Yahoo Tuning Groups Ultimate Backup tuning Speaking of 5:7:9 harmonies...

Couldn't find the post just now where Carl mentioned Pierce's octave-repeating JI scale based on 5:7:9 triads, but it looked like something doable in 19-EDO--a MODMOS of Sensi[8], IIRC. Well, I totally just stumbled on a scale in 19-EDO that's even better than that one--it really is a 5:7:9 analogue to meantone. It's a 7-note scale generated by a slightly sharp 10/9 (or flat 9/5), and it gives 6 consonant triads--3 otonal, 3 utonal, each on different roots, and even better, it puts 7/5 and 9/7 into the same interval class (a 4th), so you can do parallel harmony in 4ths that switches from otonal to utonal just like meantone. The only difference is how the chords are laid out in the scale: in the 3 3 3 3 3 1 3 mode, it's o-o-o-u-u-u-aug-(o).

If you look at the scale in terms of approximate ratios, it looks like this:
1/1
10/9 (or 9/8?)
5/4
7/5
14/9
7/4
9/5
2/1

So, lots of consonant dyads formed with the root!

I've played around with it a bit and it sounds good--the tempering in 19-EDO is mild enough that the chords sound smooth, even played low. Looks like you even get two full 4:5:7:9 tetrads, too, so if you took it up to the 13-note MOS, you'd get a lot more. 25-EDO is a better tuning, as is (of course) 31-EDO. Has this temperament been named? I'm gonna guess the important commas here are 5103/5000 and...81/80?

-Igs

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 1:44 AM, cityoftheasleep
<igliashon@...> wrote:
>
> Couldn't find the post just now where Carl mentioned Pierce's octave-repeating JI scale based on 5:7:9 triads, but it looked like something doable in 19-EDO--a MODMOS of Sensi[8], IIRC. Well, I totally just stumbled on a scale in 19-EDO that's even better than that one--it really is a 5:7:9 analogue to meantone. It's a 7-note scale generated by a slightly sharp 10/9 (or flat 9/5), and it gives 6 consonant triads--3 otonal, 3 utonal, each on different roots, and even better, it puts 7/5 and 9/7 into the same interval class (a 4th), so you can do parallel harmony in 4ths that switches from otonal to utonal just like meantone. The only difference is how the chords are laid out in the scale: in the 3 3 3 3 3 1 3 mode, it's o-o-o-u-u-u-aug-(o).

I was playing around with this one the other day in 13-tet. The 13-tet
version is truly magical, although it makes things line up differently
than the ratios you have listed.

> I've played around with it a bit and it sounds good--the tempering in 19-EDO is mild enough that the chords sound smooth, even played low. Looks like you even get two full 4:5:7:9 tetrads, too, so if you took it up to the 13-note MOS, you'd get a lot more. 25-EDO is a better tuning, as is (of course) 31-EDO. Has this temperament been named? I'm gonna guess the important commas here are 5103/5000 and...81/80?

This would end up being in the 2.5.7.9 subgroup - there's no 3/2
involved, and the generator is 9/8, two of which make a 3/2. So 81/80
still vanishes. If we're saying that four of those brings you to 7/5,
then 225/224 also vanishes, so this temperament is basically the
2.5.7.9 version of septimal meantone. Or you could think of this as a
MODMOS of meantone - C D E F# G# A# Bb C, which is lydian #5 #6 b7.

This brings me to another point, though, that I've been thinking much
about - this is another MOS of meantone, really. It just happens to be
formed by using 2x the generator instead of the normal 1x the
generator. A similarly awesome scale occurs if you use double the
generator in porcupine, which is a minor third, three of which get you
to 16/9 or 7/4. Scales like that are well worth looking into.

-Mike

🔗Graham Breed <gbreed@...>

On 23 April 2011 09:51, Mike Battaglia <battaglia01@...> wrote:

> This brings me to another point, though, that I've been thinking much
> about - this is another MOS of meantone, really. It just happens to be
> formed by using 2x the generator instead of the normal 1x the
> generator. A similarly awesome scale occurs if you use double the
> generator in porcupine, which is a minor third, three of which get you
> to 16/9 or 7/4. Scales like that are well worth looking into.

