Yahoo Tuning Groups Ultimate Backup tuning Some new 5-limit microtemperaments

Let's call these "syntonic-chromatic equivalence" temperaments:
.
..
...
Mavila: (81/80)^-1 = 25/24
Dicot: (81/80)^0 = 25/24
Porcupine: (81/80)^1 = 25/24
Tetracot: (81/80)^2 = 25/24
Amity: (81/80)^3 = 25/24
...
..
.

Petr's comma pumps indirectly make use of this relation (perhaps
unintentionally, or at least by my analysis), it might as well be put
out in the open. Modulation by 81/80 is an extremely awkward thing if
you're used to meantone; you get the unsettling feeling that you
haven't quite "come back home." However, if this 81/80 "pitch drift"
ends up being tempered equal to another useful pitch in the system, in
this case 25/24, then you can make use of this equivalence such that
the pitch drift becomes "meaningful" in a larger context.

For example, modulation by 81/80 in porcupine is actually just
modulation by 25/24, which you can use musically if you're clever. I
advocated using porcupine this way in my "functional" examples,
although it is a departure from the MOS-based approach that Petr is
also having success exploring. The pattern continues beyond Porcupine
- Tetracot makes it so that two modulations by 81/80 is modulation by
25/24, and Amity makes it so that three modulations is modulation by
25/24.

Temperaments like these are important because they enable us to turn
something unfamiliar (movement by 81/80) into something familiar
(movement by 25/24). It seems obvious to continue the pattern, so
let's see what comes out. Keep in mind that the concept of stacking
81/80 and turning it into something else will lead us to exponentially
larger and larger commas, so there's no way to avoid getting huge
commas - it is best to not think about MOS's here at all and just
modulate around the lattice. So here's what we get:

Gravity: (81/80)^4 = 25/24
5&67 temperament, eliminates 129140163/128000000, so named because the
generator is a "Grave" fifth (or 27/20)
http://x31eq.com/cgi-bin/rt.cgi?ets=65_7&limit=5

Absurdity: (81/80)^5 = 25/24
5&84 temperament, eliminates 10460353203/10240000000, so named because
this is just an absurd temperament. The generator is 81/80 and the
period is 800/729, which is (10/9) / (81/80).
http://x31eq.com/cgi-bin/rt.cgi?ets=7_84&limit=5

This is where I'll stop. Keep in mind here that the approach is to
come up with a "lego block" chord progression that shoots you up by
81/80, and then to just repeat this a few times until you end up at
25/24, at which point you can go back down. So really huge commas and
temperaments thought to be unusably complex might have some use if you
use this approach.

I'll explore (81/80)^n = 16/15 next.

-Mike

🔗lobawad <lobawad@...>

(81/80)^3 is the "naturally occuring" instance of course- the first three you've got there are pretty heavy-duty. I know you also like 34-edo, and this is one of the reasons it's so nice- the tempered "81/80" is wide and works out to half the very good tempered "25/24". Drift, drift, shift back with sultry Romantic semitone.

Using Wuerschmidt temperament, you can do this same kind of thing by using the coincidence of twice the tempered 128/125 giving you the tempered 25/24 (that is, three M3 fall short of the octave by half a chromatic semitone, in that temperament). That would be a "diesis > chroma equivalence).

