new dim7 renderings

I went back to MIDI so I could do inversions, but I made the
starting pitch a little lower than before and switched patches
to "clarinet", which sounds better on my system (YMMV).

I collected all of the chords discussed in this thread, and
expressed them as harmonic series segments in the inversion
with no factors of 2 in the largest number. They are...
9-11-13-15
10-12-14-17
15-18-21-25
20-24-28-33
25-30-35-42
30-35-42-49
75-90-108-125
...and of course, 4-tET and Gene's planar tempering.

http://lumma.org/stuff/dim7.zip

Also in this file are the .seq and .scl files I used.

I deleted all previous files pertaining to this thread from
my serever, so the above is file to get.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> 9-11-13-15
> 10-12-14-17
> 15-18-21-25
> 20-24-28-33
> 25-30-35-42
> 30-35-42-49
> 75-90-108-125
> ...and of course, 4-tET and Gene's planar tempering.

I think, perhaps surprisingly, the tempered chords sound distinctly
different. The JI chords still have that locked quality to them. The
most consonat sounding is the 126/125 planar tempered one. The one with
the least character, but not surprisingly, is 4-et, which is pretty
tasteless compared to the others. Despite having some of the same kind
of ambiguity, the planar tempered chord does not have the cardboard
quality of 4-et, and of course does not sound the same in all
inversions.

> > 9-11-13-15
> > 10-12-14-17
> > 15-18-21-25
> > 20-24-28-33
> > 25-30-35-42
> > 30-35-42-49
> > 75-90-108-125
> > ...and of course, 4-tET and Gene's planar tempering.
>
> I think, perhaps surprisingly, the tempered chords sound
> distinctly different. The JI chords still have that locked
> quality to them. The most consonat sounding is the 126/125
> planar tempered one. The one with the least character, but
> not surprisingly, is 4-et, which is pretty tasteless compared
> to the others. Despite having some of the same kind of
> ambiguity, the planar tempered chord does not have the
> cardboard quality of 4-et, and of course does not sound the
> same in all inversions.

I think 25:30:35:42 now sounds different from the planar-
tempered chord because they now begin on different inversions.
Even though all inversions are played, the first one played
influences the others surprisingly much (I should perhaps
install some rests between the chords).

Other than that I mostly agree with what you say. I think
9:11:13:15 is the most consonant chord, but it doesn't sound
much like dim7. After that, I think 10:12:15:17 is probably
most consonant. To be followed by the planar one. I could
see how one might rate it as more consonant than these
other two, though -- depends what you're listening for.
The planar chord has an 'open' quality, whereas the "locked"
sound of the 10:12:15:17 is a bit more involved and pinched.
I agree that the least pleasant of the lot is 4-tET.
Boo!

-Carl

Fine!

However... yet another MIDI-related niggle. If my acoustics are
correct, the clarinet timbre should have only ODD harmonics.

Therefore, the tuning of intervals with an even number in their
ratios is not subject to beating! ... 6/5, 10/7, 12/7 are just not
there. Perhaps not surprising that many chords sound smoother.

This might also account for an odd phenomenon I heard: some tunings
(eg 25-30-35-42) beat only in certain inversions! Since inversion of
an interval means multiplying one frequency by 2, this takes an odd-
odd ratio to an odd-even one and vice versa - and with this timbre,
either one or the other interval beats, but not both. Thus 42/25
doesn't beat but 25/21 does...

Incidentally I hear a strong, rapid beating in 10-12-14-17 'root
position' - maybe this is due to 17/12 sounding as an out-of-tune
7/5, with comma 85/84. (It is a better fit to 10/7, but that is one
of the intervals that do not generate beats when mistuned with
clarinets...) This is one of my gripes against that chord, the other
being 17/14 heard as a 6/5 out by 85/84... both solved by 25-30-35-42
(&inversions), at the expense of the rather less mistuned 25/21.

