diminished 7th chords

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<<< You only list one function, that of vii-7 in some key...but what about
"neighbor" vii. For example, in C Major, C-D#-F#-A as a "neighbor" to C-E-G.
(I suppose this could be considered a re-spacing of vii-7/V, but I think
there could be other tuning schemes to apply to this as well). >>>

I've been messing with this question in my just-intonation notes. For the
neighboring diminished there are basically four different enharmonic
spellings based on which note is which in the ambiguous diminished chord.
I've been racking my brain over the question of whether there is an
"official" choice of this. You can think in terms of which of the four
triads the particular chord spelling would resolve to. It would also
determine which of the four notes in the dim note you would lower a half
step to create the dominant on the "missing root" that resolves to the same
chord. The other enharmonic choices are a diminished that resolves to a
triad a step below the tonic, one that resolves to a triad a half step above
the tonic, and one that resolves to the chord a major third above the tonic.

I think of the diminished, in a just-intononation environment, as the upper
four parts of a flatted-ninth, and which has the ratio of 10:12:14:17. You
can form the corresponding dominant by lowering the 17th harmonic to 16.

Billy

This was Helmholtz's conclusion of the full diminished 7th, so you are in good company

I think of the diminished, in a just-intononation environment, as the upper
four parts of a flatted-ninth, and which has the ratio of 10:12:14:17. You
can form the corresponding dominant by lowering the 17th harmonic to 16.

Billy
,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

-----Original Message-----
From: Billy Gard [mailto:billygard@comcast.net]
Sent: Monday, February 4, 2008 06:53 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re:diminished 7th chords

<<< You only list one function, that of vii-7 in some key...but what about
"neighbor" vii. For example, in C Major, C-D#-F#-A as a "neighbor" to C-E-G.
(I suppose this could be considered a re-spacing of vii-7/V, but I think
there could be other tuning schemes to apply to this as well). >>>

I've been messing with this question in my just-intonation notes. For the
neighboring diminished there are basically four different enharmonic
spellings based on which note is which in the ambiguous diminished chord.
I've been racking my brain over the question of whether there is an
"official" choice of this. You can think in terms of which of the four
triads the particular chord spelling would resolve to. It would also
determine which of the four notes in the dim note you would lower a half
step to create the dominant on the "missing root" that resolves to the same
chord. The other enharmonic choices are a diminished that resolves to a
triad a step below the tonic, one that resolves to a triad a half step above
the tonic, and one that resolves to the chord a major third above the tonic.

I think of the diminished, in a just-intononation environment, as the upper
four parts of a flatted-ninth, and which has the ratio of 10:12:14:17. You
can form the corresponding dominant by lowering the 17th harmonic to 16.

Billy

I think that the pure diminished 7th chord is a product of equal
temperament. It's a chord based on 12-eq symmetry, and rehashing it in
JI is an afterthought. I don't see it as having an acoustical basis,
JI-wise, as the major & minor triads, etc. do, even though it can be
done as Kraig suggest by for instance 10:12:14:17.

To my ears, although I like this sound, I like the 12-eq version
better--it's one of the things 12-eq excels at, as well as the
augmented triad.

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> This was Helmholtz's conclusion of the full diminished 7th, so you
are in good company
>
>
> I think of the diminished, in a just-intononation environment, as
the upper
> four parts of a flatted-ninth, and which has the ratio of
10:12:14:17. You
> can form the corresponding dominant by lowering the 17th harmonic to 16.
>
> Billy
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
> -----Original Message-----
> From: Billy Gard [mailto:billygard@...]
> Sent: Monday, February 4, 2008 06:53 PM
> To: tuning@yahoogroups.com
> Subject: [tuning] Re:diminished 7th chords
>
> <<< You only list one function, that of vii-7 in some key...but what
about
> "neighbor" vii. For example, in C Major, C-D#-F#-A as a "neighbor"
to C-E-G.
> (I suppose this could be considered a re-spacing of vii-7/V, but I think
> there could be other tuning schemes to apply to this as well). >>>
>
> I've been messing with this question in my just-intonation notes.
For the
> neighboring diminished there are basically four different enharmonic
> spellings based on which note is which in the ambiguous diminished
chord.
> I've been racking my brain over the question of whether there is an
> "official" choice of this. You can think in terms of which of the four
> triads the particular chord spelling would resolve to. It would also
> determine which of the four notes in the dim note you would lower a half
> step to create the dominant on the "missing root" that resolves to
the same
> chord. The other enharmonic choices are a diminished that resolves to a
> triad a step below the tonic, one that resolves to a triad a half
step above
> the tonic, and one that resolves to the chord a major third above
the tonic.
>
> I think of the diminished, in a just-intononation environment, as
the upper
> four parts of a flatted-ninth, and which has the ratio of
10:12:14:17. You
> can form the corresponding dominant by lowering the 17th harmonic to 16.
>
> Billy
>

AKJ wrote:

> I think that the pure diminished 7th chord is a product of equal
> temperament. It's a chord based on 12-eq symmetry, and rehashing it in
> JI is an afterthought.

