A proposed list of 5-limit not-quite-Just-thingies

I propose we go with either these 19 (in weighted-complexity order)

neutral thirds
meantone
pelogic
augmented
porcupine
diminished
diaschismic
small diesic
quarter fourths
kleismic
twin meantone
half meantone-fourth
half meantone-fifth
minimal diesic
schismic
wuerschmidt
tiny diesic
orwell
amt

or if you want a bit more complexity add

seventh of major third
semisuper
parakleismic
twin schismic
half schismic-fourth
half schismic-fifth

bringing the total up to 26.

To see what they mean and see how they might be arrived at using
Gene's badness, modified to use Paul's weighted complexity, see the
latest incarnation of my spreadsheet.

http://dkeenan.com/Music/5LimitTemp.xls.zip

They require an error cutoff of 35 cents and the short list has a
weighted complexity cutoff of 10 while the long list has 12.

Any objections? Short list or long?

>neutral thirds
>meantone
>pelogic
>augmented
>porcupine
>diminished
>diaschismic
>small diesic
>quarter fourths
>kleismic
>twin meantone
>half meantone-fourth
>half meantone-fifth
>minimal diesic
>schismic
>wuerschmidt
>tiny diesic
>orwell
>amt

What you're calling small diesic is magic. I wonder why 5-limit
miracle didn't make Gene's original list... otherwise the short
list has everything I care about.

-Carl

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:

> What you're calling small diesic is magic.

Sorry, my mistake.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> I propose we go with either these 19 (in weighted-complexity order)

> neutral thirds
> meantone
> pelogic
> augmented
> porcupine
> diminished
> diaschismic
> small diesic = magic
> quarter fourths = quadrafourths
> kleismic
> twin meantone = garbage
> half meantone-fourth = ditto
> half meantone-fifth = ditto
> minimal diesic
> schismic
> wuerschmidt
> tiny diesic
> orwell
> amt

What happened to chromic?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> What happened to chromic?

I don't need it on the list, and by your badness it is _much_ worse
than any of those on the list. You'd have to include lots of others if
you included chromic. I should never have tried to name it. Sorry.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> What happened to chromic?

Oh I think I realise why you ask, since you wouldn't want it by your
own badness measure. You think I'm violating my own badness measure.
The thing is, I've changed my parameters from 7.4 cents and 0.5 power,
to 5.5 cents and 0.43 power to make it agree better with your list.

See my latest version spreadsheet
http://dkeenan.com/Music/5LimitTemp.xls.zip

I've fixed the names.

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

> See my latest version spreadsheet
> http://dkeenan.com/Music/5LimitTemp.xls.zip
>
> I've fixed the names.

i don't think this is correct, dave:

"Flat_badness = complexity^2*error with cutoffs"

that's not flat at all, right, gene?

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

> Oh I think I realise why you ask, since you wouldn't want it by your
> own badness measure. You think I'm violating my own badness measure.

Nope. My 5-limit badness is 774, not something to set the world on fire, but it did pass your measure, as you say. I want a name for it because my 7-limit badness score is 173, so we have a decent 5-limit
temperament which heats up in the 7-limit.

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

> i don't think this is correct, dave:
>
> "Flat_badness = complexity^2*error with cutoffs"
>
> that's not flat at all, right, gene?

Not for 5-limit.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Oh I think I realise why you ask, since you wouldn't want it by
your
> > own badness measure. You think I'm violating my own badness
measure.
>
> Nope. My 5-limit badness is 774, not something to set the world on
fire, but it did pass your measure, as you say.
>

That was before I went back to using Paul's weighted complexity. Now
it comes out badder than pelogic and semisuper (and with my new
parameters 5.5c and 0.43 power, badder than seventh of major third
[-9,7] and parakleismic).

> I want a name for it because my 7-limit badness score is 173, so we
> have a decent 5-limit temperament which heats up in the 7-limit.

Being good at 7-limit is a good reason to have a name for it, but
obviously should have no bearing on whether or not it's included in
the 5-limit list.

If you modify your badness to use log-of-odd-limit weighted complexity
you will see that its requirement of 9 generators to the fifth
combined with its so-so 3 cent rms error, gives it a badness of 1113
that pushes it way down the list. The worst on my proposed list so far
is only 625 by this badness (quadrafourths).

But I don't mind including it so long as it doesn't require you to
include anything else. By your badness with weighted complexity, there
may well be some others in between quadrafourths and "chrome" that I
will find objectionable. There are certainly some that _you_ will find
objectionable: the half and twin neutral thirds and kleismics (that
I'm happy to omit).

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > twin meantone = garbage
> > half meantone-fourth = ditto
> > half meantone-fifth = ditto

Paul, you say they are not temperaments because they do not map to a
_the_ JI lattice, and are therefore merely "tuning systems". But of
course they map to two (or more) disconnected JI lattices which is a
hell of a lot better than a mere "tuning system". How about we call
them degenerate temperaments?

Gene, since you feel so strongly about it I'll go with Paul's earlier
suggested compromise. Leave them out of the lists. Mention somewhere
in the paper that these modifications are possible and result in a
degenerate temperament having the same errors as the parent
temperament but twice the complexity, and a different pair of
(generator, period) and therefore different MOS. Use these meantone
ones as the example, then say no more about them.

Agreed?

Anyone else object?

I just added degeneracy detection to my spreadsheet.
http://dkeenan.com/Music/5LimitTemp.xls.zip

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

> Paul, you say they are not temperaments because they do not map to a
> _the_ JI lattice, and are therefore merely "tuning systems". But of
> course they map to two (or more) disconnected JI lattices which is a
> hell of a lot better than a mere "tuning system".

Eh? Why is that? So does the 7-et+5-et system, which is the sort of thing I want to avoid; these degenerate temperaments are headed in that direction.

How about we call
> them degenerate temperaments?

Sounds OK; I don't want to give the impression that they are the same kind of thing as a "regular" regular temperament. Mentioning them makes sense to me, dwelling on them doesn't.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > > twin meantone = garbage
> > > half meantone-fourth = ditto
> > > half meantone-fifth = ditto
>
> Paul, you say they are not temperaments because they do not map to
a
> _the_ JI lattice, and are therefore merely "tuning systems". But of
> course they map to two (or more) disconnected JI lattices which is
a
> hell of a lot better than a mere "tuning system". How about we call
> them degenerate temperaments?

decent, though torsion, rather than contorsion, would seem closer to
the usage of 'degeneracy' that i'm familiar with from physics and
math, when applied to temperaments.

>> hell of a lot better than a mere "tuning system". How about we call
>> them degenerate temperaments?
>
>decent, though torsion, rather than contorsion, would seem closer to
>the usage of 'degeneracy' that i'm familiar with from physics and
>math, when applied to temperaments.

Agree. They're not instances of degenerate temperament. They're
instances of superposed, perfectly legitimate temperament_s_.

-Carl