Slendric has double the generator of miracle, semaphore has double the
generator of negri. Semaphore and mohajira each double their
generator to get meantone, as well.

Graham

🔗Carl Lumma <carl@...>

--- "cityoftheasleep" <igliashon@...> wrote:
>
> Couldn't find the post just now where Carl mentioned Pierce's
> octave-repeating JI scale based on 5:7:9 triads,

Bohlen's. Bohlen was the real innovator with all of this
stuff.

/tuning/topicId_98256.html#98355

> It's a 7-note scale generated by a slightly sharp 10/9
> (or flat 9/5), and it gives 6 consonant triads--3 otonal,
> 3 utonal, each on different roots, and even better,
> it puts 7/5 and 9/7 into the same interval class (a 4th),
> so you can do parallel harmony in 4ths that switches from
> otonal to utonal just like meantone. The only difference
> is how the chords are laid out in the scale: in
> the 3 3 3 3 3 1 3 mode, it's o-o-o-u-u-u-aug-(o).
> If you look at the scale in terms of approximate ratios,
> it looks like this:
> 1/1
> 10/9 (or 9/8?)
> 5/4
> 7/5
> 14/9
> 7/4
> 9/5
> 2/1

Neat scale. Care to name it?

> I've played around with it a bit and it sounds good--the
> tempering in 19-EDO is mild enough that the chords sound
> smooth, even played low. Looks like you even get two full
> 4:5:7:9 tetrads, too, so if you took it up to the 13-note
> MOS, you'd get a lot more. 25-EDO is a better tuning, as
> is (of course) 31-EDO. Has this temperament been named?

I call it septimal meantone.

> I'm gonna guess the important commas here are 5103/5000
> and...81/80?

Commas are 81/80 and 126/125. Ideal ET is 31, and Graham's
machine suggests an interesting stretched 7-ET too.

-Carl

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Bohlen's. Bohlen was the real innovator with all of this
> stuff.
>
> /tuning/topicId_98256.html#98355

Sorry, thanks for the refresher.

> Neat scale. Care to name it?

Is the name "minortone" taken?

> I call it septimal meantone.

Not in 25-EDO, it's not. I like thinking of it as 2.5.7.9'. Yes, it shows up everywhere you find a septimal meantone, but it shows up in other places too, suggesting something else happening.

> > I'm gonna guess the important commas here are 5103/5000
> > and...81/80?
>
> Commas are 81/80 and 126/125. Ideal ET is 31, and Graham's
> machine suggests an interesting stretched 7-ET too.

Why not 5103/5000? That's the difference between three 10/9's and 7/5, right?

-Igs

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> This would end up being in the 2.5.7.9 subgroup - there's no 3/2
> involved, and the generator is 9/8, two of which make a 3/2.

Two 9/8's make a 3/2? I think you're ready for sleep, Mike.

>If we're saying that four of those brings you to 7/5,

Nope, we're saying three. Three 10/9's (or 9/8's, I guess) gets you to a 7/5 here.

> Or you could think of this as a
> MODMOS of meantone - C D E F# G# A# Bb C, which is lydian #5 #6 b7.

That had occurred to me as I was playing it.

Except it also occurs in non-Meantone temperaments, like 25-EDO, 32-EDO, and 37-EDO. So what's the deal?

> A similarly awesome scale occurs if you use double the
> generator in porcupine, which is a minor third, three of which get you
> to 16/9 or 7/4.

Isn't that "amity" or something?

-Igs

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 3:10 AM, cityoftheasleep
<igliashon@...> wrote:
>
> Two 9/8's make a 3/2? I think you're ready for sleep, Mike.

Haha... whoops. But I still got the commas right though.

> >If we're saying that four of those brings you to 7/5,
>
> Nope, we're saying three. Three 10/9's (or 9/8's, I guess) gets you to a 7/5 here.

