Tuning of diminished triads and dominant 7ths

I believe this has been a holy war in the past, but what the heck, it's my
turn. :-)

I have come to the conclusion, based on how good they sound in 19-tET or
1/3 comma meantone, that in general, the "ideal" or "correct" tuning of
diminished triads and dominant 7ths is not 5:6:7 and 4:5:6:7, despite their
use in the clever adaptive JI of barbershop, but 25:30:36 and 20:25:30:36.
Note that these are simply stacked minor thirds and a minor third stacked
on a major triad, as classical theory would have them.

When tempered, the only ratios that matter are the following:

diminished
B D F
5 : 6
5 : 6

dominant 7th
G B D F
4 : 5 : 6
5 : 6
5 : 9

I believe that the ratio of the diminished fifth (B:F) in these chords is
simply irrelevant, in the same way that the ratio of the major 7th interval
is irrelevant in the major 7th chord. They are simply dissonances.

We have the names "harmonic 7th" or "augmented sixth" to distinguish the
4:5:6:7 from the dominant 7th, and the name "supermajor 7th" (correct me if
I'm wrong) for the 1/9:1/7:1/6:1/5. But what should we call the 5:6:7 and
1/7:1/6:1/5 to distinguish them from the diminished? Note that all these
chords _are_ distinct in 19-tET.

I also argue from the presence of a 5:9 in the dom 7th (and minor 7th
chord) to the fact that the diatonic scale ought to be considered as
approximating 9-limit-with-no-7's (and maybe no 4:9?) rather than merely
5-limit. This would shift the various optimal diatonic meantones somewhat,
with the direction depending on whether 4:9's are included. Towards 1/4
comma if 4:9 and 5:9 are included, towards 1/3 comma if only 5:9 is included.

Regards,
-- Dave Keenan
http://dkeenan.com

[Dave Keenan, TD 244.9:]
> I believe this has been a holy war in the past, but what the heck,
> it's my turn. :-)

> I have come to the conclusion, based on how good they sound in 19-tET
> or 1/3 comma meantone, that in general, the "ideal" or "correct"
> tuning of diminished triads and dominant 7ths is not 5:6:7 and
> 4:5:6:7, despite their use in the clever adaptive JI of barbershop,
> but 25:30:36 and 20:25:30:36. Note that these are simply stacked
> minor thirds and a minor third stacked on a major triad, as classical
> theory would have them.

Heathen! (-: Going back to Mark Nowitzky's discussion, with charts, at

http://www.pacificnet.net/~nowitzky/justint/dom7.htm

We now have advocates on this list for everything but 12-tET!

Dave, have you given an actual 4:5:6:7 a chance? In 19-tET, 7:4 is
represented by 15 microtones, which makes it flatter than an actual
7:4 by 12.9 cents (put another way, 15 microtones span 947 cents, less
than half-way between C-A and C-Bb in 12-tET). A Just 7:4 already
sounds flat to our 12-tET-burned ears, so it's no wonder that a 19-tET
representation of 4:5:6:7 is hard to swallow.

JdL

Correction! In an earlier post I stated that the 19-tET 7:4,
represented by 15 microtones, is 12.9 cents shy of a Just 7:4. It's
actually worse than that, 21.46 cents shy.

JdL

Dave Keenan wrote,

>I have come to the conclusion, based on how good they sound in 19-tET or
>1/3 comma meantone, that in general, the "ideal" or "correct" tuning of
>diminished triads and dominant 7ths is not 5:6:7 and 4:5:6:7, despite their
>use in the clever adaptive JI of barbershop, but 25:30:36 and 20:25:30:36.
>Note that these are simply stacked minor thirds and a minor third stacked
>on a major triad, as classical theory would have them.

