7-limit optimal et vals

Here is a list of all of them which are not already standard vals, for
n from 1 to 1 100. No torsion issues arise. In some cases other vals
scored nearly as well.

<1 2 3 3]

<3 5 7 9]

<8 13 19 23]

<11 18 26 31]

<13 20 30 36]

<14 22 32 39]

<17 27 40 48]

<20 31 46 56]

<23 36 53 64]

<28 44 65 78]

<30 47 69 84]

<33 52 76 92]

<34 54 79 96]

<39 62 91 110]

<48 76 112 135]

<52 83 121 146]

<54 85 125 151]

<64 102 149 180]

<65 103 151 183]

<66 104 153 185]

<67 106 155 188]

<71 112 165 199]

<85 135 198 239]

<86 136 199 241]

<96 152 223 269]

<98 155 227 275]

<100 159 232 281]

what is the optimality criterion?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here is a list of all of them which are not already standard vals,
for
> n from 1 to 1 100. No torsion issues arise. In some cases other vals
> scored nearly as well.
>
> <1 2 3 3]
>
> <3 5 7 9]
>
> <8 13 19 23]
>
> <11 18 26 31]
>
> <13 20 30 36]
>
> <14 22 32 39]
>
> <17 27 40 48]
>
> <20 31 46 56]
>
> <23 36 53 64]
>
> <28 44 65 78]
>
> <30 47 69 84]
>
> <33 52 76 92]
>
> <34 54 79 96]
>
> <39 62 91 110]
>
> <48 76 112 135]
>
> <52 83 121 146]
>
> <54 85 125 151]
>
> <64 102 149 180]
>
> <65 103 151 183]
>
> <66 104 153 185]
>
> <67 106 155 188]
>
> <71 112 165 199]
>
> <85 135 198 239]
>
> <86 136 199 241]
>
> <96 152 223 269]
>
> <98 155 227 275]
>
> <100 159 232 281]

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> what is the optimality criterion?

Minimax error in the 7-limit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > what is the optimality criterion?
>
> Minimax error in the 7-limit.

any differences if you use rms?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > what is the optimality criterion?
> >
> > Minimax error in the 7-limit.
>
> any differences if you use rms?

and are you allowing the octaves to be tempered? i.e. Do they apply
strictly to EDOs or to ET's generally?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > > wrote:
> > > > what is the optimality criterion?
> > >
> > > Minimax error in the 7-limit.
> >
> > any differences if you use rms?
>
> and are you allowing the octaves to be tempered? i.e. Do they apply
> strictly to EDOs or to ET's generally?

EDOs only, but I didn't know you were calling non=integer mappings ets.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > > > wrote:
> > > > > what is the optimality criterion?
> > > >
> > > > Minimax error in the 7-limit.
> > >
> > > any differences if you use rms?
> >
> > and are you allowing the octaves to be tempered? i.e. Do they apply
> > strictly to EDOs or to ET's generally?
>
> EDOs only, but I didn't know you were calling non=integer mappings ets.

Aha. It seems Graham might be missing that too.

We've always had tET's and cET's. The step of n-tET is 1/n of an
octave, while the step of n-cET is n cents. Nowadays we seem to be
using EDO more for what used to be tET. And we have ED3 for the BP
tunings.

But I've certainly been guilty in the past, of being sloppy about this
and calling things ET's when I should have been more specific and said
tET's or EDO's.

Monz's definition agrees.
http://sonic-arts.org/dict/eqtemp.htm

But with non-EDO ETs the mapping still contains only integers, it's
just that the optimum step size (generator) is allowed to be something
other than an integer fraction of an octave. I assume that's what you
meant too.

>And we have ED3 for the BP tunings.

Who's we? I, for one, reject any and all EDx terminology
with the Iron Fist of Discountenance...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >And we have ED3 for the BP tunings.
>
> Who's we? I, for one, reject any and all EDx terminology
> with the Iron Fist of Discountenance...

That's fine, but we still _have_ EDO and ED3 whether we want to use
them or not. Or are you able to erase them from your memory? :-)

If so, sorry to remind you of them again, and don't ever look at the
index to Monz's dictionary. At least keep away from the E's, OK. ;-)

>That's fine, but we still _have_ EDO and ED3 whether we want to use
>them or not. Or are you able to erase them from your memory? :-)

Actually, only a tiny fraction of the tiny fraction of theorists
who use these lists use this terminology.

>If so, sorry to remind you of them again, and don't ever look at the
>index to Monz's dictionary. At least keep away from the E's, OK. ;-)

There are ways of attacking a terminology. Publishing papers with
similar but subtly different terminology, for example. I don't think
this will be necessary, though, as the worthlessness of "EDO" should
be readily apparent to most onlookers.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> There are ways of attacking a terminology. Publishing papers with
> similar but subtly different terminology, for example.

How about rational argument?

> I don't think
> this will be necessary, though, as the worthlessness of "EDO" should
> be readily apparent to most onlookers.

I'm afraid it's worthlessness isn't apparent to me. I'd appreciate it
if you could take the time to explain.

>> There are ways of attacking a terminology. Publishing papers with
>> similar but subtly different terminology, for example.
>
>How about rational argument?

I tried in msgs. 7372 and 7523 to the big list.

>> I don't think
>> this will be necessary, though, as the worthlessness of "EDO"
>> should be readily apparent to most onlookers.
>
>I'm afraid it's worthlessness isn't apparent to me. I'd appreciate
>it if you could take the time to explain.

We already have very well-accepted terminology for EDO. It seems
perverse to have a problem with it but not with the term "octave".
EDx isn't as flexible as -CET when x isn't a whole number.
Otherwise, I'd suggest using ET not ED and omitting the 2 instead
of additng an O, thus arriving at the standard usage for things
like 12 ET, and at 13 ET3 for things like Bohlen-Pierce.

-Carl

hi Carl,

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >And we have ED3 for the BP tunings.
>
> Who's we? I, for one, reject any and all EDx terminology
> with the Iron Fist of Discountenance...

why?

-monz