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Let's call these "syntonic-chromatic equivalence" temperaments:
> .
> ..
> ...
> Mavila: (81/80)^-1 = 25/24
> Dicot: (81/80)^0 = 25/24
> Porcupine: (81/80)^1 = 25/24
> Tetracot: (81/80)^2 = 25/24
> Amity: (81/80)^3 = 25/24
> ...
> ..
> .
>
> Petr's comma pumps indirectly make use of this relation (perhaps
> unintentionally, or at least by my analysis), it might as well be put
> out in the open. Modulation by 81/80 is an extremely awkward thing if
> you're used to meantone; you get the unsettling feeling that you
> haven't quite "come back home." However, if this 81/80 "pitch drift"
> ends up being tempered equal to another useful pitch in the system, in
> this case 25/24, then you can make use of this equivalence such that
> the pitch drift becomes "meaningful" in a larger context.
>
> For example, modulation by 81/80 in porcupine is actually just
> modulation by 25/24, which you can use musically if you're clever. I
> advocated using porcupine this way in my "functional" examples,
> although it is a departure from the MOS-based approach that Petr is
> also having success exploring. The pattern continues beyond Porcupine
> - Tetracot makes it so that two modulations by 81/80 is modulation by
> 25/24, and Amity makes it so that three modulations is modulation by
> 25/24.
>
> Temperaments like these are important because they enable us to turn
> something unfamiliar (movement by 81/80) into something familiar
> (movement by 25/24). It seems obvious to continue the pattern, so
> let's see what comes out. Keep in mind that the concept of stacking
> 81/80 and turning it into something else will lead us to exponentially
> larger and larger commas, so there's no way to avoid getting huge
> commas - it is best to not think about MOS's here at all and just
> modulate around the lattice. So here's what we get:
>
> Gravity: (81/80)^4 = 25/24
> 5&67 temperament, eliminates 129140163/128000000, so named because the
> generator is a "Grave" fifth (or 27/20)
> http://x31eq.com/cgi-bin/rt.cgi?ets=65_7&limit=5
>
> Absurdity: (81/80)^5 = 25/24
> 5&84 temperament, eliminates 10460353203/10240000000, so named because
> this is just an absurd temperament. The generator is 81/80 and the
> period is 800/729, which is (10/9) / (81/80).
> http://x31eq.com/cgi-bin/rt.cgi?ets=7_84&limit=5
>
> This is where I'll stop. Keep in mind here that the approach is to
> come up with a "lego block" chord progression that shoots you up by
> 81/80, and then to just repeat this a few times until you end up at
> 25/24, at which point you can go back down. So really huge commas and
> temperaments thought to be unusably complex might have some use if you
> use this approach.
>
> I'll explore (81/80)^n = 16/15 next.
>
> -Mike
>

🔗lobawad <lobawad@...>

Oh, and in a tuning or temperament in which 36/35 and 25/24 are equivalent (say 41-edo), of course you automatically get this action for ratios of 7 as well. That is, for example, the step from minor third to major third and the step from subminor third to minor third are tempered to the same size, which is twice the step from the major third to the ditone.

--- In tuning@yahoogroups.com, "lobawad" <lobawad@...> wrote:
>
> (81/80)^3 is the "naturally occuring" instance of course- the first three you've got there are pretty heavy-duty. I know you also like 34-edo, and this is one of the reasons it's so nice- the tempered "81/80" is wide and works out to half the very good tempered "25/24". Drift, drift, shift back with sultry Romantic semitone.
>
> Using Wuerschmidt temperament, you can do this same kind of thing by using the coincidence of twice the tempered 128/125 giving you the tempered 25/24 (that is, three M3 fall short of the octave by half a chromatic semitone, in that temperament). That would be a "diesis > chroma equivalence).
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Let's call these "syntonic-chromatic equivalence" temperaments:
> > .
> > ..
> > ...
> > Mavila: (81/80)^-1 = 25/24
> > Dicot: (81/80)^0 = 25/24
> > Porcupine: (81/80)^1 = 25/24
> > Tetracot: (81/80)^2 = 25/24
> > Amity: (81/80)^3 = 25/24
> > ...
> > ..
> > .
> >
> > Petr's comma pumps indirectly make use of this relation (perhaps
> > unintentionally, or at least by my analysis), it might as well be put
> > out in the open. Modulation by 81/80 is an extremely awkward thing if
> > you're used to meantone; you get the unsettling feeling that you
> > haven't quite "come back home." However, if this 81/80 "pitch drift"
> > ends up being tempered equal to another useful pitch in the system, in
> > this case 25/24, then you can make use of this equivalence such that
> > the pitch drift becomes "meaningful" in a larger context.
> >
> > For example, modulation by 81/80 in porcupine is actually just
> > modulation by 25/24, which you can use musically if you're clever. I
> > advocated using porcupine this way in my "functional" examples,
> > although it is a departure from the MOS-based approach that Petr is
> > also having success exploring. The pattern continues beyond Porcupine
> > - Tetracot makes it so that two modulations by 81/80 is modulation by
> > 25/24, and Amity makes it so that three modulations is modulation by
> > 25/24.
> >
> > Temperaments like these are important because they enable us to turn
> > something unfamiliar (movement by 81/80) into something familiar
> > (movement by 25/24). It seems obvious to continue the pattern, so
> > let's see what comes out. Keep in mind that the concept of stacking
> > 81/80 and turning it into something else will lead us to exponentially
> > larger and larger commas, so there's no way to avoid getting huge
> > commas - it is best to not think about MOS's here at all and just
> > modulate around the lattice. So here's what we get:
> >
> > Gravity: (81/80)^4 = 25/24
> > 5&67 temperament, eliminates 129140163/128000000, so named because the
> > generator is a "Grave" fifth (or 27/20)
> > http://x31eq.com/cgi-bin/rt.cgi?ets=65_7&limit=5
> >
> > Absurdity: (81/80)^5 = 25/24
> > 5&84 temperament, eliminates 10460353203/10240000000, so named because
> > this is just an absurd temperament. The generator is 81/80 and the
> > period is 800/729, which is (10/9) / (81/80).
> > http://x31eq.com/cgi-bin/rt.cgi?ets=7_84&limit=5
> >
> > This is where I'll stop. Keep in mind here that the approach is to
> > come up with a "lego block" chord progression that shoots you up by
> > 81/80, and then to just repeat this a few times until you end up at
> > 25/24, at which point you can go back down. So really huge commas and
> > temperaments thought to be unusably complex might have some use if you
> > use this approach.
> >
> > I'll explore (81/80)^n = 16/15 next.
> >
> > -Mike
> >
>

