_The_ 31-limit temperament?

Just for the record, since I worked it out while looking at notation
issues, here's what might be the only 31-limit temperament of any
musical interest, and even that interest is extremely doubtful.

It is consistent with 311-ET and 388-ET, the only two
31-limit-consistent ETs less than 1200.

It has an octave period and a generator of

MA optimum 405.14866 c 1.177 c max-abs error
RMS optimum 405.15025 c 0.528 c rms error

The generator is the temperament's approximate enneadecimal major
third (19:24).

The mapping is

prime gens
------------
3 -19
5 75
7 -45
11 -126
13 8
17 98
19 -20
23 -111
29 -33
31 -166

Weighted rms complexity is 109.2 generators.
Max-absolute complexity is 316 generators.

All 108 31-limit ratios are represented uniquely.

--- I wrote:
> Just for the record, since I worked it out while looking at notation
> issues, here's what might be the only 31-limit temperament of any
> musical interest, and even that interest is extremely doubtful.

That should have been "might be the only 31-limit _linear_
temperament".

31-limit rational is 11D (in the sense that meantone is 2D) and George
Secor, as part of the notation effort, has apparently found an 8D
31-limit temperament whose unison vectors are all smaller than 0.5 c,
but I'd like to be sure this is the best we can do.

So here's the challenge:

Find the lowest dimensioned 31-limit temperament that has no unison
vector larger than 0.5 cent. I think the 0.5 c limit must apply to any
possible set of unison vectors for the temperament. Is this a coherent
requirement?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I think the 0.5 c limit must apply to any
> possible set of unison vectors for the temperament. Is this a
coherent
> requirement?

no.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I think the 0.5 c limit must apply to any
> > possible set of unison vectors for the temperament. Is this a
> coherent
> > requirement?
>
> no.

Ok. Forget it. Maybe one set of unison vectors, all less than 0.5 c
will do.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Find the lowest dimensioned 31-limit temperament that has no unison
> vector larger than 0.5 cent.

There will always be 31-limit temperaments for each dimension up to
pi(31)=11 such that they have a basis consisting of commas no larger that half a cent, there will never be one such that all the commas are less than half a cent. This does not seem to be a well-defined question, so I think I'll just go ponder some 31-limit temperaments.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> There will always be 31-limit temperaments for each dimension up to
> pi(31)=11 such that they have a basis consisting of commas no larger that half a cent, there will never be one such that all the commas are less than half a cent.

Well...except for in codimension 1, with one comma!

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > There will always be 31-limit temperaments for each dimension up
to
> > pi(31)=11 such that they have a basis consisting of commas no
larger that half a cent, there will never be one such that all the
commas are less than half a cent.
>
> Well...except for in codimension 1, with one comma!

you got me there!

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Find the lowest dimensioned 31-limit temperament that has no
unison
> > vector larger than 0.5 cent.
>
> There will always be 31-limit temperaments for each dimension up to
> pi(31)=11 such that they have a basis consisting of commas no larger
that half a cent, there will never be one such that all the commas are
less than half a cent. This does not seem to be a well-defined
question, so I think I'll just go ponder some 31-limit temperaments.
>

Thanks for convincing me that I haven't supplied enough constraints.

I think this is what we want:

Find the lowest dimensioned 31-limit temperament having a basis
consisting of commas no larger that half a cent, where the absolute
value of the exponent of each prime in each comma of the basis is no
greater than:

Prime Exponent limit
---------------------
2 unbounded (but because of the other constraints it won't be
bigger than 57)
3 12 (because of Pythagorean-12 based notation)
5 2 (because 25 is in the 31 odd-limit)
7 1
11 1
13 1
17 1
19 1
23 1
29 1
31 1

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Thanks for convincing me that I haven't supplied enough constraints.
>
> I think this is what we want:
>
> Find the lowest dimensioned 31-limit temperament having a basis
> consisting of commas no larger that half a cent, where the absolute
> value of the exponent of each prime in each comma of the basis is
no
> greater than:
>
> Prime Exponent limit
> ---------------------
> 2 unbounded (but because of the other constraints it won't be
> bigger than 57)
> 3 12 (because of Pythagorean-12 based notation)
> 5 2 (because 25 is in the 31 odd-limit)
> 7 1
> 11 1
> 13 1
> 17 1
> 19 1
> 23 1
> 29 1
> 31 1

i don't like this kind of constraint because it makes 11/7 seem as
complex as 77/64.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Thanks for convincing me that I haven't supplied enough
constraints.
> >
> > I think this is what we want:
> >
> > Find the lowest dimensioned 31-limit temperament having a basis
> > consisting of commas no larger that half a cent, where the
absolute
> > value of the exponent of each prime in each comma of the basis is
> no
> > greater than:
> >
> > Prime Exponent limit
> > ---------------------
> > 2 unbounded (but because of the other constraints it won't be
> > bigger than 57)
> > 3 12 (because of Pythagorean-12 based notation)
> > 5 2 (because 25 is in the 31 odd-limit)
> > 7 1
> > 11 1
> > 13 1
> > 17 1
> > 19 1
> > 23 1
> > 29 1
> > 31 1
>
> i don't like this kind of constraint because it makes 11/7 seem as
> complex as 77/64.