I might agree for Renaissance and Baroque music, but the dominant sevenths
of Wagner and especially of jazz require a less sour tuning. Also, I agree
with Mark Nowitzky that one would prefer (on a melodic basis) to have a
near-just perfect fourth between the seventh of the dominant and the tonic,
rather than the 27:20 that your version would imply in JI (assuming the
dominant is a just 3/2 of the tonic). 19-tET or 1/3-comma meantone, of
course, make the 27:20 into a melodically unobjectionable fourth.

[John A. deLaubenfels TD245.1]
>Dave, have you given an actual 4:5:6:7 a chance? In 19-tET, 7:4 is
>represented by 15 microtones, which makes it flatter than an actual
>7:4 by 21.46 cents (put another way, 15 microtones span 947 cents, less
>than half-way between C-A and C-Bb in 12-tET). A Just 7:4 already
>sounds flat to our 12-tET-burned ears, so it's no wonder that a 19-tET
>representation of 4:5:6:7 is hard to swallow.

Oh no! I was quite aware of how bad the 4:7's are in 19-tET. I was
comparing the 19-tET dominant 7th to a just 4:5:6:7.

To me, despite its 21.5c error in the 4:7, a 19-tET augmented 6th chord
isn't too hard to swallow as an approximation of a just 4:5:6:7, but it is
impossible to swallow as an approximate dominant 7th. Whereas the 19-tET
dominant 7th sounds just fine *as a dominant 7th* but is absolutely nothing
like a just 4:5:6:7.

Thanks for pointing me to Mark Nowitzky's excellent discussion of the
matter. I refer to that below.

[Paul H. Erlich TD245.7]
>I might agree for Renaissance and Baroque music, but the dominant sevenths
>of Wagner and especially of jazz require a less sour tuning.

I must plead ignorance here. Do you mean merely one closer to 9:16, or one
even further towards 4:7. Here are the candidates:

5:9 1018c ---
|
1/3 comma 1010c |
1/4 comma 1007c 22c (syntonic comma)
|
12-tET 1000c |
9:16 996c ---
|
|
|
27c (septimal comma)
|
|
|
4:7 969c ---

>Also, I agree
>with Mark Nowitzky that one would prefer (on a melodic basis) to have a
>near-just perfect fourth between the seventh of the dominant and the tonic,
>rather than the 27:20 that your version would imply in JI (assuming the
>dominant is a just 3/2 of the tonic). 19-tET or 1/3-comma meantone, of
>course, make the 27:20 into a melodically unobjectionable fourth.

Ah yes. I was not considering JI, where the dominant seventh cannot be both
the subdominant and a 5:9 from the dominant. I agree with Mark Nowitzky
too. In this case I would use 9:16 but I still tend to think of this as
playing the role harmonically, of an approximate 5:9 rather than an
approximate 4:7 (and the upper interval as an approximate 5:6 rather than
6:7).

I was rather considering how one should rate a temperament as to how good
its dominant 7ths are, or whether in fact it has any. I concluded that
proximity to 4:5:6:7 is completely wrong as a measure of this, but agree
that proximity to 9:16 is important melodically. However in temperaments
where the syntonic comma is distributed (i.e. many of interest) 5:9 and
9:16 are the same thing.

Of course in 22-tET 4:7 and 9:16 are the same thing (and 5:9 is practically
nonexistent), but as far as the sound of the chord itself, is this a good
dominant 7th chord or rather a really bad dominant 7th chord but a
reasonable harmonic 7th chord?

I want to return to another question:
We have names that recognise 4:5:6:7 as different from a dominant 7th. i.e.
harmonic 7th or augmented 6th (which is what I'd say jazz and blues often
use the 12-tET 7th chord to approximate). But what names are there that
distinguish 5:6:7 from 1/7:1/6:1/5 from 5:6:36/5 = 25/6:5:6?

How about respectively:
minor aug 4th (no 5th)
subminor aug 4th (no 5th)
diminished

It's just that those first two are a bit of a mouthful and don't
necessarily translate to tunings other than meantones.

-- Dave Keenan
http://dkeenan.com