🔗Mike Battaglia <battaglia01@...>

Alright, so then we'll call these "diatonic-chromatic equivalence" temperaments:
.
..
...
Bug: (81/80)^-1 = 16/15 (27/25 vanishes)
Father: (81/80)^0 = 16/15 (16/15 vanishes)
Blackwood: (81/80)^1 = 16/15 (256/243 vanishes)
Superpyth: (81/80)^2 = 16/15 (20480/19683 vanishes)
...
..
.

These might be even more useful than the other ones mentioned, because
modulation by 16/15 is even easier than 25/24. So to continue the
pattern we have:

!!!! - Immunity: (81/80)^3 = 16/15 (1638400/1594323 vanishes) -
Microtempered "Semaphore"
This is a great temperament - I don't know why it's never been
explored before! Great pure harmonies with a beautifully slightly
sharp fifth. Right there in 34-equal.
The generator is the difference between the literal Pythagorean
tritone (as in (9/8)^3) and 5/4, meaning it's 729/640, two of which
get you to 4/3. Looks to me to be more useful than Amity. You have to
ignore the MOS's completely and go right for the comma pumps.
I would suggest "Semapure", but doesn't seem to do this justice.
Perhaps "Immunity," because it's a counterpart to Amity or something
like that. There is definitely some subgroup variant that's in the
Archipelago. Please feel free to suggest a better name.
http://x31eq.com/cgi-bin/rt.cgi?ets=34_5&limit=5

5-limit Rodan: (81/80)^4 = 16/15 (131072000/129140163 vanishes) -
Microtempered "Gamelan"
Also very pure harmonies. The generator is once again 729/640 (!!),
but this time set up so that three of them get you to 3/2. Interesting
how 729/640 keeps coming up. You will also notice that so far, these
temperaments are interestingly microtempered versions of other
temperaments. This is in 41-equal.
http://x31eq.com/cgi-bin/rt.cgi?ets=46_41&limit=5

5-limit Vulture: (81/80)^5 = 16/15 (10485760000/10460353203 vanishes)
Generator this time is the difference between a Pythagorean major 7th
(243/128) and a 5/4, which is 243/160 (or 320/243). Supported by
53-equal.

Gene 5&60 temperament - (81/80)^6 = 16/15 (838860800000/847288609443
vanishes) - Microtempered "Blackwood"
This will be the last one. Just like the last list had one where the
period splits into 7, this time the period splits into 5. The 7-limit
version of this temperament makes more sense and vanishes 245/243 and
16807/16384.

That's it for now. It's interesting that despite that these commas are
absolutely enormous, the 7-limit versions are often much more
digestible. This "immunity" temperament seems like the best of the
lot, a huge gaping hole of lower complexity than the rest below and of
purer harmony than anything above. I'd like to see some comma pumps
that deal with that. I'll come up with some myself once I have more
time.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Fri, May 20, 2011 at 12:05 PM, lobawad <lobawad@...> wrote:
>
> (81/80)^3 is the "naturally occuring" instance of course- the first three you've got there are pretty heavy-duty. I know you also like 34-edo, and this is one of the reasons it's so nice- the tempered "81/80" is wide and works out to half the very good tempered "25/24". Drift, drift, shift back with sultry Romantic semitone.

