Yahoo Tuning Groups Ultimate Backup tuning Diminished 7th and augmented triad

126/125-planar

Tuning of 2, 3, 5, and 7 in cents
[1200., 1899.984322, 2789.269735, 3367.840561]

Mapping
[<1, 0, 0, -1], <0, 1, 0, -2], <0, 0, 1, 3]]

Diminished 7th chord
[0, 310.714587, 621.429174, 932.143761]

225/224-planar

Tuning of 2, 3, 5, and 7 in cents
[1200., 1899.812912, 2784.171625, 3367.969074]

Mapping
[<1, 0, 0, -5], <0, 1, 0, 2], <0, 0, 1, 2]]

Augmented triad
[0, 384.171625, 768.343250]

Carl is invited to experiment.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> 225/224-planar
>
> Tuning of 2, 3, 5, and 7 in cents
> [1200., 1899.812912, 2784.171625, 3367.969074]
>
> Mapping
> [<1, 0, 0, -5], <0, 1, 0, 2], <0, 0, 1, 2]]
>
> Augmented triad
> [0, 384.171625, 768.343250]

Presumably the idea is that the outer interval approximates 14:9,
since 5/4*5/4/(14/9) = 225/224. Carl, I included this chord and its
inversions, all capped off with an outer octave, in the big blackjack
tetrad list I posted for Joseph Pehrson, using the name "augmagic" if
I recall correctly.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > Presumably the idea is that the outer interval approximates 14:9,
> > since 5/4*5/4/(14/9) = 225/224. Carl, I included this chord and
its
> > inversions, all capped off with an outer octave, in the big
> blackjack
> > tetrad list I posted for Joseph Pehrson, using the
name "augmagic"
> if
> > I recall correctly.
>
> If 5/4-5/4-9/7 in 225/224 is augmagic, what is 6/5-6/5-6/5-7/6 in
> 126/125?

Oh. Actually, I wouldn't have called that any kind of "magic" chord
at all, since the 25:18 isn't approximating a consonance (unless you
assume 25-limit -- fuggeddabouddit!).

🔗Carl Lumma <ekin@lumma.org>

>126/125-planar
>
>Tuning of 2, 3, 5, and 7 in cents
>[1200., 1899.984322, 2789.269735, 3367.840561]
>
>Mapping
>[<1, 0, 0, -1], <0, 1, 0, -2], <0, 0, 1, 3]]
>
>Diminished 7th chord
>[0, 310.714587, 621.429174, 932.143761]

http://lumma.org/tuning/planar-lower.mid

Now we're cookin' with gas!

compared to:
http://lumma.org/tuning/300-lower.mid (12-tET)
http://lumam.org/tuning/6-5-lower.mid (5-prime-limit JI)

and:
http://lumma.org/tuning/10-12-14-17-lower.mid
http://lumma.org/tuning/9-11-13-15-lower.mid

I'll have to do more listening, but it seems the planar
version has an ambiguity the JI versions lack, and it
seems noticeably smoother than the 12-equal or
chain-of-6/5s chord (though the chain-of-6/5s has a
nice 'pinch' to my ear).

>225/224-planar
>
>Tuning of 2, 3, 5, and 7 in cents
>[1200., 1899.812912, 2784.171625, 3367.969074]
>
>Mapping
>[<1, 0, 0, -5], <0, 1, 0, 2], <0, 0, 1, 2]]
>
>Augmented triad
>[0, 384.171625, 768.343250]

http://lumma.org/tuning/aug-planar.mid

compared to:
http://lumma.org/tuning/7-9-11.mid
http://lumma.org/tuning/8-10-13.mid
http://lumma.org/tuning/12-15-19.mid
http://lumma.org/tuning/16-20-25.mid

Here, the 7-9-11 and 8-10-13 are clearly not aug
triads in the traditional sense, though they do
rock the house.

Among traditional aug sounds, the 12-15-19 sounds
best to me, while the planar and 16-20-25 versions
are nearly indistinguishable.