I do admit 30-35-42-49 is a good deal worse! Scratch that one, unless
you're planning on writing for a horror movie.

~~~T~~~

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> I went back to MIDI so I could do inversions, but I made the
> starting pitch a little lower than before and switched patches
> to "clarinet", which sounds better on my system (YMMV).
>
> I collected all of the chords discussed in this thread, and
> expressed them as harmonic series segments in the inversion
> with no factors of 2 in the largest number. They are...
> 9-11-13-15
> 10-12-14-17
> 15-18-21-25
> 20-24-28-33
> 25-30-35-42
> 30-35-42-49
> 75-90-108-125
> ...and of course, 4-tET and Gene's planar tempering.
>
> http://lumma.org/stuff/dim7.zip
>
> Also in this file are the .seq and .scl files I used.
>
> I deleted all previous files pertaining to this thread from
> my serever, so the above is file to get.
>
> -Carl
>

> Fine!
>
> However... yet another MIDI-related niggle. If my acoustics are
> correct, the clarinet timbre should have only ODD harmonics.

Heh... not on my synth/speakers. Depends on the quality of
your synth.

> Therefore, the tuning of intervals with an even number in their
> ratios is not subject to beating! ... 6/5, 10/7, 12/7 are just
> not there. Perhaps not surprising that many chords sound smoother.

There's much more to this than beating, but I don't even think
this is correct (don't have time to figure it out at the moment).

> Incidentally I hear a strong, rapid beating in 10-12-14-17 'root
> position' - maybe this is due to 17/12 sounding as an out-of-tune
> 7/5, with comma 85/84.

One hears beating because partials fall inside the critical
band, not because one interval is within the field of attraction
of another.
Other types of dissonance can arise from this, but in a 17-limit
tetrad like this I don't think an inner dyad being too close
to something like 7/5 is a problem.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> I think 25:30:35:42 now sounds different from the planar-
> tempered chord because they now begin on different inversions.
> Even though all inversions are played, the first one played
> influences the others surprisingly much (I should perhaps
> install some rests between the chords).

It doesn't have the detuned sound of tempering in the same way that
the planar chord does.

> Other than that I mostly agree with what you say. I think
> 9:11:13:15 is the most consonant chord, but it doesn't sound
> much like dim7. After that, I think 10:12:15:17 is probably
> most consonant. To be followed by the planar one. I could
> see how one might rate it as more consonant than these
> other two, though -- depends what you're listening for.

I thought it had the sweetest sound; the others are harsher. It also
has a detuning shimmer or quiver to it, but that is another issue--I
think.

> The planar chord has an 'open' quality, whereas the "locked"
> sound of the 10:12:15:17 is a bit more involved and pinched.
> I agree that the least pleasant of the lot is 4-tET.
> Boo!

It's not at all harsh, but it just isn't pleasing to my ear. It has a
curiously cardboardy, boring sound to it. It makes me wonder if the
dim7 chord shouldn't be counted as another of the harmonic caualties
of moving to strict 12-et. What's a little strange is that the planar
chord, which has three identical thirds, to my ear completely lacks
this cardboard quality.

<<< Incidentally I hear a strong, rapid beating in 10-12-14-17 'root
position' - maybe this is due to 17/12 sounding as an out-of-tune 7/5, with
comma 85/84. (It is a better fit to 10/7, but that is one of the intervals
that do not generate beats when mistuned with clarinets...) This is one of
my gripes against that chord, the other being 17/14 heard as a 6/5 out by
85/84... both solved by 25-30-35-42 (&inversions), at the expense of the
rather less mistuned 25/21. >>>

When I listen to my tuned MIDI file of this chord ratio, I hear something
down there too, but I think I know what it is. Try pitching the chord
higher, at least an octave above middle C. The beat should now be a note
with a discernable pitch. If it is 3 octaves and a major third below the
bass note of the chord, you are hearing the fundamental. This is a good
sign, for it is a hallmark of the just-tuned chord, and can be heard in just
chords with low ratio values.

Also listen to the 5:6:7:9 half-diminished 7th chord based on the same note.
You should hear a fundamental that is an octave higher than the one you
heard in the diminished 7th.

Billy

> > I think 25:30:35:42 now sounds different from the planar-
> > tempered chord because they now begin on different inversions.
> > Even though all inversions are played, the first one played
> > influences the others surprisingly much (I should perhaps
> > install some rests between the chords).
>
> It doesn't have the detuned sound of tempering in the same way
> that the planar chord does.

I hear that in the latest demo, though I don't remember hearing
it in the wave files. And it seems a subtle thing compared to
the other kinds of differences in this group of chords.

> > Other than that I mostly agree with what you say. I think
> > 9:11:13:15 is the most consonant chord, but it doesn't sound
> > much like dim7. After that, I think 10:12:15:17 is probably
> > most consonant. To be followed by the planar one. I could
> > see how one might rate it as more consonant than these
> > other two, though -- depends what you're listening for.
>
> I thought it had the sweetest sound; the others are harsher. It
> also has a detuning shimmer or quiver to it, but that is another
> issue--I think.

It does have a sweet 5:3-like quality to it, which may not
be wanted in a dim7 chord actually.

> > The planar chord has an 'open' quality, whereas the "locked"
> > sound of the 10:12:15:17 is a bit more involved and pinched.
> > I agree that the least pleasant of the lot is 4-tET.
> > Boo!
>
> It's not at all harsh, but it just isn't pleasing to my ear.
> It has a curiously cardboardy, boring sound to it.

When I first started listening to microtonal music, I described
12-equal chords as cardboardy. The beat rates and relationships
somehow manage to sound uniform no matter what you're playing,
and just fast enough to be truly corrupt the sound. A major
third of 395 cents already sounds much better. It's one of the
reasons I've liked JdL's notion of pain increasing as the square
of error.

> What's a little strange is that the planar chord, which has
> three identical thirds, to my ear completely lacks this
> cardboard quality.

Agree.

-Carl

> > Therefore, the tuning of intervals with an even number in their
> > ratios is not subject to beating! ... 6/5, 10/7, 12/7 are just
> > not there. Perhaps not surprising that many chords sound smoother.
>
> There's much more to this than beating, but I don't even think
> this is correct (don't have time to figure it out at the moment).

Sorry Tom, you're right. Odd harmonics only means no roughness
minima at ratios with an even number in them (thanks, Bill
Sethares).

Bill says measurements of clarinets show they produce "mostly"
odd harmonics. But I still insist the Microsoft synth doesn't
capture this aspect on my machine... and even if it did, I feel
like there's considerable distortion in my laptop speakers
and/or headphone amp.

-Carl