Harmonically yes, but not melodically. When I listen to meantone or 5-limit JI, for example, I find it very important that there's a clear difference between B-D-F-G#, B-D-F-Ab, B-D-E#-G#, and Cb-D-F-Ab.

Petr

My experience of it is different. It occurs naturally in some JI
scales. See tuning post #67885 for a discussion of such a scale.
The JI interpretation in that scale is 25:30:35:42, however.

Robin

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
>
> I think that the pure diminished 7th chord is a product of equal
> temperament. It's a chord based on 12-eq symmetry, and rehashing it in
> JI is an afterthought. I don't see it as having an acoustical basis,
> JI-wise, as the major & minor triads, etc. do, even though it can be
> done as Kraig suggest by for instance 10:12:14:17.
>
> To my ears, although I like this sound, I like the 12-eq version
> better--it's one of the things 12-eq excels at, as well as the
> augmented triad.
>
> --- In tuning@yahoogroups.com, kraiggrady@ wrote:
> >
> > This was Helmholtz's conclusion of the full diminished 7th, so you
> are in good company
> >
> >
> > I think of the diminished, in a just-intononation environment, as
> the upper
> > four parts of a flatted-ninth, and which has the ratio of
> 10:12:14:17. You
> > can form the corresponding dominant by lowering the 17th harmonic
to 16.
> >
> > Billy
> > ,',',',Kraig Grady,',',',
> > '''''''North/Western Hemisphere:
> > North American Embassy of Anaphoria island
> > '''''''South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> > -----Original Message-----
> > From: Billy Gard [mailto:billygard@]
> > Sent: Monday, February 4, 2008 06:53 PM
> > To: tuning@yahoogroups.com
> > Subject: [tuning] Re:diminished 7th chords
> >
> > <<< You only list one function, that of vii-7 in some key...but what
> about
> > "neighbor" vii. For example, in C Major, C-D#-F#-A as a "neighbor"
> to C-E-G.
> > (I suppose this could be considered a re-spacing of vii-7/V, but I
think
> > there could be other tuning schemes to apply to this as well). >>>
> >
> > I've been messing with this question in my just-intonation notes.
> For the
> > neighboring diminished there are basically four different enharmonic
> > spellings based on which note is which in the ambiguous diminished
> chord.
> > I've been racking my brain over the question of whether there is an
> > "official" choice of this. You can think in terms of which of the four
> > triads the particular chord spelling would resolve to. It would also
> > determine which of the four notes in the dim note you would lower
a half
> > step to create the dominant on the "missing root" that resolves to
> the same
> > chord. The other enharmonic choices are a diminished that resolves
to a
> > triad a step below the tonic, one that resolves to a triad a half
> step above
> > the tonic, and one that resolves to the chord a major third above
> the tonic.
> >
> > I think of the diminished, in a just-intononation environment, as
> the upper
> > four parts of a flatted-ninth, and which has the ratio of
> 10:12:14:17. You
> > can form the corresponding dominant by lowering the 17th harmonic
to 16.
> >
> > Billy
> >
>

The problem with microtuning the dim7th is that unlike 12edo; it does
not repeat obviously octave by octave.

Assuming that it has three consistent intervals of a bIIIrd; it would
give you C-Eb-Gb-Bbb in a meantone tuning.

Part of the "magic of the diminshed used in 12edo is that it
introduces an harmonic ambiguity, as the "tonic" can be considered as
any of the four notes, and hence is useful for "lazy" modulations and
transpositions.

In 88edo you could consider it as a series of intervals of 23 steps.

23*4 = 92, so after the first four notes you sharpen the pitches by 4
steps of 88edo for each ocatve.

On Feb 8, 2008, at 6:29 PM, Petr Parízek wrote:

>
> AKJ wrote:
>
>
>
> > I think that the pure diminished 7th chord is a product of equal
> > temperament. It's a chord based on 12-eq symmetry, and rehashing
> it in
> > JI is an afterthought.
>
>
>
> Harmonically yes, but not melodically. When I listen to meantone or
> 5-limit JI, for example, I find it very important that there’s a
> clear difference between B-D-F-G#, B-D-F-Ab, B-D-E#-G#, and Cb-D-F-Ab.
>
>
>
> Petr
>
>
>
>
>
>
>

Charles Lucy
lucy@lucytune.com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Hi Robin, Petr, Aaron, Christopher, and everyone else,

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> AKJ wrote:
>
>
>
> > I think that the pure diminished 7th chord is
> > a product of equal temperament. It's a chord based
> > on 12-eq symmetry, and rehashing it in JI is an
> > afterthought.
>
> Harmonically yes, but not melodically. When I listen
> to meantone or 5-limit JI, for example, I find it
> very important that there's a clear difference
> between B-D-F-G#, B-D-F-Ab, B-D-E#-G#, and Cb-D-F-Ab.

Thanks, guys. Comments like this are exactly what i'd
like to add to what i already have on my webpage.

I've been really busy, but will include all of these
responses onto the page as soon as i can.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

The dim7 has been used occasionally in late Baroque music, both
keyboard and ensemble, where one would not particularly expect ET. Its
function is not as an ambiguous 'pivot' but simply an intensification
of the dominant 7th - counting as a dissonance. One might also
consider non-ET diminished chords appearing in Romantic chamber music
or choral music.