You mean (81/80)^2, I think, because it looks like you're talking
about tetracot. (81/80)^3 ~= 16/15 also pops up in 34-equal, which
means 81/80 = 128/125, which means 34-equal is a diaschismatic
temperament. The (81/80)^3 = 16/15 one looks like a killer tuning
because it's something of a counterpart to Amity, however three
iterations this time get you to 16/15 which is even easier to deal
with than 25/24. I suggested "Immunity" as a name.

> Using Wuerschmidt temperament, you can do this same kind of thing by using the coincidence of twice the tempered 128/125 giving you the tempered 25/24 (that is, three M3 fall short of the octave by half a chromatic semitone, in that temperament). That would be a "diesis > chroma equivalence).

Ah, that makes a lot of sense. So in Magic, 128/125 = 25/24, and the
chain of major thirds meets 3/1. In Wuerschmidt, (128/125)^2 = 25/24,
and the chain of major thirds meets 6/1. I'm going to venture a guess
that (128/125)^3 = 25/24 means the chain of major thirds meets 12/1...
and upon checking, it does. 50331648/48828125 vanishes there and we
end up with http://x31eq.com/cgi-bin/rt.cgi?ets=3_46&limit=5

I was going to do (81/80)^n equals 128/125 next, which should also
probably give some interesting results. Looks like first you have
diaschismatic, then schismatic, then compton/Aristoxenean/Pythagorean
comma vanishing/etc, then it's this "double schismatic" temperament:
http://x31eq.com/cgi-bin/rt.cgi?ets=12_79&limit=5

Look at the mapping for 5 and you'll see why I say it's double schismatic.

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Let's call these "syntonic-chromatic equivalence" temperaments:
> > .
> > ..
> > ...
> > Mavila: (81/80)^-1 = 25/24
> > Dicot: (81/80)^0 = 25/24
> > Porcupine: (81/80)^1 = 25/24
> > Tetracot: (81/80)^2 = 25/24
> > Amity: (81/80)^3 = 25/24
> > ...
> > ..
> > .
> >
> > Petr's comma pumps indirectly make use of this relation (perhaps
> > unintentionally, or at least by my analysis), it might as well be put
> > out in the open. Modulation by 81/80 is an extremely awkward thing if
> > you're used to meantone; you get the unsettling feeling that you
> > haven't quite "come back home." However, if this 81/80 "pitch drift"
> > ends up being tempered equal to another useful pitch in the system, in
> > this case 25/24, then you can make use of this equivalence such that
> > the pitch drift becomes "meaningful" in a larger context.
> >
> > For example, modulation by 81/80 in porcupine is actually just
> > modulation by 25/24, which you can use musically if you're clever. I
> > advocated using porcupine this way in my "functional" examples,
> > although it is a departure from the MOS-based approach that Petr is
> > also having success exploring. The pattern continues beyond Porcupine
> > - Tetracot makes it so that two modulations by 81/80 is modulation by
> > 25/24, and Amity makes it so that three modulations is modulation by
> > 25/24.
> >
> > Temperaments like these are important because they enable us to turn
> > something unfamiliar (movement by 81/80) into something familiar
> > (movement by 25/24). It seems obvious to continue the pattern, so
> > let's see what comes out. Keep in mind that the concept of stacking
> > 81/80 and turning it into something else will lead us to exponentially
> > larger and larger commas, so there's no way to avoid getting huge
> > commas - it is best to not think about MOS's here at all and just
> > modulate around the lattice. So here's what we get:
> >
> > Gravity: (81/80)^4 = 25/24
> > 5&67 temperament, eliminates 129140163/128000000, so named because the
> > generator is a "Grave" fifth (or 27/20)
> > http://x31eq.com/cgi-bin/rt.cgi?ets=65_7&limit=5
> >
> > Absurdity: (81/80)^5 = 25/24
> > 5&84 temperament, eliminates 10460353203/10240000000, so named because
> > this is just an absurd temperament. The generator is 81/80 and the
> > period is 800/729, which is (10/9) / (81/80).
> > http://x31eq.com/cgi-bin/rt.cgi?ets=7_84&limit=5
> >
> > This is where I'll stop. Keep in mind here that the approach is to
> > come up with a "lego block" chord progression that shoots you up by
> > 81/80, and then to just repeat this a few times until you end up at
> > 25/24, at which point you can go back down. So really huge commas and
> > temperaments thought to be unusably complex might have some use if you
> > use this approach.
> >
> > I'll explore (81/80)^n = 16/15 next.
> >
> > -Mike
> >
>
>

