@Tim: Oops. You're right, I meant "major (C-E-G)".

@Tim: Despite my initial reservations, I'm coming around to the viewpoint you expressed in your answer: a 9th sounds like a major 2nd, an 11th sound like a perfect 4th, a ♭9 sounds like a minor 2nd, etc., but slightly less dissonant (for the reason I gave in my reply above to @Dekkadeci).

@Tim: Certainly the overtone mix of a triad is much more complicated than that of an interval: C E G is a mixture of a M3, m3, and P5. But I am asking only about intervals. By the way, don't conflate the meanings of the word "minor" in "minor interval" and "minor chord". A minor 3rd interval, by itself, does not compel the ear to hear a minor chord sound. Only when a third pitch is added does the minor (E-G-B) or major (C-G-B) chord sound emerge. However, I wouldn't say that adding a pitch changes "the relationship between" E and G; it just creates two more intervals.

The paper to which you refer concerns a question such as: For the major triad C-E-G, why is C the root of the chord? Helmholtz answers by observing that E & G occur in the overtone series for C, but C and G are absent from the overtone series of E while C and E are absent from the overtone series of G. The problem with this explanation is that it fails to justify C as the root of the minor triad C-Eb-G. The article reviews several attempts to provide an explanation, but none are very successful. What do you see as the relationship between this question and the one I raised?

I agree with your assessment. To me, a major 9th sounds more consonant than a major 2nd and even a major seventh; perhaps it's a soft consonance. A minor 9th is definitely a sharp dissonance. I have difficulty deciding about the 11th and 13th.

Dekkadeci; I agree with your first comment, but not so much your second. Dissonance arises when two pitches are close and the resulting beat frequency becomes high enough and loud enough to be audible. This beating is evident for a minor second. For the minor ninth, say C-Db', the Db' is combining not with C but rather with its overtone C', which is much less loud. Nonetheless, if you play a C and a Db' together, it sounds quite harsh, much more than, say, the M7 with C and B. A C7b9 chord appears frequently in music, especially jazz, but only where its harshness is desired.

My question is about intervals, but the answer has implications about chords as well as "avoid notes" in scales.

@Albrecht Hügli: Right, but the consonance/dissonance of a chord results from that of each interval in the chord. For example, the 11th (F') sounds awful when added to a C major 9th chord (C E G B D) because the interval E-F' is a minor 9th, which sounds awful.

Reference: 20th Century Harmony by Vincent Persichetti, p. 8, available at [vdocuments.site/persichetti-notes.html]. Also [en.wikipedia.org/wiki/Guitar_chord].

I've never encountered the phrases "Aeolian (harmonic) mode" and "Dorian harmonic mode". What do they mean? Anyway, the statement made on the website concerns the Aeolian (6th) mode of the major scale, and I asked about the Dorian (2nd) mode of the major scale. Let's not get distracted by "harmonic" modes.

@Matt L.: You're right, of course. I just sent e-mail to the author.

@Tim: Granted, but this web site concerns quite technical issues and is well written and thorough. It may contain mistakes, but I doubt it is promulgating music theory conspiracies.

It does seem reasonable to require a dominant-tonic relationship in defining "tonally effective" modes. Indeed, for the C Ionian mode, in the authentic cadence (V I), the V chord (the G major triad) contains the leading tone (B) for the tonic C, whereas for the D Dorian mode, the V/ii = vi is the A minor triad, which does not contain the leading tone C# for the tonic D. However, the same can be said for the A Aeolian mode: the V/vi = iii is the E minor triad, which likewise omits the leading tone G# for the tonic A. So again: Why is A Aeolian "tonally effective" whereas D Dorian is not?