My JI version inspired by septimal meantone is 1:6/5:10/7:12/7, which
I think is just an inversion of the chord already mentioned with
alternating 6/5's and 7/6. As a dissonance it is a pretty mild one...
The dim7 of 12/7 resolves to 3/2 via two semitones of 16/15 and 15/14,
both very close to the meantone diatonic semitone.

~~~T~~~

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> AKJ wrote:
>
>
>
> > I think that the pure diminished 7th chord is a product of equal
> > temperament. It's a chord based on 12-eq symmetry, and rehashing it in
> > JI is an afterthought.
>
>
> Harmonically yes, but not melodically. When I listen to meantone or
5-limit JI, for example, I find it very important that there's a clear
difference between B-D-F-G#, B-D-F-Ab, B-D-E#-G#, and Cb-D-F-Ab.

Petr,

You are quite correct in pointing this out, however, isn't it
interesting that historically, anyway, we don't tend to see examples
of meantone-tuned diminished 7ths, or even triads? They were not
sounds that were recognized as in anyway forming a structural music
basis...where, in the 12-eq dimished, we see them used as 'bridges' of
modulation, etc., where their ambiguity serves a structural purpose.

-AKJ.

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:

> interesting that historically, anyway, we don't tend to see examples
> of meantone-tuned diminished 7ths, or even triads? They were not
> sounds that were recognized as in anyway forming a structural music
> basis...where, in the 12-eq dimished, we see them used as 'bridges' of
> modulation, etc., where their ambiguity serves a structural purpose.
>
> -AKJ.
>

Are you saying that diminished triads didn't occur much historically
in meantone contexts? But I think they did (eg frequently in Byrd, in
first inversion - d f b - or Froberger), but generally in the role of
dissonances.

~~~T~~~

The diminished 7th chords are spectacular in regular 1/6 comma, since they're built of two pure tritones interlocked...and since it still works strongly even if they get inverted or misspelled. Play around with it in regular 1/6 on keyboards, to understand what I mean by that.

Maybe that strong sound contributed to the heavy use of diminished 7th chords by 18th century composers; or vice versa, their use of those is some evidence (perhaps weak, but there nonetheless) of a regular 1/6 layout or something near it? Handel was certainly a heavy user, for whatever reason.

Brad Lehman

Hi Brad,

--- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
>
> The diminished 7th chords are spectacular in
> regular 1/6 comma, since they're built of two
> pure tritones interlocked...and since it
> still works strongly even if they get inverted
> or misspelled. Play around with it in regular 1/6
> on keyboards, to understand what I mean by that.
>
> Maybe that strong sound contributed to the
> heavy use of diminished 7th chords by 18th century
> composers; or vice versa, their use of those is
> some evidence (perhaps weak, but there nonetheless)
> of a regular 1/6 layout or something near it?
> Handel was certainly a heavy user, for whatever reason.

Thanks for that.

I never noticed before that Handel used diminished-7th
chords so much -- i've always been of the impression
that Bach was the first composer to really exploit them,
and then it seemed to me that Beethoven was the first
really "heavy user" of them.

I've made several Tonescape files of Beethoven's music
tuned in 1/6-comma meantone, and sometimes he juxtaposes
different groupings of instruments playing the same chord,
but spells it differently among the different instruments.
Of course, in 1/6-comma meantone these different spellings
produce different-sounding chords ... and a word i would
use to describe that is "spectacular", indeed.

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

I could argue with equal validity that the tritone of 31-edo is an
audibly pure 7/5 - which is much clearer to the ear than whatever
definition of 'pure tritone' Brad wants to use (can 45/32 be
meaningfully called 'pure'?).

'Logically' then by the same token, any composer who uses a lot of
diminished sevenths is providing evidence for the use of a tuning
approximating to 31-edo, e.g. 1/4- or 2/9-comma meantone.