Remember that the purpose of this temperament is to make a notation
with a minimum number of symbols (or sagittal flags) that can notate
rational scales so even Johnny Reinhard can't tell the difference, and
notate all ETs below 100-ET and many above it.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > Thanks for convincing me that I haven't supplied enough
> constraints.
> > >
> > > I think this is what we want:
> > >
> > > Find the lowest dimensioned 31-limit temperament having a basis
> > > consisting of commas no larger that half a cent, where the
> absolute
> > > value of the exponent of each prime in each comma of the basis
is
> > no
> > > greater than:
> > >
> > > Prime Exponent limit
> > > ---------------------
> > > 2 unbounded (but because of the other constraints it won't
be
> > > bigger than 57)
> > > 3 12 (because of Pythagorean-12 based notation)
> > > 5 2 (because 25 is in the 31 odd-limit)
> > > 7 1
> > > 11 1
> > > 13 1
> > > 17 1
> > > 19 1
> > > 23 1
> > > 29 1
> > > 31 1
> >
> > i don't like this kind of constraint because it makes 11/7 seem
as
> > complex as 77/64.
>
> Remember that the purpose of this temperament is to make a notation
> with a minimum number of symbols (or sagittal flags) that can
notate
> rational scales so even Johnny Reinhard can't tell the difference,
and
> notate all ETs below 100-ET and many above it.

even so.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > i don't like this kind of constraint because it makes 11/7 seem
> as
> > > complex as 77/64.
> >
> > Remember that the purpose of this temperament is to make a
notation
> > with a minimum number of symbols (or sagittal flags) that can
> notate
> > rational scales so even Johnny Reinhard can't tell the difference,
> and
> > notate all ETs below 100-ET and many above it.
>
> even so.

So what would you suggest?

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> So what would you suggest?

What was wrong with the plan of using 2^a 3^b p commas? I'm not at all clear why you want to abandon it.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > So what would you suggest?
>
> What was wrong with the plan of using 2^a 3^b p commas? I'm not at
all clear why you want to abandon it.

We're not abandoning that plan at all. Sorry I didn't explain. These
sub-half-cent thingies are better thought of, not as commas, but as
schismas, where a schisma is defined (for our purposes here at least)
as a small difference between commas (just as _the_ schisma is the
difference between the syntonic and pythagorean commas).

We want to reduce the number of symbols to less than the number of
primes, if possible, so we're interested in sub-half-cent schismas
which correspond to a very simple relationship between commas like x +
y ~= z, where x,y and z are some of our 2^a*3^b*p commas.

The most striking schisma proposed so far, found by George Secor, is
4095:4096 which says that the 13-comma is the 5-comma plus the
7-comma.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:

> The most striking schisma proposed so far, found by George
Secor, is
> 4095:4096 which says that the 13-comma is the 5-comma
plus the
> 7-comma.

how do you get 13 from 5 and 7?

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> wrote:
>
> > The most striking schisma proposed so far, found by George
> Secor, is
> > 4095:4096 which says that the 13-comma is the 5-comma
> plus the
> > 7-comma.
>
> how do you get 13 from 5 and 7?

5-comma 80:81 21.51 c
7-comma 63:64 27.26 c
5-comma + 7-comma = 48.77 c
13-comma 1024:1053 48.35 c

It doesn't matter for notating ETs less than 100 that the 4095:4096
doesn't always vanish, because one never needs to use all three
notational commas together (5, 7 & 13).

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...>
wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> > wrote:
> >
> > > The most striking schisma proposed so far, found by
George
> > Secor, is
> > > 4095:4096 which says that the 13-comma is the 5-comma
> > plus the
> > > 7-comma.
> >
> > how do you get 13 from 5 and 7?
>
> 5-comma 80:81 21.51 c
> 7-comma 63:64 27.26 c
> 5-comma + 7-comma = 48.77 c
> 13-comma 1024:1053 48.35 c
>
> It doesn't matter for notating ETs less than 100 that the
4095:4096
> doesn't always vanish, because one never needs to use all
three
> notational commas together (5, 7 & 13).

i'm not following.

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...>
> wrote:
> > --- In tuning-math@y..., "emotionaljourney22" <paul@s...>
> wrote:
> > > how do you get 13 from 5 and 7?
> >
> > 5-comma 80:81 21.51 c
> > 7-comma 63:64 27.26 c
> > 5-comma + 7-comma = 48.77 c
> > 13-comma 1024:1053 48.35 c
> >
> > It doesn't matter for notating ETs less than 100 that the
> 4095:4096
> > doesn't always vanish, because one never needs to use all
> three
> > notational commas together (5, 7 & 13).
>
> i'm not following.

I'm not sure what it is you're not following. Do you follow how to get
13 from 5 and 7 (within 0.5 cents)? Please ask a specific question, or
put up a specific objection.