🔗lobawad <lobawad@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, May 20, 2011 at 12:05 PM, lobawad <lobawad@...> wrote:
> >
> > (81/80)^3 is the "naturally occuring" instance of course- the first three you've got there are pretty heavy-duty. I know you also like 34-edo, and this is one of the reasons it's so nice- the tempered "81/80" is wide and works out to half the very good tempered "25/24". Drift, drift, shift back with sultry Romantic semitone.
>
> You mean (81/80)^2, I think, because it looks like you're talking
> about tetracot.

Sorry, I should have seperated the first sentence. (81/80)^3 is the "naturally occuring" in that it's the least amount of temperament
(1600000/1594323).

Then I went on to the more easily accessed (81/80)^2, and as you pointed out, (81/80)^3 is conveniently the major chroma in 34, I hadn't considered that though I've surely used it.

>81/80)^3 ~= 16/15 also pops up in 34-equal, which
> means 81/80 = 128/125, which means 34-equal is a diaschismatic
> temperament. The (81/80)^3 = 16/15 one looks like a killer tuning
> because it's something of a counterpart to Amity, however three
> iterations this time get you to 16/15 which is even easier to deal
> with than 25/24. I suggested "Immunity" as a name.

>
> I was going to do (81/80)^n equals 128/125 next, which should also
> probably give some interesting results.

That should be really interesting and useful. In their natural states, two syntonic commas are nearly identical in size to the major diesis.

🔗Mike Battaglia <battaglia01@...>

On Fri, May 20, 2011 at 1:26 PM, lobawad <lobawad@...> wrote:
>
> > I was going to do (81/80)^n equals 128/125 next, which should also
> > probably give some interesting results.
>
> That should be really interesting and useful. In their natural states, two syntonic commas are nearly identical in size to the major diesis.

Well, I ended up posting the results right there, so there's the end
of that. I could continue past the double schismatic one if anyone's
interested, but I think we'll just get diminishing returns beyond
that. Any other ideas for things we should look at? The (25/24)^n ->
(16/15) one is interesting -> first goes father -> augmented -> magic
-> dicot, and then this interesting and as far as I know undiscovered
temperament for (25/24)^3 = 16/15:

http://x31eq.com/cgi-bin/rt.cgi?ets=26_29&limit=5

Generator is 32/25 (!) and 7 of those gets you to 16/3. Strange I've
never heard of this one before! You can temper the generator equal to
14/11 if you want, which it is almost exactly to.

After that we get this weird temperament for (25/24)^4 = (16/15)

http://x31eq.com/cgi-bin/rt.cgi?ets=3_21bbc&limit=5

and then for (25/24)^5 = (16/15) you get this useless temperament

http://x31eq.com/cgi-bin/rt.cgi?ets=3_28bbcc&limit=5

The generator is a more accurate 5/4 than the actual mapping for 5/4,
so I'd say the winning streak is now over. Any other ideas?

-Mike

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Fri, May 20, 2011 at 2:49 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > !!!! - Immunity: (81/80)^3 = 16/15 (1638400/1594323 vanishes) -
> > Microtempered "Semaphore"
> > This is a great temperament - I don't know why it's never been
> > explored before!
>
> The reason is not hard to find: it's closely associated to 34et, but 34 has 5-limit commas 2048/2024, 15625/15552, 20000/19683 and these are simpler and hence have sucked all the oxygen out of (16/15)/(81/80)^3.

Did you read my rationale for considering larger commas, and how
Petr's method makes use of temperaments like that? Also, how did
people ever discover amity then?

Here's another question: the expression (16/15)/(81/80)^n traces out a
line in TE interval space, right? Except this line doesn't go through
the origin. What shapes do things like these trace out in the dual of
this space?

-Mike

🔗lobawad <lobawad@...>

🔗Mike Battaglia <battaglia01@...>

On Fri, May 20, 2011 at 3:45 PM, lobawad <lobawad@...> wrote:
>
> But looking at commas from the viewpoint of commas not tempered out but tempered "in", that is, in simple relation to each other and to scale steps within the tuning, is a very useful approach. It's certainly something I do because I don't like tempering out 81/80 but I do like working its snaking motion backing into concretely scalar degrees, same goes for the diesis (and 36/35).