Haha... whoops again. Maybe I really am ready for sleep. But if two
10/9's gets you to 5/4, then by definition 10/9 is being equated with
9/8, so it's the same thing. If three of those gets you to 7/5, then
225/224 vanishes. I guess as per Carl's last message the normal basis
for the temperament is 81/80, 126/125, which is equivalent to stating
that it's 81/80 and 225/224, and should also be equivalent to the
commas you listed in your last message as well.

> > This brings me to another point, though, that I've been thinking much
> > about - this is another MOS of meantone, really. It just happens to be
> > formed by using 2x the generator instead of the normal 1x the
> > generator.
>
> Except it also occurs in non-Meantone temperaments, like 25-EDO, 32-EDO, and 37-EDO. So what's the deal?

You have to map them as 2.3.5.7.9' temperaments, where 9 is read "9
prime." That is, there end up being two mappings for 9 - the direct,
"prime" 9, which you should use in chords like 8:9:10:11 or something,
and then the composite 9, which you should use things like major 9
chords (or just use your discretion in general). 25-EDO doesn't
eliminate 81/80, but it eliminates 81'/80. So when you say that 10/9
is the generator, you actually mean 10/9', because in 25-EDO the
actual 10/9 you'd get by mapping 3 and 5 and going from there isn't
the one you're talking about.

> > A similarly awesome scale occurs if you use double the
> > generator in porcupine, which is a minor third, three of which get you
> > to 16/9 or 7/4.
>
> Isn't that "amity" or something?

If three of them get you to 16/9, that's just normal porcupine,
because the comma that ends up vanishing is 250/243. You just end up
with the 2.5.9 version of porcupine, similar to how your temperament
gets you to the 2.5.9 version of meantone. But the resulting 3L4s
scale is awesome - it sounds like diminished[8], except cut off at 7
notes. It's basically the same thing as kleismic[7], but if you're in
porcupine temperament then there's one 4:7:9:11 chord in the scale, so
playing the mode where that's the root makes the whole thing sound
"otonal" and badass in general.

If we're saying that three of them get you to 7/4, then 875/864
vanishes, and you're in a rank-3 temperament apparently called
"supermagic," and one of its rank-2 children is porcupine.

-Mike

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > This would end up being in the 2.5.7.9 subgroup - there's no 3/2
> > involved, and the generator is 9/8, two of which make a 3/2.
>
> Two 9/8's make a 3/2? I think you're ready for sleep, Mike.
>
> >If we're saying that four of those brings you to 7/5,
>
> Nope, we're saying three. Three 10/9's (or 9/8's, I guess) gets you to a 7/5 here.

I'm not feeling so hot because of all this post-surgery medication, so I'm not following this as well as I might, but it sure sounds like tutone temperament:

http://xenharmonic.wikispaces.com/Chromatic+pairs#Tutone

I came up with that one some while back, before the recent subgroup temperament explosion.

🔗cityoftheasleep <igliashon@...>

Also, I was thinking last night while falling asleep that the 19-TET version has the advantage that it turns 1/(5:7:9) into what is basically 10:13:18, which is much simpler than 35:45:63 (what 1/(5:7:9) would be in harmonics), and since 11 isn't very well in-tune in 19, it might be sensible to treat this as a 2.9.5.7.13 subgroup instead. Not sure what the comma would be, though.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > http://xenharmonic.wikispaces.com/Chromatic+pairs#Tutone
> >
> > I came up with that one some while back, before the recent subgroup temperament
> > explosion.
>
> That looks like the one. I figured it had to have been found, it's too good and too obvious. I like the name, too.
>
> -Igs
>

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 1:13 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Also, I was thinking last night while falling asleep that the 19-TET version has the advantage that it turns 1/(5:7:9) into what is basically 10:13:18, which is much simpler than 35:45:63 (what 1/(5:7:9) would be in harmonics), and since 11 isn't very well in-tune in 19, it might be sensible to treat this as a 2.9.5.7.13 subgroup instead. Not sure what the comma would be, though.

That would be the infamous 91/90, which equates 9/7 with 13/10. We
should probably give this comma a name, as it keeps popping up. It
also turns 1/(6:7:9) into 10:13:15.

