"Accepted" Tensions for Maj, Min, Dom Chords (Fewell Book)

[randalljazz]

it is not a theory book, it is an approach to get guitarists playing lines that don't sound like scales. the dictionary is not where to look for poetry, though it is full of words...

[fumblefingers]

the #11 would be "natural" on the IV chord in major, no?

[Nuff Said]

the #11 would be "natural" on the IV chord in major, no?

Or are the tensions related to the naturally occurring Overtone Harmonic series:


BAIN: The Harmonic Series (Overtone Series)

Hmmmmmm......

Nuff Said

[JonR]

Or are the tensions related to the naturally occurring Overtone Harmonic series:


BAIN: The Harmonic Series (Overtone Series)

Hmmmmmm......

Nuff Said

Except in equal temperament those are all out of tune .
IOW, the 11th harmonic is actually midway between a tuned perfect 11th and a #11. The 13th likewise is in between a b13 and major 13.
Only octave harmonics (2nd, 4th, 8th, 16th) are exactly in tune with equal temperament.
The 3rd and 9th harmonics are very close to the equivalent ET pitches (2 and 4 cents out respectively. (And octaves of those are out by the same amount: ie, 6th and 12th harmonics are 2 cents out, and the 18th is 4 cents out.)
The 5th and 10th harmonics are 14 cents out (flat of ET).
The 7th and 14th are 34 cents out (flat of ET).
Higher numbers can occasionally be interesting. Eg the 15th is quite close to a major 7th, the 17th very close to a minor 9th and the 19th to a minor 3rd (the harmonic series approximates to semitones around this region). But this is coincidence.
(However, the point at which the harmonic steps equate to a semitone is relevant to guitar fret positioning - check out the "rule of 18" for fret calculation. http://www.musemath.com/miscellany/ruleOf18.html)

Of course, there's a good argument that it's really equal temperament that's "out of tune"!

But the point is that it's difficult to argue that all chord extensions are related to the overtone series (in the way we can argue the major triad derives from it). It's tempting, especially as the harmonic numbers tally. And the 9th is very close (as the 5th of the 5th in terms of chord tones). But the 7th harmonic is (IMO) far enough out to not relate to the ET minor 7th as we use it. And the 11th and 13th are definitely far too ambiguous to draw any conclusions.
If we tune a chord to pure intervals in a 4:5:6:7 ratio it sounds a bit like a root position dom7 chord. But it's more consonant, more "pure". Dom7 chords in equal temperament are dissonant - or rather have different kinds of consonance. Eg, the ratio between the 5th and 7th in a tuned dom7 chord is closer to a 5:6 minor 3rd than the 6:7 interval in the "pure 7th" chord.