My last list allowed in some duplicates; the following list of 81

pairs is from an improved version of the program:

{135/128, 36/35}

{36/35, 200/189}

{405/392, 25/24}

{81/80, 126/125}

{4000/3969, 1728/1715}

{64/63, 245/243}

{25/24, 21/20}

{28/27, 50/49}

{36/35, 225/224}

{160/147, 25/24}

{50/49, 81/80}

{5103/5000, 49/48}

{28/27, 49/48}

{64/63, 126/125}

{1728/1715, 3136/3125}

{128/125, 50/49}

{672/625, 28/27}

{81/80, 3136/3125}

{3200/3087, 28/27}

{1029/1024, 4375/4374}

{245/243, 2048/2025}

{3136/3125, 4375/4374}

{225/224, 1029/1024}

{50/49, 64/63}

{49/48, 392/375}

{28/27, 256/245}

{126/125, 245/243}

{50/49, 245/243}

{50/49, 875/864}

{50/49, 6144/6125}

{25/24, 49/48}

{245/243, 225/224}

{50/49, 4375/4374}

{25/24, 64/63}

{25/24, 49/45}

{126/125, 1029/1024}

{225/224, 4375/4374}

{25/24, 2401/2400}

{81/80, 6144/6125}

{81/80, 1728/1715}

{64/63, 686/675}

{36/35, 1323/1280}

{49/48, 3136/3125}

{64/63, 4375/4374}

{28/27, 126/125}

{49/48, 225/224}

{3200/3087, 25/24}

{81/80, 225/224}

{245/243, 1029/1024}

{245/243, 3136/3125}

{36/35, 50/49}

{28/27, 6144/6125}

{25/24, 243/224}

{4000/3969, 245/243}

{50/49, 525/512}

{36/35, 64/63}

{28/27, 1029/1024}

{1029/1024, 3136/3125}

{25/24, 81/80}

{49/48, 81/80}

{2401/2400, 3136/3125}

{81/80, 2401/2400}

{49/48, 126/125}

{64/63, 875/864}

{36/35, 1029/1000}

{25/24, 28/27}

{28/27, 3125/3024}

{2401/2400, 6144/6125}

{36/35, 5625/5488}

{36/35, 21/20}

{81/80, 875/864}

{36/35, 16/15}

{25/24, 360/343}

{126/125, 1728/1715}

{36/35, 15/14}

{225/224, 1728/1715}

{64/63, 225/224}

{64/63, 3136/3125}

{4375/4374, 6144/6125}

{2401/2400, 4375/4374}

{28/27, 21/20}

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

> {126/125, 1728/1715}

I selected the above at random from my list, and came to the

following conclusions:

(1) It should be discussed in conjunction with the 31+27;58 and

27+4;31, and

(2) It should be discussed in conjunction with a corresponding

11-limit linear temperament.

Is the world of JI ready for this, I wonder? Paul made the music

theory world sound a bit like the big-endians vs the little-endians.

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

>Paul gave a list previously which was a little more inclusive:

>25/24, 28/27, 36/35, 49/48, 50/49, 64/63, 81/80, 126/125, 245/243,

>225/224, 1029/1024, 1728/1715, 2401/2400, 3136/3125, 4375/4374,

>6144/6125

Well, Gene, the first four entries were intended to be used only as

chromatic unison vectors.

Actually, we should be taking three at a time, not two. If all three

(in the Minkowski reduced basis, hopefully) are commatic UVs, we get

an ET. If one is chromatic, we get an MOS. If two are chromatic, a

planar temperament. If three are chromatic, a JI block.

I think the UVs in the reduced basis have to be in Kees's list or

something like it -- if they aren't, then it seems likely that some

step in the scale will be smaller than one of the commatic unison

vectors -- which we shouldn't allow.

Just some imprecise thoughts . . .

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Actually, we should be taking three at a time, not two. If all

three

> (in the Minkowski reduced basis, hopefully) are commatic UVs, we

get

> an ET. If one is chromatic, we get an MOS. If two are chromatic, a

> planar temperament. If three are chromatic, a JI block.

This will prevent you from considering a lot of interesting

temperaments.

> I think the UVs in the reduced basis have to be in Kees's list or

> something like it -- if they aren't, then it seems likely that some

> step in the scale will be smaller than one of the commatic unison

> vectors -- which we shouldn't allow.