The resolution to this 'paradox' is that a composer's using a lot of
diminished chords does not provide evidence for any tuning in
particular. It is evidence that the composer liked the sound of
diminished chords, in whatever tuning was around (or even regardless
of the exact details of tuning!).

~~~T~~~

--- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
>
> The diminished 7th chords are spectacular in regular 1/6 comma, since
> they're built of two pure tritones interlocked...and since it still
> works strongly even if they get inverted or misspelled. Play around
> with it in regular 1/6 on keyboards, to understand what I mean by that.
>
> Maybe that strong sound contributed to the heavy use of diminished 7th
> chords by 18th century composers; or vice versa, their use of those is
> some evidence (perhaps weak, but there nonetheless) of a regular 1/6
> layout or something near it? Handel was certainly a heavy user, for
> whatever reason.
>
>
> Brad Lehman
>

In 31-edo, the Gb of A-Eb-Gb-A could be 10/7.

Oz.

----- Original Message -----
From: "Tom Dent" <stringph@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 12 �ubat 2008 Sal� 21:13
Subject: [tuning] Re: diminished 7th chords

>
> I could argue with equal validity that the tritone of 31-edo is an
> audibly pure 7/5 - which is much clearer to the ear than whatever
> definition of 'pure tritone' Brad wants to use (can 45/32 be
> meaningfully called 'pure'?).
>
> 'Logically' then by the same token, any composer who uses a lot of
> diminished sevenths is providing evidence for the use of a tuning
> approximating to 31-edo, e.g. 1/4- or 2/9-comma meantone.
>
> The resolution to this 'paradox' is that a composer's using a lot of
> diminished chords does not provide evidence for any tuning in
> particular. It is evidence that the composer liked the sound of
> diminished chords, in whatever tuning was around (or even regardless
> of the exact details of tuning!).
>
> ~~~T~~~
>
>
> --- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
> >
> > The diminished 7th chords are spectacular in regular 1/6 comma, since
> > they're built of two pure tritones interlocked...and since it still
> > works strongly even if they get inverted or misspelled. Play around
> > with it in regular 1/6 on keyboards, to understand what I mean by that.
> >
> > Maybe that strong sound contributed to the heavy use of diminished 7th
> > chords by 18th century composers; or vice versa, their use of those is
> > some evidence (perhaps weak, but there nonetheless) of a regular 1/6
> > layout or something near it? Handel was certainly a heavy user, for
> > whatever reason.
> >
> >
> > Brad Lehman
> >
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@> wrote:
>
> > interesting that historically, anyway, we don't tend to see examples
> > of meantone-tuned diminished 7ths, or even triads? They were not
> > sounds that were recognized as in anyway forming a structural music
> > basis...where, in the 12-eq dimished, we see them used as 'bridges' of
> > modulation, etc., where their ambiguity serves a structural purpose.
> >
> > -AKJ.
> >
>
> Are you saying that diminished triads didn't occur much historically
> in meantone contexts? But I think they did (eg frequently in Byrd, in
> first inversion - d f b - or Froberger), but generally in the role of
> dissonances.

exactly...they are vertical structures only as passing voice leading,
whereas in Beethoven, you see them as sustained structures (e.g.
development section of 'Moonlight sonata' 1st movement). Show me a
renaissance example of a full diminished seventh chord anywhere, and
I'll eat my hat.

BTW, on could look at the example you give (a diminished triad) as a
rootless dominant 7th, too. I would argue that this makes the most sense.

-AKJ

> I could argue with equal validity that the tritone of 31-edo is an
> audibly pure 7/5 - which is much clearer to the ear than whatever
> definition of 'pure tritone' Brad wants to use (can 45/32 be
> meaningfully called 'pure'?).
>
> 'Logically' then by the same token, any composer who uses a lot of
> diminished sevenths is providing evidence for the use of a tuning
> approximating to 31-edo, e.g. 1/4- or 2/9-comma meantone.
>
> The resolution to this 'paradox' is that a composer's using a lot of
> diminished chords does not provide evidence for any tuning in
> particular. It is evidence that the composer liked the sound of
> diminished chords, in whatever tuning was around (or even regardless
> of the exact details of tuning!).

hmm....not sure I can whole-heartedly agree with the last paragraph. I
think you do have an interesting and valid point re:31-eq, and the
7/5 though. Still, I find 45/32 relatively 'pureish' in sound...at
least, it has an interesting *qualitative difference*, as of last time
I heard it....at least it's more towards pure than good-ol' 12-eq C-F#!

If we extend this argument (from your last paragraph), we have a hard
time understanding how the history of tuning affects the history of
music in general...for instance the emergence of 5/4 instead of 81/64
definately formed a bridge from medieval to renaissance practice, and
I would argue that the occurance of plenty of harmonic major thirds
*would* stongly imply meantone!

-AKJ

> exactly...they are vertical structures only as passing voice leading,
> whereas in Beethoven, you see them as sustained structures (e.g.
> development section of 'Moonlight sonata' 1st movement). Show me a
> renaissance example of a full diminished seventh chord anywhere, and
> I'll eat my hat.

Bach's organ music uses plenty of diminished harmony (not just
as a dissonance).

-Carl

Ozan Yarman wrote:
> In 31-edo, the Gb of A-Eb-Gb-A could be 10/7.

Which means the Gb-A is a 7/6 and every interval in the chord approximates some 7-limit consonance. So there's no a priori reason why a chord with equal thirds would be preferred over this one.

Note from my graph here:

http://x31eq.com/meantone.htm#approx

the approximations to 7:5 and 10:7 are not that bad in 1/6 comma meantone (about -0.17 on the x axis). But the 7:4 isn't so good and the 7:6 is a little worse. 12-equal is that bit worse again on all counts.

Along with the symmetric chords, there's a whole family of chords that only work in certain temperaments, but not necessarily equal ones. This diminished 7th half-rationalization requires 126:125 to be tempered out. Some non-meantones that also support it are 27, 46, 58, and 65-equal.

Some other impossible chords 31 supports: augmented triads as 5:4 5:4 9:7 to an octave (because the ubiquitous 225:224 is tempered out) and neutral triads as 11:9 11:9 to a perfect fifth (because 243:242 is tempered out).

Graham

Uh, I made a typo. It should have been C-Eb-Gb-A.

A very informative page, Graham.

Oz.

----- Original Message -----
From: "Graham Breed" <gbreed@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 13 �ubat 2008 �ar�amba 13:59
Subject: Re: [tuning] Re: diminished 7th chords

> Ozan Yarman wrote:
> > In 31-edo, the Gb of A-Eb-Gb-A could be 10/7.
>
> Which means the Gb-A is a 7/6 and every interval in the
> chord approximates some 7-limit consonance. So there's no a
> priori reason why a chord with equal thirds would be
> preferred over this one.
>
> Note from my graph here:
>
> http://x31eq.com/meantone.htm#approx
>
> the approximations to 7:5 and 10:7 are not that bad in 1/6
> comma meantone (about -0.17 on the x axis). But the 7:4
> isn't so good and the 7:6 is a little worse. 12-equal is
> that bit worse again on all counts.
>
> Along with the symmetric chords, there's a whole family of
> chords that only work in certain temperaments, but not
> necessarily equal ones. This diminished 7th
> half-rationalization requires 126:125 to be tempered out.
> Some non-meantones that also support it are 27, 46, 58, and
> 65-equal.
>
> Some other impossible chords 31 supports: augmented triads
> as 5:4 5:4 9:7 to an octave (because the ubiquitous 225:224
> is tempered out) and neutral triads as 11:9 11:9 to a
> perfect fifth (because 243:242 is tempered out).
>
>
> Graham
>
>

"Carl Lumma" wrote:

"Bach's organ music uses plenty of diminished harmony (not just
as a dissonance)."

Could you given an example in Bach in which a diminished harmony is not used as a dissonance (i.e. resolves to a simple consonance)?

Daniel Wolf

Not sure about organ pieces, but the Chromatic Fantasia for clavichord
ends with a long descending chain of diminished chords over a pedal
point. The Gigue of the Bb major Partita and the D minor Prelude of
the WTC ('Book 1') both have similar proto-Wagnerian sequences of
diminished chords - but only triads in the case of the Prelude.

The opening chorus of the St. John Passion is also a reasonable
example of a long non-resolving chord sequence including dim7's.
And does the shortest chorus of the St. Matthew (the word 'Barabbam'
on a single dim7 chord) have a resolution?
~~~T~~~

--- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@...> wrote:
>
> "Carl Lumma" wrote:
>
> "Bach's organ music uses plenty of diminished harmony (not just
> as a dissonance)."
>
> Could you given an example in Bach in which a diminished harmony is
not
> used as a dissonance (i.e. resolves to a simple consonance)?
>
> Daniel Wolf
>

That's BWV903 (in Dmin). It does, like all of the organ
stuff I can think of, eventually end on a major or minor
triad. But I don't take that to mean all the previous
diminished chords were dissonances.

The opening of the BWV542 fantasia is almost entirely
diminished. Elsewhere in the piece, alternations between
dim and minor chords are heavily biased towards the former.
The sound is not one of a need to resolve.

The BWV534 prelude has a dim7 false ending, and the
subsequent cadence is rather laborious.

-C.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> Not sure about organ pieces, but the Chromatic Fantasia for
> clavichord ends with a long descending chain of diminished
> chords over a pedal point.
//
> --- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@> wrote:
> >
> > "Carl Lumma" wrote:
> >
> > "Bach's organ music uses plenty of diminished harmony (not
> > just as a dissonance)."
> >
> > Could you given an example in Bach in which a diminished
> > harmony is not used as a dissonance (i.e. resolves to a
> > simple consonance)?
> >
> > Daniel Wolf
> >
>

<<< The diminished 7th chords are spectacular in regular 1/6 comma, since
they're built of two pure tritones interlocked...and since it still works
strongly even if they get inverted or misspelled. Play around with it in
regular 1/6 on keyboards, to understand what I mean by that. >>>

Having figured out how a 1/4 comma is for getting a pure major 3rd, I've
been wondering just what they were thinking when they invented a 1/6 comma.
A recent post pointed out that chords in the diminished variety were
considered dissonances in the "meantone period". What would be the purpose
of a temperament for producing a "pure" tritone?

Actually I would think that one of the septimal tunings would be a purer
tritone anyway, since 7/5 or 10/7 is far simpler ratio than the 45/32 that a
1/6 comma meantone should produce.

Beatless in Seattle.
Billy

maybe a music secret society had a secret practice?

Billy Gard wrote:
>
> ......What would be the purpose
> of a temperament for producing a "pure" tritone?
>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Carl Lumma Wrote

"That's BWV903 (in Dmin). It does, like all of the organ
stuff I can think of, eventually end on a major or minor
triad. But I don't take that to mean all the previous
diminished chords were dissonances."

Of course it does -- the length of the prolongation in no means changes the dissonant character and function of the harmony.

"The opening of the BWV542 fantasia is almost entirely
diminished. Elsewhere in the piece, alternations between
dim and minor chords are heavily biased towards the former.
The sound is not one of a need to resolve."