-Mike

🔗genewardsmith <genewardsmith@...>

You might compare it to this:

http://xenharmonic.wikispaces.com/Chromatic+pairs#Baldy

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> That would be the infamous 91/90, which equates 9/7 with 13/10. We
> should probably give this comma a name, as it keeps popping up. It
> also turns 1/(6:7:9) into 10:13:15.

So I think this temperament is really 81/80, 126/125, and 91/90; at least, that seems to me to be the most psychoacoustically-valid interpretation of what 19 is doing with it. If it's not named, how about "narrowtone"? Or maybe "deutone", a portmanteau of "deutsch" and "tone" that phonetically reflects the relationship to tutone, as well as obliquely referencing that Heinz Bohlen (a German) indirectly inspired its discovery?

As for the comma...geez, I dunno, but I agree it seems like an important one. It works wonders in 19 and 27-EDO, anyway.

-Igs

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 2:45 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > That would be the infamous 91/90, which equates 9/7 with 13/10. We
> > should probably give this comma a name, as it keeps popping up. It
> > also turns 1/(6:7:9) into 10:13:15.
>
> So I think this temperament is really 81/80, 126/125, and 91/90; at least, that seems to me to be the most psychoacoustically-valid interpretation of what 19 is doing with it. If it's not named, how about "narrowtone"? Or maybe "deutone", a portmanteau of "deutsch" and "tone" that phonetically reflects the relationship to tutone, as well as obliquely referencing that Heinz Bohlen (a German) indirectly inspired its discovery?

I like that, very clever. Deutone. So this is in the 2.7.9.13 subgroup?

> As for the comma...geez, I dunno, but I agree it seems like an important one. It works wonders in 19 and 27-EDO, anyway.

27-EDO does some good stuff with it, although its greatest strength
seems to lie in tunings where 64/63 doesn't vanish, so you don't have
really sharp fifths. 46-EDO does this comma right, as Gene pointed out
some time ago. It can lead to very pure harmonies; the 6:7:9 triads in
46-EDO are really good, and then inversely, the 10:13:15 triads are
also really good. 24-EDO eliminates it as well, although the 9/7's
tend on the sharp side.

-Mike

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I like that, very clever. Deutone. So this is in the 2.7.9.13
> subgroup?

I like how fast things go here, but Igs' scale doesn't
use 11 or 13. At least, we need to check the notion that
1/(5:7:9) is approximating that 13-limit chord rather than
the 9-limit utonality or something else. I've got to leave
for a conference but I figured I better say something before
a whole new raft of temperaments are christened in sin.

-Carl

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 4:02 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I like that, very clever. Deutone. So this is in the 2.7.9.13
> > subgroup?
>
> I like how fast things go here, but Igs' scale doesn't
> use 11 or 13.

91/90 involves 13, but 2.7.9.13 doesn't involve 11 anyway.

> At least, we need to check the notion that
> 1/(5:7:9) is approximating that 13-limit chord rather than
> the 9-limit utonality or something else.

Wouldn't the two just be different mappings?

> I've got to leave
> for a conference but I figured I better say something before
> a whole new raft of temperaments are christened in sin.

Oh, so you're the religious type now, are you?

-Mike

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 91/90 involves 13, but 2.7.9.13 doesn't involve 11 anyway

Tutone involves 11, Deutone involves 13, and Igs' scale
involves neither.

> > At least, we need to check the notion that
> > 1/(5:7:9) is approximating that 13-limit chord rather than
> > the 9-limit utonality or something else.
>
> Wouldn't the two just be different mappings?

No. -Carl

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 4:08 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > 91/90 involves 13, but 2.7.9.13 doesn't involve 11 anyway
>
> Tutone involves 11, Deutone involves 13, and Igs' scale
> involves neither.

I would normally agree, but when you talk about things approximating
the 9-limit utonality, as if they were stable attractors of the
system, I have to jump off the train.

