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.