Again, it's not a question of quantity but function. Just check out the voice leading -- one can, in principle, jump to a diminished sonority anywhere (even from another diminished chord, as Bach sometimes does in his recitatives), but the subsequent voice leading will eventually be toward a major or minor triad.

"The BWV534 prelude has a dim7 false ending, and the
subsequent cadence is rather laborious."

Although it well may be a weak composition, Bach was clearly obliged to add a resolution.

Tom Dent wrote

"Not sure about organ pieces, but the Chromatic Fantasia for clavichord
ends with a long descending chain of diminished chords over a pedal
point. The Gigue of the Bb major Partita and the D minor Prelude of
the WTC ('Book 1') both have similar proto-Wagnerian sequences of
diminished chords - but only triads in the case of the Prelude."

One can have indefinitely long series of dissonances, and pedal points are a typical place not only to do this, but also to emphasize the dissonant character. As to the sequences, I would hesitate to call them "proto-Wagnerian" in that the formal functions of the sequences, when taken as a single compound voice leading (as the neo-Riemannians might have it) is never, to my knowledge, either a static enjoyment of the diminished sonority nor a functional aimless modulation.

"The opening chorus of the St. John Passion is also a reasonable
example of a long non-resolving chord sequence including dim7's.
And does the shortest chorus of the St. Matthew (the word 'Barabbam'
on a single dim7 chord) have a resolution?"

I asked specifically for organ works, but I think that these are clear exceptions that prove the rule, as in both Passions the dramatic need of the given moments were for unresolved dissonances -- a need which is not found in the closed forms of the keyboard works.

The function you're talking about is clearly there, but
I maintain it has been subjugated. Ultimately, this is
a subjective thing.

-Carl

--- In tuning@yahoogroups.com, "Daniel Wolf" <djwolf@...> wrote:
>
> Carl Lumma Wrote
>
> "That's BWV903 (in Dmin). It does, like all of the organ
> stuff I can think of, eventually end on a major or minor
> triad. But I don't take that to mean all the previous
> diminished chords were dissonances."
>
> Of course it does -- the length of the prolongation in no
> means changes the dissonant character and function of the
> harmony.
>
> "The opening of the BWV542 fantasia is almost entirely
> diminished. Elsewhere in the piece, alternations between
> dim and minor chords are heavily biased towards the former.
> The sound is not one of a need to resolve."
>
> Again, it's not a question of quantity but function
...

The subject quite entertains me and brings in light of the phenomenological dilemma posed by western tuning, the relationship of dissonance to ambiguity. Not an easy border to define

>
> ...
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Not quite sure what this means, ... but we have been trying to
position ourselves on obscure borders.

Every composition by Wagner, and every act of every opera, ends on a
root position major or minor chord, so dissonances are ultimately
removed, if not formally resolved. It just takes longer.

I suspect what Daniel is referring to is a more or less persistent
sense of lack of tonal anchor, which often occurs in late Wagner,
sometimes aided by a proliferation of diminished harmonies. In Bach
only parts of the G minor Fantasia, Chromatic Fantasia and some rare
parts of the WTC (A minor prelude Book II?) and Musical Offering
(Fugue a 3) could be described as tonally unanchored. Even this small
count is enough to mark him as one of the most adventurous of the
Baroque period, or actually the whole 18th century.

By the way, Daniel, how long are you going to sit on it? Did it
disappear into some obscure corner of your filing system?

~~~T~~~

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> The subject quite entertains me and brings in light of the
> phenomenological dilemma posed by western tuning, the relationship of
> dissonance to ambiguity. Not an easy border to define
>
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island
<http://anaphoria.com/index.html>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> I could argue with equal validity that the tritone of 31-edo is an
> audibly pure 7/5 - which is much clearer to the ear than whatever
> definition of 'pure tritone' Brad wants to use (can 45/32 be
> meaningfully called 'pure'?).
>
I, too, wonder what Brad is smoking when he talks about "pure"
tritones in 1/6 mean. If I play c-f# in 1/6, I instantly hear the
beating, and when I "fix" it, i.e. tune it "pure" (beatless), I find
my ear goes to either 7/5 or 10/7, both of which sound far more
"pure", though quite different in feel.

Ciao,

P

oh what is wrong with a few beats as long as it works with the rest of the scale!

Paul Poletti wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, "Tom > Dent" <stringph@...> wrote:
> >
> >
> > I could argue with equal validity that the tritone of 31-edo is an
> > audibly pure 7/5 - which is much clearer to the ear than whatever
> > definition of 'pure tritone' Brad wants to use (can 45/32 be
> > meaningfully called 'pure'?).
> >
> I, too, wonder what Brad is smoking when he talks about "pure"
> tritones in 1/6 mean. If I play c-f# in 1/6, I instantly hear the
> beating, and when I "fix" it, i.e. tune it "pure" (beatless), I find
> my ear goes to either 7/5 or 10/7, both of which sound far more
> "pure", though quite different in feel.
>
> Ciao,
>
> P
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Because words ought to have meanings, and it was claimed that 45/32 is
a 'pure' interval. If you tune it up you will find that the 7th
harmonic starts beating quite regularly and audibly. This contradicts
the common-sense practical definition that a pure interval is one that
has been adjusted to eliminate beating. You simply can't tune a 'pure
45/32' directly by ear. Same as you can't tune 81/64 directly by ear.
(Directly: in one step!)

Or do you want to redefine 'pure' to mean, whatever I happen to like
the sound of just now. Perhaps you are a member of a secret society
whose aim is to use language in an insidiously different way from most
other people.

Ross Duffin has even referred to 16/15 as a 'pure semitone'... which
is an acoustic contradiction in terms.