-Carl

🔗Paul Erlich <paul@stretch-music.com>

'This page cannot be found'

> I'll have to do more listening, but it seems the planar
> version has an ambiguity the JI versions lack, and it
> seems noticeably smoother than the 12-equal or
> chain-of-6/5s chord

It's also quite similar to what you'd get in 31-equal or 1/4-comma
meantone.

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <paul@stretch-music.com>

🔗Carl Lumma <ekin@lumma.org>

At 03:11 PM 12/3/2003, you wrote:
>--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >126/125-planar
>> >
>> >Tuning of 2, 3, 5, and 7 in cents
>> >[1200., 1899.984322, 2789.269735, 3367.840561]
>> >
>> >Mapping
>> >[<1, 0, 0, -1], <0, 1, 0, -2], <0, 0, 1, 3]]
>> >
>> >Diminished 7th chord
>> >[0, 310.714587, 621.429174, 932.143761]
>>
>> http://lumma.org/tuning/planar-lower.mid
>>
>> Now we're cookin' with gas!
>
>'This page cannot be found'

Ackgysh! Sorry, it's all fixed now.

-Carl

🔗Paul Erlich <paul@stretch-music.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> If 5/4-5/4-9/7 in 225/224 is augmagic, what is 6/5-6/5-6/5-7/6
> in
> > >> 126/125?
> > >
> > >Oh. Actually, I wouldn't have called that any kind of "magic"
> chord
> > >at all, since the 25:18 isn't approximating a consonance (unless
> you
> > >assume 25-limit -- fuggeddabouddit!).
> >
> > But 25:18 * 126:125 is 7:5.
>
> Oops! Maybe it's time for me to retire? I know I've lauded the
> meantone diminished 7th chord before; don't know how I made this slip.

It's not so much a meantone chord as a specifically 126/125 chord;
using just 126/125, (6/5)^2 ~ 10/7 and (6/5)^3 ~ 12/7, so
6/5-6/5-6/5-7/6 is a 1-6/5-10/7-12/7.

Aside from the 225/224 augmagic chord, another chord which seems magic
in more senses than one is the 8/7-8/7-8/7-8/7-7/6 in 1029/1024. The
reduction of this one gives a 1-8/7-21/16-3/2-12/7 chord. If we have
augmagic (225/224) and dimmagic (126/125) is this one quintmagic
(1029/1024)?

We might note the following:

Augmagic and dimmagic is 225/224 and 126/125, meantone

Augmagic and quintmagic is 225/224 and 1029/1024, miracle

Dimmagic and quintmagic is 126/125 and 1029/1024, quartaminorthirds

126/125, 225/224 and 1029/1024 together is septimal 31-et; that is to
say <31 49 72 87| in the notation we've recently adopted on tuning
math. This magic chord flexibility is certainly a point in favor of 31!

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <paul@stretch-music.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> >
> > > Aside from the 225/224 augmagic chord, another chord which
seems
> > magic
> > > in more senses than one is the 8/7-8/7-8/7-8/7-7/6 in 1029/1024.
> >
> > Hmm . . . maybe I'm about to repeat my goof, but this one really
> > seems to fail. 64:49 * 1029:1024 = 21:16. 21-limit dyads?
>
> Why not? Stick it up a few octaves, say to 21/4, and it might be
> better, of course.

Well, it certainly doesn't qualify as a "magic" chord under my
original definition. But following along, what voicing *are* you
suggesting for the chord?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> >
> > > Aside from the 225/224 augmagic chord, another chord which seems
> > magic
> > > in more senses than one is the 8/7-8/7-8/7-8/7-7/6 in 1029/1024.
> >
> > Hmm . . . maybe I'm about to repeat my goof, but this one really
> > seems to fail. 64:49 * 1029:1024 = 21:16. 21-limit dyads?
>
> Why not? Stick it up a few octaves, say to 21/4, and it might be
> better, of course.