> > > At least, we need to check the notion that
> > > 1/(5:7:9) is approximating that 13-limit chord rather than
> > > the 9-limit utonality or something else.
> >
> > Wouldn't the two just be different mappings?
>
> No. -Carl

??? You can come up with a mapping for either one of those possibilities.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

🔗Carl Lumma <carl@...>

--- Mike Battaglia <battaglia01@...> wrote:

> I would normally agree, but when you talk about things
> approximating the 9-limit utonality, as if they were stable
> attractors of the system, I have to jump off the train.

Certain 9-limit utonal chords are stable attractors.
Meanwhile, it's far from clear 10:13:18 means anything.

> > > Wouldn't the two just be different mappings?
> >
> > No. -Carl
>
> ??? You can come up with a mapping for either one of those
> possibilities.

Igs specified the mapping. The question is what chord
it approximates.

-Carl

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> I like how fast things go here, but Igs' scale doesn't
> use 11 or 13. At least, we need to check the notion that
> 1/(5:7:9) is approximating that 13-limit chord rather than
> the 9-limit utonality or something else.

For a chord of 0-442-1010 cents, 10:13:18 is the simplest chord in spitting distance. Justly-tuned it would be 0-454-1017. The 9-limit utonality is 35:45:63 when converted to overtones, and Justly-tuned it would be 0-435-1017. I'd strongly suspect that 3HE would tell us the 9-limit utonality is itself approximating 10:13:18. So what's the problem?

-Igs

🔗cityoftheasleep <igliashon@...>

-Igs

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 4:32 PM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > I would normally agree, but when you talk about things
> > approximating the 9-limit utonality, as if they were stable
> > attractors of the system, I have to jump off the train.
>
> Certain 9-limit utonal chords are stable attractors.
> Meanwhile, it's far from clear 10:13:18 means anything.

I hear 35:45:63 and 10:13:18 as being different. 35:45:63 sounds like
a really sharp 4:5:7, but the 7/9 on the bottom is pretty strong in
and of itself. 10:13:18 sounds different - I don't know if it's more
concordant, necessarily, but the 10:13 is sharp enough to sound less
like 5/4 and more like something new.

Likewise, I hear 10:13:15 as more concordant than 14:18:21.

> > > > Wouldn't the two just be different mappings?
> > >
> > > No. -Carl
> >
> > ??? You can come up with a mapping for either one of those
> > possibilities.
>
> Igs specified the mapping. The question is what chord
> it approximates.

Alright. Well, count me out of that discussion, as I never thought
much of the requirement that a dyad must be "resolved" to count in a
mapping. For example, in mohajira, the generator is 11/9. You can say
that it's not really 11/9 because when you play it, it sounds
ambiguous between 5/4 and 6/5 and doesn't actually sound concordant.
Or, you can just say that that's what 11/9 itself sounds like in that
context, and map it that way.

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Sat, Apr 23, 2011 at 4:28 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > That would be the infamous 91/90, which equates 9/7 with 13/10. We
> > should probably give this comma a name, as it keeps popping up. It
> > also turns 1/(6:7:9) into 10:13:15.
>
> Since battagliaisma is such a mouthful, what about battlisma?

Mmm - tempting though it may be, I think it would be better to give
this one a poetic name, since it seems to be of such fundamental
importance.

The next low-numbered triad after 4:5:6 with a 3/2 on the outside is
6:7:9, but its inversion can sound extremely discordant. On the other
hand, you also have 10:13:15, which is another standout triad of low
complexity with a fifth on the outside, but its inversion is also
pretty discordant. Since tempering 91/90 connects the two together,
and both of the above problems go away, you end up with a tonal system
that relates probably the two most xenharmonic triads that are in
existence (at least those with 3/2 on the outside).

Even better, if you want to deal with something like 4:7:9:12 instead
of just 6:7:9, to make the whole thing rooted and otonal, then the
utonal inverse of this is a 10:13:15 chord with an added 9, which is
an awesome and really "refreshing" sound. And for the big kicker, you
can link them in complex ways to make chords like 1/1 9/8 13/10 3/2
7/4 2/1 9/4 13/5, which is completely magical (at least in 46-TET).

Lastly, this still produces decently pure harmonies, so it isn't like
you need special timbres to deal with it.