~~~T~~~

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> oh what is wrong with a few beats as long as it works with the rest of
> the scale!
>
> Paul Poletti wrote:
> >
> > --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, "Tom
> > Dent" <stringph@> wrote:
> > >
> > >
> > > I could argue with equal validity that the tritone of 31-edo is an
> > > audibly pure 7/5 - which is much clearer to the ear than whatever
> > > definition of 'pure tritone' Brad wants to use (can 45/32 be
> > > meaningfully called 'pure'?).
> > >
> > I, too, wonder what Brad is smoking when he talks about "pure"
> > tritones in 1/6 mean. If I play c-f# in 1/6, I instantly hear the
> > beating, and when I "fix" it, i.e. tune it "pure" (beatless), I find
> > my ear goes to either 7/5 or 10/7, both of which sound far more
> > "pure", though quite different in feel.
> >
> > Ciao,
> >
> > P
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island
<http://anaphoria.com/index.html>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles
>

relax. i agree with you on meaning. My comment that often intervals with a few beats are useful. if someone likes tritones in 1/6 meantone i take it they are hearing something, and have an empirical reaction.

I would though say that 16/15 could be called pure to someone used to the 15th harmonic. otherwise we might have to question the 13th then the 11th.
So the meaning of pure perhaps is not clear, cause it is used in different contexts. I know your interest in intonation is historically (correct me if i am wrong) and in that context 16/15 might not satisfy your concept of pure. Take for instance a string piece by hank badings where he used the harmonic series up in the 20s to set ancient greek tunes. i would say he is using pure intervals. Perhaps pure is not a good word to be using anyway

but i will try not to be funny, since you seem to get so offended. this medium has that problem.

Tom Dent wrote:
>
>
> Because words ought to have meanings, and it was claimed that 45/32 is
> a 'pure' interval. If you tune it up you will find that the 7th
> harmonic starts beating quite regularly and audibly. This contradicts
> the common-sense practical definition that a pure interval is one that
> has been adjusted to eliminate beating. You simply can't tune a 'pure
> 45/32' directly by ear. Same as you can't tune 81/64 directly by ear.
> (Directly: in one step!)
>
> Or do you want to redefine 'pure' to mean, whatever I happen to like
> the sound of just now. Perhaps you are a member of a secret society
> whose aim is to use language in an insidiously different way from most
> other people.
>
> Ross Duffin has even referred to 16/15 as a 'pure semitone'... which
> is an acoustic contradiction in terms.
>
> ~~~T~~~
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > oh what is wrong with a few beats as long as it works with the rest of
> > the scale!
> >
> > Paul Poletti wrote:
> > >
> > > --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> > <mailto:tuning%40yahoogroups.com>, "Tom
> > > Dent" <stringph@> wrote:
> > > >
> > > >
> > > > I could argue with equal validity that the tritone of 31-edo is an
> > > > audibly pure 7/5 - which is much clearer to the ear than whatever
> > > > definition of 'pure tritone' Brad wants to use (can 45/32 be
> > > > meaningfully called 'pure'?).
> > > >
> > > I, too, wonder what Brad is smoking when he talks about "pure"
> > > tritones in 1/6 mean. If I play c-f# in 1/6, I instantly hear the
> > > beating, and when I "fix" it, i.e. tune it "pure" (beatless), I find
> > > my ear goes to either 7/5 or 10/7, both of which sound far more
> > > "pure", though quite different in feel.
> > >
> > > Ciao,
> > >
> > > P
> > >
> > >
> >
> > --
> > Kraig Grady
> > North American Embassy of Anaphoria Island
> <http://anaphoria.com/index.html <http://anaphoria.com/index.html>>
> > The Wandering Medicine Show
> > KXLU <http://www.kxlu.com/main/index.asp > <http://www.kxlu.com/main/index.asp>> 88.9 FM Wed 8-9 pm Los Angeles
> >
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> oh what is wrong with a few beats as long as it works with the rest of
> the scale!

Nothing at all, I use 'em all the time.

;-)

It's just that a beating interval is not "pure". Why use a word if it
has no meaning?

Ciao,

P

me too!

