DA Dissonanace

I think it's time we started looking at Bruckner's scores, and checking his use of ii. Note in general that iib is most prevalently used in classical harmony, which places the dissonant interval thus as a 4th.

D
A
F

traditionally (see Schoenberg Harmonielehre), we find that this goes to V

D D
A B
F G

and so to I.

I am assuming that it's the A that is flat here, not that the D is sharp.

I'll take a peek at my Mahler ed today and see if he treats the DA fifth/fourth as a dissonance. I'd rather take the evidence of usage over any 'theory articles.' Anyone got a Haas or Nowak of the Bruckner and would like to check also.

The only fly in this ointment is whether meantone piano tuning would have influenced Mahler's writing. Bruckner would almost certainly know ji from writing his masses and providing the brass parts.

Mark

hi Mark,

> From: "Mark Gould" <mark.gould@argonet.co.uk>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, April 03, 2003 10:07 PM
> Subject: [tuning] DA Dissonanace
>
>
> I think it's time we started looking at Bruckner's scores, and checking
> his use of ii. Note in general that iib is most prevalently used in
> classical harmony, which places the dissonant interval thus as a 4th.
>
> D
> A
> F
>
> traditionally (see Schoenberg Harmonielehre), we find that this goes to
> V
>
> D D
> A B
> F G
>
> and so to I.
>
> I am assuming that it's the A that is flat here, not that the D is
> sharp.

yes, that's the way Bruckner characterized it.

with C = 1/1, the lattice for Bruckner's JI conception
of the diatonic scale is thus:

A E B
5:3 ----- 5:4 ----- 15:8
/ \ / \ / \
/ \ / \ / \
F C G D
4:3 ----- 1:1 ----- 3:2 ----- 9:8

so the proportions of all the "regular" diatonic
traids are:

I 4 :5 :6
ii 27:32:40
iii 10:12:15
IV 4 :5 :6
V 4 :5 :6
vi 10:12:15
vii 45:54:64

of course all the regular major and minor chords
(I, iii, IV, V, and vi) have a "5th" of ratio 2:3
(~702 cents), a "major-3rd" of ratio 4:5 (~386 cents)
and a "minor-3rd" of ratio 5:6 (~316 cents); the
"major" triads have the "major-3rd" on the bottom
and "minor-3rd" on top, and the "minor" triads are
the reverse.

the two exceptional chords are the ii and vii, which
both contain the Pythagorean "minor-3rd" D:F of ratio
32:27 (~294 cents), and which both have a "5th" which
is not 2:3. the other "3rd" is 4:5 in the case of ii
and 5:6 in the case of vii. the ii chord has a
"wolf 5th" D:A of ratio 27:40 (~680 cents) and the vii
has a "diminished 5th" B:F of ratio 45:64 (~610 cents).

now of course, the rub in all this is that Bruckner
always played his music on an organ, which would have
been tuned possibly in some form of meantone, less
probably in some form of well-temperament, and even
less probably in 12edo ... but manifestly *not* in
the theoretical JI ratios in my description above.

apparently, Bruckner never *heard* precisely-tuned JI
until Tanaka demonstrated his "Enharmonium" to Bruckner,
who fell in love with it at once. but this was c.1890,
rather late in Bruckner's life, and the only Bruckner
compostion which could have been affected by this
revelation was the 9th Symphony.

so, Bruckner always taught that the ii chord had a
dissonant 27:40 "wolf 5th" which demanded resolution,
even tho the tunings he normally heard would not have
presented it as such.

> I'll take a peek at my Mahler ed today and see if he
> treats the DA fifth/fourth as a dissonance. I'd rather
> take the evidence of usage over any 'theory articles.'
> Anyone got a Haas or Nowak of the Bruckner and would
> like to check also.

what's your "Mahler ed"? if you're referring to the
Mahler scores as published in the "Critical Edition",
please note my quotation marks. there's more on this
below ...

> The only fly in this ointment is whether meantone
> piano tuning would have influenced Mahler's writing.

my guess is that by the time Mahler was playing piano
as a youth (1870s) meantone was pretty much out of the
picture, and the pianos on which he played were most
likely intended to be tuned in 12edo, or *possibly*
some form of well-temperament.

> Bruckner would almost certainly know ji from writing
> his masses and providing the brass parts.

that's a very intriguing comment, but also see
my remarks about Bruckner and JI above.

but here's the *real* "fly in the ointment": it's
on record that Mahler lamented the loss of meantone
as a tuning paradigm, in a statement he made to Schoenberg
presumably around 1905 (as recorded by Peter Yates in
an article published in _Soundings_). Mahler complained
about "the loss of harmonic possibilities".

Mahler often wrote in keys with a lot of sharps, and in
his manuscripts, these passages often contain double-sharps
which have been silently transformed into their 12edo
"natural" enharmonic equivalents, supposedly for "easier
reading". i consider this to be a sacrilege, in that it
distorts the subtle harmonic nuances Mahler intended if
he had meantone in mind, which i think he did in at least
some passages in his symphonies. and by far the vast
majority of his output is for orchestra, which has flexible
intonation.

so if meantone was no longer used on pianos by Mahler's
time, why was he so fascinated with that form of tuning?
the puzzle i've pieced together is that Josef Petzval was
lecturing on 31edo and meantone at Vienna University at
precisely the time Mahler attended that school, and Petzval
had even had a piano tuned in 31edo and had also built an
instrument so tuned of his own invention called a "guitharp",
both of which he used in his demonstrations of meantone.

-monz
(last post before my vacation in Miami)