To sum up: it makes supermajor triads awesome, which alone is a feat.
In some ways it's the single most xenharmonic comma I've seen yet,
especially if the goal is to morph into higher-limit music.

Since it looks like 81/80 visually, and it's so xenharmonic, maybe we
should call it the "xentonic comma," which is what it is. I was also
throwing around the idea of calling it the "inversion comma" if people
think xentonic is too cheesy.

All of this is better than the working name I had for the comma, which
was the "gintonic comma."

-Mike

🔗Carl Lumma <carl@...>

--- "cityoftheasleep" <igliashon@...> wrote:

> > At least, we need to check the notion that
> > 1/(5:7:9) is approximating that 13-limit chord rather than
> > the 9-limit utonality or something else.
>
> For a chord of 0-442-1010 cents, 10:13:18 is the simplest
> chord in spitting distance. Justly-tuned it would be
> 0-454-1017. The 9-limit utonality is 35:45:63 when converted
> to overtones, and Justly-tuned it would be 0-435-1017.
> I'd strongly suspect that 3HE would tell us the 9-limit
> utonality is itself approximating 10:13:18. So what's
> the problem? -Igs

The pure tuning is 0_435_1018. It seems to be near a 3HE
maxima with 10:13:18 the highest-probability minimum but
we should ask Steve for the distribution. And note that
3HE results so far are provisional.

19-ET tuning does bring us closer to 10:13:18. I listened
to both pure chords and the 19-ET version. 10:13:18 does
sound more 'locked' and possibly the utonal chord
approximates it. But the utonal chord seems to have more
of half-diminished feel to it ("minorness" maybe) that may
be valuable.

It seems unnecessary to add 13 to the subgroup either way,
but I guess that really comes down to whatever makes more
sense to you.

-Carl

🔗Daniel Nielsen <nielsed@...>

🔗cityoftheasleep <igliashon@...>

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

> The pure tuning is 0_435_1018.

Damn rounding errors.

> It seems to be near a 3HE maxima with 10:13:18 the highest-probability minimum but
> we should ask Steve for the distribution. And note that 3HE results so far are
> provisional.

How rigorously were the 2HE predictions tested before they were accepted?

> 19-ET tuning does bring us closer to 10:13:18. I listened
> to both pure chords and the 19-ET version. 10:13:18 does
> sound more 'locked' and possibly the utonal chord
> approximates it. But the utonal chord seems to have more
> of half-diminished feel to it ("minorness" maybe) that may
> be valuable.

I do believe the 19-EDO version sounds noticeably different than the 31-EDO version, which is probably not approximating 10:13:18. I think I like the 19-EDO version a little better.

> It seems unnecessary to add 13 to the subgroup either way,
> but I guess that really comes down to whatever makes more
> sense to you.

I think the 13 is only important in that in 19-EDO, there are some potential 13-limit implications not found in 31-EDO. 0-3-10, for instance, is pretty likely a 9:10:13, and 0-9-16-26 is pretty likely 5:7:9:13, which you don't really get until the 13-note MOS but it probably makes sense to describe as a feature of the temperament anyway. Or don't you buy the validity of those harmonies?

In any case, adding 13 to the mix makes this temperament more obviously related to 32-EDO than to 31, which I like--hey, Gene, did you catch that? That's another one for the 32-EDO xenwiki article!

-Igs

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> How rigorously were the 2HE predictions tested before they
> were accepted?

Dozens of 2HE computations were done and discussed.

So far only one 3HE computation has been publicized, and has
known flaws.

> I think the 13 is only important in that in 19-EDO, there
> are some potential 13-limit implications not found in 31-EDO.
> 0-3-10, for instance, is pretty likely a 9:10:13, and
> 0-9-16-26 is pretty likely 5:7:9:13, which you don't really
> get until the 13-note MOS but it probably makes sense to
> describe as a feature of the temperament anyway. Or don't
> you buy the validity of those harmonies?

Seems reasonable at least. -Carl

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Have pity on someone who is not feeling well and spell it out for me.