Paul Poletti wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Kraig > Grady <kraiggrady@...> wrote:
> >
> > oh what is wrong with a few beats as long as it works with the rest of
> > the scale!
>
> Nothing at all, I use 'em all the time.
>
> ;-)
>
> It's just that a beating interval is not "pure". Why use a word if it
> has no meaning?
>
> Ciao,
>
> P
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

There is a problem/ambiguity with your note naming in this posting Oz.

from C the notes (in steps of flattened thirds) should be:

C - Eb - Gb - Bbb

or from A

A - C- Eb - Gb

or from Eb

Eb - Gb - Bbb - Dbb

or from Gb

Gb - Bbb - Dbb - Fbb

On Feb 13, 2008, at 1:18 AM, Ozan Yarman wrote:

> In 31-edo, the Gb of A-Eb-Gb-A could be 10/7.
>
> Oz.
>
> ----- Original Message -----
> From: "Tom Dent" <stringph@gmail.com>
> To: <tuning@yahoogroups.com>
> Sent: 12 Şubat 2008 Salı 21:13
> Subject: [tuning] Re: diminished 7th chords
>
>
>>
>> I could argue with equal validity that the tritone of 31-edo is an
>> audibly pure 7/5 - which is much clearer to the ear than whatever
>> definition of 'pure tritone' Brad wants to use (can 45/32 be
>> meaningfully called 'pure'?).
>>
>> 'Logically' then by the same token, any composer who uses a lot of
>> diminished sevenths is providing evidence for the use of a tuning
>> approximating to 31-edo, e.g. 1/4- or 2/9-comma meantone.
>>
>> The resolution to this 'paradox' is that a composer's using a lot of
>> diminished chords does not provide evidence for any tuning in
>> particular. It is evidence that the composer liked the sound of
>> diminished chords, in whatever tuning was around (or even regardless
>> of the exact details of tuning!).
>>
>> ~~~T~~~
>>
>>
>> --- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
>>>
>>> The diminished 7th chords are spectacular in regular 1/6 comma,
>>> since
>>> they're built of two pure tritones interlocked...and since it still
>>> works strongly even if they get inverted or misspelled. Play around
>>> with it in regular 1/6 on keyboards, to understand what I mean by
>>> that.
>>>
>>> Maybe that strong sound contributed to the heavy use of diminished
>>> 7th
>>> chords by 18th century composers; or vice versa, their use of
>>> those is
>>> some evidence (perhaps weak, but there nonetheless) of a regular 1/6
>>> layout or something near it? Handel was certainly a heavy user, for
>>> whatever reason.
>>>
>>>
>>> Brad Lehman
>>>
>>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
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>
>
>

Charles Lucy
lucy@lucytune.com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

I wanted to express Gb-A as a septimal minor third.

Oz.

----- Original Message -----
From: "Charles Lucy" <lucy@harmonics.com>
To: <tuning@yahoogroups.com>
Sent: 16 �ubat 2008 Cumartesi 2:30
Subject: Re: [tuning] Re: diminished 7th chords - note naming error?

There is a problem/ambiguity with your note naming in this posting Oz.

from C the notes (in steps of flattened thirds) should be:

C - Eb - Gb - Bbb

or from A

A - C- Eb - Gb

or from Eb

Eb - Gb - Bbb - Dbb

or from Gb

Gb - Bbb - Dbb - Fbb

On Feb 13, 2008, at 1:18 AM, Ozan Yarman wrote:

> In 31-edo, the Gb of A-Eb-Gb-A could be 10/7.
>
> Oz.
>
> ----- Original Message -----
> From: "Tom Dent" <stringph@gmail.com>
> To: <tuning@yahoogroups.com>
> Sent: 12 �ubat 2008 Salı 21:13
> Subject: [tuning] Re: diminished 7th chords
>
>
>>
>> I could argue with equal validity that the tritone of 31-edo is an
>> audibly pure 7/5 - which is much clearer to the ear than whatever
>> definition of 'pure tritone' Brad wants to use (can 45/32 be
>> meaningfully called 'pure'?).
>>
>> 'Logically' then by the same token, any composer who uses a lot of
>> diminished sevenths is providing evidence for the use of a tuning
>> approximating to 31-edo, e.g. 1/4- or 2/9-comma meantone.
>>
>> The resolution to this 'paradox' is that a composer's using a lot of
>> diminished chords does not provide evidence for any tuning in
>> particular. It is evidence that the composer liked the sound of
>> diminished chords, in whatever tuning was around (or even regardless
>> of the exact details of tuning!).
>>
>> ~~~T~~~
>>
>>
>> --- In tuning@yahoogroups.com, Brad Lehman <bpl@...> wrote:
>>>
>>> The diminished 7th chords are spectacular in regular 1/6 comma,
>>> since
>>> they're built of two pure tritones interlocked...and since it still
>>> works strongly even if they get inverted or misspelled. Play around
>>> with it in regular 1/6 on keyboards, to understand what I mean by
>>> that.
>>>
>>> Maybe that strong sound contributed to the heavy use of diminished
>>> 7th
>>> chords by 18th century composers; or vice versa, their use of
>>> those is
>>> some evidence (perhaps weak, but there nonetheless) of a regular 1/6
>>> layout or something near it? Handel was certainly a heavy user, for
>>> whatever reason.
>>>
>>>
>>> Brad Lehman
>>>
>>
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
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> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>
>

Charles Lucy
lucy@lucytune.com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

You can configure your subscription by sending an empty email to one
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