rules I listed above the root tone mechanism could be deduced. I have looked into studying the wave interference and frequency ratios of intervals that make up chords but so far I haven't found an obvious answer.

-- EDIT --

I am adding an edit here to make clear the definitions I am using. I know different people have different ideas about what a "chord" and "root" really are, so I want to be as specific as possible:

Chord - A set of unique tones. (For example; CEG, DFA, ABC, ABCDEFG,...)

Root - One or more tones in the chord that are perceived as the "net" tone. The root tone(s) is therefore used for analyzing the tonality between two chords, like a chord progression. (For example: CEG -> Root is C, C E G# -> Root is C E G#)

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    Interesting tangent from this - what makes ACEG any more likely to be Am7 than C6? – Tim Jul 12 at 9:51
  • It is a minor major seventh chord (structure = 3,4,3,2), so A is the root. Why would it be a C6? – Alex Jul 12 at 22:20
  • Why not? C6 has exactly the same notes. What makes it different? – user1079505 Jul 13 at 1:20
  • @Alex - ACEG is not a minor major 7, it's a minor seventh chord.. My point is those 4 notes also constitute C6 (a major sixth chord). CEGA in root position. So what defines either. – Tim Jul 13 at 5:17
  • @Tim Depends on the lowest? – cmp Jul 14 at 8:07
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Simplistically speaking (and ignoring lots of important detail about intonation and temperament), the major third has a frequency ratio of 5:4 compared to the root, and the perfect fifth 3:2.

Let's imagine that we have a root at frequency 100 Hz, and pick out the first few partials of each note of a major chord.

Root: 100 Hz, 200 Hz, 300 Hz, 400 Hz, 500 Hz, 600 Hz

Third: 125 Hz, 250 Hz, 375 Hz, 500 Hz, 625 Hz, 750Hz...

Fifth: 150 Hz, 300Hz, 450 Hz, 600 Hz, 750 Hz, 900 Hz...

Very simplistically speaking, what the human ear does is try to find groups of harmonics that are multiples of each other to see if it they can be considered 'one sound'. (From a survival point of view, we want to know how many actual 'things' are making noise around us.) So the ear is always trying to find greatest common factors among the frequencies it's hearing.

The greatest common factor of the frequencies we've picked out there is 25 Hz. That's actually the same note as 100Hz, just 2 octaves below (i.e. the frequency is divided by 2, twice).

Another way of seeing this is that playing 3 notes at 100, 125, and 150 Hz is actually similar to playing a single note at 25Hz, albeit with the fundamental and other lower harmonics missing.

Of course as chords get more complex, these ideas don't work out quite so neatly - but then, the root of the chord isn't always heard so clearly in complex chords.

Why might C, D and E do not indicate any tone as the root tone? We could consider that the ear is going to have a harder job picking out a GCF - that GCF is going to be much lower in frequency than any of the notes actually being played. Additionally, if the notes are played in the same octaves, they will interfere with each other as the pairs (C and D) and (D and E) are close enough that critical band effects may become apparent. But depending on the timbre of the sound, it's still possible that the ear might pick out some harmonic commonality, especially with the C and the E both being present, so we can't categorically say that C, D and E do not indicate any tone as the root tone. As Peter Smith rightly points out, this could even be seen in conventional terms as a sparse C9 voicing.

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    This survival aspect is very interesting! Can you tell me some source and research? – Albrecht Hügli Jul 12 at 10:26
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    "So the ear is always trying to find greatest common factors among the frequencies it's hearing" falls apart as soon as you introduce "those important details about intonation and temperament" that you chose to "ignore". If your theory relies on fudging the numbers until it works, it's probably a bad theory. – Peter Smith Jul 12 at 18:26
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    @PeterSmith It's not really fudging the numbers; scientists have measured the range in which intervals can differ by without a perceived difference. – awe lotta Jul 12 at 19:10
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    @AlbrechtHügli damtp.cam.ac.uk/user/mem/kobe-lecture.pdf (p37 onwards) makes a similar point, though doesn't add much more detail; pnas.org/content/112/36/11155 mentions about the importance of recognizing speech; And while I presumed the question to relate to humans, you might enjoy the mention of mosquitoes in arxiv.org/html/1202.4212v2/#sec_2_1_3 ... – topo Reinstate Monica Jul 12 at 20:39
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    @topoReinstateMonica I'm not claiming that music falls apart, I'm claiming your theory falls apart because it doesn't describe music that anybody actually plays. Your theory would also tell us that a minor chord doesn't have an unambiguous root. Your post accurately describes the "phantom fundamental" phenomena. This is perceptually completely different from hearing a chord as having a root, we (or at least, I) do not hallucinate a bass note two octaves below a major chord when I recognize it as major. – Peter Smith Jul 14 at 0:41
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C-D-E doesn't have a clear root because "chords" and "roots" are concepts we use to describe a particular subset of music, a subset of music that doesn't usually include tone clusters like that. There's no "physical basis" for the "root" of a "chord" any more than there is a "physical basis" for the "fatal flaw" in a "tragic hero".

Or maybe it's a C9 chord.

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