That 19-EDO temperament I've been rambling about? The 2.5.7.9'.13 subgroup with commas 81/80, 91/90, and 126/125? Generator around 190 cents? Working title of "deutone", related to tutone but with a 13 instead of an 11 in the subgroup? Well, it works excellently in 32-EDO, too, with the 187.5-cent generator. So that's another good thing to say about 32-EDO.

I hope you feel better soon!

-Igs

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> That 19-EDO temperament I've been rambling about? The 2.5.7.9'.13 subgroup with commas 81/80, 91/90, and 126/125? Generator around 190 cents? Working title of "deutone", related to tutone but with a 13 instead of an 11 in the subgroup? Well, it works excellently in 32-EDO, too, with the 187.5-cent generator. So that's another good thing to say about 32-EDO.

If you say so. I didn't see why you put a prime on the 9; ignoring that, I get a subgroup POTE (SPOTE) generator of 191.059 cents, and 32 does not appear at all on the list of semiconvergents. 25 and 44 could be used as tunings, however.

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> If you say so. I didn't see why you put a prime on the 9; ignoring that, I get a subgroup >POTE (SPOTE) generator of 191.059 cents, and 32 does not appear at all on the list of >semiconvergents. 25 and 44 could be used as tunings, however.

I put the prime on the 9 to denote that it's not (necessarily) 3*3 in the temperament. It works out to that in 19, but not in 25, which is interesting. Looking at the scale in 32 (sorry for the lack of Scala formatting):

0
187.5
375
562.5
750
937.5
1012.5
1200

I notice that it's got a different mapping for 7 than in 25 or 31, since 5 generators gets you 12/7 instead of 7/4. It's still related to 19, since 19 conflates 7/4 and 12/7. 9/5 and 13/5 are both more in-tune in 32 than in 19...you do still get some good 7-limit harmonies, -5 generators gets a good 7/6, although 7 itself is absurdly complex (not appearing until the full 32-note scale). No idea what the important commas would be here, though.

-Igs

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

0
187.5
375
562.5
750
937.5
1012.5
1200

-Igs

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
>
> > If you say so. I didn't see why you put a prime on the 9; ignoring that, I get a subgroup >POTE (SPOTE) generator of 191.059 cents, and 32 does not appear at all on the list of >semiconvergents. 25 and 44 could be used as tunings, however.
>
> I put the prime on the 9 to denote that it's not (necessarily) 3*3 in the temperament.

It's never 3*3 in the temperament since 3 is not in the temperament.

> I notice that it's got a different mapping for 7 than in 25 or 31, since 5 generators gets you 12/7 instead of 7/4. It's still related to 19, since 19 conflates 7/4 and 12/7. 9/5 and 13/5 are both more in-tune in 32 than in 19...you do still get some good 7-limit harmonies, -5 generators gets a good 7/6, although 7 itself is absurdly complex (not appearing until the full 32-note scale). No idea what the important commas would be here, though.

65/64, 81/80 and 4394/4375 are a comma basis, though the last looks like something of a wanker of a comma.

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗martinsj013 <martinsj@...>

> --- "cityoftheasleep" <igliashon@> wrote:
> > For a chord of 0-442-1010 cents, 10:13:18 is the simplest
> > chord in spitting distance.
--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> The pure tuning [for 35:45:63] is 0_435_1018. It seems to be near a 3HE maximum with 10:13:18 the highest-probability minimum but we should ask Steve for the distribution. And note that 3HE results so far are provisional.

I looked at a grid with 10 cent resolution, with lower and upper interval between 400 and 600 cents. (This includes 5:7:9, 35:45:63 and 10:13:18. I realized too late that I would have done better to have centered it on your 0-442-1010 point i.e. (l,u)=(442,568).)

The 3HE calculation for the nearest grid points all report either 10:13:18 or 14:18:25 as most likely "simple triad". Actually 14:18:25 is slightly nearer than 10:13:18 which is slightly nearer than 35:45:63 (to your 0-442-1010).