That's heavy man. Tempering in. You mean in the sense of creating a
structure whereby 81/80 becomes useful, right? Because in that regard
I'd have to say that Immunity trumps Amity: it turns a few 81/80
shifts into a full-blown 16/15 shift, which is even easier to deal
with than 25/24. Since temperaments like these force us away from
thinking about generators or MOS's at all, the fact that you have this
weird generator doesn't mean anything, as does it not with Amity. The
fact that Amity is a few cents more accurate doesn't mean anything to
me, as I actually like the error of Immunity; it tempers the fifths a
bit sharp so they're in the really bright zone.

-Mike

🔗genewardsmith <genewardsmith@...>

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

> Here's another question: the expression (16/15)/(81/80)^n traces out a
> line in TE interval space, right? Except this line doesn't go through
> the origin. What shapes do things like these trace out in the dual of
> this space?

I don't know what you mean by "what shapes", but in general you can take the map [<7 11 16|, <5 8 12|, <3 5 7|] and transform the 5-limit into the [16/15, 25/24, 81/80] basis. In this basis you can represent any 5-limit comma:

(25/24)/(81/80)^2 = |0 1 -2>
(25/24)/(81/80)^3 = |0 1 -3>
(81/80)^4/(25/24) = |0 -1 4>
(16/15)/(81/80) = |1 0 -1>
(16/15)/(81/80)^2 = |1 0 -2>
(16/15)/(81/80)^3 = |1 0 -3>

etc.

Some familiar 5-limit commas:

81/80 = |0 0 1>
2048/2025 = |1 -1 -1>
15625/15552 = |-1 2 -1>
32805/32768 = |-1 1 2>
393216/390625 = |2 -3 0>

🔗Mike Battaglia <battaglia01@...>

Also, by "what shapes" I meant - lines in one space that go through
the origin turn into dots in the dual space, right? That's what all of
the pictures that Paul's posting on Facebook seem to mean. In this
case, (16/15)/(81/80)^n will trace out a line that doesn't go through
the origin, but goes rather through 16/15. (25/24)/(81/80)^n will
trace out a line that goes through 25/24. (1/1)/(81/80)^n will trace
out a line going through the origin. The latter should turn into a dot
in the dual (or is it the projective?) space. What about the other
ones? Do sequences like the one I've laid out, which encapsulate a lot
of different linear temperaments, trace out hyperbolae in tuning space
or something?

-Mike

On Fri, May 20, 2011 at 4:21 PM, Mike Battaglia <battaglia01@...> wrote:
> On Fri, May 20, 2011 at 4:15 PM, genewardsmith
> <genewardsmith@...> wrote:
>>
>> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>>
>> > Here's another question: the expression (16/15)/(81/80)^n traces out a
>> > line in TE interval space, right? Except this line doesn't go through
>> > the origin. What shapes do things like these trace out in the dual of
>> > this space?
>>
>> I don't know what you mean by "what shapes", but in general you can take the map [<7 11 16|, <5 8 12|, <3 5 7|] and transform the 5-limit into the [16/15, 25/24, 81/80] basis. In this basis you can represent any 5-limit comma:
>
> This is a good idea, this might well be the way to go in
> algorithmically computing these.
>
> -Mike
>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗Petr Parízek <petrparizek2000@...>

Responding to an earlier message.

Mike wrote:

> Perhaps "Immunity," because it's a counterpart to Amity or something
> like that. There is definitely some subgroup variant that's in the
> Archipelago. Please feel free to suggest a better name.

A clear counterpart to amity is not this one but unicorn.
In both unicorn and amity, 5/1 is mapped to -13 generators while the mapping for 3/1 and 5/3 is swapped (i.e. amity uses -5 and -8 generators, respectively, while unicorn has it the other way round).
This means you could happily retune an amity harmonic progression into unicorn or vice versa and, interestingly enough, pitch movements by 6/5 or 3/2 would swap while pitch movements by 5/4 would still be 5/4.
For this reason, if you make a strict comma pump in amity and in unicorn, the first four chords and the last three chords of both progressions are exactly the same. :-)
Anyway, Gene seems to have confirmed unicorn in an older message on Tuning Math, so I'd leave this name unchanged.
How to deal with a name like "immunity", I'm not sure, however.

Petr