Where is Kees's list? I was thinking of producing a list based on

some mathematical conditions, and using that, but I wondered if these

lists already represent such an effort.

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

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>

> > Actually, we should be taking three at a time, not two. If all

> three

> > (in the Minkowski reduced basis, hopefully) are commatic UVs, we

> get

> > an ET. If one is chromatic, we get an MOS. If two are chromatic,

a

> > planar temperament. If three are chromatic, a JI block.

>

> This will prevent you from considering a lot of interesting

> temperaments.

Such as?

> > I think the UVs in the reduced basis have to be in Kees's list or

> > something like it -- if they aren't, then it seems likely that

some

> > step in the scale will be smaller than one of the commatic unison

> > vectors -- which we shouldn't allow.

>

> Where is Kees's list? I was thinking of producing a list based on

> some mathematical conditions, and using that, but I wondered if

these

> lists already represent such an effort.

Start with http://www.kees.cc/tuning/perbl.html and follow the link

to http://www.kees.cc/tuning/s2357.html. What Kees doesn't tell you

(he told me) is that the unison vectors not in parentheses are the

smallest (in octaves or cents) for their level of expressibility

(i.e., for their length). The ones in parentheses are included so

that you're guaranteed to have the three smallest for any level of

expressibility. Perhaps Kees would like to chime in here?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

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

> > This will prevent you from considering a lot of interesting

> > temperaments.

> Such as?

If you take 2401/2400 and 4375/4374 by themselves, you get

ennealimmal temperament. If you try to add 25/24, 28/27, 36/35 you

get garbage. Besides, there's no point in it that I can see--why add

anything when two commas are all you need to define a 7-limit linear

temperament?

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

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

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

>

> > > This will prevent you from considering a lot of interesting

> > > temperaments.

>

> > Such as?

>

> If you take 2401/2400 and 4375/4374 by themselves, you get

> ennealimmal temperament.

I know you love that one.

> If you try to add 25/24, 28/27, 36/35 you

> get garbage.

These are not the only canditates. You left out 49/48. Also, 50/49

and 64/63 can be used as _either_ commatic _or_ chromatic -- I don't

know if I mentioned that but I think so. Other candidates may emerge

once we have a solid foundation for all this.

Anyway, I'd just like to set some reasonable bounds within which we

can flesh out the possibilities. If someone wants to use 36/35 as a

commatic unison vector, I'm all for that, but I'd like to start with

a digestible array of possibilities just for the sake of presentation.

> Besides, there's no point in it that I can see--why add

> anything when two commas are all you need to define a 7-limit

linear

> temperament?

The point is not so much infinite temperaments but rather finite

scales. At least in the approach I envision in this paper. Another

paper could more specifically address infinite temperaments.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> The point is not so much infinite temperaments but rather finite

> scales.

Given two generators, you are set to make scales; the LLL reductions

tend to give an interval of equivalence and a generator, which is

perfect. What I like about starting from two commas is your idea to

use this as a canonical scheme for classification, but certainly one

can go on to scales.

At least in the approach I envision in this paper. Another

> paper could more specifically address infinite temperaments.

The scales will presumably be in one of three things: a temperament,

an et, or a p-limit. It seems to me you can't get away from

addressing one or more of these if you are going to work with scales.

Scales are also less easy to classify than temperaments, because

there are more reasonable possibilities. Moreover, if you are willing

to restrict yourself to et scales, my a;n+m notation already does

classify them.

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

> the LLL reductions

> tend to give an interval of equivalence and a generator, which is

> perfect.

What do you mean, exactly? What do the LLL reductions give that an

unreduced basis for a linear temperament don't, in the way of an

interval of equivalence and a generator?

>

> > At least in the approach I envision in this paper. Another

> > paper could more specifically address infinite temperaments.

>

> The scales will presumably be in one of three things: a

temperament,

> an et, or a p-limit.

An ET is a temperament -- but, as I said, some might be in a linear

temperament, some might be in a planar temperament, etc.

> It seems to me you can't get away from

> addressing one or more of these if you are going to work with

scales.

Right -- but that doesn't mean that I have to talk about ennealimmal

temperament if it doesn't give me a scale with a reasonable number of

notes (as I said, I have to delimit this project somewhere!)