You might also assume 1029/1024 and 833/832 together; then
(8/7)^2 ~ 17/13. The tuning for this fits together with 1029/1024
pretty well. It's also a possible 13/10, which may make you happier
limitwise.

I don't know what harmonic entropy has to say about this business, but
I'd be interested to hear; we have 17/13 = (13/10)(170/169), which is
a difference of only ten cents anyway.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >126/125-planar
> >
> >Tuning of 2, 3, 5, and 7 in cents
> >[1200., 1899.984322, 2789.269735, 3367.840561]
> >
> >Mapping
> >[<1, 0, 0, -1], <0, 1, 0, -2], <0, 0, 1, 3]]
> >
> >Diminished 7th chord
> >[0, 310.714587, 621.429174, 932.143761]
>
> http://lumma.org/tuning/planar-lower.mid
>
> Now we're cookin' with gas!
>
> compared to:
> http://lumma.org/tuning/300-lower.mid (12-tET)
> http://lumam.org/tuning/6-5-lower.mid (5-prime-limit JI)
>
> and:
> http://lumma.org/tuning/10-12-14-17-lower.mid
> http://lumma.org/tuning/9-11-13-15-lower.mid
>
>
> I'll have to do more listening, but it seems the planar
> version has an ambiguity the JI versions lack, and it
> seems noticeably smoother than the 12-equal or
> chain-of-6/5s chord (though the chain-of-6/5s has a
> nice 'pinch' to my ear).

I think the 126/125-planar version is best, though my old favorite
from many years ago 10-12-14-17 has something to be said for it also.
9-11-13-15 seems harsh, and 6-5-lower just doesn't sound in tune--
your pinch seems overdone to me.

> http://lumma.org/tuning/aug-planar.mid
>
> compared to:
> http://lumma.org/tuning/7-9-11.mid
> http://lumma.org/tuning/8-10-13.mid
> http://lumma.org/tuning/12-15-19.mid
> http://lumma.org/tuning/16-20-25.mid
>
>
> Here, the 7-9-11 and 8-10-13 are clearly not aug
> triads in the traditional sense, though they do
> rock the house.

I like all of them! Maybe I just dig triads.

> Among traditional aug sounds, the 12-15-19 sounds
> best to me, while the planar and 16-20-25 versions
> are nearly indistinguishable.

I thought 225/224-planar was smoother, but
there was a nice JI clarity to 16-20-25. Better than either
were the 7-9-11 and 8-10-13 chords, though they sounded even
less like an augmented triad than 16-20-25. The most augmented
sounding was certainly 225/224-planar.

🔗Paul Erlich <paul@stretch-music.com>

A pretty picture to ponder:

/tuning/files/dyadic/margo.gif

> > > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> > >
> > > > Aside from the 225/224 augmagic chord, another chord which
seems
> > > magic
> > > > in more senses than one is the 8/7-8/7-8/7-8/7-7/6 in
1029/1024.
> > >
> > > Hmm . . . maybe I'm about to repeat my goof, but this one
really
> > > seems to fail. 64:49 * 1029:1024 = 21:16. 21-limit dyads?
> >
> > Why not? Stick it up a few octaves, say to 21/4, and it might be
> > better, of course.
>
> You might also assume 1029/1024 and 833/832 together; then
> (8/7)^2 ~ 17/13.

That's the most complex interval I was able to tune by ear with
sawtooth waves, back when I did that experiment. But in
this "quintmagic(?)" context, the virtual fundamentals, if any, are
fighting against this interpretation -- unless you also assume

15/13 ~ 8/7 ~ 17/15 . . .

Which then implies which tuning system?

Thanks for playing Gene, you're fun to play with.

> The tuning for this fits together with 1029/1024
> pretty well. It's also a possible 13/10, which may make you happier
> limitwise.

What's our list of possibly necessary tuning systems now?