But none of them is a local minimum of 3HE; in fact we seem to be at a saddle point, with higher ground to north and south (I mean keeping the lower interval the same and varying the upper interval) and lower ground to east and west (keeping the upper interval the same and varying the lower interval). I have a contour diagram showing this, but it is 500KB - too big to post here? The nearest local minima are at 4:5:7 and 6:8:11 (to west and east, and almost equidistant).

Note I have not stated any distances, nor have I stated the value of the "s" parameter I used - because distances are the one unresolved issue in the 3HE calculation. I am convinced that we have the correct *relative* distances, by using Erlich/Chalmers triad space, but there is debate over a multiplication factor and hence the value of "s".

Igs, I have realised that when I gave some results in message #97225, there was a bug in my code that made the 3HE values too small - if you are relying on those results, let me know and I will redo them.

Steve.

🔗Carl Lumma <carl@...>

--- "martinsj013" <martinsj@...> wrote:

> I looked at a grid with 10 cent resolution, with lower and
> upper interval between 400 and 600 cents. (This includes
> 5:7:9, 35:45:63 and 10:13:18.

Thanks Steve!

> The 3HE calculation for the nearest grid points all report
> either 10:13:18 or 14:18:25 as most likely "simple triad".
> Actually 14:18:25 is slightly nearer than 10:13:18

But the greater area of 10:13:18 should give it the higher
probability, yes?

> I have a contour diagram showing this, but it is 500KB -
> too big to post here?

My algorithm is:

Is it audio data?
..Yes: don't post it here
..No: Is it bitmap image data?
....Yes: don't post it here
....No: Do post it here.

> Note I have not stated any distances, nor have I stated the
> value of the "s" parameter I used - because distances are
> the one unresolved issue in the 3HE calculation.
> I am convinced that we have the correct *relative*
> distances, by using Erlich/Chalmers triad space, but there
> is debate over a multiplication factor and hence the
> value of "s".

In my view, RMS is too hard to visualize and 'perpendicular
distance' is the easiest to visualize. Hence I propose we
use perpendicular distance. But I don't really care which
we choose as much as I wish to heck we'd choose one already.
To that end, I note you still haven't voted!

-Carl

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > [14:18:25 is slightly nearer than 10:13:18 ...]
> But the greater area of 10:13:18 should give it the higher
> probability, yes?

Yes, I think that's what is happening.

> --- "martinsj013" <martinsj@> wrote:
> > I looked at a grid with 10 cent resolution ...
> > [... no minimum at 10:13:18 ...]
> > Note I have not stated any distances, nor have I stated the
> > value of the "s" parameter I used ...

Two refinements to this:
1) grid with 2 cent resolution does show a very shallow local minimum at 10:13:18
2) halving the value of "s" results in a definite local minimum at 10:13:18 - this is not a surprise, as smaller "s" means greater discrimination between heard triads - but does indicate that there is no single answer to the question "what does 3HE tell us?"

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Is it audio data?
> ..Yes: don't post it here
> ..No: Is it bitmap image data?
> ....Yes: don't post it here
> ....No: Do post it here.

It's a .png ...?

Steve M.

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > [14:18:25 is slightly nearer than 10:13:18 ...]
> But the greater area of 10:13:18 should give it the higher
> probability, yes?

Yes, I think that's what is happening.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Is it audio data?
> ..Yes: don't post it here
> ..No: Is it bitmap image data?
> ....Yes: don't post it here
> ....No: Do post it here.

It's a .png ...?

Steve M.

🔗Carl Lumma <carl@...>

--- "martinsj013" <martinsj@...> wrote:

> Two refinements to this:
> 1) grid with 2 cent resolution does show a very shallow local
> minimum at 10:13:18
> 2) halving the value of "s" results in a definite local minimum
> at 10:13:18 - this is not a surprise, as smaller "s" means
> greater discrimination between heard triads - but does indicate
> that there is no single answer to the question "what does 3HE
> tell us?"

s is a free parameter. It must be determined experimentally.

> > Is it audio data?
> > ..Yes: don't post it here
> > ..No: Is it [uncompressed] bitmap image data?
> > ....Yes: don't post it here
> > ....No: Do post it here. <---------
>
> It's a .png ...?

See annotation. :)

-Carl