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

> My last list allowed in some duplicates; the following list of 81

> pairs is from an improved version of the program:

I'm going to keep the ones where both UVs appear on my original list,

until we have a firmer basis for making such a list. Commas like

4000/3969 seem like unlikely choices, since there are three or more

commas that are both shorter vectors _and_ smaller musical intervals.

Meanwhile, the superparticulars with smaller numbers than 50:49 lead

to too much tempering for my taste.

> {81/80, 126/125}

> {64/63, 245/243}

> {50/49, 81/80}

> {64/63, 126/125}

> {1728/1715, 3136/3125}

> {81/80, 3136/3125}

> {1029/1024, 4375/4374}

> {3136/3125, 4375/4374}

> {225/224, 1029/1024}

> {50/49, 64/63}

> {126/125, 245/243}

> {50/49, 245/243}

> {50/49, 6144/6125}

> {245/243, 225/224}

> {50/49, 4375/4374}

> {126/125, 1029/1024}

> {225/224, 4375/4374}

> {81/80, 6144/6125}

> {81/80, 1728/1715}

> {64/63, 4375/4374}

> {49/48, 225/224}

> {81/80, 225/224}

> {245/243, 1029/1024}

> {245/243, 3136/3125}

> {1029/1024, 3136/3125}

> {2401/2400, 3136/3125}

> {81/80, 2401/2400}

> {2401/2400, 6144/6125}

> {126/125, 1728/1715}

> {225/224, 1728/1715}

> {64/63, 225/224}

> {64/63, 3136/3125}

> {4375/4374, 6144/6125}

> {2401/2400, 4375/4374}

Still a whopping 34 possibilities! Who can be quickest to the draw to

give the generator, period, and 7-limit complexity for all 34 linear

temperaments? If Gene did this right, they should all be

distinct . . . and this very well might be all the "interesting" ones

for my present purposes.

I wrote,

> If Gene did this right, they should all be

> distinct . . .

Well, the least-squares generators should be distinct . . . some of

the minimax generators might turn out to be the same (?)

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I'm going to keep the ones where both UVs appear on my original

list,

> until we have a firmer basis for making such a list. Commas like

> 4000/3969 seem like unlikely choices, since there are three or more

> commas that are both shorter vectors _and_ smaller musical

intervals.

However, 4000/3969 shows up as part of the reduced basis for 41, 53

and 68, which are all important ets. I conclude therefore it's

significant, and in any case this gives us a way of picking them.

The alternative approach, of course, is to start from pairs of ets; I

think perhaps we should do it both ways as a check.

> Meanwhile, the superparticulars with smaller numbers than 50:49

lead

> to too much tempering for my taste.

I think drawing the line between 50/49 and 49/48 is a little absurd;

why not between 49/48 and 36/35? The schisma showed up in the reduced

basis for 171; perhaps we should include that, then look at 130 and

140 and call it a day?

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

> I think drawing the line between 50/49 and 49/48 is a little

absurd;

> why not between 49/48 and 36/35? The schisma showed up in the

reduced

> basis for 171; perhaps we should include that, then look at 130 and

> 140 and call it a day?

I get

130: <2401/2400, 3136/3125, 19683/19600>

140: <2401/2400, 5120/5103, 15625/15552>

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

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>

> > I'm going to keep the ones where both UVs appear on my original

> list,

> > until we have a firmer basis for making such a list. Commas like

> > 4000/3969 seem like unlikely choices, since there are three or

more

> > commas that are both shorter vectors _and_ smaller musical

> intervals.

>

> However, 4000/3969 shows up as part of the reduced basis for 41, 53

> and 68, which are all important ets.

Maybe not for the conditions I have in mind.

> I conclude therefore it's

> significant, and in any case this gives us a way of picking them.

>

> The alternative approach, of course, is to start from pairs of ets;

I

> think perhaps we should do it both ways as a check.

>

> > Meanwhile, the superparticulars with smaller numbers than 50:49

> lead

> > to too much tempering for my taste.

>

> I think drawing the line between 50/49 and 49/48 is a little

absurd;

> why not between 49/48 and 36/35?

I wouldn't have a big problem with that -- I'm trying to keep the set

of outcomes as small as possible, without introducing a larger number

of conditions than is reasonable.