> I don't know what harmonic entropy has to say about this business,

for dyads, at least, harmonic entropy doesn't seem to give local
minima for n*d<106, unless exceptional hearing is assumed. Tuning
17/13 by ear was a matter of beats, not harmonic entropy (which is
presumably more important when a piece of music is going by).

> but
> I'd be interested to hear; we have 17/13 = (13/10)(170/169), which
is
> a difference of only ten cents anyway.

That's the problem, in a way -- and I don't think any larger otonal
constructs are going to help the situation given the chord in
question:

Unless you go Vos* (as in margo.gif), H.E. says the closer other
simple ratios are to the ratio in question, the harder it will be to
survive with its identity intact; and if 13/10 does squeak by in JI
(which would seem to require exceptional hearing), at 17/13 you have
pretty much the maximum ambiguity between 13/10 and 4/3, and no local
minimum of harmonic entropy. You certainly wouldn't see both of these
ratios in a single harmonic entropy well.

*Meaning, assume hearing errors are distributed like exp(-|d|)
instead of exp(-d^2/2). margo.gif suggests that this could result in
narrow local minima for both 13/10 and 17/13. 10 cents is a huge leap
on this landscape.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> That's the most complex interval I was able to tune by ear with
> sawtooth waves, back when I did that experiment. But in
> this "quintmagic(?)" context, the virtual fundamentals, if any, are
> fighting against this interpretation -- unless you also assume
>
> 15/13 ~ 8/7 ~ 17/15 . . .
>
> Which then implies which tuning system?

The 17-limit system with commas 105/104 and 120/119. One one of the
many linear temperaments this can be extended to is a goofy 17-limit
version of miracle, with mapping
[<1 1 3 3 2 4 4|, <0 6 -7 -2 15 -3 1|].

The 72 val covering this is <72 114 167 202 249 267 295|.
There doesn't seem to be much to recommed the mappings for 13 and 17
beyond the fact that they make both 105/104 and 120/119 into commas;
the "standard" mapping, with everything flat, makes more sense and
presumably is optimal.

🔗Carl Lumma <ekin@lumma.org>

All the files for this thread are now conveniently located
at:

http://lumma.org/tuning/sss.zip

-Carl

>>Diminished 7th chord
>>[0, 310.714587, 621.429174, 932.143761]
>
>http://lumma.org/tuning/planar-lower.mid
>
>Now we're cookin' with gas!
>
>compared to:
>http://lumma.org/tuning/300-lower.mid (12-tET)
>http://lumam.org/tuning/6-5-lower.mid (5-prime-limit JI)
>
>and:
>http://lumma.org/tuning/10-12-14-17-lower.mid
>http://lumma.org/tuning/9-11-13-15-lower.mid
>
>I'll have to do more listening, but it seems the planar
>version has an ambiguity the JI versions lack, and it
>seems noticeably smoother than the 12-equal or
>chain-of-6/5s chord (though the chain-of-6/5s has a
>nice 'pinch' to my ear).
>
>>225/224-planar
>>
>>Tuning of 2, 3, 5, and 7 in cents
>>[1200., 1899.812912, 2784.171625, 3367.969074]
>>
>>Mapping
>>[<1, 0, 0, -5], <0, 1, 0, 2], <0, 0, 1, 2]]
>>
>>Augmented triad
>>[0, 384.171625, 768.343250]
>
>http://lumma.org/tuning/aug-planar.mid
>
>compared to:
>http://lumma.org/tuning/7-9-11.mid
>http://lumma.org/tuning/8-10-13.mid
>http://lumma.org/tuning/12-15-19.mid
>http://lumma.org/tuning/16-20-25.mid
>
>Here, the 7-9-11 and 8-10-13 are clearly not aug
>triads in the traditional sense, though they do
>rock the house.
>
>Among traditional aug sounds, the 12-15-19 sounds
>best to me, while the planar and 16-20-25 versions
>are nearly indistinguishable.
>
>-Carl